A Deep Dive into the Distribution Function: Understanding Phase Space Dynamics with Continuum Vlasov-Maxwell Simulations
AABSTRACT
Title of dissertation:
A Deep Dive into the Distribution Function:Understanding Phase Space Dynamicswith Continuum Vlasov–Maxwell SimulationsJames JunoDoctor of Philosophy, 2020
Dissertation directed by:
Professor William DorlandDepartment of Physics
In collisionless and weakly collisional plasmas, the particle distribution func-tion is a rich tapestry of the underlying physics. However, actually leveraging theparticle distribution function to understand the dynamics of a weakly collisionalplasma is challenging. The equation system of relevance, the Vlasov–Maxwell–Fokker–Planck (VM-FP) system of equations, is difficult to numerically integrate,and traditional methods such as the particle-in-cell method introduce counting noiseinto the distribution function.In this thesis, we present a new algorithm for the discretization of VM-FPsystem of equations for the study of plasmas in the kinetic regime. Using thediscontinuous Galerkin (DG) finite element method for the spatial discretization anda third order strong-stability preserving Runge–Kutta for the time discretization,we obtain an accurate solution for the plasma’s distribution function in space andtime. We both prove the numerical method retains key physical properties of the a r X i v : . [ phy s i c s . p l a s m - ph ] M a y M-FP system, such as the conservation of energy and the second law of thermo-dynamics, and demonstrate these properties numerically. These results are con-textualized in the history of the DG method. We discuss the importance of thealgorithm being alias-free , a necessary condition for deriving stable DG schemes ofkinetic equations so as to retain the implicit conservation relations embedded inthe particle distribution function, and the computational favorable implementationusing a modal, orthonormal basis in comparison to traditional DG methods appliedin computational fluid dynamics.A diverse array of simulations are performed which exploit the advantages ofour approach over competing numerical methods. We demonstrate how the highfidelity representation of the distribution function, combined with novel diagnostics,permits detailed analysis of the energization mechanisms in fundamental plasmaprocesses such as collisionless shocks. Likewise, we show the undesirable effectparticle noise can have on both solution quality, and ease of analysis, with a studyof kinetic instabilities with both our continuum VM-FP method and a particle-in-cellmethod.Our VM-FP solver is implemented in the
Gkyell framework , a modularframework for the solution to a variety of equation systems in plasma physics andfluid dynamics. https://github.com/ammarhakim/gkyl Deep Dive into the Distribution Function: Understanding PhaseSpace Dynamics using Continuum Vlasov–Maxwell SimulationsbyJames Juno
Dissertation submitted to the Faculty of the Graduate School of theUniversity of Maryland, College Park in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy2020Advisory Committee:Professor William Dorland, Chair/AdvisorDr. Jason TenBarge, Co-AdvisorProfessor James DrakeProfessor Adil HassamProfessor Jacob Bedrossian (cid:13)
Copyright byJames Juno2020edication
To the memory of my father, Jim Juno, and to my wife, Annaii reface
This thesis was an enormous labor of love, and if you are reading it now withthe intention of learning about what I, and the
Gkeyll project, accomplished, fromthe bottom of my heart: thank you. The length of this thesis requires a prefaceabout my goals and what I hope a reader comes away with after reading it.At every turn, we in the
Gkeyll project have attempted to make the codeaccessible and user-friendly, and I think we have broadly accomplished this goal. Ifeel blessed to have had numerous conversations with fellow graduate students, postdoctoral scientists, and more senior members of our community that have found
Gkeyll to be an excellent tool, not just in the breadth of plasma physics that can bestudied, but in the ease with which they have found downloading the code, buildingit, and running simulations everywhere from their laptops to supercomputers.But, there is more that can be done in making a tool accessible, especially tothose just entering the field of plasma physics. While the equation system of interestin this thesis, the Vlasov–Maxwell–Fokker–Planck system of equations, is one of themost fundamental equation systems in all of plasma physics, it is not always thecase that a budding new plasma physicist has immediate exposure to the equationsystem, its derivation, and the wealth of physics content within the equation system.The few universities that offer rigorous courses in kinetic theory often break up thediscussions of this equation system over the course of a full year. In addition, somebeloved textbooks that offer clear explanations of plasma kinetic theory are out ofprint, such as Nicholson [1983], and may only become harder to find with time.iii do not claim to have rigorously derived the foundations of plasma physicsin this thesis. But it is my wish to impart physical intuition about plasma kinetictheory, thinking about a many-body system like a plasma in a statistical sense, andthe rich physics buried in the Vlasov–Maxwell–Fokker–Planck system of equationsthat ultimately made the derivation and implementation of novel numerical methodssuch a rewarding project. In this vein, I hope to proceed pedagogically through theintuition that forces us to develop kinetic theory, what kinetic theory means, andhow we obtain workable equations for the physics of a plasma so that when weultimately work to discretize the equation system and numerically integrate thediscrete system to model plasma phenomena, we have a sense of what properties ofthe continuous system of equations we would like our discretization to respect.This thesis is not intended as a user manual for the code, at least not ifa reader’s goal is to find installation instructions and assistance in building the
Gkeyll simulation framework. I refer an interested reader in this regard to our
GitHub and documentation website . It is the goal of this thesis to explain everyaspect of our numerical method, how it works, and how we can leverage this par-ticular algorithm to perform simulations of kinetic plasmas. In this way, this thesisis intended as much to be an introduction to the algorithms in the Gkeyll sim-ulation framework as it is to kinetic theory, especially the difference between themathematical formulation of an algorithm, and the translation of this algorithm tocode. https://github.com/ammarhakim/gkyl https://gkyl.readthedocs.io/en/latest/ iv have attempted to organize this thesis in a logical fashion for an aspiringplasma physicist interested in diving into the details of plasma dynamics. Chap-ter 1 provides an introduction to plasma physics and kinetic theory and attemptsto motivate both why we need the Vlasov–Maxwell–Fokker–Planck system of equa-tions, and from where this equation system ultimately comes. Importantly, whilethe discussion of the Vlasov–Maxwell–Fokker–Planck system of equations may notbe wholly rigorous, we will in detail work through many of the properties of thecontinuous system in anticipation of what properties we desire a numerical methodto respect in the process of discretizing the equation system of interest.Chapter 2 will introduce our numerical method, the discontinuous Galerkinfinite element method, and attempt to build intuition for how the method worksand how we can apply the method generally to partial differential equations. Wewill then in detail discretize the Vlasov–Maxwell–Fokker–Planck system of equationsand mathematically determine the properties our discrete scheme retains from thecontinuous system. Chapter 2 will form a mathematically complete description ofour method, before we turn to Chapter 3, where we will translate this mathematicsinto an algorithm which can be implemented in code. This conversion to code isequally nontrivial to the mathematical formulation of the algorithm, but it is mygoal that after reading Chapter 3, a reader may dive into Gkeyll with newfoundunderstanding of how to put all the pieces together into a discrete scheme that canbe used for performing numerical experiments.Chapter 4 will involve taxing testing of the implemented numerical methodfor the Vlasov–Maxwell–Fokker–Planck system of equations, and attempt to demon-vtrate to the reader that the scheme outlined in this thesis is on firm foundation; youmay trust both that the scheme discussed in this thesis is a valid one, and that thecode will work for whatever you envision doing with it. We will conclude in Chap-ter 5 with a number of applications of my implementation of the DG discretizationof the Vlasov–Maxwell–Fokker–Planck system of equations to demonstrate the fullutility of this approach, leveraging the code to understand the details of energizationprocesses and nonlinear plasma instabilities.Because this thesis is intended to live beyond my graduate career, I would askfuture readers that find typos or issues to contact me at my personal email: [email protected]. At every stage of my career, I will attempt to keep this thesis in astate of maximum utility by updating it as necessary on the
Gkeyll documentationwebsite. Readers interested in reproducing the simulations presented in this thesiscan do so by running the input files available through a
GitHub repository . Thechangesets used to produce the data are documented in the input file, and whereappropriate the scripts used to produce the figures in this thesis can be found along-side the input files. In addition, if any readers are interested in the publicationswhich formed the basis for this thesis, I refer them to Juno et al. [2018], Hakim,Francisquez, Juno, and Hammett [2019], Hakim and Juno [2020], Juno et al. [2020],and Skoutnev, Hakim, Juno, and TenBarge [2019].Without further ado, let us begin. I hope you ultimately find this thesis asmuch fun to read as I had writing it. To quote Robert Louis Stevenson, “It is onething to mortify curiosity, another to conquer it.” https://github.com/ammarhakim/gkyl-paper-inp vi cknowledgments A long thesis requires a long acknowledgements, as this thesis would not existwithout the support I have received from many people over the years.I have been blessed to have received a great deal of mentorship throughoutmy graduate career. It is not possible to thank just one person for serving as anadvisor to my graduate career, so let me instead thank them all. I am gratefulto Prof. William Dorland, who took me on as his student before I even officiallystarted at the University of Maryland and who graciously accepted my ambition formyself by allowing me to pitch him the project that this thesis became. Bill hasbeen a constant source of guidance, and his own experiences in developing one of theleading computational plasma physics tools provided invaluable perspective that ledto some of the most strenuous tests of
Gkeyll . The confidence I have in the code isborne of many discussions with him on software development and verification andvalidation, and I think we got some exciting science out of this project to boot.I would like to thank Dr. Jason TenBarge, my co-advisor, for being a constantsource of inspiration and a valuable scientific confidant. Jason probably did notknow what he was getting into having an open door policy and me as a student,but I learned so much from him throughout my graduate career. If I have gained areputation for asking questions at conferences, it is only because of the sheer volumeof knowledge Jason has imparted to me. I hope for a long career of collaborationsbetween the two of us as I move to the next stage of my career. I am especiallygrateful to Jason and his wife Helen for their friendship through the challenges ofviiraduate school. I have treasured immensely the good company, good food, andboard games more than I can say.Finally, I must thank Dr. Ammar Hakim for his tutelage and guidance from thevery beginning of me joining the
Gkeyll project. Ammar has followed and mentoredmy growth since I was an undergraduate summer student at the Princeton PlasmaPhysics Lab, and he must have known he made a strong impression when I pitchedBill on continuing to work on
Gkeyll as a University of Maryland graduate student.I am proud of all the work I, and the
Gkeyll team, have accomplished. Serving as acoding apprentice to a computational physicist as outstanding as Ammar has beenmore valuable than any single course I have taken in my entire education. I hopeto only grow more as a programmer through continued collaboration with him as amember of the
Gkeyll team.Speaking of the
Gkeyll team, this thesis owes a tremendous debt of gratitudeto all of you. To the members past and present, Petr Cagas, Manaure Francisquez,Tess Bernard, Valentin Skoutnev, Noah Mandell, Eric Shi, Liang Wang, JonathanNg, and Rupak Mukherjee, thank you. I am blessed to have had such an amazingteam to code alongside. Thank you especially to Petr Cagas for all his work writing postgkyl , which this thesis leverages extensively. We have accomplished so much,and I look forward to many more years of working together with you all.Thank you to the other members of my committee, Prof. Jim Drake, Prof.Adil Hassam, and Prof. Jacob Bedrossian. Most especially thank you to Jim, whowas generous enough to offer me a post doctoral position that I happily accepted. Ilook forward to the work we will do together.viiihank you to all of my plasma compatriots at the University of Maryland,George Wilkie, Joel Dahlin, Wrick Sengupta, Lora Price, Elizabeth Paul, MichaelMartin, Gareth Roberg-Clark, Harry Arnold, Qile Zhang, Rahul Gaur, MichaelNastac, Rog´erio Jorge, and Alessandro Geraldini. I would like to especially thankElizabeth and Wrick for being such incredible and hospitable friends. This thesiswould not exist without your support.Every scientist needs a break from science, and I have been fortunate enoughto have had the very best of friends in Sandy and Clarissa Craddock, JonathanVannucci, Zachary Eldredge, Katie Goff, Rodney Snyder, Molly Carpenter, SteffiRathe, Humberto Gilmer, Carl Mitchell, Sheehan Ahmed, and Jesse Rivera. I re-main eternally grateful for the drinks and laughs we have shared. A special thankyou to Sandy, Jon, Zach, and Rodney for your companionship through the trials ofqualifying exams.Thank you to all members of The Shed, past and present, who have comealong with me on adventures at the table. I have never played with a better dungeonmaster than Zak Schooley, who rekindled my love for tabletop games. RecordingAdventures in Hyperborea with James Upton, James Wiley, and Jonathan Hill hasbeen a singular joy for me over these past few years. I cannot wait to see what thedice have in store for us next.There are many others to thank. Thank you to Dr. Gregory Hammett forassisting in supervising me with Ammar when I was but an undergraduate on the
Gkeyll team, for continuing to support the whole
Gkeyll project, and for beinga constant source of physics insight. Thank you to Prof. Matt Kunz and Prof.ixnatoly Spitkovsky for all of our physics discussions in my home away from home inPrinceton. A special thank you to Matt for allowing me to sit in on his IrreversibleProcesses in Plasmas course so I could refresh my knowledge of plasma kinetictheory in anticipation of writing this thesis. Thank you to Dr. Ian Abel, who isas much a plasma compatriot as my fellow graduate students and post docs. Ihave Ian to thank for developing the intuition I have on asymptotic methods, andI am so grateful for his thoughtful responses to my long emails in the early daysof my graduate career when I was still learning gyrokinetics. Thank you to Dr.Marc Swisdak, who graciously agreed to collaborate on work that appears in thisthesis. I am thrilled that the post doctoral position with Jim will allow us continuecollaborating more in the coming years. Thank you to Dr. Matt Landreman andProf. Tom Antonsen for an uncountable number of enlightening physics discussionsand for the deep, engaging questions you both asked of me during group meetings.Thank you to the entire Chalmer’s group for being such early adopters of
Gkeyll and seeing so much potential in us, especially T¨unde F¨ul¨op, Istv´an Pusztai,and Andr´eas Sundstr¨om. It has been an absolute pleasure to collaborate with youall. I recall being worried that you had caught a subtle bug for the electron Landaudamping and dynamo paper, and I was ready to send you an email discouraging theuse of
Gkeyll for this project, only for Istv´an to email within 24 hours that he hadfigured out a physics explanation for the code’s behavior. It is no exaggeration thatI am overwhelmed with excitement for our future research endeavors.Thank you to the colleagues I have made through the Solar, Heliospheric,and INterplanetary Environment (SHINE) conference, Prof. Gregory Howes andxrof. Kristopher Klein. You both have been so encouraging of the developmentof
Gkeyll , and I am elated that your encouragement has led to multiple ongoingprojects, the results of which partially appear in this thesis. Amongst my fellowgraduate students at SHINE, I am grateful to have served alongside Doˇga Can Su¨Ozt¨urk, Samaiyah Farid, and Emily Lichko as SHINE Student Representatives. Ifeel strongly that we made the SHINE conference a better experience for students,and I hope our initiatives are a fixture in the conference for years to come. Specialthanks to Emily for taking the time to discuss her work on magnetic pumping withme, informing a rigorous test of
Gkeyll .Thank you to those that supported me from the very beginning. Thank youto my family, most especially my mom, Constance Lynn, who has been an endlesswellspring of support. Thank you to the teachers who lit the spark of curiosity andnurtured the flame, Kristy Elam, Richard McGowan, and Jeff Peden in high school,and Prof. Frank Toffoletto and Prof. Anthony Chan at Rice University.And thank you most of all to my wife, Anna Wright, who this thesis is dedi-cated too alongside the memory of my father. Anna, you are the light of my life, andwords do no exist to describe the extent of your support for me, and the way youinspire me each and every day. This thesis has been a highly collaborative effort,and your name is right alongside all of my
Gkeyll teammates. It was the greatest ofprivileges to have you by my side through this journey, and I look forward to whatwe do next together.This work was supported by a NASA Earth and Space Science Fellowship,grant no. 80NSSC17K0428. xi able of Contents
Dedication iiPreface iiiAcknowledgements viiList of Tables xvList of Figures xviList of Abbreviations xxviii1 Introduction 11.1 What is a plasma? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The Debye length and the Plasma Parameter . . . . . . . . . . . . . . 61.3 The challenge in modeling plasmas . . . . . . . . . . . . . . . . . . . 91.4 An introduction to kinetic theory . . . . . . . . . . . . . . . . . . . . 151.5 Bogoliubov’s Timescale Hierarchy and theVlasov–Maxwell–Fokker–Planck System of Equations . . . . . . . . . 231.6 Properties of the Vlasov–Maxwell–Fokker–PlanckSystem of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.7 A Brief History of Kinetic Numerical Methodsand the Objectives of This Thesis . . . . . . . . . . . . . . . . . . . . 392 The Discontinuous Galerkin Finite Element Method 482.1 L Minimization of the Error . . . . . . . . . . . . . . . . . . . . . . 492.2 The Semi-Discrete Vlasov–Maxwell System of Equations . . . . . . . 582.3 Properties of the Semi-Discrete Vlasov–MaxwellSystem of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.4 An Interlude on Weak Equality and Weak Operators . . . . . . . . . 832.5 The Semi-Discrete Fokker–Planck Equation . . . . . . . . . . . . . . 942.6 Properties of the Semi-Discrete Fokker–Planck Equation . . . . . . . 98xii.7 The Time Discretization of theVlasov–Maxwell–Fokker–Planck System of Equations . . . . . . . . . 1112.8 Summary of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 1193 From Math to Code: Efficient Implementation of DG forthe Vlasov–Maxwell–Fokker–Planck System of Equations 1313.1 Polynomial Bases in 1D: Nodal versus Modal . . . . . . . . . . . . . . 1323.2 Polynomial Bases in Higher Dimensions:The “Curse of Dimensionality” and Serendipitous Basis Choices . . . 1413.3 Transforming from Computational Space to Physical Space . . . . . . 1483.4 Evaluating the Integrals: The Importance of anAlias-Free Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1533.5 Extending the Recovery Scheme to Higher Dimensions . . . . . . . . 1683.6 Computing the Coupling Moments . . . . . . . . . . . . . . . . . . . 1723.7 A Computational Complexity Experiment . . . . . . . . . . . . . . . 1773.8 Summary of the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 1834 Benchmarking our DG Vlasov–Maxwell–Fokker–Planck Solver in
Gkeyll ist of Tables P N q ( x )and the weights W i = 2 / [(1 − x i )( P (cid:48) N q ( x i )) ] [Abramowitz and Stegun,1985]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553.5 Number of quadrature points required to integrate the volume termfor the advection of the distribution function in velocity space as afunction of dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . 1613.6 Reduction in the number of quadrature points, relative to isotropicquadrature, required to integrate the volume term for the advectionof the distribution function in velocity space. . . . . . . . . . . . . . . 1613.7 Summary of the parameters for the numerical experiment to comparethe full cost of an alias-free nodal and orthonormal, modal algorithm. 183xv ist of Figures d p = c/ω pp . We plotthe reduced proton distribution function in x − v x (second from topplot) and slices of the proton distribution function in v x − v y (bottomplots) at the specified lines in the x − v x plots, x = 19 . , . , . , and22 . d p . We will discuss this structure and the specific energizationmechanisms of this collisionless shock in Chapter 5, but for now wedraw attention to the quality of the solution from a continuum rep-resentation of the distribution function using a phase space grid. Bydirectly discretizing the VM-FP system of equations in phase space,we can represent fine-scale structure in velocity space which we canleverage to dive in to the details of the energization of the protons. . 452.1 The projection of f ( x ) = x + sin(5 x ) onto piecewise constant (left),piecewise linear (middle), and piecewise quadratice (right) functions.The domain from [ − ,
1] is divided into non-overlapping cells and theprojection is done within each cell to minimize the L error. We beginto see some of the connection between the discontinuous Galerkinmethod and finite element methods, as moving to higher polynomialorder manifestly reduces the L error between the exact solution andprojected solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.2 Annotated piecewise linear representation to make our notation moreclear, most especially superscript plus-minus, where the solution isevaluated just inside, − , or just outside +, the cell interface. . . . . . 54xvi.3 Comparison of advection of a Gaussian pulse one period with a piece-wise constant (left) and piecewise linear (right) basis function expan-sion and upwind fluxes. While the piecewise constant solution suf-fers from numerical diffusion which leads to poor agreement betweenthe analytic solution (red) and the numerical solution (black), thepiecewise linear solution agrees to a reasonably high degree with theexpected result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.4 Weak division for the p = 1 basis, Eq. (2.117), to compute u from M u . = M . In this plot, M = 1 and the effect of changing M (toprow) on the flow u (bottom row) is shown. As the density steepens,the velocity becomes larger. If the density becomes too steep, if weincrease the slope of the density further so that the density has a zerocrossing at x = ± / √ u wouldblows in the sense that the flow becomes an unrealizable function. Im-portantly, this blow-up condition corresponds to the situation wherethe slope of M becomes too steep to represent M with a positivedefinite function, which physically corresponds to a situation wherethe representation of the density is producing negative density func-tions. Since the value of the particle density can only be positive,this blow-up is highly undesirable. . . . . . . . . . . . . . . . . . . . . 913.1 The simple monomial basis (left) defined in Eq. (3.1) and the or-thonormal basis obtained by a Gram-Schmidt orthogonolization (or-thonormalization) applied to the monomial basis (right). We cansee that in the limit of high polynomial order, the monomial basisbecomes less linearly independent, i.e., the higher order polynomialsare essentially indistinguishable. On the other hand, the orthonormalbasis maintains better “coverage” of the space on the interval from[ − ,
1] so that it is easy to imagine why higher order orthonormalpolynomials do actually improve the accuracy of the representation. . 1393.2 Schematic drawing of the nodal locations for the Serendipity basis in1D (top), 2D (middle), and 3D (bottom) for polynomial orders one(far left), two (middle left), three (middle right), and four (far right). 147xvii.3 The computational kernel for the volume integral, Eq. (3.36), for thecollisionless advection in phase space of the particle distribution func-tion in one spatial dimension and two velocity dimensions (1X2V) forthe piecewise linear tensor product basis. Note that this compu-tational kernel takes the form of a C++ kernel that can be calledrepeatedly for each grid cell K j depending on the local cell centercoordinate and the local grid spacing. Here, the local cell coordi-nate is the input “const double w” and the local grid spacing is theinput “const double dxv”. The out array is the increment to theright hand side due this volume integral contribution in a forwardEuler time-step, i.e., a piece of Eq. (2.165) for the Vlasov–Fokker–Planck equation. To complete the right hand side of Eq. (2.165) forthe evolution of the particle distribution function, for a given phasespace cell, we require the surface contributions for the collisionlessadvection, as well as the computational kernels for the correspond-ing tensors encoding the spatial discretization of the Fokker–Planckequation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1653.4 Example computational kernel for the calculation of the zeroth throughsecond moments using weak equality in one spatial dimension andtwo velocity dimensions (1X2V) with piecewise linear, tensor prod-uct, modal, orthonormal polynomials. Note that this computationalkernel is called inside a loop over velocity space for a given configu-ration space cell, as we are integrating over velocity space. . . . . . . 1743.5 A C++ computational kernel for the construction and inversion of thematrix to solve the coupled linear system for the discrete flow andtemperature, u h and T h . Here, we show the form of the matrix in onespatial and one velocity dimension (1X1V) using piecewise quadraticSerendipity polynomials. Since both u h and T h have three degrees offreedom, i.e., three basis functions, which describe their projection,the coupled linear system is six by six. We construct the individualterms in the matrix using a combination of weak multiplication, weakdivision, and the corrections at the boundary due to our finite velocityspace extents. We can then use a linear algebra library, in this caseEigen, to solve the linear system and determine the discrete flowand temperature required in the evaluation of the drag and diffusioncoefficients in the discrete Fokker–Planck equation. . . . . . . . . . . 1773.6 Scaling, i.e., the time to evaluate the update versus the number de-grees of freedom, N p , in a cell, of just the streaming term, α xh = v ,(left) and the total, streaming and acceleration, update (right) for theVlasov solver. The dimensionality of the solve is denoted by the rele-vant marker, and the three colors correspond to three different basisexpansions: black:maximal-order, blue:Serendipity, and red:tensor.Importantly, this is the scaling of the full update, for every dimen-sion, i.e., the 3x3v points include the six dimensional volume integraland all twelve five dimensional surface integrals. . . . . . . . . . . . 178xviii.1 (a) Relative change in energy, ∆ M /M ( t = 0) = [ M ( t ) − M ( t =0)] /M ( t = 0), for p = 1, N = 16 (solid and dashed blue) and p = 2, N = 8 (dotted and dash-dot orange) cases for relaxation of asquare distribution to a discrete Maxwellian. The decrease in energyin our conservative scheme is close to machine precision. The curveslabeled ‘no conservation’ omit the boundary correction terms anduse regular moments instead of “star moments” (for p = 1) neededfor momentum and energy conservation. (b) Time-history of relativechange in entropy. When using the conservative scheme, the entropyrapidly increases and remains constant once the distribution functionbecomes a discrete Maxwellian. . . . . . . . . . . . . . . . . . . . . . 1924.2 The initial (a), relaxed (b) distribution function in a 1X2V relaxationtest. Conservation (c) of energy (orange) and momentum (green, pur-ple) is at machine precision for our conservative scheme. Neglectingboundary corrections breaks conservation by more than 8 orders ofmagnitude. Purple and green curves overlay each other on this scale.(d) The entropy increases rapidly and then remains constant once thediscrete Maxwellian is obtained. . . . . . . . . . . . . . . . . . . . . . 1944.3 Density (a), velocity (b), temperature (c) and gas frame, or kinetic,heat-flux (d) from a Sod-Shock problem. Plotted are results withKnudsen numbers of 1 /
10 (red), 1 /
100 (magenta), and 1 /
500 (blue),with the inviscid Euler results (black dashed) shown for comparison.As the gas becomes more collisional, i.e., decreasing Knudsen number,the solutions tend to the Euler result. Note that there is no heat-fluxin the inviscid limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1974.4 The density (a), velocity (b), and distribution function (c) for theSod-shock problem with a sonic point in the rarefaction. Complicatedshock structures are formed and are visible both in the moments aswell as the distribution function. . . . . . . . . . . . . . . . . . . . . . 1994.5 The relative change in the momentum (a) and energy (b) for p = 1(blue) and p = 2 (orange) cases for the Sod-shock problem with asonic point in the rarefaction. Our conservative scheme gives us ma-chine precision errors in momentum and energy errors that are nearlyindependent of polynomial order and only depend on the number oftime-steps taken in each simulation. However, neglecting the bound-ary corrections needed for conservation leads to errors orders of mag-nitude greater. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200xix.6 The change in the total, electron plus proton and electromagnetic,energy for a number of simulations to demonstrate the robustness ofour energy conserving scheme. The scheme’s energy conservation isindependent of the polynomial order (top left/right), with the caveatthat the choice of polynomial order 1 requires sufficient velocity res-olution to reduce the projection errors in projecting | v | . The lattercaveat of projection errors in the polynomial order 1 simulations isalso the reason for the dip in the most resolved polynomial order 1calculation, where the computation of errors is the most sensitive andwe must be careful about finite precision effects. We note though thatfor fixed time-step we recover the energy conservation result of p = 2and p = 3 if we use enough velocity space resolution with the p = 1simulations. Likewise, the scheme’s energy conservation depends onlythe size of the time-step, not the configuration space resolution (bot-tom left/right). The convergence of the energy errors in the top leftplot match our expectations for a third order time-stepping method,2.5 and 2.9 for p = 2, and 2.0 and 2.9 for p = 3. . . . . . . . . . . . . 2024.7 The change in the total, electron plus proton, momentum in a num-ber of simulations. Simulations with polynomial order 2 (left) andpolynomial order 3 (right) are performed with increasing configura-tion space and velocity space resolution to demonstrate that errorsin the total momentum decrease with increasing configuration spaceresolution, while only weakly depending on velocity space resolution.The convergence orders of the polynomial order 2 simulations are1.35, 2.55, 2.93, and 3.14, and the convergence orders of the poly-nomial order 3 simulations are 2.83, 3.32, 3.38, and 4.76, and theseconvergence orders are calculated using the higher velocity resolutionresults. We note the convergence orders are largely unaffected byusing the lower velocity resolution simulations to compute them. . . . 2044.8 The change in the L norm of the electron (left) and proton (right)distribution function with increasing resolution and polynomial order.As expected, the behavior of the L norm of the distribution functionis monotonic and decays in time. We note as well that increasingthe polynomial order from 2 to 3 corresponds extremely well witha doubling of the resolution, providing direct evidence for the oftenassumed benefit of a high order method. . . . . . . . . . . . . . . . . 2054.9 Comparison of the divergence of the electric field (dashed line) andthe charge density (stars) for polynomial order 2 (left) and polynomialorder 3 (right) simulations at the end of the simulation, t = 1000 ω − pe .We can see that the two quantities agree reasonably well, especiallyas we refine the grid. Even as higher amplitude, smaller scale, electricfields are excited in the higher resolution simulations, the two quan-tities track each other well, despite the fact that we do not enforcethis condition, and the charge density does not appear anywhere inevolved system of equations. . . . . . . . . . . . . . . . . . . . . . . . 207xx.10 The non-resonant (top) and resonant (bottom) advection of a dis-tribution of electrons in phase space, over-plotted with the analyti-cal solution. The electron distribution function is plotted at f ( x = π, v x , v y ). We can see that in both cases the distribution function’sevolution is well described by our derived analytical solution, and thatin the non-resonant case, where the distribution function is advectedfor a large number of inverse cyclotron periods, there is no noticeablediffusion of the distribution function in phase space. We emphasizethat these simulations are performed with polynomial order 2 on arelatively coarse velocity space mesh, N v x = N v y = 16 with veloc-ity space extents [ − v th e , v th e ] in both the v x and v y dimensions, so∆ v x = ∆ v y = 1 v th e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2114.11 The value of the flow computed from the simulations (red dots) over-plotted with the analytic solution (black line) for non-resonant (top)and resonant (bottom) cases. The values of the flow are plotted at u x ( x = π ) , u y ( x = π ). . . . . . . . . . . . . . . . . . . . . . . . . . . . 2124.12 Comparison of a polynomial order 2 (left) and polynomial order 3(right) simulation of the non-resonant case at t = 1000Ω − c . The elec-tron distribution function is plotted at f ( x = π, v x , v y ). On this coarsemesh, N v x = N v y = 16 with velocity space extents [ − v th e , v th e ] inboth the v x and v y dimensions, so ∆ v x = ∆ v y = 1 v th e , the diffusion ofthe distribution function in phase space starts to become noticeablefor the polynomial order 2 case after running the simulation for along enough time. But, we note that for the same coarse mesh, thedistribution function in the polynomial order 3 simulation remainspristine at this late time. . . . . . . . . . . . . . . . . . . . . . . . . . 2134.13 Prototypical evolution of the electromagnetic energy (blue), (cid:15) (cid:82) | E | dx ,for the damping of a Langmuir wave, in this case kλ D = 0 .
5, for anumber of plasma periods (left), and the evolution of various compo-nents of the energy for the full length of the simulation (right). Theright plot is the relative change in the energy component comparedto the total energy at t = 0, i.e, ∆ E comp /E . The local maxima (redcircles) of the evolution in the left plot are used to determine both thedamping rate and frequency of the excited wave via linear regression,with the black line being our reference fit for the damping rate. Wenote that energy is very well conserved, and, as expected, the plasmawaves damp on the electrons, converting electromagnetic energy toelectron thermal energy. . . . . . . . . . . . . . . . . . . . . . . . . . 2164.14 Damping rates (left) and frequencies (right) of Langmuir waves fromtheory (solid line) and for a number of Vlasov–Maxwell simulations(red circles). The solid lines are obtained using a root finding tech-nique applied to Eq. (4.26). The x-axis of both figures is normalizedto the Debye length, λ D , and the y-axis of both figures is normalizedto the plasma frequency, ω pe . . . . . . . . . . . . . . . . . . . . . . . . 217xxi.15 The aluminum (left) and proton impurity (right) distribution func-tions in the vicinity of the shock at t = 35 (cid:112) m e /m p ω − pe ∼ ω − pe .Over-plotted in white are contours of constant H ( x, v ) = m s v + q s φ ( x ), the Hamiltonian. We note that the Hamiltonian has beentransformed to the rest frame of the shock, ˆ v = v − V shock , and thereis some freedom in computing φ ( x ) from the electric field in our sim-ulations. We choose φ ( x = 0) = 0 on the left edge of the domain,and then integrate E x along the 1D domain to determine the electro-static potential. We draw attention to the trapped particle regionsin the proton distribution function just down-stream of the shock,which amplify the cross-shock potential and lead to a large reflectedpopulation of protons. Note that we are plotting a normalized valuefor the distribution function, as in Pusztai et al. [2018], and that thev-axes are different for the two species. . . . . . . . . . . . . . . . . . 2204.16 The exponential growth of the LHDI electric field (left) and the LHDIelectric field visualized in configuration space late in the linear stageat t = 6Ω − ci (right). The growth rate, γ ∼ . ci , compares well withlinear theory and the results presented in Ng et al. [2019]. Likewise,the mode structure in a snapshot of the LHDI electric field corre-sponds to the typical LHDI electric field for an m = 8 perturbation,with the electric field localized to the edge of the current sheet wherethe density gradient is largest. The LHDI electric field magnitudeis normalized to B v A = B / √ µ n m p where B is the asymptoticmagnetic field and n is the density in the current layer. . . . . . . . 2254.17 The distribution function for the protons plotted at f ( x, y = − . ρ p , v x , v y =0 . v th p ), at the edge of the current sheet (left), and a further cut of the2D distribution function, f ( x = 2 . ρ p , y = − . ρ p , v x , v y = 0 . v th p )(right). The mode structure for an m = 8 perturbation is again easilyseen in the 2D visualization of the proton distribution function, as theprotons at the edge of the current sheet are resonant with the grow-ing electric field from the LHDI. We have over-plotted the initial driftvelocity (red solid) and the phase velocity for the resonance condition(green dashed) on top of the 1D cut of the distribution function at x = 2 . ρ p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2264.18 Comparison of linear theory (solid line) calculated from the disper-sion relation in Eq. (4.44) after rotation to the coordinate systemaligned with k , Eqns. (4.47–4.49), with a number of Gkeyll simula-tions (stars) for the filamentation limit, θ = 0 ◦ , an oblique mode at θ = 45 ◦ , and the two-stream limit, θ = 90 ◦ . We observe good agree-ment between the linear theory and our DG Vlasov-Maxwell solver. . 230xxii.19 The evolution of the electromagnetic fields, E x (top left), E y (topmiddle), and B z (top right), as well as the electron distribution func-tion at ( y = L y / , v y = 0) (bottom left), ( x = L x / , v x = 0) (bottommiddle), and ( x = L x / , y = L y /
2) (bottom right) at t = 125 ω − pe as the oblique mode, θ = 45 ◦ , instability is going nonlinear. We ob-serve the growth of all three components of the initial electromagneticfields, with standard signatures of both two-stream- and filamenta-tion modes in the distribution function: the phase space vortices inthe x − v x and y − v y plane, and the deflection of the beams in the v x − v y plane respectively. . . . . . . . . . . . . . . . . . . . . . . . . 2324.20 The evolution of the electromagnetic fields, E x (top left), E y (topmiddle), and B z (top right), as well as the electron distribution func-tion at ( y = L y / , v y = 0) (bottom left), ( x = L x / , v x = 0) (bottommiddle), and ( x = L x / , y = L y /
2) (bottom right) at t = 500 ω − pe ofthe oblique mode, θ = 45 ◦ , instability deep in the nonlinear phaseof the dynamics. Here, we observe little, if any, magnetic field, asthe electrostatic wells forming in the electric field components scatterparticles to a nearly isotropic state in the v x − v y plane and depletethe phase space structure required to support the magnetic field. . . . 2334.21 Field energy as a function of time for the linear collisional Landaudamping problem with varying collisionality. Similar to Figure 4.13,we compute the damping rate of each simulation by fitting to thepeaks of the field energy. The collision frequency ν is normalized tothe electron plasma frequency. . . . . . . . . . . . . . . . . . . . . . . 2354.22 Damping rate versus collisionality computed from simulations such asthose shown in Figure 4.21. As expected, the damping rate shuts offwith increasing collisionality due to the particles being scattered bycollisions before they can resonate with the wave. The black dashedline shows an analytical estimate of the damping rate computed fromexpressions found in Anderson and O’Neil [2007b] and agrees wellwith the results computed here. . . . . . . . . . . . . . . . . . . . . . 2354.23 Time evolution of the magnetic field (top) in the middle of the domainfrom the magnetic pumping problem. As the antenna currents rampup, an oscillating field is created that then transfers energy, via pitch-angle scattering, to the plasma, leading to an increase in the thermalenergy (bottom). With zero collisionality (bottom, green), the energyexchange is completely reversible, and no net heating is observed, butas the collision frequency is made finite, magnetic pumping begins toheat the plasma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238xxiii.24 Heating rate via magnetic pumping, plus an additional viscous heat-ing mechanism, as a function of normalized collision frequency. Thecode agrees well with the theoretical prediction (black line) magneticpumping at lower collision frequency, but shows an additional heat-ing mechanism at higher collisionalty due to the viscous damping ofout-of-plane flows, which are included in the Braginskii-based theory(red line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2405.1 The x-electric field (top), y-electric field (second from top), z-magneticfield (middle), reduced proton distribution function (second from bot-tom), and reduced electron distribution function (bottom), both in-tegrated in v y , after the perpendicular shock has formed and propa-gated through the simulation domain. We have marked an approx-imate transition from upstream of the shock to the shocked plasma(dashed-dotted lines), and likewise an approximate transition fromthe shock to the downstream region (dashed lines). To mark themean values of the oscillating downstream electromagnetic fields, wehave used a solid black line to mark the approximate compression ofthe magnetic field, along with E = 0. . . . . . . . . . . . . . . . . . . 2545.2 The proton (top two rows) and electron (bottom row) distributionfunctions plotted through the shock at t = 11Ω − cp . As we movefrom upstream, x = 24 . d p , through the shock ramp centered at x = 21 . d p , we can identify the reflected proton population as well asa broadening of the electron distribution function. . . . . . . . . . . . 2555.3 Proton distribution functions (top row), C v x field-particle correlations(middle row), and C v y field-particle correlations (bottom row) in theshock foot and ramp region, where the shock has begun energizingthe plasma. We see clear evidence in the proton distribution functionof a high energy tail in v x − v y . Further, we note that the energizationof the plasma is localized to this high energy tail. This energizationis due to the component of the proton distribution function whichreturns upstream via its gyromotion, and is thus able to gain energyalong the motional electric field, E y , which supports the E × B drift. 2585.4 Proton distribution functions (top row), C v x field-particle correla-tions (middle row), and C v y field-particle correlations (bottom row)in the overshoot and transition regions of the shock, after much ofthe secular energization has been completed by the shock. We seethat the magnitude of the field-particle correlation has decreased incomparison to Figure 5.3, and that the correlation has become moreunstructured. By this point in the shock, protons in the plasmaare almost downstream, and thus no long experience the gradient inthe magnetic field off which the protons reflected, preventing themfrom gaining further energy along the motional electric field. Whatremains is oscillatory energy exchange between the plasma and theelectromagnetic fields. . . . . . . . . . . . . . . . . . . . . . . . . . . 259xxiv.5 (a) Real space trajectory of a proton as it traverses the shock frontand (b) the corresponding velocity space trajectory. Note that themagnetic gradient is assumed to be a discontinuity in this simplepicture of the perpendicular shock. The colors of the particle trajec-tories in real space (a) correspond to the particle’s location in phasespace (b). Black is upstream, blue corresponds to a proton crossingthe magnetic discontinuity before returning upstream, gaining energyalong the red trajectory, and then returning downstream and follow-ing the green trajectory. . . . . . . . . . . . . . . . . . . . . . . . . . 2615.6 Electron distribution functions (top row), C v x field-particle correla-tions (middle row), and C v y field-particle correlations (bottom row)in the shock foot and ramp region. The field-particle correlationhas a slight asymmetry that corresponds to an energy gain to the x field-particle correlation and an energy loss due to the y field-particlecorrelation. The gain in energy due to E x exceeds the loss in energydue to E y , corresponding to a net energization of the electrons. . . . . 2655.7 Electron distribution functions (top row), C v x field-particle correla-tions (middle row), and C v y field-particle correlations (bottom row)in the overshoot and transition regions of the shock. Here, we observethe opposite behavior to Figure 5.6, where now the asymmetry in thefield particle correlation is such that the particles gain energy dueto E y and lose energy due to E x . The gain in energy due to E y stillexceeds the loss in energy due to E x , so the electrons continue to gainenergy in this region of the shock. This particular energization sig-nature in the y field particle correlation arises from alignment of the ∇ x B drift and the motional electric field, E y , and relies on conserva-tion of the electron’s magnetic moment, the first adiabatic invariant.Because of the relationship between this energization mechanism andthe electron’s first adiabatic invariant, we call this adiabatic heating. 2665.8 (Top panel) Profiles along the shock normal direction of the perpen-dicular magnetic field B z (blue) and the motional electric field E y (red), (Middle panel) trajectory of an electron in the ( x, y ) plane,and (Bottom panel) the rate of work done by the electric field on theelectron j y E y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2685.9 Contour plot of the angle of the fastest-growing mode in the param-eter space of v th e /u d and u d /c (top panel). θ = 90 ◦ corresponds to apure two-stream mode, and θ = 0 ◦ corresponds to a pure filamenta-tion mode. Red crosses correspond to the four simulations presented.Growth rates versus wavenumber (bottom panels) of different modesfor the hot (right panel) and cold (left panel) cases for u d = 0 . c .We can see in the hot case, v th e /u d = 0 .
5, that the two-stream in-stability is the fastest growing mode, while when we make the beamscolder, v th e /u d = 0 .
1, the oblique modes for a variety of angles havecomparable growth rates to the pure two-stream instability. . . . . . . 274xxv.10 Growth and saturation of magnetic field (top panel) and electric field(bottom panel) energies normalized by the initial total electron en-ergy for beams with drift velocity u d = 0 . c at different tempera-tures. Solid lines correspond to 2X2V simulations with initial ran-dom modes which drive two-stream, oblique and filamentation modes,while dashed lines correspond to 1X2V simulations which only sup-port pure filamentation modes. We can see clearly the effect of thehigher dimensionality and competition between the different modes,since for all 1X2V simulations, regardless of the ratio of v th e /u d , amagnetic field grows and saturates, whereas the growth of a mag-netic field is sensitive to this ratio of v th e /u d when the two-stream,oblique, and filamentation modes are allowed to compete with eachother in two configuration space dimensions. . . . . . . . . . . . . . . 2765.11 t = 60 ω − pe and t = 100 ω − pe snapshots of the evolution of the hot case.We see the initial development of the two-stream instability and roll-up of the distribution function, before the electron tubes formed bythe two-stream instability are destroyed by the more slowly growingfilamentation instability. . . . . . . . . . . . . . . . . . . . . . . . . . 2785.12 t = 150 ω − pe and t = 300 ω − pe snapshots of the evolution of the hotcase. In the deep nonlinear phase we observe the development ofa temperature anisotropy in the distribution function, which pro-vides a secondary free energy source for the secular Weibel instabil-ity. The growth of the secular Weibel instability from the temperatureanisotropy ultimately supports a saturated magnetic field. . . . . . . 2795.13 t = 30 ω − pe and t = 50 ω − pe snapshots of the evolution of the coldcase. We observe significantly more structure in the electromagneticfields compared to the hot case in Figure 5.11, as a variety of obliquemodes all growth in tandem with the two-stream instability. Theseadditional modes also lead to additional phase space structure, incontrast to the simple plateaus in v y which formed in the hot case. . . 2805.14 t = 100 ω − pe and t = 175 ω − pe snapshots of the evolution of the coldcase. The saturated oblique modes have now given their energy backto the electrons in a much more isotropic fashion than a pure two-stream mode, leading to almost zero temperature anisotropy. With-out a temperature anisotropy to provide free energy to the Weibelinstability, the magnetic field collapses, and we observe no saturatedmagnetic field structure. . . . . . . . . . . . . . . . . . . . . . . . . . 281xxvi.15 Effective temperature anisotropy of the hot case (red) and cold case(blue) over time. The effective temperature anisotropy starts at afinite value because of the initial beams in v y and then decreasesas the beam-driven instabilities are excited. For the hot case, thetemperature anisotropy reduces to a finite value, off which the secularWeibel instability can ultimately feed. In the cold case, the effectivetemperature anisotropy decreases to a value close to one, i.e., closeto isotropy, and thus there is no free energy source for the secularWeibel instability to grow and support a saturated magnetic field. . . 2845.16 Comparison of the integrated magnetic field energy between a num-ber of particle-in-cell simulations, varying the particles-per-cell, andthe Gkeyll
VM-FP simulation of the cold case. In the limit of largeparticle-per-cell counts, the particle-in-cell simulations agree with thecontinuum kinetic result, but as the number of particles-per-cell is de-creased, a saturated magnetic field appears. . . . . . . . . . . . . . . 2865.17 Comparison of the integrated magnetic field energy between the largestand smallest particle-per-cell counts, with and without a low pass fil-ter, and the
Gkeyll
VM-FP simulation of the cold case. We can seethat the filter does allow for the recovery of the collapsing magneticfield in the low particle-per-cell count, adding credibility to the inter-pretation that the saturated magnetic field is due to noise. . . . . . . 287xxvii ist of Abbreviations
VM-FP Vlasov–Maxwell–Fokker–Planck (system of equations)DG discontinuous Galerkin (finite element method)xxviii ome of the material in thischapter has been adapted fromJuno et al. [2018], Hakim,Francisquez, Juno, andHammett [2019], and Hakimand Juno [2020].
Chapter 1: Introduction
Plasmas are ubiquitous in nature, and the study of plasmas has application toa wide variety of problems, from the development of nuclear fusion, to understandingthe dynamic interaction between the solar wind and the Earth’s magnetosphere, toelucidating the mysteries of astrophysical phenomena such as binary star collisionsor the accretion disks of black holes. Unfortunately, many plasmas of interest areonly weakly collisional and far from equilibrium, making the system best describedby kinetic theory. The use of kinetic theory significantly complicates the theoreticalanalysis and simulation of the plasma’s dynamics due to the increased dimensionalityof the corresponding equations, which are solved in a combined position and velocityphase space, along with the large collection of waves and instabilities that the kineticsystem supports. 1hile there are many avenues for tackling the numerical solution of the kineticequation, popular approaches such as the particle-in-cell method have deficienciesdue to the counting noise inherent to the algorithm. This noise can significantlydegrade the quality of the solution, in addition to making the ultimate analysis ofsimulations more challenging, especially for problems requiring high signal-to-noiseratio. In this thesis, we outline and demonstrate the utility of an approach thatdirectly discretizes the kinetic equation on a phase space grid.This approach requires care, as we must consider both the cost, since the par-tial differential equation is defined in a six dimensional phase space, alongside thechallenges which arise from the wealth of physics buried within the equation systemof interest. For example, important conservation relations, such as the conserva-tion of energy, are implicit to the kinetic equation, leading to additional difficultiesin ensuring a discrete scheme satisfies these properties. But this same wealth ofphysics contained in the kinetic equation motivates a direct discretization of thekinetic equation. We can leverage the uncontaminated phase space from a contin-uum discretization to diagnose energization processes directly in phase space andcarefully ascertain the nonlinear saturation mechanisms of unstable plasmas.Some readers may be left wondering right from the beginning why the numer-ical solution of a plasma system is at all challenging. Before diving deeper into thedetails of the algorithm and the verification of this approach, let us take a momentto address the paradoxically simple yet subtle question of what makes plasmas sorich in their underlying physics. We will then define some of the terminology usedin this brief introduction, most importantly kinetic theory , and how we use kinetic2heory to derive a useful equation system for modeling a plasma. This brief overviewwill serve as the foundation from which we will build intuition for what we wantfrom a numerical model of a kinetic plasma, most especially the fundamental physicsproperties of a plasma we would like our discretization to respect.It is the goal of this introduction to proceed in a pedagogical fashion. Wewill assume no prior plasma physics knowledge, much less knowledge about thesubtleties described so far concerning particle versus continuum methods. We willconnect this holistic introduction to plasma physics to these questions regardingour choice of numerical method in the final section of this introduction, Section 1.7,when we outline the objectives of this thesis.
Formally, a plasma is a collection of mobile, or “free,” charged particles. Col-lection in this case refers to the fact that a plasma is an N -body system, where N (cid:29)
1. By mobile, or “free,” we mean that the particles in a plasma are notconfined by inter-particle forces and the individual particles in a plasma behavesimilarly to a gas, as opposed to a solid or crystalline structure, albeit with theadded complication of the particles being charged. And in this case, the fact thatthe particles are charged means that the particles are subject to the Lorentz forceand can interact with each other via microscopic electromagnetic fields governed byMaxwell’s equations.This definition of a plasma is somewhat restrictive. In this case, we limit3urselves to what are commonly referred to as weakly coupled plasmas, as the mobilecomponent of our definition implies the kinetic energy of the particles is much, muchgreater than the potential energy of the particles. Likewise, we restrict our attentionto plasmas which are fully ionized.Let us be a bit more concrete about our definition of a plasma, so that we cangain more intuition for what it means to limit ourselves to this subset of so-calledweakly coupled plasmas. Consider a gas of some number of charged species, suchas a gas of protons and electrons, where each charged species has density n . Since n , the density, is the number of particles in a given volume, the average distancebetween two charged particles is roughly n − / . This rough estimate for the averagedistance between two particles can be used to approximate the average potentialenergy per particle in this sample plasma,Φ ∼ π(cid:15) e r ∼ π(cid:15) n e , (1.1)where e is the elementary charge, i.e., the charge carried by a proton. Likewise,we can estimate the average kinetic energy of a particle using the equi-partitiontheorem, 12 m s (cid:104) v (cid:105) ∼ k B T s , (1.2)where m s and T s are the mass and temperature of the particles of species s , re-spectively, and (cid:104)·(cid:105) denotes an average over all particle velocities at a given point inspace. Here, k B is Boltzmann’s constant.Thus, our definition of a plasma, that the average kinetic energy of the particles4s much larger than the average potential energy of the particles, implies32 k B T s (cid:29) π(cid:15) n e , (1.3)or 6 πn (cid:18) (cid:15) k B T s n e (cid:19) (cid:29) . (1.4)This expression at first glance looks somewhat unremarkable, but upon raising bothsides to the 3 / λ D s = (cid:114) (cid:15) k B T s n e , (1.5)we obtain (6 π ) n λ D (cid:29) . (1.6)Note that in the definition of the Debye length we could have a species dependentdensity, but since we have assumed that both the protons and electrons have thesame density we have set n p = n e = n .Ignoring the constant for a moment, we may gain a bit of intuition for what wehave just found. n λ D is the number of particles in a cube with side lengths equal tothe Debye length. We will gain a deeper understanding of the physical significanceof this expression which follows from our definition of the a plasma in the followingsection. 5 .2 The Debye length and the Plasma Parameter Because plasmas are a large collection of charged particles, inevitably, theparticles will rearrange themselves in response to each other’s charges. Considerone particular particle with a positive charge. Since the particle’s charge is positive,the electrons in the plasma will be attracted to the particle, while the positivelycharged ions will be repelled, creating a local area where the density of the electronshas increased, while the density of the positively charged ions has decreased.Without loss of generality, let us take the positively charged ions to be protons.Then, if the electrons have density n e , and the protons have density n p , Poisson’sequation tells us that the electric potential for the plasma is ∇ φ = − ρ c (cid:15) = e(cid:15) ( n e − n p ) − q T δ ( r ) , (1.7)where we denoted the charge of the particular particle as q P and used the Diracdelta function, δ ( r ), to denote the position of the particle in space.We need to determine how the density of electrons and protons has been mod-ified by the presence of this particular charge. If we assume that we have waitedlong enough for electrons and protons to come into thermodynamic equilibrium withthe particular charge, i.e., that we wait long enough that the temperature becomesa well-defined quantity, we can use equilibrium statistical mechanics. Without in-sisting that the electrons and protons have the same temperature, only that wecan define temperatures, the densities of the electrons and protons are given by the6oltzmann distribution, n e = n exp (cid:18) eφk B T e (cid:19) , (1.8) n p = n exp (cid:18) − eφk B T p (cid:19) , (1.9)where n is the density of the electrons and protons far away from the particularcharged particle of interest, i.e., far enough away so that electric potential from theparticular charged particle of interest is zero.But, recall what we have continually reiterated from our definition of a plasma:the average potential energy of the particles is much less than the average kineticenergy. Therefore, eφ (cid:28) k B T s , and the exponential function can be Taylor expandedfar away from r = 0, the location of the particular charged particle, so that ∇ φ = 1 r ddr (cid:18) r dφdr (cid:19) = e n (cid:15) k B (cid:18) T e + 1 T p (cid:19) φ. (1.10)Using Eq. (1.5), we can see that the above equation simplifies to1 r ddr (cid:18) r dφdr (cid:19) = (cid:32) λ D e + 1 λ D p (cid:33) φ. (1.11)If one waits longer for the protons and electrons to come in to thermodynamicequilibrium with each other so the temperatures of the two species are equal, T e = T p = T , then (cid:32) λ D e + 1 λ D p (cid:33) = 2 λ D . (1.12)The solution to this differential equation then follows from trying functions of the7orm φ = ˜ φ/r , so that d ˜ φdr = 2 λ D ˜ φ. (1.13)The only solution which respects the boundary condition that the electric potential, φ , not blow up as r → ∞ is a solution of the form φ ( r ) = A exp (cid:32) − √ λ D r (cid:33) , (1.14)where A is a constant of integration. This constant of integration can be found byconsidering the boundary condition at r = 0, where the electric potential will bedominated by the particular charged particle of interest. We know from Gauss’ lawthat the electric potential of an individual charged particle is simply 1 / (4 π(cid:15) ) q P /r so that the complete solution for the electric potential of an individual chargedparticle in a plasma is φ ( r ) = 14 π(cid:15) q P r exp (cid:32) − √ λ D r (cid:33) . (1.15)This functional form for the potential implies that the electric potential, andthus the charge, of a particle falls off much faster than just the inverse of the distance.It thus follows from this solution that the charged particles in a plasma rearrangethemselves to cancel the charges of their neighbors, and that the characteristic lengthscale on which a plasma’s charged particles are screened is the Debye length.We return now to Eq. (1.6) with newfound understanding of the physical sig-nificance of the Debye length. If the number of particles in a Debye cube is verylarge, then it becomes a bit more apparent why these plasmas are often referred8o as weakly coupled. When the number of particles in a Debye cube is large, noindividual electrostatic interaction between particles is of dynamical importance.Because a single particle is feeling the electrostatic potential of a large number ofparticles in its immediate vicinity, the individual electrostatic interactions betweenparticles are dwarfed by the accumulation of all of the electrostatic interactions. Weneed not discuss the electric field one particle exerts on another; rather, what werequire is the net electric field of all of the particles in a Debye cube, so that we mayobtain the aggregated response of the particles in the plasma.In this regard, we should avoid being dismissive of the individual electrostaticinteractions occurring within a Debye cube in a plasma. It is true that the sum isgreater than the individual parts in a weakly coupled plasma where there are manyparticles in a Debye cube. But in this vein, we must distinguish between individual and collective effects. Eq. (1.15) shows us that the electrostatic potential of an individual particle falls off exponentially on scales larger than the Debye length,but the collective effects of all the individual particles within the Debye cube canbe of critical importance for the plasma’s dynamics. The collective response of theplasma, on scales above and below the Debye length, is a crucial consideration inthe derivation of the resulting equations of interest in the forthcoming sections. For now, let us use this discussion as a segue into the original question whichgalvanized defining a plasma: “Why is modeling a plasma hard?” Regardless of9hether individual particle-particle interactions are important or irrelevant in amany-body plasma, it does not change the fact that the equations of motion forparticles in electromagnetic fields are well-known and easy enough to solve numer-ically. So why not model all the particle-particle interactions in the many-bodysystem?The answer may be obvious just from the description of a plasma as a many-body system. Since the challenge inherent in modeling a plasma can be seen evenwithout considering the magnetic field, for readability, we will ignore the magneticfield for now and only consider the particle-particle interactions from the plasma’sself-consistent electric field. In this case, one could evolve a single particle underthe equations of motion, d x k dt = v k , (1.16) d v k dt = q k m k N (cid:88) i =1 ,i (cid:54) = k E i , (1.17)where k is the label for the particle being evolved. One would then proceed to solvethese equations for each k = 1 , . . . , N . The electric field in this system of equationsis given by E i = 14 π(cid:15) q i ( x i − x k ) ˆ x ik . (1.18)Here, E i is the electric field particle i exerts on particle k , so that in this notation, x i is the i th particle’s position, x k is the position of the particle currently beingevolved, i.e. the particle the electric field E i is acting on, and ˆ x ik is the unit vectorpointing from x i to x k . 10hat is the computational cost to solve these two coupled sets of ordinarydifferential equations? For each of the N particles, we require at least N − N (cid:88) i =1 ( N − i ) = N − N . (1.19)Thus, the computational complexity of such an algorithm is O ( N ), meaningif we double the number of particles we are evolving numerically, we quadruple thecost to compute the solution. In addition, this argument implies that, at minimum,this method requires on the order of N operations to perform a single time step.But, modern supercomputers are already fast and will only continue to speedup with time. As of the completion of this thesis, we have achieved exascale com-puting . Is this enough to make this approach feasible?We require a concrete example. Let us consider the ITER (International Ther-monuclear Experimental Reactor) Tokamak currently being built to demonstrate thefeasibility of nuclear fusion as a power source. According to the website for ITER(ITER 2020), the vacuum vessel is 840 m in volume, and the average density ofthe electrons in the plasma will be ∼ m − . The plasma will be quasi-neutral,so a conservative estimate of the number of particles, protons, electrons, and alpha With the caveat that this is only for reduced precision, i.e., the supercomputers at the writingof this thesis could achieve an exaflop, 10 floating point operations per second, if one only requiredsingle precision. ∼ particles. A single “shot,” or run of the experiment,is expected to last anywhere from 100 to 1000 seconds. So, could we model ITERthrough a full experimental shot, tracking every particle in the experiment?To answer this question, we require one final piece of information: the fastesttime scale in the system, so that we know how many time-steps we would requireto evolve all the particles for 100 to 1000 seconds. The fastest time scale in aplasma can be found by considering how particles jostle about. We have found acharacteristic length scale, the Debye length, Eq. (1.5), and it is simple enough todefine a characteristic velocity from the equipartition theorem12 mv rms ∼ N k B T, (1.20)where N is the number of degrees of freedom. Each degree of freedom thus hasroot-mean-square velocity v th s = (cid:114) k B T s m s . (1.21)This speed is called the thermal velocity, and approximates the average speed ofparticles with temperature T s (or energy k B T s ). Note that Eq. (1.21) is typicallyreferred to as the thermal velocity even though it is not a vector quantity, and thoughwe will maintain this nomenclature throughout our discussion, we will attempt tominimize confusion by emphasizing Eq. (1.21) is a speed where appropriate. Theratio of these two quantities for the electrons, v th e λ D e = (cid:115) e n e (cid:15) m e , (1.22)12as units of inverse seconds and defines a frequency, ω pe = (cid:115) e n e (cid:15) m e . (1.23)Although we have not demonstrated this here, this frequency is roughly thehighest frequency in the system. With the Debye length as our length scale, and thisfrequency, ω pe , called the plasma frequency, setting a time scale, we have a roughestimate of the cost of numerical integration of Eqns. 1.16–1.17. For the purposesof numerical integration, we wish to avoid particles moving distances greater thanthe Debye length on time scales shorter than the inverse plasma frequency, as thismay introduce numerical instabilities into the integration of the particle orbits.In ITER, the plasma frequency for the electrons is ∼ – 10 Hz, so evenif our numerical method is very robust and requires only one time-step per inverseplasma frequency, we require a large number of time steps per second. In total,assuming N operations per time-step and 10 time steps to model a single run ofthe experiment, a computer simulation of all the particle dynamics would need to doapproximately 10 floating point operations. If current supercomputers, even withthe trade-offs in terms of floating point precision, can only perform 10 floating pointoperations per second, an exa-flop, we would still require 10 seconds of simulationtime. For reference, the universe has only existed for just over 10 seconds, so asimulation like this requires quite a few universe lifetimes with modern computerarchitecture.It is worth taking a moment to go further and try to improve this algorithmbefore we give up on tracking every single particle-particle interaction. For example,13here are algorithms which would reduce the cost of computing the electric field,and thus the algorithm, from O ( N ) to O ( N ), by using a multipole expansionof the electrostatic potential [Greengard and Rokhlin, 1987]. Such algorithms arecommonly employed in computational cosmology for solving for the gravitationalpotential of a large number of dark matter particles and simulating galactic dynamicsand evolution [Stadel, 2001]. Even with a multipole expansion of the electric field,the total number of operations would reduce from 10 to only ∼ , and thetotal time to 10 seconds. Unfortunately, this is not a large enough reduction, andone would have to track a lot fewer particles for a lot less time, to say nothingof the added complexity of the magnetic field acting on individual particles. Forexample, it would be quite a large simulation to run on a modern supercomputerfor 4 continuous months, ∼ seconds, so one would have to eliminate 12 orders ofmagnitude in some combination of the amount of time being simulated and numberof particles being evolved, again to say nothing of the assumptions which made thisback-of-the-envelope calculation remotely reasonable.So, what is one to do? All hope is not lost for the reason we have emphasizedthroughout these introduction sections. That is, the individual particle-particle dy-namics are of minimal importance in a weakly coupled plasma, and in fact whatis principally important to the plasma’s dynamics is its collective response to elec-tromagnetic fields. In other words, a weakly coupled plasma is an ideal system forwhich a mean-field theory may arise, one which allows for the study of the plasmaof interest in a statistical sense. We will be careful to define both what we mean bya mean-field theory and what we mean by thinking about the particle dynamics in14 statistical sense in the next section. Up until now, we have concerned ourselves with the microscopic properties ofthe plasma, and, as demonstrated in the previous section, this limits our ability tomodel the plasma. We would like to still respect the fact that the plasma is madeof discrete particles though, and so we turn to kinetic theory. “Kinetic” in thiscase means, “pertaining to motion,” and kinetic theory provides the foundation toconsider the motion of all of the particles in the plasma, but without the stringentrequirement to track individual particle dynamics and interactions.Consider the density of particles of species s , N s , in a combined position andvelocity space. This density is simply a sum of Dirac delta functions denoting theindividual positions and velocities of every particle in the plasma, N s ( x , v , t ) = N (cid:88) i =1 δ ( x − X i ) δ ( v − V i ) , (1.24)where we have used capital X i and V i to specify the individual particle positionsand velocities in the x – v phase space. The motions of the particles in this plasmain space and time are governed by the particle characteristics , d X i dt = V i , (1.25) d V i dt = q s m s [ E m ( X i , t ) + V i × B m ( X i , t )] , (1.26) Assuming the particles are traveling at velocities much less than the speed of light, | v | (cid:28) c ,so we can ignore the Lorentz boost factors, and further that the self-force due to radiation is ofminimal dynamical importance. m . We can see that change in velocity doesnot couple to the acceleration in such a way as to require a third set of equations forthe time derivative of the acceleration of the particles and thus the two equationsEqns. (1.25) and (1.26) are closed once we specify evolution equations for the elec-tromagnetic fields. In this case, the evolution of the electromagnetic fields is givenby Maxwell’s equations, ∂ B m ( x , t ) ∂t + ∇ x × E m ( x , t ) = 0 , (1.27) (cid:15) µ ∂ E m ( x , t ) ∂t − ∇ x × B m ( x , t ) = − µ J m ( x , t ) , (1.28) ∇ x · E m ( x , t ) = (cid:37) mc ( x , t ) (cid:15) , (1.29) ∇ x · B m ( x , t ) = 0 , (1.30)where the microscopic charge density and current density are given by (cid:37) mc ( x , t ) = (cid:88) s q s (cid:90) N s ( x , v , t ) d v , (1.31)and J m ( x , t ) = (cid:88) s q s (cid:90) v N s ( x , v , t ) d v , (1.32)respectively.This density of particles of species s , N s , can neither be created nor destroyedbecause the number of particles cannot change in time, assuming the system isclosed. This attribute implies N s obeys a continuity equation. For a reader unfa-miliar with the concept of a conservation equation, consider a quantity f ( r , t ), a16unction of some space r and time, which in the process of its motion in space andtime can neither be created nor destroyed. Then, this quantity f ( r , t ) obeys (cid:90) Ω ∂f ( r , t ) ∂t d r = 0 , (1.33)where Ω is the domain the function f ( r , t ) is defined in. But this quantity f ( r , t ) canstill be transported throughout the domain Ω. Let us define the flux function for thefunction f ( r , t ) as G , where G could be as simple as a constant, or as complex as anonlinear function of the quantity of interest, G = G ( f ). Then, the flux of f ( r , t ) is G f ( r , t ). We have argued in Eq. (1.33) that the time derivative of the integral overthe whole domain of the function f ( r , t ) is zero, which means that the flux of thefunction f ( r , t ) out the boundary of the domain must also be zero, (cid:73) ∂ Ω f ( r , t ) G · d S = 0 , (1.34)so that we can say (cid:90) Ω ∂f ( r , t ) ∂t = − (cid:73) ∂ Ω f ( r , t ) G · d S . (1.35)But using the divergence theorem, (cid:73) ∂ Ω f ( r , t ) G · d S = (cid:90) Ω ∇ r · [ G f ( r , t )] d r , (1.36)so that we can argue (cid:90) Ω ∂f ( r , t ) ∂t + ∇ r · [ G f ( r , t )] d r = 0 , (1.37)allows us to attain an evolution equation for the function f ( r , t ) using the fact that17he integrand itself must also be equal to zero, ∂f ( r , t ) ∂t + ∇ r · [ G f ( r , t )] = 0 . (1.38)Because the time rate of change of the quantity, f ( r , t ), integrated over the wholedomain, is zero, i.e., f ( r , t ) is not appearing or disappearing over time, f ( r , t ) willinevitably obey an equation of the form Eq. (1.38).If the density of particles of species s , N s , obeys a similar equation, whatis the flux function to advect N s in the combined position-velocity phase space?It is simply the characteristics defined in Eqns. 1.25–1.26, but importantly, with achange of variables from the individual particles’ physical locations and velocitiesto the phase space coordinates. This change follows from the fact that N s is afunction of the phase space variables x and v , not a function of the individualparticle positions. Thus, the conservation equation governing the evolution of thedensity of particles of species s is ∂N s ( x , v , t ) ∂t + ∇ x · [ v N s ( x , v , t )]+ ∇ v · (cid:26) q s m s [ E m ( x , t ) + v × B m ( x , t )] N s ( x , v , t ) (cid:27) = 0 . (1.39)This equation is more commonly known as the Klimontovich equation, or Klimon-tovich’s equation [Klimontovich, 1967, Nicholson, 1983]. Oftentimes, Eq. (1.39) isrearranged to emphasize the connection between the particle characteristics, and18ow N s advects in phase space, ∂N s ( x , v , t ) ∂t + v · ∇ x N s ( x , v , t )+ q s m s [ E m ( x , t ) + v × B m ( x , t )] · ∇ v N s ( x , v , t ) = 0 , (1.40)where we have exploited the fact that ∇ x · v = 0 , (1.41) ∇ v · [ E m ( x , t ) + v × B m ( x , t )] = 0 , (1.42)in the rearrangement of Eq. (1.39) to Eq. (1.40). Eq. (1.41) likely seems intuitive,the velocity coordinate v of course does not depend on the configuration spacecoordinate x , and Eq. (1.42) follows from properties of the cross product, in additionto the fact that the electromagnetic fields themselves do not depend on velocity. Justas Eq. (1.39) follows from the fact that the density of particles of species s cannotbe created or destroyed, Eq. (1.40) shows that N s is constant along characteristics,i.e., DN s ( x , v , t ) Dt = 0 , (1.43)where D/Dt is a convective derivative,
DDt = ∂∂t + v · ∇ x + q s m s [ E m ( x , t ) + v × B m ( x , t )] · ∇ v , (1.44)a time derivative with respect to a moving coordinate system.The Klimontovich equation is essentially an alternative way of expressing themotion of every particle in phase space, and it suffers from the same issues discussed19n Section 1.3. We do not want to track the motion of every particle in phasespace, especially if we can prioritize collective effects over individual particle-particleinteractions and microscopic fields in a weakly coupled plasma. But how does onego from the Klimontovich equation to a more suitable representation of a weaklycoupled plasma’s dynamics? How does one obtain an equation which contains theaccumulated physics of the many individual particle interactions in our many-bodysystem?We now leverage a mathematical technique known as an ensemble average.An ensemble average is an average over realizations of the solution, i.e., an aver-age of the results of different initial conditions. Imagine, if one could, solving theKlimontovich equation many times and finding with different initial conditions thecollective motion of the plasma was similar while the details of the individual particleinteractions varied. A concrete example: imagine solving the Klimontovich equa-tion repeatedly for the plasma system considered in Section 1.2. While the detailsof the relaxation to a Debye-shielded charged particle may vary from realization torealization depending on how exactly we initialize the electrons around the partic-ular positively charged particle, we still end up at the same place: a distribution ofelectrons moving around a positively charged particle, shielding its charge stronglybeyond this characteristic length scale of the Debye length.So what would this mean for the collective behavior, a Debye shielded chargedparticle for example, to be roughly similar between different realizations of theplasma’s dynamics? We turn now to the language of statistics to lay a solid founda-tion for the next derivation. This roughly similar collective behavior is an example20f the average response of the plasma to its internal, individual particle-particle,dynamics, likely with some standard deviation or variance across different realiza-tions. While every realization of the Klimontovich equation is deterministic, thereis also some stochasticity between different realizations. We now argue that a moreappropriate, and ultimately more useful, way to characterize the plasma’s dynamicsis by focusing on this stochasticity, so as to obtain a probabilistic description of theplasma’s dynamics.We define the particle distribution function for species s as f s ( x , v , t ) = (cid:104) N s ( x , v , t ) (cid:105) , (1.45)where (cid:104)·(cid:105) defines the ensemble average, the average over many (formally an infinitenumber) realizations of the plasma. The particle distribution function tells us howmany particles are likely to be found in a small volume ∆ x ∆ v . Before, the densityof particles of species s could only take the value of 0 or 1—it was a simply a sumof Dirac delta functions for the exact location in configuration and velocity spaceof each particle. We have now shifted perspective to focusing on the probability offinding a particle at a particular location in position-velocity phase space.To obtain an equation for the evolution of the particle distribution functionwe ensemble average Eq. (1.39), the Klimontovich equation, ∂f s ( x , v , t ) ∂t + ∇ x · [ v f s ( x , v , t )] + ∇ v · (cid:26) q s m s [ E ( x , t ) + v × B ( x , t )] f s ( x , v , t ) (cid:27) = − (cid:28) q s m s ∇ v · { [ δ E ( x , t ) + v × δ B ( x , t )] δN s ( x , v , t ) } (cid:29) , (1.46)21here δN s ( x , v , t ) = N s ( x , v , t ) − f s ( x , v , t ) , (1.47) δ E ( x , t ) = E m ( x , t ) − E ( x , t ) , (1.48) δ B ( x , t ) = B m ( x , t ) − B ( x , t ) , (1.49)and we have used the shorthand E = (cid:104) E m (cid:105) and B = (cid:104) B m (cid:105) for the ensemble-averaged fields. By definition, the ensemble average of the fluctuating quantities (cid:104) δN s (cid:105) = (cid:104) δ E (cid:105) = (cid:104) δ B (cid:105) = 0. Thus, in the process of ensemble averaging the Klimon-tovich equation, terms proportional to (cid:104) N s δ E (cid:105) = N s (cid:104) δ E (cid:105) and their permutationswill vanish, leaving only the term which is quadratic in the fluctuating quantities.Eq. (1.46) is the plasma kinetic equation. We are close to a more useful equa-tion, as we have replaced a deterministic equation with a probabilistic equation,which will allow us to understand the plasma’s collective behavior irrespective ofthe details of the discrete particle dynamics. Importantly, in the process of ensembleaveraging, we now have on the left hand side of Eq. (1.46) how the plasma responds toensemble-averaged electromagnetic fields, i.e., effective electromagnetic fields fromthe collective motions of the entire plasma instead of individual particle-particleelectromagnetic interactions. But we have retained the effects of the discrete parti-cle interactions on the right-hand side, or at least the accumulation of many discreteparticle interactions. We need one final simplification, to complete the derivation ofthe equation, and equation system, which is of principal interest in this thesis.22 .5 Bogoliubov’s Timescale Hierarchy and theVlasov–Maxwell–Fokker–Planck System of Equations To complete the probabilistic picture of a plasma, we need to know the physicsof the right hand side of the plasma kinetic equation, Eq. (1.46). We have alreadyshown in Section 1.2 that the electric field from an individual particle in the plasmafalls off exponentially at length scales larger than the Debye length, so we mightexpect the physics of these fluctuating fields to be at scales smaller than the De-bye length. Indeed, that must be the case, as the fluctuating electromagnetic fieldsbecome vanishingly small on scales larger than the Debye length, i.e., the “mi-croscopic” electromagnetic fields and ensemble-averaged electromagnetic fields areindistinguishable when one is no longer considering “microscopic” scales. This jus-tification may seem like a tautology, that once we consider length and time scales inthe plasma on which collective effects arise, we no longer have to concern ourselveswith these fluctuating quantities. In fact, it can be shown that the term on the righthand side of Eq. (1.46) scales like Λ − , the inverse of the plasma parameter, so it is One can see this scaling with a thought experiment. Imagine breaking an electron into aninfinite number of pieces, so that n e → ∞ , m e → , e → n e e, e/m e , v th e all remain constant. Note that in this thoughtexperiment, the electron temperature T e → ω pe , λ D are both constant through the break up ofthe electron. Importantly, this means the plasma parameter Λ = nλ D → ∞ . Now, any volume,no matter how small contains an infinite number of point particles with an infinitesimal charge.Statistical mechanics tells us that the fluctuations in the density will scale like the square rootof the density, δN s ∼ N / ∼ Λ / , but the electromagnetic fields, for example the electric fieldfrom Poisson’s equation, scales like δ E ∼ eδN ∼ N − N / ∼ N − / , because the charge density isconstant, meaning e ∼ N − . Thus, the right hand side of the plasma kinetic equation, Eq. (1.46),is constant in this thought experiment. But on the left hand side of Eq. (1.46), the distributionfunction becomes infinite in this thought experiment, f e → ∞ , so the right hand side vanisheswith the scaling of the left hand side, N ∼ Λ. The contribution of the fluctuating fields is thusΛ − smaller in scaling for the evolution of the particle distribution function. (cid:29) ∼ Λ Coulomb collisions the particles are experiencingwithin a Debye cube is slightly more subtle. While each individual Coulomb collisiona particle experiences is a small effect, a small deviation to its trajectory, the cu-mulative effect of many Coulomb collisions can significantly perturb the path of theparticle. One may have to wait an exceedingly long time for the cumulative effectof many Coulomb collisions to noticeably affect the plasma’s dynamics comparedto the collective motion of the plasma contained in the left hand side of Eq. (1.46),especially given the scaling of the right hand side compared to the left hand sideof Λ − . But, wait long enough, and small deviations will accumulate to make animpact on the dynamics of these plasma particles.How long is long enough to wait for Coloumb collisions to be of dynamical im-portance? Bogoliubov’s timescale hierarchy [Nicholson, 1983] tells us that a plasma’sdynamical evolution consists of the following stages:1. Pair correlations are established, leading to shielded Coloumb potentials onDebye scales. These correlations are established on the time scale of the in-verse electron plasma frequency, Eq. (1.23), and once these correlations areestablished, for tω pe (cid:38)
1, collective behavior dominates over individual parti-cle interactions.2. The plasma relaxes to local thermodynamic equilibrium. We will show in the24ext section, Section 1.6, that this relaxation is contained in the physics ofcollisions, the right hand side of Eq. (1.46). If we define a collision frequency ν , we expect the plasma to relax to local thermodynamic equilibrium on timescales νt (cid:38)
1, a much longer time scale than the plasma frequency ν/ω pe ∼ Λ − , given the scaling of the terms in Eq. (1.46).
3. On time scales νt (cid:29)
1, the plasma attempts to relax to global thermodynamicequilibrium. The plasma’s boundary conditions or sources may prevent thisglobal relaxation from occurring, but on these time scales, we would seekalternative means of describing the plasma so as to capture its transport.We have engaged in a small amount of circumlocution as we attempted to notget too far ahead of ourselves in a heuristic derivation of the equation system ofinterest. A detailed derivation of the collisional response of the plasma, valid for all We can also argue for the difference in the time scale of collisions versus the plasma frequencyby estimating the size of the mean free path, the average distance a particle travels before itexperiences a significant deflection due to a binary inter-particle Coulomb collision, compared tothe Debye length. Here significant deflection could mean the accumulation of many small angleCoulomb collisions, i.e., small deviations due to individual electrostatic interactions, or by onelarge angle collision due to a close fly-by of one plasma particle of another. The mean free pathcan be estimated from the collisional cross section σ , λ mfp ∼ nσ ∼ T ne , (1.50)where we have estimated the collisional cross section σ ∼ d by balancing the potential energy ata distance d with the average kinetic energy of the particle, e /d ∼ T . Comparing the mean freepath and the Debye length, we have, λ mfp λ D ∼ T ne (cid:114) e nT ∼ nλ D , (1.51)which is the plasma parameter Λ ∼ nλ D (cid:29)
1. But if the mean free path is much larger than theDebye length, than considering a thermal particle moving with velocity v th , v th v th λ mfp λ D = νω pe ∼ Λ . (1.52) (cid:28) q s m s ∇ v · ([ δ E + v × δ B ] δN s ) (cid:29) ∼ ∇ v · (cid:20) − ( A f s ) + 12 ∇ v · (cid:16) ←→ D f s (cid:17)(cid:21) , (1.53)where we have dropped the spatial dependence temporarily for notational conve-nience. We note that the details of the derivation of Eq. (1.53) can be found inChapter 3 and Appendix A of Nicholson [1983], where the author performs the fullBBGKY (Bogoliubov-Born-Green-Kirkwood-Yvon) hierarchy to derive the equationsystem of interest, including the Fokker–Planck equation.Each individual Coulomb collision has a small effect on the trajectory of a par-ticle in a plasma, so in analogy with Brownian motion in a gas, the cumulative effectof many Coulomb collisions is a diffusive process in velocity space. The exact expres-sions for the drag coefficient, A , and the diffusion tensor, ←→ D , in Eq. (1.53) requiremore careful treatment, and a more in depth discussion and derivation [Rosenbluthet al., 1957]. We choose, in this thesis, a simplified form for the drag and diffusioncoefficients, A = u − v , (1.54) ←→ D = 2 Tm ←→ I , (1.55)where ←→ I is the identity tensor. These simplified drag and diffusion coefficients are26elated to the velocity moments of the particle distribution function, u ( x , t ) = (cid:82) v f ( x , v , t ) d v (cid:82) f ( x , v , t ) d v , (1.56) T ( x , t ) m = 13 (cid:82) | v − u ( x , t ) | f ( x , v , t ) d v (cid:82) f ( x , v , t ) d v , (1.57)where the factor of 1 / as the diffusion coefficient, v th = T /m ,but we want to emphasize the connection between the drag and diffusion coefficientsand the velocity moments of the particle distribution function, so we will switch no-tation and define the diffusion coefficient with respect to the temperature. Whilethese expressions for the drag and diffusion coefficients may appear unintuitive atfirst glance, there is a rich history for their use as a lowest order approximation tothe Fokker-Planck behavior we expect the plasma to have due to Coloumb collisions.The “full” Fokker–Planck operator [Rosenbluth et al., 1957] includes addi-tional physics, most importantly that collisions should be velocity dependent andthat faster particles experience fewer collisions. However, the solution to the com-plete Fokker–Planck operator is more computationally demanding. For the purposesof the algorithms and physics presented in this thesis, the most important compo-nent of modeling collisions is that collisions are a Fokker–Planck operator modelingthe drag and diffusion in velocity space particles should experience from the accu- Note that in this definition, Boltzmann’s constant has been absorbed into the temperature, k B T s → T s , so that the units of temperature are an energy, e.g. electron-volts or Joules. s in phase space, ∂f s ( x , v , t ) ∂t + ∇ x · [ v f s ( x , v , t )] + ∇ v · (cid:26) q s m s [ E ( x , t ) + v × B ( x , t )] f s ( x , v , t ) (cid:27) = ν s ∇ v · (cid:26) [ v − u s ( x , t )] f s ( x , v , t ) + T s ( x , t ) m s ∇ v f s ( x , v , t ) (cid:27) , (1.58)where we have added the collision frequency ν s , which will allow us to accuratelycharacterize the contribution of the collision operator to the dynamics in comparisonto the collisionless evolution from the macroscopic electromagnetic fields. In otherwords, we can pick the collision frequency ν s to be Λ − smaller than the electronplasma frequency, ω pe , as it should be. This equation is coupled to the ensemble-averaged Maxwell’s equations for the evolution of the macroscopic electromagnetic28elds, ∂ B ( x , t ) ∂t + ∇ x × E ( x , t ) = 0 , (1.59) (cid:15) µ ∂ E ( x , t ) ∂t − ∇ x × B ( x , t ) = − µ J ( x , t ) , (1.60) ∇ x · E ( x , t ) = (cid:37) c ( x , t ) (cid:15) , (1.61) ∇ x · B ( x , t ) = 0 , (1.62)where the current density J and charge density ρ c are related to velocity momentsof the particle distribution function, ρ c ( x , t ) = (cid:88) s q s (cid:90) f s ( x , v , t ) d v , (1.63) J ( x , t ) = (cid:88) s q s (cid:90) v f s ( x , v , t ) d v . (1.64)Having closed the equation system with the coupling between the electro-magnetic fields and the particle distribution function, the Vlasov–Maxwell–Fokker–Planck system of equations is complete. This equation system forms the foundationfor the theory of weakly coupled plasmas and will be the principal focus for the re-mainder of this thesis. The particle distribution function contains a wealth of data,and we are strongly motivated by the veritable treasure trove of information theparticle distribution function holds. We thus want to make sure however we chooseto numerically integrate the Vlasov–Maxwell–Fokker–Planck system of equations,we can still leverage the particle distribution function to understand the plasma’sdynamics.While we now have an equation system we can actually use the computer to29olve, having simplified the N-body dynamics of the plasma to a probabilistic equa-tion system in a six dimensional phase space, we must still be careful in our nextsteps for how we discretize the Vlasov–Maxwell–Fokker-Planck system of equations.In this next section, we will review many of the most important properties of theVlasov–Maxwell–Fokker-Planck system of equations. These properties of the con-tinuous system of equations will help us ultimately make an informed decision inboth our choice, and implementation, of the numerical method for constructing the discrete
Vlasov–Maxwell–Fokker-Planck system of equations.
Before we begin this discussion of the properties of the continuous Vlasov–Maxwell–Fokker–Planck (VM-FP) system of equations, we want to simplify some ofour notation for readability. Firstly, we will separate the collisionless and collisionalcomponents of the system of equations, ∂f collisionlesss ∂t = −∇ x · ( v f s ) − ∇ v · (cid:20) q s m s ( E + v × B ) f s (cid:21) , (1.65) ∂f cs ∂t = ν s ∇ v · (cid:20) ( v − u s ) f s + T s m s ∇ v f s (cid:21) , (1.66) ∂f s ∂t = ∂f collisionlesss ∂t + ∂f cs ∂t , (1.67)and we will drop the notation for the explicit dependence on configuration spaceand phase space. Importantly, this separation is for readability of the coming dis-cussion of the properties of the VM-FP system of equations, and is not due to any30xplicit need to separate the collisionless and collisional components of the plasma’sevolution. These contributions to the plasma’s dynamics are on equal footing, andwe could just as easily demonstrate the properties of VM-FP system of equationsholistically, but we feel this makes the subsequent discussion unnecessarily dense.Suffice to say, if, for example both the collisionless and collisional components of theVM-FP system of equations conserve the total energy in the system, we know thattogether the whole system conserves the energy.For brevity of notation, we will introduce the full phase space variable z =( x , v ) so that the collisionless component of the VM-FP system of equations can bewritten as, ∂f collisionlesss ∂t = −∇ z · ( α s f s ) , (1.68)a conservation equation in the full phase space with phase space flux, α s = (cid:18) v , q s m s [ E + v × B ] (cid:19) . (1.69)We will also use the notation K to define the phase space domain that the distri-bution function is defined on and Ω to define the configuration space domain thatvelocity space moments and electromagnetic fields are defined on.We have hinted at the connection between the bulk properties of the plasmaand the velocity moments of the particle distribution function. Both the compo-nents of the drag and diffusion coefficients, Eqns. 1.56–1.57, and the charge densityand current density that couple the particle dynamics to the electromagnetic fields,Eqns. 1.63–1.64, are defined with integrals over velocity space of the particle dis-31ribution function. We should solidify this connection with a number of definitionswhich will prove critical to our discussion of the properties of the continuous VM-FPsystem of equations. We will focus on the first few velocity moments, ρ s = m s n s = m s (cid:90) f s d v , (1.70) M s = m s n s u s = m s (cid:90) v f s d v , (1.71) E s = 32 n s T s + 12 m s n s | u s | = 12 m s (cid:90) | v | f s d v , (1.72)i.e., the mass density, momentum density, and energy density of the plasma specieswith label s .To gain intuition for why these quantities can be defined this way, recall whatthe particle distribution function is: the probability of finding a particle in a givenvolume ∆ x ∆ v . Thus, if we integrate the particle distribution in velocity space,Eq. (1.70), we are computing the number density of the particles (the number ofparticles per unit volume) at a given configuration space location, or the mass densityat a given configuration space location. For the higher velocity moments, we canmake similar connections. The velocity weighted moment, which includes a factor ofthe particle mass, Eq. (1.71), tells us the amount of momentum per unit volume, themomentum density, at a particular configuration space location. We might have alsoguessed this physical interpretation for Eq. (1.71) by considering what statistics tellsus the first velocity moment is: the average velocity of the particles. This same logiccan be applied to Eq. (1.72); the second velocity moment, weighted by m s /
2, givesus the total energy density—internal, 3 / n s T s , plus kinetic, 1 / m s n s | u s | —of the32articles at a given configuration space location. And, in the language of statistics,the second velocity moment is related to the spread, or standard deviation, of theparticle velocities.In both velocity space moment cases, we should be careful not to make theconnection between our physical intuition and our knowledge of statistics superficial.The average velocity is only u , not the full definition of the momentum density inEq. (1.71), and we have to account for this average velocity when computing thereal standard deviation, i.e., σ ∝ (cid:115)(cid:90) | v − u s | f s d v , (1.73)where we have used the standard notation of the variable σ for the standard devi-ation. It is not the energy density, Eq. (1.72), that is the variance of the particledistribution function. Only the square root of the internal energy, 3 / n s T s , will en-ter into the definition of the variance, because in subtracting off the average velocitywe are eliminating the kinetic energy, 1 / m s n s | u s | , component. If we recall our def-initions for the various components of the drag and diffusion coefficients, Eq. (1.56)and Eq. (1.57), we can make the parallels concrete, and drive home some of theintuition for our choice of simplified drag and diffusion coefficients. The particledistribution function is centered around some velocity u , the average velocity of theparticles, with some variance in velocity space quantifying the thermal spread of theparticles , (cid:112) T s /m s . Thus, we naturally have links between our physical intuition In the diffusion coefficient Eq. (1.57), T s /m s is the standard deviation squared. The variancemust have the same units as velocity and thus the actual spread in velocity space of the distribu-tion function is (cid:112) T s /m s , the thermal velocity, again noting that we have absorbed Boltzmann’sconstant into our definition of the temperature, k B T s → T s . x ∆ v .Let us now move on to properties of the continuous VM-FP system of equa-tions. Since one set of properties we wish to quantify are the conservation relationsinherent to the system of equations, we will need to assume specific boundary con-ditions for the distribution function and electromagnetic fields. In particular, wewill assume the distribution function f ( x , v → ±∞ , t ) → f s ). Note that in this assumption, it naturally follows that f ( x , v → ±∞ , t ) → v n for finite n . Likewise, we will take configura-tion space to be either periodic or some similar self-contained boundary condition,such as a reflecting wall for E , B , and the distribution function at the edge of con-figuration space.We wish to be rigorous at this point and prove many of these properties,but to avoid the discussion becoming overly cumbersome here in the introduction,we prove all of the forthcoming properties of the VM-FP system of equations inAppendix A. Here, we will only state the properties to foreshadow the work wewill do in the upcoming chapters on retaining properties of the continuous VM-FP system of equations when we discretize and numerically integrate the equationsystem. We will first focus on the collisionless component, Eq. (1.65), of the VM-FPsystem of equations, often referred to as the Vlasov–Maxwell part. The Vlasov–Maxwell system of equations has the following properties:34 roposition 1. The Vlasov–Maxwell system conserves mass, ddt (cid:18) m s (cid:90) K f s d z (cid:19) = 0 . (1.74) Proposition 2.
The collisionless Vlasov–Maxwell system conserves the L norm ofthe distribution function, i.e., ddt (cid:18) (cid:90) K f s d z (cid:19) = 0 . (1.75) Proposition 3.
The collisionless Vlasov–Maxwell system conserves the entropy den-sity S = − f ln( f ) of the system , ddt (cid:20)(cid:90) K − f s ln( f s ) d z (cid:21) = 0 . (1.76) Proposition 4.
The Vlasov-Maxwell system conserves the total, particles plus fields,momentum, ddt (cid:32)(cid:90) Ω (cid:88) s M s + (cid:15) E × B d x (cid:33) = 0 . (1.77) The first term is the total particle momentum, and the second term is the momentumcarried by the electromagnetic fields.
Proposition 5.
The Vlasov-Maxwell system conserves the total, particles plus fields, Note that it is the physicists’ convention to include a minus sign in the definition of theentropy, thus making the entropy a non-decreasing quantity and the Maxwellian the maximumentropy state. The minus sign could be dropped, as is often done in the theory of hyperbolicconservation laws, and then the entropy would be a non-increasing quantity and the Maxwellianwould minimize the entropy. For a discussion of the Maxwellian velocity distribution as the entropymaximizing particle distribution function, see Proposition 9 and Corollary 1. nergy, ddt (cid:32)(cid:90) Ω (cid:88) s E s + (cid:15) | E | + 12 µ | B | d x (cid:33) = 0 . (1.78) The first term is the total particle energy, and the second two terms are the energycontained in the electromagnetic fields.
So, the collisionless Vlasov–Maxwell system of equations conserves mass, to-tal momentum, and total energy, and additionally the entropy of the particles isunchanged by the collisionless component of the VM-FP system of equations. Thelatter property of entropy conservation in the collisionless system naturally leads usto a discussion of collisions. We alluded to the effect collisions would have on thethermodynamics of the plasma with Bogoliubov’s timescale hierarchy in Section 1.5.We will now make the connection concrete with a discussion of the properties of theFokker-Planck collision operator, Eq. (1.66). We first focus on the conservation prop-erties of the Fokker-Planck collision operator, and then we will discuss the effect ofthe collision operator on the thermodynamics of the system. As with our discussionof the collisionless Vlasov-Maxwell system of equations, the proofs for the propertiesof the continuous Fokker–Planck collision operator can be found in Appendix A.
Proposition 6.
The Fokker–Planck equation conserves mass, ddt (cid:18) m s (cid:90) K f cs d z (cid:19) = 0 . (1.79) Proposition 7.
The Fokker–Planck equation conserves the particle momentum, ddt (cid:18)(cid:90) K m s v f cs d z (cid:19) = 0 . (1.80)36 roposition 8. The Fokker–Planck equation conserves the particle energy, ddt (cid:18)(cid:90) K m s | v | f cs d z (cid:19) = 0 . (1.81) Proposition 9.
The Fokker–Planck equation leads to a non-decreasing entropy den-sity, S = − f ln( f ) , of the system, ddt (cid:20)(cid:90) K − f cs ln( f cs ) d z (cid:21) ≥ . (1.82) Thus, the Vlasov–Maxwell–Fokker–Planck system of equations satisfies the SecondLaw of Thermodynamics, ∆ S ≥ . Corollary 1.
The maximum entropy solution to the Fokker–Planck collision oper-ator is attained by the Maxwellian velocity distribution, f s = n s (cid:18) m s πT s (cid:19) exp (cid:18) − m s | v − u s | T s (cid:19) . (1.83) Thus, the Vlasov–Maxwell–Fokker–Planck system of equations satisfies Boltzmann’sH-theorem, and a plasma in local thermodynamic equilibrium is described by theMaxwellian velocity distribution.
So, the Fokker–Planck component of the VM-FP system of equations alsoconserves mass, momentum, and energy, so that the complete equation system pos-sesses these properties. And the Fokker–Planck component is a critical piece of theevolution of the thermodynamics of the plasma, governing both entropy productionand providing us the form of the distribution function which maximizes the entropyand describes local thermodynamic equilibrium—see Appendix A for further dis-cussions of the connection between the Maxwellian velocity distribution and local37hermodynamic equilibrium. We should reiterate that our discussion of the collisionoperator in the VM-FP system of equations utilizes simplified drag and diffusioncoefficients [Lenard and Bernstein, 1958, Dougherty, 1964], Eqns. (1.56) and (1.57),and that, while collisions in a plasma are well approximated by a Fokker–Planck op-erator, the real drag and diffusion coefficients are more complex [Rosenbluth et al.,1957]. Nonetheless, this equation system contains all the ingredients required tocharacterize a weakly coupled plasma, a plasma whose collective motions dominateover individual particle-particle interactions. This equation system is simultaneouslymore computationally tractable than integrating all the particle trajectories, whilealso still containing the properties our physical intuition tells us the plasma shouldhave despite this perspective shift to a probabilistic picture from the deterministicpicture of individual particle motions.This discussion naturally leads us into the next section. We have presented anequation system for modeling a myriad of plasma systems, relevant everywhere fromlaboratories, to the heliosphere, to astrophysical systems such as the interstellar andintracluster medium. We want to now utilize the computer to understand the dy-namics of weakly coupled plasmas. But just because we have made the problemof simulating plasma dynamics computationally tractable, shifting our perspectivefrom integrating every single particle’s equations of motion to focusing on the col-lective behavior we know to be of critical importance, does not imply we have madethe problem easy. There is a rich history in tackling the numerical integration ofthe Vlasov–Maxwell–Fokker–Planck system of equations, and it is worth reviewingthis history to motivate the novel approach derived and implemented in this thesis.38 .7 A Brief History of Kinetic Numerical Methodsand the Objectives of This Thesis
We restate here, in its entirety, the VM-FP, or Vlasov–Maxwell–Fokker–Planck,system of equations, ∂f s ∂t = −∇ z · ( α s f s ) + ν s ∇ v · (cid:20) ( v − u s ) f s + T s m s ∇ v f s (cid:21) ,∂ B ∂t + ∇ x × E = 0 , (cid:15) µ ∂ E ∂t − ∇ x × B = − µ J , ∇ x · E = (cid:37) c (cid:15) , ∇ x · B = 0 , where, α s = (cid:18) v , q s m s [ E + v × B ] (cid:19) , u s = (cid:82) v f s d v (cid:82) f s d v , T s m s = 13 (cid:82) | v − u s | f s d v (cid:82) f s d v ,ρ c = (cid:88) s q s (cid:90) f s d v , J = (cid:88) s q s (cid:90) v f s d v , define the phase space flux, flow and temperature per mass, and charge density andcurrent density, which close the system of equations and couple the electromagneticfields to the motion of the particles. This equation system provides an alternative,ultimately more useful, perspective on the evolution of the plasma by shifting froma purely deterministic picture to a probabilistic picture; we track the evolution ofthe particle distribution function for the probability of finding particles in a phasespace volume ∆ x ∆ v instead of every individual particle in the plasma.Given the discussion in Section 1.6, we would like however we ultimately dis-39retize the VM-FP system of equations to retain some of these properties of thecontinuous system of equations. But, we also want to weigh the computational fea-sibility of our approach. The VM-FP system of equations involves the solution of ahigh dimensional, up to six dimensions plus time, partial differential equation, andthis presents its own challenges numerically.Because of the high dimensionality of the Vlasov–Fokker–Planck equation forthe dynamics of the particle distribution function, the most common numerical tech-niques historically have been Monte Carlo methods, principally the particle-in-cell(PIC) method [Dawson, 1962, Langdon and Birdsall, 1970, Dawson, 1983, Birdsalland Langdon, 1990]. This approach attempts to alleviate the computational chal-lenge in integrating the Vlasov–Fokker–Planck equation in the six dimensional phasespace by discretizing the particle distribution function as a collection of “macropar-ticles,” i.e., particles of finite size [see, e.g, Lapenta, 2012, and references therein].Maxwell’s equations are then discretized on a grid, and the charge and current den-sity of the “macroparticles” are deposited on the grid for the coupling. By makingthe particles have finite size, the scheme essentially smooths over the spatial scalesof the particle size, eliminating discrete particle effects. Thus, despite the numericalmethod involving the integration of particle trajectories, the PIC method really isa discretization of the VM-FP system of equations. There are additional subtletiesfor the Fokker–Planck component of the equation system since the collisional com-ponent of the dynamics occurs inside the macroparticle’s finite size; thus, numerical,unphysical, collisions can arise [Hockney, 1968, Okuda and Birdsall, 1970, Okuda,1972, Hockney, 1971, Langdon, 1979, Krommes, 2007], and the implementation of40 physical collision operator requires modifications to the underlying particle-in-cellalgorithm [Lemons et al., 2009].As a consequence of discretizing the particle distribution function as a collec-tion of macroparticles, the numerical method only requires a configuration spacegrid—the velocity space discretization is implicit in the sampling of the particles tocompute quantities such as the charge and current density. Thus, the dimensional-ity of the problem is reduced from six to three, with the freedom to use as many,or as few, particles per configuration space grid cell as deemed necessary to resolvethe kinetic plasma physics encompassed in the VM-FP system of equations. Thisreduction in dimensionality, combined with modern algorithms for particle-sortingand sampling, allows one to construct efficient schemes for the complete particle-in-cell algorithm which perform well on the largest supercomputers in the world [e.g.,Fonseca et al., 2008, Bowers et al., 2009, Germaschewski et al., 2016].Discretizing the particle distribution function as a collection of macroparticleshas its disadvantages though, chief among them the particle noise that is introducedvia the particle’s finite size.This pollution of the solution of the VM-FP systemof equations is a real travesty, as the particle distribution function is such a richtapestry of the underlying physics of the weakly coupled plasma. One can alwaysmollify this concern by increasing the number of particles in the simulation, but thecounting noise decreases like 1 / √ N , where N is the number of particles per gridcell. In addition to degrading the quality of the solution and potentially making theultimate analysis more challenging, the particle noise inherent to the PIC algorithm41an have more severe consequences, potentially giving incorrect or deceptive answersin situations requiring high signal to noise ratios. For example, Camporeale et al.[2016] have demonstrated that a large number of particles-per-cell is required tocorrectly identify wave-particle resonances and compare well with linear theory.There are means of reducing noise in PIC methods, such as the delta-f PIC method[Parker and Lee, 1993, Hu and Krommes, 1994, Denton and Kotschenreuther, 1995,Belova et al., 1997, Cheng et al., 2013a, Kunz et al., 2014], but noise mitigationtechniques like the delta-f PIC method can break down if the distribution functiondeviates significantly from its initial value. Further noise mitigation techniques,such as very high order particle shapes, e.g., particle-in-wavelets [Nguyen van yenet al., 2010, 2011] and von Mises distributions based on Kernel Density Estimationtheory [Wu and Qin, 2018], and time-dependent deformable shape functions forthe particles [Coppa et al., 1996, Abel et al., 2012, Hahn and Angulo, 2015, Kates-Harbeck et al., 2016] are active areas of research. However, these more sophisticatedparticle shape functions add significant computational complexity to the algorithm.Thus, preliminary application of some of these techniques is done in post-processingto assist in analysis [Totorica et al., 2018], and not in situ during a simulation, soany issues due to noise that arise during the course of a simulation are not mitigated.We thus have strong motivation, both from a desire to eliminate noise and adesire to fully leverage the particle distribution function in our analyses, to directlydiscretize the VM-FP system of equations on a phase space grid. But as we havesaid before, direct discretization of a six dimensional, plus time, partial differentialequation, presents its own challenges. To mitigate the cost, much of the current42ody of research on direct discretization of the VM-FP system of equations hasfocused on the hybrid approximation [Valentini et al., 2007, Valentini et al., 2010,Greco et al., 2012, Perrone et al., 2012, Servidio et al., 2014, Valentini et al., 2016,Kempf, 2012, Kempf et al., 2013, Pokhotelov et al., 2013, Palmroth et al., 2018].In this approximation, proton species are treated with the Vlasov–Maxwell systemof equations, with potentially a Fokker–Planck equation for the ion-ion collisions[Pezzi et al., 2015, 2019], while the electrons are taken to be a massless, isothermalbackground. This approximation still requires the solution of the VM-FP system ofequations on a high dimensional phase space grid, but the challenges in multi-scalemodeling of a plasma, from the electron to the proton scales to the macroscopicdynamics, are alleviated. There are exceptions in recent years [Vencels et al., 2016,Wettervik et al., 2017, Roytershteyn and Delzanno, 2018, Roytershteyn et al., 2019],but the direct discretization approach for the full VM-FP system of equations for thesolution of a multi-species weakly coupled plasma, including the effects of collisions,is not common.It is the objective of this thesis to outline, derive, and implement a novelscheme for the numerical integration of the multi-species VM-FP system of equa-tions. Such a scheme should, as much as possible, respect the properties derivedin Section 1.6. But, in order for our scheme to accomplish this goal, we must becareful to respect the fact that many of these properties, most especially the con-servation properties, are implicit to the equation system being evolved. In otherwords, we must, for example, encode the fact that the second velocity space mo-ment is a conserved quantity in our evolution of the particle distribution function.43specially for Fokker–Planck collision operators, such schemes are an active areaof research [Taitano et al., 2015, Hirvijoki and Adams, 2017, Hirvijoki et al., 2018],but the task of a robust, accurate, conservative, and cost effective numerical methodfor the full VM-FP system of equations is a tall task. We have tackled this task inthis thesis, and applied the resulting algorithm to a wide variety of plasma systemsto solve outstanding questions about the energization mechanisms in fundamen-tal plasma processes and the nonlinear dynamics of saturated plasma instabilitiesusing the pristine, noise-free, distribution function granted to us by a continuumdiscretization of the VM-FP system of equations.As an example of the power of this approach of direct discretization, we showin Figure 1.1 the results of a simulation we will discuss in Chapter 5. Figure 1.1shows the proton distribution function undergoing energization due to a collision-less shock, a shock wave which forms on scales smaller than the particle’s mean-freepath. The conversion of energy in collisionless shocks, from the kinetic energy of theincoming supersonic flow to other forms of energy, e.g., thermal energy, thus occursdue to kinetic processes such as wave-particle interactions and small-scale instabili-ties rather than inter-particle collisions. We will study this system in greater detailwhen we discuss analysis techniques for extracting data from such a pristine repre-sentation of the distribution function. Suffice to say, the quality of the distributionfunction from the continuum approach discussed in this thesis is made manifest byinspection of the structure the algorithm can resolve on a phase space grid.Having motivated our wish to directly discretize the VM-FP system of equa-tions, and briefly demonstrated the capability to resolve detailed particle distribu-44igure 1.1: The electromagnetic fields (top plot) and proton distribution functiondue to a collisionless shock, where the kinetic energy of an incoming supersonicflow is dissipated and converted into other forms of energy, e.g., thermal energy, onscales smaller than the particle mean-free path, such as the proton inertial length d p = c/ω pp . We plot the reduced proton distribution function in x − v x (second fromtop plot) and slices of the proton distribution function in v x − v y (bottom plots)at the specified lines in the x − v x plots, x = 19 . , . , . , and 22 . d p . We willdiscuss this structure and the specific energization mechanisms of this collisionlessshock in Chapter 5, but for now we draw attention to the quality of the solutionfrom a continuum representation of the distribution function using a phase spacegrid. By directly discretizing the VM-FP system of equations in phase space, wecan represent fine-scale structure in velocity space which we can leverage to dive into the details of the energization of the protons.45ion function structure in kinetic plasma processes like collisionless shocks with thisapproach, we now discuss the organization of the rest of the thesis. We will de-scribe the numerical method, the discontinuous Galerkin finite element method, inChapter 2. Chapter 2 will form a complete mathematical description of our discretesystem, including what properties the discrete VM-FP system of equations retainscompared to the continuous VM-FP system of equations, and the stability prop-erties of the algorithm. We will then move to a discussion of the implementationof the algorithm in Chapter 3. This discussion will detail two of the major break-throughs in this thesis: the requirement that the algorithm be alias-free so it retainsthe properties of the discrete scheme, most especially the stability and conservationproperties, and the specific choice of an orthonormal, modal basis expansion in thediscontinuous Galerkin method to optimize the computational complexity of thealgorithm.Chapter 4 will numerically demonstrate the accuracy and robustness of theimplemented scheme. We will show via a variety of numerical tests the provenproperties of the discrete scheme, and compare a number of numerical experiementsto known analytic solutions. Chapter 5 will be a tour-de-force showcase of thepower of the implemented scheme. With access to a high fidelity representationof the particle distribution function from our direct discretization, we will examineenergization mechanisms in fundamental plasma processes directly in phase space,such as the collisionless shock shown in Figure 1.1, and conclude with an applicationcomparison between the particle-in-cell method and our continuum approach thatshows explicitly where particle noise can pollute the simulation of plasma kinetic46ystems.The scheme is implemented within the Gkeyll framework.
Gkeyll is a generalpurpose, open-source, simulation framework with support for five- [Hakim et al.,2006] and ten-moment multi-fluid [Hakim, 2008, Wang et al., 2015, Ng et al., 2015,Wang et al., 2019], full-f gyrokinetic [Shi et al., 2015, Shi, 2017, Mandell et al., 2020],and Vlasov–Maxwell–Fokker–Planck systems [Juno et al., 2018, Hakim et al., 2019,Hakim and Juno, 2020]. For the purposes of reproducibility, the source code for
Gkeyll is available through
GitHub , and all input files for the simulations run inthis thesis are available through a GitHub repository , with the changesets used toproduce the data documented in the input file. Additional documentation can befound through the Gkeyll documentation website . https://github.com/ammarhakim/gkyl https://github.com/ammarhakim/gkyl-paper-inp https://gkyl.readthedocs.io/en/latest/ ome of the material in thischapter has been adapted fromJuno et al. [2018], Hakim,Francisquez, Juno, andHammett [2019], and Hakimand Juno [2020]. Chapter 2: The Discontinuous Galerkin Finite Element Method
The method we will employ to discretize the Vlasov–Maxwell–Fokker–Plancksystem of equations is called the discontinuous Galerkin finite element method, orDG for short. DG was first introduced to study neutron transport [Reed and Hill,1973] and became an active area of study in numerical methods after the generalformulation of the algorithm by Cockburn and Shu [1998b, 2001]. DG has becomean enticing method for a variety of problems, from computational fluid dynamicsto seismology and wave equations[see, e.g., Hesthaven and Warburton, 2007, andreferences therein], because DG methods are constructed to combine advantages ofboth finite element methods and finite volume methods. By combining the powerof the finite element method, principally the high order accuracy and flexibility inthe chosen basis expansion, with the benefits of a finite volume method, such as48ocality of data and the ability to construct conservative discretizations, one candesign robust, physically-motivated, numerical methods for the chosen equation orequation system of interest. In fact, DG has become a particularly active area ofresearch in recent years for kinetic equations such as the Vlasov–Maxwell–Fokker–Planck system of equations, and its subsidiaries Vlasov–Poisson and Vlasov–Ampere[Cheng et al., 2011, 2013b, 2014a,b]It is worth taking a moment to give some intuition for the construction ofthe DG method in a more general context before diving in to our discretization ofthe VM-FP system of equations. We will define what we mean by a “Galerkin”method, and then apply DG to a simple hyperbolic partial differential equation. Indoing so, we will be able to connect with our knowledge of other numerical methods,and see why DG is often discussed as a hybrid finite volume-finite element method,combining the strengths of both numerical methods into a singular, powerful, meansof discretizing a partial differential equation. L Minimization of the Error
The two essential ingredients of a Galerkin method are the definition of somefinite dimensional space of functions and a definition of errors. The former allowsus to connect the function space the continuous equation, or equation system, livesin, to a discrete representation of the solution to our equation or equation system.The latter gives us a unique way of finding the discrete representation, as we wouldlike to minimize the errors of our discrete representation of our solution.49onsider an interval [ − ,
1] and the function space of polynomials of order p , P p . The particular space of polynomials will form a complete basis on our interval .On this interval, we will employ the inner product, (cid:104) f, g (cid:105) L = (cid:90) − f ( x ) g ( x ) dx, (2.1)with the following norm, (cid:104) f, f (cid:105) L = (cid:90) − f ( x ) dx, (2.2)the L norm.In general, we want to solve problems of the form ∂f ( x, t ) ∂t = G [ f ] , (2.3)where G [ f ] is some operator for f . G [ f ] may be a very general operator, suchas in the VM-FP system of equations wherein we have first order terms, e.g., thecollisionless advection in phase space, and second order terms, e.g., the collisionoperator. In seeking an approximation of our solution f ( x, t ), we will expand f ( x, t )in our basis set, f ( x, t ) ≈ f h ( x, t ) .. = N (cid:88) k =1 f k ( t ) φ k ( x ) , (2.4)where φ k ( x ) ∈ P p , for k = 1 , . . . , N . Thus, the problem of interest is approximated A good example of such a complete basis would be the Legendre polynomials up to some order n , P n ( x ). N (cid:88) k =1 df k ( t ) dt φ k ( x ) = G [ f h ] , (2.5)and we need to determine the time evolution of the coefficients f k ( t ). Note that wehave changed notation from ∂/∂ t to d/dt to emphasize that the coefficients f k areonly a function of time.We defined a norm in Eq. (2.2), so let us minimize the error with respect tothis norm, E L = (cid:90) − (cid:32) N (cid:88) k =1 df k ( t ) dt φ k ( x ) − G [ f h ] (cid:33) dx, (2.6)by taking the derivative of the error with respect to each time-dependent coefficient, ∂E L ∂f (cid:48) (cid:96) = 2 (cid:90) − φ (cid:96) ( x ) (cid:32) N (cid:88) k =1 df k ( t ) dt φ k ( x ) − G [ f h ] (cid:33) dx. (2.7)Here, we have used the shorthand f (cid:48) (cid:96) = df (cid:96) /dt . To minimize the error with respectto the time derivative of the coefficients, we set Eq. (2.7) equal to 0, (cid:90) − (cid:88) k df k ( t ) dt φ k ( x ) φ (cid:96) ( x ) dx = (cid:90) − G [ f h ] φ (cid:96) ( x ) dx. (2.8)To give a bit more insight into how one could then evaluate this expression to findeach of the time dependent coefficients, consider what this expression reduces to ifthe polynomials φ k ( x ) ∈ P p for k = 1 , . . . , N are an orthonormal basis set such that (cid:90) − φ k ( x ) φ (cid:96) ( x ) dx = δ k(cid:96) , (2.9)where δ k(cid:96) = 1 if k = (cid:96) and zero otherwise. Then our equation for the time evolution51f the coefficients would reduce to df (cid:96) dt = (cid:90) − G [ f h ] φ (cid:96) ( x ) dx, (2.10)for (cid:96) = 1 , . . . , N , and we would then have a system of ordinary differential equationsto solve for each of df (cid:96) /dt .The discussion up to this point has been somewhat abstract, so we would liketo make this concrete in two ways. First, let us perform the L minimization ofthe error on a non-polynomial function. In doing so, we would like to show whatit means to take a function in some infinite dimensional space, since it would takean infinite number of polynomials to represent this function normally, and projectit to a finite dimensional subspace.We plot in Figure 2.1 the projection of the function f ( x ) = x + sin(5 x ) onto anumber of different basis expansions. Here, we have a further generalization of theprevious discussion for the Galerkin method, where the domain of [ − ,
1] is furthersubdivided into non-overlapping cells, and the projection is done within each cell.As we move to higher and higher polynomial order, we can see the reduction, evenjust visually, of the error between the exact solution and our discrete representationof the solution. This reduction in the error with higher polynomial order is our firstevidence of the connection between the discontinuous Galerkin method and finiteelement methods, where higher order basis sets correspond to higher accuracy.The second way we will make our discussion of the Galerkin minimization ofthe L error less abstract is by considering the full discretization of the constant52igure 2.1: The projection of f ( x ) = x + sin(5 x ) onto piecewise constant (left),piecewise linear (middle), and piecewise quadratice (right) functions. The domainfrom [ − ,
1] is divided into non-overlapping cells and the projection is done withineach cell to minimize the L error. We begin to see some of the connection betweenthe discontinuous Galerkin method and finite element methods, as moving to higherpolynomial order manifestly reduces the L error between the exact solution andprojected solution.advection equation in one dimension, ∂f ( x, t ) ∂t + λ ∂f ( x, t ) ∂x = 0 . (2.11)Define the domain of the advection equation as Ω, which we will divide into non-overlapping cells I j ∈ Ω j , for j = 1 , ..., N j . Plugging − λ∂f /∂x into Eq. (2.8) for theoperator G [ f h ], and integrating by parts we obtain (cid:90) I j df h,j dt φ (cid:96) dx = − λφ (cid:96),j +1 / ˆ F j +1 / + λφ (cid:96),j − / ˆ F j − / + λ (cid:90) I j dφ (cid:96) dx f h,j dx, (2.12)where the subscripts j ± / − , sides of the cellrespectively, and f h,j is the projection of the solution in each cell I j as defined byEq. (2.4). Note that the solution in each cell requires a minimization of the error forevery φ (cid:96) , (cid:96) = 1 , . . . , N , for however many basis functions in each cell one has, andfurther that the full solution is a direct sum over all cells I j ∈ Ω j , f h ( x, t ) = N j (cid:77) j =1 f h,j ( x, t ) . (2.13)53ince we have a solution in each cell I j , the integration by parts gives us a meansto connect the solution within each cell to its neighbors, but we need to prescribethe numerical flux function, ˆ F j ± / . A natural choice for the constant advectionequation is known as upwind fluxes,ˆ F ( f + h , f − h ) = f − h if λ > f + h if λ < , (2.14)where the superscript plus-minus is the solution evaluated just inside, − , or justoutside +, the cell interface—see Figure 2.2 for a visualization of this notation.Figure 2.2: Annotated piecewise linear representation to make our notation moreclear, most especially superscript plus-minus, where the solution is evaluated justinside, − , or just outside +, the cell interface.To make further progress, let us consider two cases. The first case is one in54hich our basis expansion is just the set of piecewise constant basis functions, φ = { } . (2.15)Substituting the piecewise constant basis function into Eq. (2.12), we obtain, df j dt ∆ x = − λ ( f j − f j − ) , (2.16)since the derivative of a constant function is 0, and the integral of the left hand sidein Eq. (2.12) when the basis function is a constant is the volume of the cell, ∆ x .We can immediately recognize this formula as a first order finite volume method, oran upwind finite difference method, if you prefer. We can then discretize the timederivative with a forward Euler method to obtain f n +1 j = f nj − λ ∆ t ∆ x ( f j − f j − ) , (2.17)and should we choose, we could combine multiple forward Euler steps into a multi-stage method, such as a Runge–Kutta method.The second case is one in which our basis functions are a piecewise linearexpansion, φ , = { , x − x j ) / ∆ x } , (2.18)where x j is the cell center value of cell I j . We can obtain update formulas fora forward Euler step for the constant and linear coefficients when employing the55iecewise linear basis, f n +11 ,j = f n ,j − λ ∆ t ∆ x (cid:16) ˆ F j +1 / − ˆ F j − / (cid:17) , (2.19) f n +12 ,j = f n ,j − λ ∆ t ∆ x (cid:16) ˆ F j +1 / + ˆ F j − / (cid:17) + 6 λ ∆ t ∆ x f n ,j , (2.20)which again, can be combined into a general multi-stage time-stepping method.Note that the numerical flux function ˆ F j ± / is still given by Eq. (2.14), but dueto the piecewise linear representation within a cell, we will need to evaluate thenumerical flux function at the corresponding cell interfaces when implementing themethod.So the switch from piecewise constant basis functions, which produced a stan-dard first order finite volume method, to piecewise linear basis functions, led to moregeneral update formulas. As we might expect, the accuracy of the method has alsoimproved as a result of switching to a higher order set of basis functions. To seethis, we plot in Figure 2.3 the result of advecting a Gaussian pulse on a domain [0 , N j = 32 (32 cells) and periodic boundary conditions one full period. The sizeof the time-step is chosen to satisfy stability constraints for a forward Euler time-step. We expect that after one period, the initial condition and the final solutionshould be identical, since the exact solution of the linear advection equation is sim-ply f ( x − λt, t ), where f is the initial condition at t = 0. However, the first orderfinite volume method has significant numerical diffusion, leading to a less accuraterepresentation of the solution than the piecewise linear basis function solution.Based on the results of this numerical experiment, we now want to morestrongly connect the discontinuous Galerkin method to finite volume methods. It56igure 2.3: Comparison of advection of a Gaussian pulse one period with a piecewiseconstant (left) and piecewise linear (right) basis function expansion and upwindfluxes. While the piecewise constant solution suffers from numerical diffusion whichleads to poor agreement between the analytic solution (red) and the numericalsolution (black), the piecewise linear solution agrees to a reasonably high degreewith the expected result.is natural to think of DG as a generalization of finite volume methods. In finitevolume methods, one only tracks the evolution of a single quantity in each cell, thecell average, just like with our piecewise constant representation. But, we now seethere is no reason to restrict ourselves. We can evolve higher “moments,” coefficientscorresponding to a higher order representation of our solution, within a cell, and indoing so, obtain a higher accuracy numerical method.A useful analogy is to connect DG with higher order finite volume methodssuch as MUSCL schemes [van Leer, 1979] or the piecewise parabolic method [Colellaand Woodward, 1984]. In these higher order finite volume methods, one is still onlytracking the evolution of the cell average, but a reconstruction of the solution is doneat every time-step to increase the order of accuracy of the scheme, e.g., a linear orquadratic reconstruction of the solution. In the DG method, instead of generatinga reconstruction, we are explicitly evolving something like a reconstruction—we are57volving the higher order representation of the solution inside the cell! With new-found intuition about how the DG method works, let us now turn to the equationsystem of interest in this thesis, the Vlasov–Maxwell–Fokker–Planck system of equa-tions. We will proceed in stages just as with the properties of the VM-FP system ofequations in Chapter 1, first focusing on the collisionless component of the equationsystem, the Vlasov–Maxwell system of equations. We seek a discretization of the Vlasov–Maxwell system of equations usingthe discontinuous Galerkin method in all of phase space. To discretize the Vlasovequation, we introduce a phase space mesh T with cells K j ∈ T , j = 1 , . . . , N , anda piecewise polynomial approximation space for the distribution function, f s ( z , t ), V ph = { w : w | K j ∈ P p , ∀ K j ∈ T } , (2.21)where P p is some space of polynomials of order p . We then seek f h ∈ V ph such that,for all K j ∈ T , (cid:90) K j w ∂f h ∂t d z + (cid:73) ∂K j w − n · ˆ F dS − (cid:90) K j ∇ z w · α h f h d z = 0 , (2.22)for all test functions w ∈ V ph . Eq. (2.22) is commonly referred to as the discrete-weak form of the Vlasov equation. In the derivation of the discrete-weak form ofthe Vlasov equation, we have used integration by parts on the operator for the fluxin phase space, thus producing the surface and volume integrals in Eq. (2.22).The pieces of the discrete-weak form of the Vlasov equation again evoke the58omparison to finite element and finite volume methods. The third term, the volumeintegral, calls to mind the integrals over a cell one performs in a finite elementmethod, while the second term, the surface integral, involves the prescription of anumerical flux function, ˆF , exactly as in a finite volume method. The subscript h indicates the discrete solution, the notation w − ( w + ) indicates that the function isevaluated just inside (outside) the location on the surface ∂K j , and n is an outwardunit vector on the surface of the cell K j .The discrete distribution function is represented as f h ( t, z ) = (cid:88) i f i ( t ) w i ( z ) , (2.23)where w i ( z ) are a set of polynomials chosen such that they lie in the aforementionedspace of polynomials P p , i.e., we are employing a Galerkin method where the testfunctions and basis functions are one and the same. We will avoid specifying theexact polynomial space P p for now, as the specific form of the polynomials is nota necessary component of the mathematical formulation of the algorithm. All thatwe will require in our mathematical formulation is that the basis set is made up ofpolynomials.There are many choices for the numerical flux function, ˆF , which can be em-ployed for the Vlasov equation. We will pick the numerical flux function mostimportantly to be a Godunov flux, (cid:73) ∂K j w − n · ˆ F dS = − (cid:73) ∂K j w + n · ˆ F dS. (2.24)In other words, the flux into the cell K j along some surface ∂K j is equal and opposite59n sign to the flux out of its neighbor cell along the shared interface. This propertylikely reads like a sensible and obvious property one would desire of a numerical fluxfunction, as it means that the flux is conserved across the interface, i.e., there isno creation or destruction of the distribution function as it advects in phase space.Example Godunov fluxes include central fluxes, n · ˆ F ( α + h f + h , α − h f − h ) = 12 n · (cid:0) α + h f + h + α − h f − h (cid:1) , (2.25)the local Lax-Friedrichs flux, n · ˆ F ( α + h f + h , α − h f − h ) = 12 n · (cid:0) α + h f + h + α − h f − h (cid:1) − c f + − f − ) , (2.26)where c = max ∂K j ( | n · α + h | , | n · α − h | ), and the global Lax-Friedrichs flux , n · ˆ F ( α + h f + h , α − h f − h ) = 12 n · (cid:0) α + h f + h + α − h f − h (cid:1) − τ f + − f − ) , (2.27)where τ = max T | n · α h | . Note the difference between the local and global Lax-Friedrichs fluxes, where in the local Lax-Friedrichs flux, Eq. (2.26), the max of thephase space flux is taken along the specific surface ∂K j , while for the global Lax-Friedrichs flux, Eq. (2.27), the max of the phase space flux is taken over the entiredomain T . Both Eqns. (2.26) and (2.27) are defined with the motivation to penalizethe size of the jumps in the flux so that the discontinuities can be controlled in somefashion. We will see in Proposition 11 that this penalization naturally leads to somenumerical diffusion, thus why we refer to the penalty term as controlling the size ofthe jumps in the flux. Note that global Lax-Friedrichs flux applies to a general class of numerical flux functions inwhich the parameter, τ , is a globally calculated quantity. restriction ofthe phase-space mesh, T , to configuration space by T Ω . The cells in configurationspace are denoted by Ω j ∈ T Ω , for i = 1 , . . . , N Ω , where N Ω are the number ofconfiguration space cells, and we introduce the solution space X ph = { ϕ : ϕ | Ω j ∈ P p , ∀ Ω j ∈ T Ω } . (2.28)These basis, and test, functions are defined only on the configuration space domainΩ and thus contain only dependence on the configuration space variable x . As withthe discrete distribution function, we seek, E h , B h ∈ X ph such that, for all Ω j ∈ T Ω , (cid:90) Ω j ϕ ∂ B h ∂t d x + (cid:73) ∂ Ω j d s × ( ϕ − ˆ E h ) − (cid:90) Ω j ∇ x ϕ × E h d x = 0 , (2.29) (cid:15) µ (cid:90) Ω j ϕ ∂ E h ∂t d x − (cid:73) ∂ Ω j d s × ( ϕ − ˆ B h ) + (cid:90) Ω j ∇ x ϕ × B h d x = − µ (cid:90) Ω j ϕ J h d x . (2.30)Note in the derivation of Eqns. (2.29–2.30), we needed to evaluate volume integralswhich include terms of the form ϕ ∇ x × E h , for ϕ ∈ X ph and likewise for the magneticfield, B h . We have made use of the fact that (cid:90) Ω j ϕ ∇ x × E h (cid:124) (cid:123)(cid:122) (cid:125) ∇ x × ( ϕ E h ) −∇ x ϕ × E h d x . (2.31)Gauss’ law can then be used to convert one volume integral into a surface integral (cid:90) Ω j ∇ x × ( ϕ E h ) d x = (cid:73) ∂ Ω j d s × ( ϕ E h ) , (2.32)where d s is the (vector) area-element that points in the direction of the outwardnormal to the configuration space cell Ω j .61s with the discrete-weak form for the Vlasov equation, Eq. (2.22), we requirea prescription for the numerical flux functions ˆE h , ˆB h . We consider two methodsof obtaining the cell interface fields needed in the discrete weak-form of Maxwell’sequations: central fluxes and upwind fluxes. As we will see later, both numericalflux functions have advantages and disadvantages, particularly in terms of the con-servation properties the discrete system retains from the continuous system. Forcentral fluxes, we use averages of values just across the interface, i.e.,ˆ E h = (cid:74) E (cid:75) , (2.33)ˆ B h = (cid:74) B (cid:75) , (2.34)where (cid:74) · (cid:75) represents the averaging operator, (cid:74) g (cid:75) ≡ ( g + + g − ) / , (2.35)for any function g .On the other hand, using upwind fluxes requires solving a Riemann problem ina coordinate system local to that face. Consider a local coordinate system ( s , τ , τ )on the configuration space cell face, i.e., on ∂ Ω j . Here, s is a unit vector normal to ∂ Ω j , and τ and τ are tangent vectors such that τ × τ = s . Let ( E , E , E )and ( B , B , B ) be electric and magnetic fields in this coordinate system. Then,assuming variations only along direction s , Maxwell’s equations reduce to ∂B /∂t =0, ∂E /∂t = 0, and the following uncoupled set of two equations for the tangential62eld components, ∂B ∂t − ∂E ∂x = 0; ∂E ∂t − c ∂B ∂x = 0 , (2.36)and ∂B ∂t + ∂E ∂x = 0; ∂E ∂t + c ∂B ∂x = 0 . (2.37)Multiplying the first of each pair by c and adding and subtracting from the secondof that pair we obtain a set of four uncoupled constant advection equations exactlylike the constant advection equation considered in Section 2.1, ∂∂t ( E + cB ) − c ∂∂x ( E + cB ) = 0 , (2.38) ∂∂t ( E − cB ) + c ∂∂x ( E − cB ) = 0 , (2.39)and ∂∂t ( E + cB ) + c ∂∂x ( E + cB ) = 0 , (2.40) ∂∂t ( E − cB ) − c ∂∂x ( E − cB ) = 0 . (2.41)Hence, the solution to the Riemann problem with initial conditions is( E , E ) = ( E − , E − ); ( B , B ) = ( B − , B − ) , (2.42)for x <
0, and ( E , E ) = ( E +2 , E +3 ); ( B , B ) = ( B +2 , B +3 ) , (2.43)63or x >
0. At x = 0, the solution isˆ E + c ˆ B = E +3 + cB +2 , (2.44)ˆ E − c ˆ B = E − − cB − , (2.45)and ˆ E + c ˆ B = E − + cB − , (2.46)ˆ E − c ˆ B = E +2 − cB +3 . (2.47)Rearranging these expressions shows that the upwind fields in the local face coordi-nate system are ˆ E = (cid:74) E (cid:75) − c { B } (2.48)ˆ E = (cid:74) E (cid:75) + c { B } (2.49)and ˆ B = (cid:74) B (cid:75) + { E } /c (2.50)ˆ B = (cid:74) B (cid:75) − { E } /c (2.51)where {·} is the jump operator, { g } ≡ ( g + − g − ) / g , and subscripts 2 and 3 denote the two directions tangent tothe surface normal. Note that we require the two directions tangent to the surfacenormal since the surface integral involves a cross product for the discrete version64f Maxwell’s equations, Eqns. (2.29)-(2.30). The solutions to the Riemann problemgiven by Eqns. (2.48)-(2.51) are identical to those presented in previous studies ofMaxwell’s equations [Barbas and Velarde, 2015].Eqns. (2.22) and (2.29)-(2.30) define the semi-discrete Vlasov–Maxwell systemof equations, i.e., a discretization in phase and configuration space, with the timediscretization not yet specified. Before proceeding to the properties of our semi-discrete system, we note that the discretization of Maxwell’s equations given byEqns. (2.29) and (2.30) does not include the constraints given by Eqns. (1.61) and(1.62), i.e., the divergence constraints in Maxwell’s equations, ∇ x · E = ρ c /(cid:15) and ∇ x · B = 0. Thus, our algorithm may violate these constraints over the course ofthe simulation. Where appropriate in Chapter 4 as part of the benchmarking of thescheme, we will discuss how the violation of the divergence constraints in Maxwell’sequations manifests. We proceed as we did with the continuous system, first considering whetherthe discrete system conserves mass (or number) density, and then moving throughthe subsequent conservation properties we studied for the continuous system inSection 1.6. An important consideration for the discrete scheme, just like with thecontinuous system, will be our boundary conditions in configuration and velocityspace. While we can employ similar boundary conditions in configuration space for65he discrete system as we did with the continuous system, i.e., periodic or some sortof self-contained boundary like a reflecting wall, velocity space is slightly more subtle.Since the continuous distribution function was defined on v ∈ [ −∞ , ∞ ], we coulduse “half-open” cells, where a grid cell in velocity space could span | v | > v max , wherethe absolute value encompasses both positive and negative values for the velocity ofthe particles. However, we will instead employ a fixed boundary in velocity space, v ∈ [ v min , v max ], and at the velocity space boundary employ zero-flux boundaryconditions, n · ˆ F ( x , v max ) = n · ˆ F ( x , v min ) = 0 . (2.53)Note that Eq. (2.53) corresponds to a homogeneous Neumann boundary conditionin velocity space. This velocity space boundary condition, along with appropriateboundary conditions in configuration space, will allow us to prove the followingproperties for the discrete scheme. Proposition 10.
The discrete scheme conserves mass, ddt (cid:88) j (cid:90) K j m s f h d z = 0 . (2.54) Proof.
Choosing w = m s , a constant, in the discrete weak-form, Eq. (2.22), andsumming over all phase-space cells K j , (cid:88) j (cid:90) K j m s ∂f h ∂t d z + (cid:88) j (cid:73) ∂K j m s n · ˆ F dS = 0 , (2.55)where the volume term vanishes since it involves the gradient of a constant function.If the appropriate boundary conditions are chosen, i.e., zero-flux boundary condition66n velocity space and periodic boundary conditions in configuration space, or asimilar self-contained boundary condition such as a reflecting wall, then the sumover surface integrals is a telescopic sum and vanishes. This pairwise cancellationof the surface integrals requires no special knowledge of the form of the numericalflux function n · ˆ F = n · ˆ F ( α − h f − h , α + h f + h ); we only require that the numerical fluxfunction is Godunov, Eq. (2.24), and that the flux at both configuration space andvelocity space boundaries vanishes as it does with zero flux boundary conditions invelocity space, plus an appropriate boundary condition in configuration space. Weare then left with (cid:88) j (cid:90) K j m s ∂f h ∂t d z = 0 , (2.56)and it is thus shown that the semi-discrete scheme in the continuous time limitconserves the total (mass) density.Before we move on to the L norm, we consider the following Lemma on thecompressibility of phase space. Lemma 1.
Phase space incompressibility holds for the discrete system, i.e., ∇ z · α h = 0 . (2.57) Proof.
For the specific discrete phase space flow in the Vlasov-Maxwell system, α h = ( v , q s /m s [ E h + v × B h ]). Within a cell, Eq. (2.57) is zero since, as with thecontinuous system, v has no configuration space dependence, and q s /m s ( E h + v × B h )has no divergence in velocity space. The question is whether the jumps in α h across67ell interfaces in phase space are accounted for by the scheme. Integrating Eq. (2.57)over a phase space cell K j , employing the divergence theorem, and summing overcells, (cid:88) j (cid:73) ∂K j n · α − h dS = 0 . (2.58)This result follows for the simple reason that the phase space flow is in fact con-tinuous with respect to the surfaces considered, allowing us to pairwise cancel theintegrand upon summation. For example, consider the configuration space compo-nent of the flow α h , v . The velocity, v , is continuous across configuration spacesurfaces because v has no configuration space dependence. Likewise, the velocityspace component of α h , q s /m s ( E h + v × B h ), is continuous across velocity spacesurfaces because E h and B h have no velocity space dependence, and v in the v × B h term is the velocity coordinate, and thus is continuous. We note that this proof isspecific to the phase space flow for the Vlasov-Maxwell system and in general maynot hold for all systems.Using Lemma 1, we can examine the behavior of the L norm of the distri-bution function. The exact behavior of the L norm will depend on the choice ofnumerical flux function, and importantly, the fact that the phase space flux, α h , iscontinuous at the corresponding surface interfaces allows us to simplify the numer-68cal flux functions previously defined, n · ˆ F ( α h f + h , α h f − h ) = 12 n · α h (cid:0) f + h + f − h (cid:1) , (2.59) n · ˆ F ( α h f − h , α h f + h ) = n · α h f − if sign( α h ) > , n · α h f + if sign( α h ) < , (2.60) n · ˆ F ( α h f − h , α h f + h ) = 12 n · α h (cid:0) f + h + f − h (cid:1) − τ f + − f − ) , (2.61)with τ = max T | n · α h | , the global maximum of the phase space flux over the entiredomain T as before. Importantly, Eq. (2.26) has simplified to an upwind flux because α h is continuous at the corresponding surface interfaces. An additional consequenceof α h being continuous at the corresponding surface interfaces: Eqns. (2.60) and(2.61) are now solely penalizing the jump in the distribution function, f h , as opposedto the jump in the flux. Connecting to our earlier discussion in Section 2.2, we nowexamine the L norm of the distribution function in our semi-discrete scheme forthe Vlasov equation and determine what effect these numerical flux functions haveon the time evolution of the L norm. Proposition 11.
The discrete scheme conserves the L norm of the distributionfunction when central fluxes are employed and decays the L norm of the distributionfunction monotonically when using either upwind fluxes or global Lax-Friedrichsfluxes.Proof. Since the distribution function itself lies in the test space, we can set w = f h
69n Eq. (2.22). We then have, (cid:90) K j f h ∂f h ∂t d z + (cid:73) ∂K j f − h n · ˆ F dS − (cid:90) K j ∇ z f h · α h f h d z =12 (cid:90) K j ∂f h ∂t d z + (cid:73) ∂K j f − h n · (cid:18) ˆ F − α h f − h (cid:19) dS = 0 , (2.62)where we have used Lemma 1 to rewrite, ∇ z f h · α h f h = 12 ∇ z · (cid:0) α h f h (cid:1) , (2.63)since phase space is incompressible, even in our discrete system, and then used thedivergence theorem. First, consider the case where ˆF is given by Eq. (2.59), centralfluxes. If we sum over all cells, and group cells pairwise by their common interface,we find, (cid:88) j (cid:73) ∂K j f − h n · (cid:18) ˆ F − α h f − h (cid:19) dS = (cid:88) j (cid:73) ∂K j n · (cid:18) f − h (cid:18) ˆ F − α h f − h (cid:19) − f + h (cid:18) ˆ F − α h f + h (cid:19)(cid:19) dS = (cid:88) j (cid:73) ∂K j n · α h (cid:0) f − h f + h − f + h f − h (cid:1) = 0 . (2.64)Thus, central fluxes do not change the L norm of the distribution function in oursemi-discrete scheme. 70e can proceed in a similar fashion for upwind fluxes, Eq. (2.60), (cid:88) j (cid:73) ∂K j f − h n · (cid:18) ˆ F − α h f − h (cid:19) dS = (cid:88) j (cid:73) ∂K j n · (cid:18) f − h (cid:18) ˆ F − α h f − h (cid:19) − f + h (cid:18) ˆ F − α h f + h (cid:19)(cid:19) dS = (cid:88) j (cid:73) ∂K j | n · α h | (cid:0) ( f − h ) − f − h f + h + ( f + h ) (cid:1) dS = (cid:88) j (cid:73) ∂K j | n · α h | (cid:0) f − h − f + h (cid:1) dS, (2.65)where we have used the fact that if α h > (cid:88) j (cid:73) ∂K j n · (cid:18) f − h (cid:18) ˆ F − α h f − h (cid:19) − f + h (cid:18) ˆ F − α h f + h (cid:19)(cid:19) dS = (cid:88) j (cid:73) ∂K j n · α h (cid:0) f − h − f + h (cid:1) dS, (2.66)and if α h < (cid:88) j (cid:73) ∂K j n · (cid:18) f − h (cid:18) ˆ F − α h f − h (cid:19) − f + h (cid:18) ˆ F − α h f + h (cid:19)(cid:19) dS = − (cid:88) j (cid:73) ∂K j n · α h (cid:0) f − h − f + h (cid:1) dS, (2.67)so we can simplify the behavior of the L norm irrespective of the sign of α h byabsorbing the minus sign into the α h < (cid:90) K j ∂f h ∂t d z = − (cid:88) j (cid:73) ∂K j | n · α h | (cid:0) f − h − f + h (cid:1) dS, (2.68)a negative definite quantity. Thus, the L norm is a monotonically decaying quantitywhen using upwind fluxes.We can proceed in a similar fashion to the two previous derivations for the71lobal Lax-Friedrichs flux. Since one component of the global Lax-Friedrichs flux isexactly equivalent to central fluxes, we know that this component of the global Lax-Friedrichs flux will not contribute to the time evolution of the L norm. Followinga similar procedure to what we used for upwind fluxes, we find12 (cid:90) K j ∂f h ∂t d z = − (cid:88) j (cid:73) ∂K j τ (cid:0) f − h − f + h (cid:1) dS, (2.69)a negative definite quantity. So, global Lax-Friedrichs fluxes also monotonicallydecay the L norm, and they further decay the L norm more strongly since, τ = max T | n · α h | ≥ | n · α h | , (2.70)at every surface interface ∂K j . We can then say that the penalization of the sizeof the jumps in the distribution function, whether by the use of upwind fluxes,Eq. (2.60), or by the use of a global Lax-Friedrichs flux, Eq. (2.61), introduces nu-merical diffusion into the scheme by decaying the L norm of the distribution func-tion. Corollary 2.
If the discrete distribution function f h remains positive definite, thenthe discrete scheme conserves the entropy if the L norm is conserved, and thediscrete scheme grows the discrete entropy monotonically if the L norm is a mono-tonically decaying function , ddt (cid:88) j (cid:90) K j − f h ln( f h ) d z ≥ The behavior of the discrete entropy is due to our convention in the definition of the entropy.If one drops the minus sign in the definition of the entropy, then the discrete entropy is a mono-tonically decreasing function when the L norm is a monotonically decreasing function if thediscrete distribution function f h remains positive definite. roof. Using the well known bound,ln( x ) ≤ x − , (2.72)we can see that ln( f h ) ≤ f h −
1, so long as f h remains a positive definite quantity,and thus ln( f h ) is well-defined. Multiplying by − f h then gives us the inequality, − f h ln( f h ) ≥ − f h + f h . (2.73)But, the left-hand side is just the discrete entropy. Integrating over a phase spacecell K j , summing over cells, and taking the time-derivative of both sides gives us anexpression for the time evolution of the discrete entropy in our scheme, ddt (cid:88) j (cid:90) K j − f h ln( f h ) d z ≥ ddt (cid:88) j (cid:90) K j − f h + f h d z . (2.74)Now, we note that in Proposition 11 we have already proved that the L norm ofthe discrete distribution function is either a conserved quantity or a monotonicallydecaying function, depending on which numerical flux function we employ. Thus,the negative of the L norm is either exactly conserved or a monotonically increas-ing function, and by Proposition 10, the semi-discrete scheme conserves particles.Therefore, the discrete entropy is either conserved or a monotonically increasingfunction depending on our choice of numerical flux function.It is worth taking a moment to reflect on the practical consequences of Propo-sition 11 and Corollary 2. These choices of numerical flux functions, Eqns. (2.59–2.61), lead to L stable schemes, schemes which do not grow the L norm. Inaddition, if we employ a numerical flux function that leads to the decay of the L L norm leads naturally to the growth of the dis-crete entropy. In other words, numerical diffusion can manifest in our scheme in theform of the growth of the discrete entropy. Importantly, as of yet, the numerical fluxfunction only affects the discrete entropy. We will now examine the conservation ofenergy in our semi-discrete scheme, first in Maxwell’s equations, and then for thecomplete system. Lemma 2.
The semi-discrete scheme for Maxwell’s equations conserves electromag-netic energy exactly when using central fluxes and monotonically decays when usingupwind fluxes, ddt (cid:88) k (cid:90) Ω k (cid:18) (cid:15) | E h | + 12 µ | B h | (cid:19) d x ≤ − (cid:88) k (cid:90) Ω k J h · E h d x . (2.75) Note that because J h · E h can have either sign, by monotonic decay when using upwindfluxes, we mean that when the right hand side is positive, the electromagnetic energywill increase less than (cid:12)(cid:12)(cid:12)(cid:80) k (cid:82) Ω k J h · E h d x (cid:12)(cid:12)(cid:12) , and when the right hand side is negativethe electromagnetic energy will decay more than − (cid:12)(cid:12)(cid:12)(cid:80) k (cid:82) Ω k J h · E h d x (cid:12)(cid:12)(cid:12) .Proof. From the discrete weak-form of Maxwell’s equations, we need to computeequations for | E h | and | B h | . Since each component of the field lies in the selectedtest space, we take the i th -component of Eq. (2.29) and use B hi as a test function,e.g., choose ϕ = B hx . Summing these three equations will give us an expressionfor the time-derivative of | B h | . We follow the same procedure for Eq. (2.30), whichgives an expression for the time-derivative of | E h | . With a bit of algebra, we obtain ddt (cid:90) Ω j | B h | d x + (cid:73) ∂ Ω j d s · ˆ E h × B − h + (cid:90) Ω j E h · ∇ x × B h d x = 0 , (2.76)74nd (cid:15) µ ddt (cid:90) Ω j | E h | d x − (cid:73) ∂ Ω j d s · ˆ B h × E − h − (cid:90) Ω j B h · ∇ x × E h d x = − (cid:90) Ω j J h · E h d x . (2.77)We now multiply both equations by 1 /µ and add them. Since E h · ∇ x × B h − B h · ∇ x × E h = ∇ x · ( B h × E h ) , (2.78)we can combine the third terms of Eqns. (2.76) and (2.77), (cid:90) Ω j ∇ x · ( B h × E h ) d x = (cid:73) ∂ Ω j d s · B − h × E − h . (2.79)In the above result, note that upon integration by parts, we must use the field justinside the face of cell Ω j . Hence, the evolution of the electromagnetic energy in asingle cell becomes ddt (cid:90) Ω j (cid:18) (cid:15) | E h | + 12 µ | B h | (cid:19) d x + (cid:73) ∂ Ω j d s · (cid:16) ˆ E h × B − h + E − h × ˆ B h − E − h × B − h (cid:17) = − (cid:90) Ω j J h · E h d x . (2.80) Exact Energy Conservation With Central Flux.
Using central-fluxes to de-termine the interface fields, i.e., setting ˆ E h = (cid:74) E (cid:75) and ˆ B h = (cid:74) B (cid:75) , gives us, ddt (cid:90) Ω j (cid:18) (cid:15) | E h | + 12 µ | B h | (cid:19) d x + 12 (cid:73) ∂ Ω j d s · (cid:0) E + h × B − h + E − h × B + h (cid:1) = − (cid:90) Ω j J h · E h d x , (2.81)where the E − h × B − h terms cancel upon substitution of central fluxes for the interfacefields. Summing over all configuration space cells and assuming appropriate bound-75ry conditions in configuration space, we see that the surface term vanishes becauseit is symmetric and has opposite signs for the two cells sharing an interface. Thiscancellation of the surface term leads to the desired discrete electromagnetic energyconservation equation, ddt (cid:88) k (cid:90) Ω k (cid:18) (cid:15) | E h | + 12 µ | B h | (cid:19) d x = − (cid:88) k (cid:90) Ω k J h · E h d x . (2.82) Monotonic Decay With Upwind Flux.
To see what happens when using up-wind fluxes, we transform the fields appearing in surface integral into the ( s , τ , τ )coordinate system. We can then write the third term in Eq. (2.80) as, d s · (cid:16) ˆ E h × B − h + E − h × ˆ B h − E − h × B − h (cid:17) = ds (cid:104) ( ˆ E B − − ˆ E B − ) + ( E − ˆ B − E − ˆ B ) − ( E − B − − E − B − ) (cid:105) . (2.83)Using Eqns. (2.48)-(2.51) for the interface fields, assuming appropriate boundaryconditions, and summing over all configuration space cells, we then obtain ddt (cid:88) k (cid:90) Ω k (cid:18) (cid:15) | E h | + 12 µ | B h | (cid:19) d x = − (cid:88) k (cid:90) Ω k J h · E h d x + (cid:88) j (cid:73) ∂ Ω j ds (cid:0) { E } E − /c + { E } E − /c + c { B } B − + c { B } B − (cid:1) . (2.84)Note that due to the symmetry of the terms, the central flux terms in Eqns. (2.48)-(2.51) have vanished on summing over all cells. Now consider the contribution of theterm { E } E − to the two cells adjoining some face. This term will be ( E +2 − E − ) E − / E − − E +2 ) E +2 /
2. On summing over the two cells, this contribution will become − ( E +2 − E − ) /
2. Similar results are achieved for the other electric and magneticfield coordinates. Hence, the surface terms, on summation, contribute non-positive76uantities to the right-hand side, implying that ddt (cid:88) k (cid:90) Ω k (cid:18) (cid:15) | E h | + 12 µ | B h | (cid:19) d x < − (cid:88) k (cid:90) Ω k J h · E h d x . (2.85)Note that because the resulting surface terms contribute non-positive quantities,we can say that, despite the sign of J h · E h being undetermined, the electromag-netic energy still monotonically decays, i.e., when the right hand side is positive,the electromagnetic energy will increase less than (cid:12)(cid:12)(cid:12)(cid:80) k (cid:82) Ω k J h · E h d x (cid:12)(cid:12)(cid:12) , and whenthe right hand side is negative, the electromagnetic energy will decay more than − (cid:12)(cid:12)(cid:12)(cid:80) k (cid:82) Ω k J h · E h d x (cid:12)(cid:12)(cid:12) . Lemma 3. If | v | belongs to the approximation space V ph , then the semi-discretescheme satisfies ddt (cid:88) j (cid:88) s (cid:90) K j m | v | f h d z − (cid:88) k (cid:90) Ω k J h · E h d x = 0 . (2.86) Note that the species index is implied, the sum over j in the first term is over allphase space cells, and the sum over k in the second term is over all configurationspace cells.Proof. If | v | ∈ V ph , we can set w = m | v | / (cid:90) K j m | v | ∂f h ∂t d z + (cid:73) ∂K j m | v | n · ˆ F dS − (cid:90) K j ∇ z (cid:18) m | v | (cid:19) · α h (cid:124) (cid:123)(cid:122) (cid:125) q v · E h f h d z = 0 . (2.87)Since | v | is continuous at cell interfaces, there is no distinction between the ba-sis function w evaluated just inside and outside the cell surface interface. Upon77umming over all cells and the number of species, and the use of appropriate bound-ary conditions in velocity space and configuration space as in Proposition 10, weare again able to exploit the fact that the numerical flux function is Godunov andcancel the telescopic sum to obtain, ddt (cid:88) j (cid:88) s (cid:90) K j m | v | f h d z − (cid:88) k (cid:90) Ω k J h · E h d x = 0 . (2.88)Note that we have performed the integration in velocity space and substituted inthe current density, leaving an integration and sum over only configuration space.This operation is somewhat subtle, and we will discuss this operation and operationssimilar in the next section, Section 2.4. Corollary 3.
Even if only using piecewise linear polynomials and | v | does not belong to the approximation space V ph , then the semi-discrete scheme satisfies ddt (cid:88) j (cid:88) s (cid:90) K j m | v | f h d z − (cid:88) k (cid:90) Ω k J h · E h d x = 0 . (2.89) We again note that the species index is implied, the sum over j in the first termis over all phase space cells, and the sum over k in the second term is over allconfiguration space cells. In this case, g refers to the projection of the prescribedfunction onto a lower order basis set.Proof. We define the projection of | v | onto piecewise linear basis functions as | v | .78ubstituting this in for our test function, w , in Eq. (2.22) we obtain, (cid:90) K j m | v | ∂f h ∂t d z + (cid:73) ∂K j m | v | n · ˆ F dS − (cid:90) K j ∇ z (cid:18) m | v | (cid:19) · α h (cid:124) (cid:123)(cid:122) (cid:125) q v · E h f h d z = 0 , (2.90)where v is the derivative of the piecewise linear representation of 1 / | v | and isa piecewise constant in each cell . We note that because | v | is also continuousat cell interfaces, we can again exploit the fact that the numerical flux function isGodunov, and upon summing over all cells and species, and employing appropriateboundary conditions in velocity and configuration space as in Proposition 10, cancelthe surface integral since it is a telescopic sum. We are then left with, ddt (cid:88) j (cid:88) s (cid:90) K j m | v | f h d z − (cid:90) K j q v · E h f h d z = 0 . (2.92)Upon substitution of the projected current, J h , after performing the velocity inte-gration first, we obtain the desired analogous expression to Lemma 3 for piecewiselinear polynomials. Proposition 12.
If central-fluxes are used for Maxwell’s equations, and if | v | ∈ V ph ,the semi-discrete scheme conserves total (particles plus field) energy exactly, ddt (cid:88) j (cid:88) s (cid:90) K j m | v | f h d z + ddt (cid:88) k (cid:90) Ω k (cid:18) (cid:15) | E h | + 12 µ | B h | (cid:19) d x = 0 . (2.93) We can show that v is the cell center velocity, ∇ v (cid:18) | v | (cid:19) = 12 ( v left + v right ) = v center , (2.91)since | v | is continuous. Here v left/right is the value of the velocity on the left (right) edge of thecell, so the average value of the two quantities is the cell center velocity. f upwind fluxes are used for Maxwell’s equations, and if | v | ∈ V ph , the semi-discretescheme decays the total (particles plus field) energy, ddt (cid:88) j (cid:88) s (cid:90) K j m | v | f h d z + ddt (cid:88) k (cid:90) Ω k (cid:18) (cid:15) | E h | + 12 µ | B h | (cid:19) d x < . (2.94) And if only piecewise linear polynomials are used and thus | v | / ∈ V ph , then theprojected energy will either be conserved or decaying depending on the choice offluxes for Maxwell’s equations, ddt (cid:88) j (cid:88) s (cid:90) K j m | v | f h d z + ddt (cid:88) k (cid:90) Ω k (cid:18) (cid:15) | E h | + 12 µ | B h | (cid:19) d x ≤ , (2.95) so long as the scheme is consistent and the appropriate current J h is incrementedon to the electric field in Maxwell’s equations.Proof. The proof of this proposition follows from the substitution of the results ofLemma 2 into the results of Lemma 3, or Corollary 3 if | v | is not in the solutionspace.We wish to make a few remarks about the results of this section. Firstly,we emphasize that energy conservation for the Vlasov equation was agnostic onthe specific form of the numerical flux function, central, upwind, or global Lax-Friedrichs, so long as the numerical flux is Godunov. Secondly, we want to pointout a subtlety in comparison between the continuous proof of energy conservation,Proposition 5, and the proof of energy conservation for our semi-discrete system,Proposition 12. The continuous proof involves the manipulation of terms which arehigher order than | v | , but we note that the higher order terms in the continuous80roof come from the substitution of the explicit expressions for α , the phase spaceflow, whereas in the discrete proof presented here, we have left the discrete phasespace flow α h as is to stress the fact that α h has its own basis function expansion.Thus, the higher order terms which are explicit in the continuous energy conservationproof are implicit here in the discrete energy conservation proof. If | v | ∈ V ph , then v ∈ V ph as well, and terms in the discrete phase space flow such as v , the configurationspace component of the phase space flow, can be exactly represented in terms of ourbasis function expansion.Finally, we note that, although the total energy decays when using upwindfluxes for Maxwell’s equations, this decay is small due to the high order natureof the scheme. We will demonstrate this explicitly in Chapter 4 as part of thebenchmarking of the algorithms. Other authors have also demonstrated that thisloss of energy is small for higher order schemes such as the DG method employedhere [Balsara and K¨appeli, 2017].Before we conclude this section on the properties of the semi-discrete Vlasov–Maxwell system of equations, we would be remiss not to discuss the evolution ofthe total momentum. The total momentum, particles plus fields, is conserved inthe continuous system of equations, but what about our semi-discrete system? Ourformulation of the DG method for the Vlasov–Maxwell system of equations does not conserve momentum.We can show momentum non-conservation by choosing w = m s v and proceed-81ng as we did with the continuous system, (cid:90) K j m v ∂f h ∂t d z + (cid:73) ∂K j m vn · ˆ F dS − (cid:90) K j ∇ z ( m v ) · α h f h d z = 0 . (2.96)Since v is continuous, upon summation over all phase space cells and species, weobtain (cid:88) j (cid:88) s (cid:90) K j m s v ∂f h ∂t d z − (cid:88) k (cid:90) Ω k ρ c h E h + J h × B h d x = 0 . (2.97)We can proceed exactly as we did with the continuous proof, but we note a keysubtlety, ddt (cid:88) j (cid:88) s (cid:90) K j m s v f h d z + ddt (cid:88) k (cid:90) Ω k ( (cid:15) E × B ) d x + (cid:90) Ω j ∇ x (cid:18) (cid:15) | E h | + 12 µ | B h | (cid:19) − ∇ x · (cid:18) (cid:15) E h E h + 1 µ B h B h (cid:19) d x = 0 . (2.98)Since the electric and magnetic fields are discontinuous across configuration spacecell interfaces, we cannot use integration by parts to eliminate the latter two terms.In other words, integration by parts holds only locally and not over the whole domaindue to the jumps in the fields across surfaces. However, it is important to note fromthe form of this equation that momentum conservation depends only weakly onvelocity space resolution. Since the size of the discontinuities in the electric andmagnetic fields decrease with increasing configuration space resolution, we can morestrongly conserve momentum by increasing configuration space resolution.So, our semi-discrete Vlasov–Maxwell system of equations using the discontin-uous Galerkin finite element method conserves mass, and can conserve the energy, L norm, and entropy depending on our choice of numerical flux function, while82ncurring errors in the total momentum due to our discretization of Maxwell’s equa-tions. We still need to discretize the system in time, but we will delay this discussionfor a moment as we move to the semi-discrete discretization of the Fokker–Planckcollision operator and the discrete Fokker–Planck collision operator properties. Be-fore we derive the semi-discrete form of the Fokker–Planck operator, it is useful togo into more detail on a concept we have been surreptitiously employing throughoutour discussion of the discontinous Galerkin method: the concept of weak equality.Weak equality underlies all of our discussion up to this point, but we have not madeexplicit what it means for two functions to be weakly equal, nor how we can use weakequality to actually compute quantities we require in our algorithm, such as velocitymoments and the drag and diffusion coefficients in the Fokker–Planck equation. Consider some interval I and some function space P spanned by basis set ψ (cid:96) , (cid:96) = 1 , . . . , N . We will define two functions f and g to be weakly equal if (cid:90) I ( f − g ) ψ (cid:96) dx = 0 , ∀ (cid:96) = 1 , . . . , N. (2.99)We will denote weakly equal functions by f . = g . Unlikely strong equality, in whichfunctions agree at all points in the interval, weak equality only assures us that theprojection of the functions on a chosen basis set is the same. However, the functionsthemselves may be quite different from each other with respect to their behaviour,e.g, each function’s positivity or monotonicity in the interval.83he connection between weak equality and the minimization of the error inthe L norm in Section 2.1 is immediately clear. In constructing a DG discretizationof some operator G [ f ], we are saying, ∂f∂t . = G [ f ] , (2.100)and then we construct a projection of the solution f in the space P , which we choseto be the space of piecewise polynomials of order p, , i.e., P p . This concept of weakequality is another means of deriving Eqns. (2.22) and (2.29)-(2.30) for the semi-discrete Vlasov–Maxwell system of equations, and why these forms for the Vlasovequation and Maxwell’s equation are referred to as the discrete weak forms for theseequations.The real power in the concept of weak equality is the ability to connect func-tions defined in different spaces. Consider an operation we performed as part of ourproof of energy conservation for the Vlasov equation, Lemma 3, (cid:88) j (cid:88) s (cid:90) K j q s v · E h f h d z .. = (cid:88) k (cid:90) Ω k J h · E h d x . (2.101)Note that we are using the .. = symbol here to emphasize that in the process ofproving Lemma 3, we took Eq. (2.101) as a definition.While Eq. (2.101) may seem to follow naturally from our definition of thecontinuous current density in Eq. (1.64), the subtlety here is that the distributionfunction projection is defined over the full phase space, f h ∈ V ph , while the currentdensity is defined only in the solution space for configuration space, J h ∈ X ph . But,84ere is where we can leverage weak equality, J h . = (cid:88) j (cid:88) s (cid:90) K j \ Ω k q s v f h d v , (2.102)i.e., we project the integral over velocity space of the distribution function, weightedby q s v in this case, onto configuration space basis functions in the space X ph . Notethe change of subscript between the phase space cell K j and configuration space cellΩ k since for the purposes of this operation, we need to sum the contributions fromall the velocity space cells for a given configuration space cell. The full computationfor this expression would be (cid:88) m J m (cid:90) Ω k ϕ m ϕ (cid:96) d x = (cid:88) j (cid:88) s (cid:90) Ω k (cid:90) K j \ Ω k q s v f h ϕ (cid:96) d v d x , (2.103)upon plugging in the phase space expansion of the distribution function. Note thatthis operation is performed for all ϕ (cid:96) ∈ X ph . This procedure gives us a general meansof defining the velocity space moments, such as the current density, which couplethe particle dynamics and the electromagnetic fields.So, the actual operation for proving Lemma 3 is (cid:88) j (cid:88) s (cid:90) K j q s v · E h f h d z . = (cid:88) k (cid:90) Ω k J h · E h d x . (2.104)Importantly, for the purposes of using the weak operation to compute the currentdensity in Eq. (2.104), we should have technically substituted w = 1 / m s | v | ϕ (cid:96) ( x ),where ϕ (cid:96) are each of our (cid:96) configuration space basis functions, as our test function w when proving Lemma 3 (and Corollary 3). In other words, to actually convertthe integral over velocity space of v · E h f h to the discrete analog of J h · E h , we must85nsure we are projecting the velocity integral of v · E h f h onto the full configurationspace expansion.These procedures, such as the operation defined in Eq. (2.103), are sometimesreferred to as weighted L projections, or more generally weighted projections, if thenorm of choice is not the L norm. A more mathematically complete discussion ofthese types of projection operators can be found in textbooks on the foundations offinite element methods, such as Brenner and Scott [2008], and these operators arecommon throughout the literature [Cockburn and Dawson, 2000]. In fact, there hasbeen growing interest in leveraging weighted L projections in novel ways, especiallyfor wave propagation in heterogeneous media, so that the complexities of the mediathe wave is propagating in are directly encoded within the discretization [Chan et al.,2017, Chan and Wilcox, 2019, Guo and Chan, 2020, Shukla et al., 2020].We will use the concept of weak equality in a similar fashion to the construc-tion of these weighted L projections to define other types of weak operators inanticipation of the needed machinery to discretize the Fokker–Planck equation inthe VM-FP system of equations. We will define a new set of notation to make thesubsequent discussion a bit more clear, M h . = (cid:88) j (cid:90) K j \ Ω k f h d v , (2.105) M h . = (cid:88) j (cid:90) K j \ Ω k v f h d v , (2.106) M h . = (cid:88) j (cid:90) K j \ Ω k | v | f h d v , (2.107)which are related to discrete representations of Eqns. (1.70–1.72), but without factors86f mass and the relevant constants. For example, we can compute the discrete chargedensity and discrete current density from Eqns. (2.105–2.106), ρ c h = (cid:88) s q s M hs , (2.108) J h = (cid:88) s q s M hs . (2.109)For the drag and diffusion coefficients in the Fokker–Planck equation, Eqns.(1.54) and (1.55), we require the flow and temperature, Eqns. (1.56) and (1.57),which involve a number of different operations applied to the velocity moments,such as the division of two velocity moments in Eq. (1.56). We might naively expectthe discrete representation for the flow to be u h = (cid:80) j (cid:82) K j \ Ω k v f h d v (cid:80) j (cid:82) K j \ Ω k f h d v . (2.110)But, we only know the projections of the moments, not the actual functions, so sim-ple division like in Eq. (2.110) is ill-defined. To make this point more concrete, con-sider what constructing a polynomial expansion of u h defined in Eq. (2.110) wouldrequire: a polynomial expansion of a rational function, since both the numeratorand denominator have their own polynomial expansions in Eq. (2.110). We cannotproject a rational function onto a polynomial as we would incur aliasing errors inthe construction of the polynomial because a rational function requires an infinitenumber of polynomials to represent, and we are already limiting ourselves to a finitesubspace of polynomials.By aliasing errors, we mean errors that arise due to being unable to uniquelydetermine the representation of the quantity of interest. Because the flow u h in87q. (2.110) has no unique representation in a finite subspace of polynomials, theultimate computation of Eq. (2.110) can lead to uncontrolled and unbounded errors .We will find later in Chapter 3 that the elimination of aliasing errors will prove acritical component of constructing stable discretizations of the VM-FP system ofequations.So, in anticipation of this requirement for our algorithm, how do we eliminatealiasing errors in the computation of the flow and temperature required by theFokker–Planck equations? To find u h , we need to invert the weak-operator equation, M h u h . = M h . (2.111)Using the definition of weak-equality in Eq. (2.99) extended to multiple dimensions,this expression means, (cid:90) I ( M h u h − M h ) ψ (cid:96) d x = 0 , (2.112)where in general, for our algorithms, the space I is a configuration space cell Ω j (or K j ) and the basis expansion is ϕ ∈ X ph (or w ∈ V ph ). This procedure shoulddetermine u h , i.e., the projection of the flow in the function space, so we can write u h = (cid:80) m u m ψ m , leading to the linear system of equations (cid:88) m u m (cid:90) I M h ψ (cid:96) ψ m d x = (cid:90) I M h ψ (cid:96) d x , (2.113) A suitable analogy would be the aliasing that arises in the context of Fourier transforms,where an undersampled signal, a signal which would require a higher sampling rate to resolvethe Nyquist frequency, will produce an inaccurate Fourier transform due to power in the higherfrequencies being “aliased” into the signal[see, e.g., Press et al., 2007]. This power aliased into thesignal is the same manifestation of the unbounded and uncontrolled errors that arise in trying toconstruct a polynomial representation of a rational function like the rational function in our naivedefinition of the flow u h in Eq. (2.110). (cid:96) = 1 , . . . , N . Inverting this linear system determines u m and hence the pro-jection of u h in the function space. We call this process weak-division . Note thatweak-division only determines u h to an equivalence class as we can replace the spe-cific u h in the function space with any other function that is weakly equal to it.In addition, note that M h and M h are determined by Eq. (2.105) and Eq. (2.106)and thus themselves have expansions that must be included in this computation.Therefore, the weight in the weighted L projection is the basis expansion of themoment M h .We can follow a similar procedure for the temperature, M h T h m . = 13 ( M h − M h · u h ) , (2.114)where the factor of 1 / weak-division and what can be referred to as weak-multiplication ,because we require the expansion of M h · u h , K h . = M h · u h , (2.115) (cid:88) m K m (cid:90) I ψ (cid:96) ψ m d x = (cid:90) I M h · u h ψ (cid:96) d x , (2.116)where both M h and u h themselves have expansions which must be included in thecomputation. These sorts of “polynomial operations,” where division and multi-plication are extended to act on quantities which have expansions in some basis,have been exploited previously in the literature [Atkins and Shu, 1998, Lockard andAtkins, 1999]. 89aving generalized certain operations such as division and multiplication tosituations where all the quantities of interest are projections, we can ask the ques-tion: do these operations have similar rules to their elementary counterparts? Forexample, does weak-division have the equivalent of divide-by-zero issues? Considerthe interval [ − ,
1] and the orthonormal linear basis set ψ = 1 √ ψ = √ √ x. (2.117)In one dimension, where M h and u h have just a single component, let M = 1 and M = n ψ + n ψ . For this simple case, the result of weak-division is u = √ n − n ( n − √ n x ) . (2.118)Hence, the weak-division is not defined for n = ± n . This calculation showsthat, even if the mean density is positive, the slope cannot become too steep—seeFigure 2.4 where we plot the trend of steepening the slope of density, M , and theeffect of the steepening on the calculation of the flow, u . When the “blow-up”occurs, i.e., n = ± n , M has a zero-crossing at either x = ± / √
3. Althoughthe function for the flow, u , appears well behaved through the steepening of thedensity, this blow-up corresponds to the situation where the density itself becomesunrealizable with a positive definite function.In this regard, there is nothing necessarily unphysical with a piecewise lin-ear reconstruction M ( x ) having a zero crossing within the domain, and the DGalgorithm can result in such solutions. In principle, there is a physically realizable Formally, the mean density n is a positive definite quantity, so this constraint should simplybe n = n p = 1 basis, Eq. (2.117), to compute u from M u . = M . In this plot, M = 1 and the effect of changing M (top row) onthe flow u (bottom row) is shown. As the density steepens, the velocity becomeslarger. If the density becomes too steep, if we increase the slope of the densityfurther so that the density has a zero crossing at x = ± / √ u would blows in the sense that the flow becomes an unrealizablefunction. Importantly, this blow-up condition corresponds to the situation wherethe slope of M becomes too steep to represent M with a positive definite function,which physically corresponds to a situation where the representation of the densityis producing negative density functions. Since the value of the particle density canonly be positive, this blow-up is highly undesirable.function that is weakly equivalent ˜ M ( x ) . = M ( x ) but positive everywhere, as longas n < √ n . However, if the slope of M becomes too steep, if the density variestoo rapidly within a cell, we lose the ability to construct a physically realizablerepresentation for the density, and thus the computation of the flow u would alsobecome physically unrealizable. In practice we can use constraints like these tolimit the slope of the density and thus make the weak division operator always wellposed. This idea of using these constraints to limit the higher order moments in91ur DG expansion is similar to the philosophy of limiters in high order finite-volumemethods. In smooth regions, we use can the standard calculations and so retainhigh-order accuracy there, while introducing limiters where the solution is locallyvarying too quickly to be accurately resolved, in order to robustly preserve certainproperties of the solution.Another application of weak-equality is to recover a continuous function froma discontinuous one. Say we want to construct a continuous representation ˆ f on theinterval I = [ − , f , which has a single discontinuity at x = 0.We can choose some function spaces P L and P R on the interval I L = [ − ,
0] and I R = [0 ,
1] respectively. Then, we can reconstruct a continuous function ˆ f such thatˆ f . = f L x ∈ I L on P L (2.119)ˆ f . = f R x ∈ I R on P R . (2.120)where f = f L for x ∈ I L and f = f R for x ∈ I R .As with all our previous discussions about weak equality, this procedure onlydetermines ˆ f up to its projections in the left and right intervals. To determine ˆ f uniquely, we use the fact that given the N pieces of information in I L and N piecesof information in I R , where N is the number of basis functions in P L,R , we canconstruct a polynomial of maximum order 2 N −
1. We can hence writeˆ f ( x ) = N − (cid:88) m =0 ˆ f m x m . (2.121)Using this expression in Eqn. (2.119) and (2.120) completely determines ˆ f . In acertain sense, the recovery procedure is a special case of a more general method92o go from one basis to another under the restriction of weak equality, just as weconstructed the velocity moments, defined only in configuration space, from anoperation over the full phase space.And just as we leveraged weak equality to give us a prescription for the com-putation of the components of the drag and diffusion coefficients, this procedure torecover a continuous function from discontinuous function foreshadows an additionalneed we have when discretizing the Fokker–Planck equation: the ability to computesecond derivatives. As an example in one dimension, we wish to compute g . = f xx where we know f on a mesh with cells I j = [ x j − / , x j +1 / ]. Multiply by some testfunction ψ ∈ P j , where P j is the function space in cell I j and integrate to get thefollowing weak-form, (cid:90) I j ψg dx = ψ ˆ f x (cid:12)(cid:12)(cid:12)(cid:12) x j +1 / x j − / − (cid:90) I j ψ x f x dx. (2.122)Where we have replaced f by the reconstructed function ˆ f in the surface term. Notethat we need two reconstructions , one using data in cells I j − , I j and the other usingdata in cells I j , I j +1 . In the volume term, we continue to use f itself and not theleft/right reconstructions as the latter are weakly-equal to the former and can bereplaced without changing the volume term. Once the function space is selected, wehave completely determined g .Notice that one more integration by parts can be performed in Eq. (2.122) toobtain another weak-form, (cid:90) I j ψg dx = ( ψ ˆ f x − ψ x ˆ f ) (cid:12)(cid:12)(cid:12)(cid:12) x j +1 / x j − / + (cid:90) I j ψ xx f dx. (2.123)93n this form, we need to use both the value and first derivative of the reconstructedfunctions at the cell interfaces. Numerically, each of these weak-forms will lead todifferent update formulas. For example, for piecewise linear basis functions, thevolume term drops out in Eq. (2.123). We will find two integration by parts allowsus to retain more properties of the continuous Fokker–Planck equation in our semi-discrete formulation of the Fokker–Planck equation in Section 2.6.The procedure outlined above is essentially the recovery discontinuous Galerkin(RDG) scheme first proposed in van Leer and Nomura [2005] and van Leer and Lo[2007]. Extensive study of the properties of the RDG scheme to compute secondderivatives is presented in Hakim et al. [2014], where it is shown that the RDGscheme has some advantages compared to the standard local discontinuous Galerkin(LDG) schemes [Cockburn and Shu, 1998a, Cockburn and Dawson, 2000] tradition-ally used to discretize diffusion operators in DG. The formulation in terms of weakequality allows systematic extension to higher dimensions just as we developed gen-eral formulas for velocity moments irrespective of dimensionality, and we turn now toa semi-discrete formulation of the Fokker–Planck equation given the tools outlinedin this section. We now want to derive the semi-discrete form of the Fokker–Planck equationusing a DG method. Since the Fokker–Planck equation is solved in tandem withthe Vlasov–Maxwell portion of the VM-FP system of equations, we will consider94he same phase space mesh, T , with cells K j , and solution space V ph defined inEq. (2.21). For the Fokker–Planck component, we can integrate by parts once toobtain a discrete weak form, analogous to the collisionless component of the VM-FP system of equations in Eq. (2.22), (cid:90) K j w ∂f h ∂t d z = (cid:73) ∂K j ν w − n · ˆ G dS − (cid:90) K j ν ∇ v w · (cid:20) ( v − u h ) f h + T h m ∇ v f h (cid:21) d z . (2.124)Here, the numerical flux function ˆ G includes both the drag and diffusion terms, n · ˆ G = n · (cid:18) ˆ F drag + T h m ∇ v ˆ f (cid:19) , (2.125)where ˆ F drag is a numerical flux function for the drag term. Our only requirementfor the numerical flux function for the drag term will be that, like the collisionlessflux in phase space, this numerical flux function for the drag term is a Godunovflux, Eq. (2.24). The latter term involves the recovery of the distribution function ata velocity space cell as described in Section 2.4. Note that T h is unchanged by therecovery process, as the temperature is only a function of configuration space, andthus is continuous across velocity space interfaces. As with the collisionless phasespace flux, example Godunov fluxes for the drag term include, n · ˆ F drag = 12 n · ( v − u h )( f + + f − ) , (2.126) n · ˆ F drag = n · ( v − u h ) f − if sign( v − u h ) > , n · ( v − u h ) f + if sign( v − u h ) < , (2.127) n · ˆ F drag = 12 n · ( v − u h )( f + + f − ) − max T | v − u h | f + − f − ) , (2.128)95here we have already exploited the fact that v − u h is continuous at velocity spaceinterfaces to simplify a central flux, upwind flux, and global Lax-Friedrichs flux tothe forms shown in Eqns. (2.126–2.128).While Eq. (2.124) may seem like a perfectly fine DG method for the Fokker–Planck equation, the method as written in Eq. (2.124) does not retain some of theimportant properties of the continuous system. For example, we can show that themethod as written in Eq. (2.124) does not conserve momentum. To see this lackof conservation, substitute w = m v , where we have dropped the species subscriptbecause, as we showed in Section 1.6, the Fokker–Planck equation conserves themomentum of each species individually. Upon substitution of w = m v and summingover all cells, we obtain (cid:88) j (cid:90) K j m v ∂f h ∂t d z = − (cid:88) j (cid:90) K j mν ∇ v v · (cid:20) ( v − u h ) f h + T h m ∇ v f h (cid:21) d z , (2.129)where we have already eliminated the surface term due to the assumption of the fluxbeing Godunov and the fact that v is continuous at velocity space interfaces. Notethat implicit in the cancellation of the surface terms is the fact that we are againemploying zero-flux boundary conditions in velocity space, similar to Eq. (2.53), n · ˆ G ( x , v max ) = n · ˆ G ( x , v min ) = 0 . (2.130)Additionally, while the numerical flux due to the drag is a Godunov flux, and thuswhy it can be eliminated upon summation over cells, the reason the ∇ v ˆ f term, thegradient of the recovered distribution function, vanishes is for the simple reasonthat ∇ v ˆ f can also be constructed to be continuous at the corresponding interfaces.96hen constructing the recovered distribution function, we can make sure that boththe value, and the slope, are continuous at the shared interface, a desirable propertyfor the discretization of a diffusion operator!Substitution of ∇ v v = ←→ I allows us to determine under what conditions ourdiscrete scheme conserves momentum. Firstly, we require that (cid:88) j (cid:90) K j ( v − u h ) f h = 0 , (2.131)but this is simply M h u h . = M h once the integrals are separated into their con-figuration space and velocity space components . So, if we ensure computation ofthe discrete flow exactly as we described in Section 2.4, i.e., that the projection ofthe discrete flow is consistent and incurs no aliasing errors, this term will vanish.Unfortunately, the final term, (cid:88) j (cid:90) K j T h m ∇ v f h = (cid:88) j (cid:73) ∂K j T h m f − h dS (cid:54) = 0 , (2.132)since the distribution function is not continuous at cell interfaces. Thus, in this for-mulation of the semi-discrete Fokker–Planck equation, we cannot expect to conservemomentum.A similar argument shows that the semi-discrete Fokker-Planck equation de-scribed in Eq. (2.124) does not conserve energy either. The lack of conservation ofboth momentum and energy can be traced to the gradient term, T h /m ∇ v f h , which Note that Eq. (2.111) is a stronger statement than Eq. (2.131), because Eq. (2.111) is the fullprojection of the flow onto the configuration space basis expansion. As we mentioned in Section 2.4when discussing the equality in Eq. (2.104), for the purposes of discussing conservation relations,we have substituted expressions such as w = v or w = | v | , but we could have just as easilysubstituted w = v ϕ (cid:96) ( x ) or w = | v | ϕ (cid:96) ( x ), where ϕ (cid:96) ∈ X ph are each of the (cid:96) basis functionsspanning configuration space. Doing so would not change the algebra and the subsequent proofsand would make the connection between Eq. (2.111) and Eq. (2.131) concrete. (cid:90) K j w ∂f h ∂t d z = (cid:73) ∂K j ν w − n · ˆ G dS − (cid:73) ∂K j ν n · ∇ v w − T h m ˆ f dS − (cid:90) K j ν (cid:20) ∇ v w · ( v − u h ) f h − ∇ v w (cid:18) T h m f h (cid:19)(cid:21) d z . (2.133)For this scheme, we require both the value and the slope of the recovered distributionfunction at the cell interfaces. Eq. (2.133) will be the form whose properties weexamine in the next section as we determine what we have retained compared tothe continuous Fokker–Planck equation. We now proceed as we did in Section 2.3, but for the semi-Discrete Fokker–Planck equation, to determine what properties the semi-discrete formulation retainsin comparison to the continuous equation. As with the semi-discrete Vlasov equa-tion, we will assume the boundary conditions in velocity space are zero flux, n · ˆ G ( x , v max ) = n · ˆ G ( x , v min ) = 0 .
98n addition, we note that there is an additional boundary term due to our secondintegration by parts, (cid:73) ∂K j ν n · ∇ v w − T h m ˆ f dS = (cid:90) Ω j (cid:73) ∂V max/min ν n · ∇ v w − T h m f h ( x , v max/min , t ) d x dS V max/min , (2.134)where we have separated the surface integral over the edge of velocity space intoan integral over configuration space and the specific edge of velocity space surface,and we have substituted for the recovery polynomial at the edge of velocity spacethe distribution function evaluated at the edge of velocity space. Since we haveno information beyond the edge of velocity space due to the zero flux boundarycondition on the numerical flux function ˆ G , choosing the recovery polynomial atthe edge of velocity space to be simply the distribution function evaluated at theedge is the most natural choice. This vector notation may seem somewhat strange,so as a concrete example, this operation for the v x derivative is (cid:73) ∂K j ν ˆ x · ∇ v w − T h m ˆ f dS = (cid:90) Ω j (cid:73) ∂V max/min ν ∇ v x w − T h m f h ( x , v x max/min , v y , v z , t ) d x dv y dv z , (2.135)i.e., for the edge of v x in velocity space, we evaluate the distribution function atthe maximum or minimum v x and leave the other dependencies (all of x and v y , v z )intact to be integrated over. This particular boundary term will turn out to beimportant for the conservation properties of the semi-discrete system, in addition tobeing an explicit boundary term required as part of the complete update formula.99ote that, because the Fokker–Planck equation only involves derivatives in velocityspace, our semi-discrete formulation of the Fokker–Planck equation is agnostic tothe boundary conditions in configuration space for the following properties. We willalso drop the species subscript from the mass, as we know from Section 1.6 thatthe continuous Fokker–Planck equation conserves mass, momentum, and energyindividually for each species. Proposition 13.
The discrete scheme in Eq. (2.133) conserves mass, ddt (cid:88) j (cid:90) K j mf h d z = 0 . (2.136) Proof.
Substituting w = m into Eq. (2.133) and summing over all cells, we obtain (cid:88) j (cid:90) K j m ∂f h ∂t d z = (cid:88) j (cid:73) ∂K j νm n · ˆ G dS = 0 , (2.137)since the gradient of a constant function is zero, and we have chosen the numeri-cal flux function ˆ G to be a Godunov flux so that the sum over surfaces pairwisecancels the flux. Combined with a zero flux boundary condition in velocity space,the proof of mass conservation is complete. Note that because the Fokker–Planckequation only contains derivatives in velocity space, just like the continuous proofin Section 1.6, we can construct the time evolution of the zeroth moment due to thesemi-discrete Fokker–Planck equation, (cid:88) j (cid:90) K j \ Ω j m ∂f h ∂t d z . = ∂ρ m h ∂t = 0 , (2.138)where ρ m h is the projection of the mass density onto configuration space basis func-tions. 100 roposition 14. The discrete scheme in Eq. (2.133) conserves momentum, ddt (cid:88) j (cid:90) K j m v f h d z = 0 , (2.139) if T h (cid:32)(cid:88) j (cid:73) ∂V maxj f h dS V max − (cid:88) j (cid:73) ∂V minj f h dS V min (cid:33) + m M h − mM h u h . = 0 , (2.140) i.e., for each velocity component, for example the x component, we have T h (cid:34)(cid:88) j (cid:73) ∂V maxj f h ( x , v x max , v y , v z ) dv y dv z − (cid:88) j (cid:73) ∂V minj f h ( x , v x min , v y , v z ) dv y dv z (cid:35) + mM xh − mM h u x h . = 0 , (2.141) where we have temporarily dropped the time dependence from f h for ease of notation.Proof. Substituting w = m v into Eq. (2.133) and summing over all cells, we obtain (cid:88) j (cid:90) K j m v ∂f h ∂t d z = − (cid:88) j (cid:73) ∂K j νT h ˆ f dS − (cid:88) j (cid:90) K j νm ( v − u h ) f h d z , (2.142)where we have eliminated the sum over the surface integral involving the numericalflux function ˆ G since it involves the Godunov flux for the drag and the gradient ofthe recovered distribution function, both of which cancel upon pairwise summationover the shared surfaces. Likewise, ∇ v v = 0, so the second volume term vanishes.These simplifications leave the surface term involving the value of the recovereddistribution function, plus the volume term for the drag. Since the recovered distri-bution function is continuous at the shared interface, this term also pairwise cancelsupon execution of the sum over the surfaces, with the exeception of the contribution101t the edge of velocity space. Thus, to conserve momentum, we require, (cid:90) Ω k T h (cid:32)(cid:88) j (cid:73) ∂V maxj f h dS V max − (cid:88) j (cid:73) ∂V minj f h dS V min (cid:33) d x + (cid:90) Ω k (cid:34)(cid:88) j (cid:90) K j \ Ω k m ( v − u h ) f h d v (cid:35) d x = 0 , (2.143)where we have used the fact that the Fokker–Planck equation only involves deriva-tives in velocity space to explicitly separate the configuration space and velocityspace integrals, i.e., we have made the conservation of momentum local to a config-uration space cell as it must be given the continuous proof in Proposition 7. Butwe note that this constraint is simply T h (cid:32)(cid:88) j (cid:73) ∂V maxj f h dS V max − (cid:88) j (cid:73) ∂V minj f h dS V min (cid:33) + m M h − mM h u h . = 0 , with the caveat that the weak equality will involve a projection over the entireconfiguration space basis expansion. To complete the proof, we just redo this cal-culation with w = m v ϕ (cid:96) ( x ) for each of our (cid:96) configuration space basis functions, ϕ (cid:96) ∈ X ph , so that we can substitute (cid:90) Ω k (cid:34)(cid:88) j (cid:90) K j \ Ω k m ( v − u h ) f h d v (cid:35) d x = 0 , with m M h − mM h u h . = 0 . (2.144)For clarity, we note that the constraint equation for the flow and temperature re-quired for momentum conservation, Eq. (2.141), in one spatial dimension and one102elocity dimension, 1X1V, is T h (cid:2) f h ( v max ) − f h ( v min ) (cid:3) + mM h − mM h u h . = 0 . (2.145) Proposition 15.
The discrete scheme in Eq. (2.133) conserves energy, ddt (cid:88) j (cid:90) K j m | v | f h d z = 0 , (2.146) if | v | ∈ V ph , and T h (cid:34)(cid:88) j (cid:73) ∂V maxj ( n · v max ) f h dS V max − (cid:88) j (cid:73) ∂V minj ( n · v min ) f h dS V min (cid:35) + mM h − m M h · u h − M h T h . = 0 , (2.147) where the n · v max/min involves a sum over the contribution from each velocity spacesurface, i.e., T h (cid:34)(cid:88) j (cid:73) ∂V maxj v x max f h ( x , v x max , v y , v z ) dv y dv z − (cid:88) j (cid:73) ∂V minj v x min f h ( x , v x min , v y , v z ) dv y dv z (cid:35) + T h (cid:34)(cid:88) j (cid:73) ∂V maxj v y max f h ( x , v x , v y max , v z ) dv x dv z − (cid:88) j (cid:73) ∂V minj v y min f h ( x , v x , v y min , v z ) dv x dv z (cid:35) + T h (cid:34)(cid:88) j (cid:73) ∂V maxj v z max f h ( x , v x , v y , v z max ) dv x dv y − (cid:88) j (cid:73) ∂V minj v z min f h ( x , v x , v y , v z min ) dv x dv y (cid:35) + mM h − m M h · u h − M h T h . = 0 , (2.148) where we have temporarily dropped the time dependence from f h for ease of notation.Proof. Since | v | is in our approximation space V ph , we can substitute w = 1 / m | v | (cid:88) j (cid:90) K j m | v | ∂f h ∂t d z = − (cid:88) j (cid:73) ∂K j νT h ( n · v ) ˆ f dS − (cid:88) j (cid:90) K j νm v · ( v − u h ) f h − νT h f h d z , (2.149)where we have again leveraged the fact that the surface integral involving the nu-merical flux function, ˆ G , is a Godunov flux for the drag, and the gradient of therecovered distribution function is continuous, so that both terms cancel upon pair-wise summation over the shared surfaces. We have also substituted ∇ v | v | = 2 v and ∇ v | v | = 6. As with our proof of discrete momentum conservation, Proposi-tion 14, the interior summation of the remaining surface terms vanishes since therecovered distribution function is continuous at velocity space surfaces, leaving onlythe integrals along the surfaces at the edge of velocity space. To conserve energy,we then must satisfy (cid:90) Ω k T h (cid:34)(cid:88) j (cid:73) ∂V maxj ( n · v max ) f h dS V max − (cid:88) j (cid:73) ∂V minj ( n · v min ) f h dS V min (cid:35) d x + (cid:90) Ω k (cid:34)(cid:88) j (cid:90) K j \ Ω k m (cid:0) | v | − v · u h (cid:1) f h − T h f h d v (cid:35) d x = 0 , (2.150)where we have again used the fact that the Fokker–Planck equation only involvesderivative in velocity space to explicitly separate the configuration space and velocityspace integrals, i.e., we have made the conservation of energy local to a configurationspace cell as it must be given the continuous proof in Proposition 8. This constraint104s simply T h (cid:34)(cid:88) j (cid:73) ∂V maxj ( n · v max ) f h dS V max − (cid:88) j (cid:73) ∂V minj ( n · v min ) f h dS V min (cid:35) + mM h − m M h · u h − M h T h . = 0 , so long as we recognize that weak equality is a stronger statement than the constraintin Eq. (2.150), and we repeat our calculation with w = 1 / m | v | ϕ (cid:96) ( x ) for each ofour (cid:96) configuration space basis functions, ϕ (cid:96) ∈ X ph , so that we can substitute (cid:90) Ω k (cid:34)(cid:88) j (cid:90) K j \ Ω k m (cid:0) | v | − v · u h (cid:1) f h − T h f h d v (cid:35) d x = 0 , (2.151)with mM h − m M h · u h − M h T h . = 0 . (2.152)We note that in one spatial dimension and one velocity dimension, 1X1V, thisconstraint in Eq. (2.147) is simply T h (cid:2) v max f h ( v max ) − v min f h ( v min ) (cid:3) + mM h − mM h u h − T h M h . = 0 , (2.153)where the coefficient multiplying T h M h has reduced from three to one becausewe are now only integrating over one velocity dimension, instead of three velocitydimensions.One of the most important consequences of Propositions 14 and 15 is that theconstraints, Eqns. 2.141 and 2.147, provide a closed set of equations to determinethe components of the drag and diffusion coefficients. Collecting our constraint105quations, T h (cid:32)(cid:88) j (cid:73) ∂V maxj f h dS V max − (cid:88) j (cid:73) ∂V minj f h dS V min (cid:33) + m M h − mM h u h . = 0 ,T h (cid:34)(cid:88) j (cid:73) ∂V maxj ( n · v max ) f h dS V max − (cid:88) j (cid:73) ∂V minj ( n · v min ) f h dS V min (cid:35) + mM h − m M h · u h − M h T h . = 0 , or in one spatial dimension and one velocity dimension (1X1V), T h (cid:2) f h ( v max ) − f h ( v min ) (cid:3) + mM h − mM h u h . = 0 ,T h (cid:2) v max f h ( v max ) − v min f h ( v min ) (cid:3) + mM h − mM h u h − T h M h . = 0 , we have a system of linear equations which allow us to uniquely determine the tem-perature, T h , and flow, u h , which can then be substituted into our discrete weakform, Eq. (2.133). These expressions may at first seem surprising, as they are acoupled set of linear equations, involving corrections to the temperature, T h , andflow, u h , based on the value of the distribution function at the boundary of veloc-ity space. If the distribution function vanishes at the boundary, one can eliminatethese boundary conditions and recover what we might naively expect for the con-straint equations for the temperature and flow, e.g., Eq. (2.111) for the flow. Butcritically, because we are using a zero-flux boundary condition in velocity space,the distribution function is not exactly zero at the boundary, and one must ac-count for this correction, however small it may be, to ensure the discrete scheme forthe Fokker–Planck equation conserves momentum and energy, both locally withina configuration space cell, and globally. 106dditionally, we note that we only discussed the case when | v | ∈ V ph whenexamining whether the semi-discrete Fokker–Planck equation conserved energy inProposition 8. Since we showed in Corollary 3 and Proposition 12 that the semi-discrete Vlasov–Maxwell system of equations conserves energy, even if only employ-ing piecewise linear polynomials and thus | v | / ∈ V ph , we can examine a similar case,but for the semi-discrete Fokker–Planck equation. We note that we can at thispoint connect our discussion about the projection of | v | onto piecewise linear basisfunctions using the language of weak equality, i.e., | v | . = | v | . (2.154)We emphasize again an important property of this projection: just like | v | , | v | iscontinuous in velocity space, so that we do not have to worry about discontinuitiesin the projection of | v | onto piecewise linear basis functions. Proposition 16.
The discrete scheme in Eq. (2.133) conserves energy when usingpiecewise linear polynomials, ddt (cid:88) j (cid:90) K j m | v | f h d z = 0 , (2.155) where | v | is the projection of | v | onto piecewise linear basis functions, if T h (cid:34)(cid:88) j (cid:73) ∂V maxj ( n · v max ) f h dS V max − (cid:88) j (cid:73) ∂V minj ( n · v min ) f h dS V min (cid:35) + mM ∗ h − m M ∗ h · u h − M ∗ h T h . = 0 . (2.156) Here, n · v max/min involves a sum over the contribution from each velocity space urface, i.e., T h (cid:34)(cid:88) j (cid:73) ∂V maxj v x max f h ( x , v x max , v y , v z ) dv y dv z − (cid:88) j (cid:73) ∂V minj v x min f h ( x , v x min , v y , v z ) dv y dv z (cid:35) + T h (cid:34)(cid:88) j (cid:73) ∂V maxj v y max f h ( x , v x , v y max , v z ) dv x dv z − (cid:88) j (cid:73) ∂V minj v y min f h ( x , v x , v y min , v z ) dv x dv z (cid:35) + T h (cid:34)(cid:88) j (cid:73) ∂V maxj v z max f h ( x , v x , v y , v z max ) dv x dv y − (cid:88) j (cid:73) ∂V minj v z min f h ( x , v x , v y , v z min ) dv x dv y (cid:35) + mM ∗ h − m M ∗ h · u h − M ∗ h T h . = 0 , (2.157) and the “star moments” are defined as follows, M ∗ h . = (cid:88) j (cid:54) = j max (cid:73) ∂K j \ Ω k ( n · ∆ v ) ˆ f dS, (2.158) M ∗ h . = (cid:88) j (cid:90) K j \ Ω k v f h d v , (2.159) M ∗ h . = (cid:88) j (cid:90) K j \ Ω k v · v f h d v , (2.160) where ˆ f is the recovery polynomial, v is / ∇ v | v | and equal to the cell centervelocity, as previously shown in Corollary 3, and ∆ v is the 1D grid spacing alongthe direction v . Note that ∆ v in the j th cell is related to the cell center velocity, ∆ v j = v j +1 − v j , (2.161) and the sum in Eq. (2.158) is over all surfaces except the edges of velocity space,i.e., the last index j max .Proof. Since we are restricting ourselves to only using piecewise linear polynomials, | v | / ∈ V ph , and we must project | v | onto our basis set using Eq. (2.154). We can108hen substitute w = 1 / m | v | into Eq. (2.133) and sum over cells to obtain (cid:88) j (cid:90) K j m | v | ∂f h ∂t d z = − (cid:88) j (cid:73) ∂K j νT h ( n · v ) ˆ f dS − (cid:88) j (cid:90) K j νm v · ( v − u h ) f h d z , (2.162)where 1 / ∇ v | v | = v , the cell center velocity. Note the differences in Eq. (2.162)compared to Eq. (2.149) in Proposition 15, when we were employing at least quadraticpolynomials and | v | ∈ V ph . The sum over surface integrals involving the numericalflux function, ˆ G , still vanishes because | v | , despite being a projection, is continuousacross velocity space interfaces, and we can thus still leverage the fact that the fluxfor the drag term is a Godunov flux and the gradient of the recovered distributionfunction is continuous to pairwise cancel the surface integrals in the sum. Impor-tantly, the volume term for the diffusion has vanished, since ∇ v | v | = 0. Likewise,we cannot cancel the interior sums over the surface in the remaining surface inte-grals like we did in Proposition 15 because v is not continuous at velocity spacesurfaces— v is a piecewise constant function! To have energy conservation, we thenmust have (cid:90) Ω k T h (cid:34)(cid:88) j (cid:73) ∂V maxj ( n · v max ) f h dS V max − (cid:88) j (cid:73) ∂V minj ( n · v min ) f h dS V min (cid:35) d x + (cid:90) Ω k (cid:34)(cid:88) j (cid:90) K j \ Ω k m v · ( v − u h ) f h d v − T h (cid:88) j (cid:54) = j max (cid:73) ∂K j \ Ω j ( n · ∆ v ) ˆ f dS (cid:35) d x = 0 , (2.163)where we have used Eq. (2.161) to simplify the interior surface integrals of the re-covered distribution function. Repeating our calculation for w = 1 / m | v | ϕ (cid:96) ( x ) for109ach of our (cid:96) configuration space basis functions, ϕ (cid:96) ∈ X ph , and using Eqns. (2.158–2.160), we then have T h (cid:34)(cid:88) j (cid:73) ∂V maxj ( n · v max ) f h dS V max − (cid:88) j (cid:73) ∂V minj ( n · v min ) f h dS V min (cid:35) + mM ∗ h − m M ∗ h · u h − M ∗ h T h . = 0 , exactly the constraint we expect for energy to be conserved. We note that in onespatial dimension and one velocity dimension, this constraint simplifies to T h (cid:2) v max f h ( v max ) − v min f h ( v min ) (cid:3) + M ∗ h − M ∗ h u h − T h M ∗ h . = 0 , (2.164)where, like in Eq. (2.153) in Proposition 15, the 1X1V constraint does not have afactor of three multiplying T h .So, the semi-discrete Fokker–Planck equation also retains conservation of en-ergy with piecewise linear polynomials, provided one modifies the constraint equa-tions, T h (cid:32)(cid:88) j (cid:73) ∂V maxj f h dS V max − (cid:88) j (cid:73) ∂V minj f h dS V min (cid:33) + m M h − mM h u h . = 0 ,T h (cid:34)(cid:88) j (cid:73) ∂V maxj ( n · v max ) f h dS V max − (cid:88) j (cid:73) ∂V minj ( n · v min ) f h dS V min (cid:35) + mM ∗ h − m M ∗ h · u h − M ∗ h T h . = 0 , or in one spatial dimension and one velocity dimension (1X1V), T h (cid:2) f h ( v max ) − f h ( v min ) (cid:3) + mM h − mM h u h . = 0 ,T h (cid:2) v max f h ( v max ) − v min f h ( v min ) (cid:3) + M ∗ h − M ∗ h u h − T h M ∗ h . = 0 . M h and M h , the standard moments given in Eqns. (2.105) and (2.106), but also the “starmoments” given by Eqns. (2.158 – 2.160). Using these coupled constraint equations,we can then uniquely compute the temperature, T h , and flow, u h , for use in oursemi-discrete Fokker–Planck equation.Before we conclude this section, we note that we have not discussed a discreteanalogy to the continuous system’s Second Law of Thermodynamics, Proposition 9,and H-theorem, Corollary 1. Unfortunately the finite velocity space extents requiredby our continuum approach complicate the requisite proofs, along with the requiredgradients of the expansion of ln( f h ). We will instead defer until Chapter 4, whenwe demonstrate numerically that the scheme still respects these essential physicsproperties. Having now constructed a semi-discrete scheme for the VM-FP system of equa-tions for the discretization of the equation system in phase space and configurationspace, we seek to complete the discretization with a discussion of how best to nu-merically integrate the semi-discrete system in time. We note that the result ofthe semi-discrete system is a set of ordinary differential equations. For the Vlasov–111okker–Planck equation we have ∂f h ∂t . = L ( f h , E h , B h , t ) , (2.165)and likewise for Maxwell’s equations, where L is a linear operator encompassing theevaluation of the integrals in the discrete weak forms, Eqns. (2.22) and (2.133), forall basis functions w ∈ V ph and all cells K j ∈ T . We will show in Chapter 3 howwe actually construct and evaluate L in Eq. (2.165). For now, we imagine that wehave evaluated L for the Vlasov–Fokker–Planck equation, and likewise Maxwell’sequations, and now need to solve the system of ordinary differential equations forthe time derivative of the discrete distribution function and electromagnetic fields.We consider in this thesis a class of strong stability preserving Runge–Kutta(SSP-RK) methods [Shu, 2002, Durran, 2010]. These methods are all multi-stageRunge–Kutta methods. Defining a forward Euler step as F ( f, t ) = f + ∆ t L ( f, t ) , (2.166)we can construct, for example, the second order SSP-RK, f (1) = F ( f n , t n ) ,f n +1 = 12 f n + 12 F (cid:0) f (1) , t n + ∆ t (cid:1) , (2.167)the third order SSP-RK, f (1) = F ( f n , t n ) ,f (2) = 34 f n + 14 F (cid:0) f (1) , t n + ∆ t (cid:1) ,f n +1 = 13 f n + 23 F (cid:0) f (2) , t n + ∆ t/ (cid:1) , (2.168)112nd the four stage third order SSP-RK: f (1) = F ( f n , t n ) ,f (2) = 12 f (1) + 12 F (cid:0) f (1) , t n + ∆ t/ (cid:1) ,f (3) = 23 f n + 16 f (2) + 16 F (cid:0) f (2) , t n + ∆ t (cid:1) ,f n +1 = 12 f (3) + 12 F (cid:0) f (3) , t n + ∆ t/ (cid:1) . (2.169)There are SSP-RK methods with more stages, as well as higher order, than themethods shown here [Shu, 2002]. Multi-stage Runge–Kutta methods require a bal-ance between the order of the scheme and the number of stages, and thus the amountof computations required. Especially for very high order multi-stage Runge–Kuttamethods, it can require increasingly large numbers of intermediate stages to attainmarginal improvements to the order of the scheme. We will most often employ thethree-stage, third order SSP-RK method, Eq. (2.168), as a balance between accuracy,computation, and memory footprint for the storage of the intermediate stages.The result of the SSP-RK-DG space-time discretization for the VM-FP systemof equations is a fully explicit scheme, and thus we expect to be restricted in thesize of our time-step by a Courant-Friedrich-Lewy (CFL) condition. CFL conditionsarise due to the restriction that we must be able to integrate the system of ordinarydifferential equations along the characteristics of the partial differential equation.In practical terms, imagine propagating a wave with velocity v in a discrete system.In order to propagate the wave along a discrete grid with some cell spacing ∆ x , wemust be careful not to take too large of a time-step, lest the wave move multiple gridcells in a single time-step and thus potentially lose amplitude and phase information.113hus, we require ∆ t (cid:46) ∆ xv . (2.170)CFL conditions can be expressed simply in terms of the CFL frequency, thefastest signal in the discrete system, d (cid:88) i =1 ω i ∆ t ≤ C, (2.171)where d is the dimensionality of the problem, ω i is the fastest frequency in each ofthe i dimensions, and C is some additional safety factor which may be required forstability. One CFL condition will come from solving Maxwell’s equations, where wemust be able to stably propagate light waves, CDIM (cid:88) i =1 c ∆ t ∆ x i ≤ p + 1 . (2.172)Here, c is the speed of light, CDIM is the number of configuration space dimensions,and p is the polynomial order of our basis expansion. We recognize c/ ∆ x i as thelargest discrete frequency in the system given some cell spacing ∆ x i in each of the i configuration space dimensions, since the speed of light is unequivocally the fastestvelocity in the system.Note that we have plugged in for the safety factor C = 1 / (2 p + 1). This CFLcondition is similar to the constraint for the finite-difference-time-domain (FDTD)discretization of Maxwell’s equations [Yee, 1966], but with this additional safetyfactor for stability which depends upon the polynomial order of our basis expan-sion[Cockburn and Shu, 2001]. In fact, Cockburn and Shu [2001] explicitly calculated114he required safety factor arising from the polynomial order of the basis expansionfor L stability, and although it is not exactly 1 / (2 p + 1), the safety factor is ap-proximately this value for a wide variety of polynomial orders, at least to within fiveto ten percent. As such, we use 1 / (2 p + 1) for the safety factor in our computationof the size of the time-step. In the limit that the grid spacing in each configurationspace dimension is equal, ∆ x = ∆ y = ∆ z , we can simplify the Maxwell’s equationCFL condition to c ∆ t ∆ x ≤ /CDIM p + 1 . (2.173)We likewise have a CFL condition for the Vlasov–Fokker–Planck equation. Wefirst note that the Vlasov equation CFL condition can be written as,∆ t CDIM + V DIM (cid:88) i =1 max T (cid:12)(cid:12)(cid:12)(cid:12) α i ∆ z i (cid:12)(cid:12)(cid:12)(cid:12) ≤ p + 1 , (2.174)where | · | is the absolute value. It will give us better intuition for this time stepconstraint by separating the configuration space and velocity space CFL conditions,∆ t (cid:34) CDIM (cid:88) i =1 max T (cid:12)(cid:12)(cid:12)(cid:12) v i ∆ x i (cid:12)(cid:12)(cid:12)(cid:12) + V DIM (cid:88) j =1 max T (cid:12)(cid:12)(cid:12)(cid:12) q s /m s ( E h + v × B h ) j ∆ v j (cid:12)(cid:12)(cid:12)(cid:12)(cid:35) ≤ p + 1 , (2.175)where we have abbreviated the number of configuration space dimensions as CDIM ,as before in Eq. (2.172), and the number of velocity space dimensions as
V DIM .The first term on the left-hand side of Eq. (2.175) uses the maximum velocity ineach direction, i.e., the velocity space edge in each direction, to determine the largestfrequency in configuration space from the local configuration space grid spacing ∆ x i .The second term on the left-hand side of Eq. (2.175) uses the maximum acceleration115ue to the electromagnetic fields measured in the phase space domain T to computethe largest frequency in velocity space from the local velocity space grid spacing∆ v j . Likewise, for the Fokker–Planck equation we have∆ t (cid:34) V DIM (cid:88) j =1 max T (cid:12)(cid:12)(cid:12)(cid:12) ν ( v − u h ) j ∆ v j (cid:12)(cid:12)(cid:12)(cid:12) + V DIM (cid:88) j =1 max T (cid:12)(cid:12)(cid:12)(cid:12) ν T h m s v j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:35) ≤ p + 1 , (2.176)where the first term on the left-hand side of Eq. (2.176) is the maximum frequencydue to the drag term, and the second term on the left-hand side of Eq. (2.176) is theCFL frequency due to the diffusion operator. Note that the CFL frequency of thediffusive term scales like (∆ v j ) − , the inverse square of the grid spacing, as it mustbecause the diffusion operator involves two derivatives of the distribution functionin velocity space. Defining CF L collisionless = CDIM (cid:88) i =1 max T (cid:12)(cid:12)(cid:12)(cid:12) v i ∆ x i (cid:12)(cid:12)(cid:12)(cid:12) + V DIM (cid:88) j =1 max T (cid:12)(cid:12)(cid:12)(cid:12) q s /m s ( E h + v × B h ) j ∆ v j (cid:12)(cid:12)(cid:12)(cid:12) , (2.177) CF L c = V DIM (cid:88) j =1 max T (cid:12)(cid:12)(cid:12)(cid:12) ν ( v − u h ) j ∆ v j (cid:12)(cid:12)(cid:12)(cid:12) + V DIM (cid:88) j =1 max T (cid:12)(cid:12)(cid:12)(cid:12) ν T h m s v j ) (cid:12)(cid:12)(cid:12)(cid:12) , (2.178)we can then say that the total CFL condition for the Vlasov–Fokker–Planck equationis ∆ t ( CF L collisionless + CF L c ) ≤ p + 1 . (2.179)A few remarks on the CFL condition for the Vlasov–Fokker–Planck equationare in order. The first remark is that we are being careful to determine the max-imum frequency in each dimension. For Maxwell’s equation, the CFL condition,Eq. (2.172), could naturally be simplified because the speed of light is the same in116ach direction. We could presume a similar restriction for the Vlasov–Fokker–Planckequation, find the maximum characteristic of each of the phase space dimensions,find the maximum of those maximum characteristics, and then include an addi-tional safety factor of 1 /d z where d z is the number of phase space dimensions. Inother words, presuming the acceleration in the x direction, E x + v y B z − v z B y , isthe maximum characteristic in the system, we can use that acceleration divided bythe grid spacing ∆ v x to calculate the CFL frequency, and then divide that CFLfrequency by six if one is evolving the Vlasov–Fokker–Planck equation in the fullsix dimensional phase space. Of course, this approach could lead to a quite restric-tive time-step compared to the combination of CFL frequencies in Eqns. (2.175) and(2.176), depending on how anisotropic the characteristics are. For example, even ifthe acceleration is quite large in the x direction, leading to a large CFL frequencyin the v x direction, the acceleration in the other two velocity dimensions, along withthe maximum velocity in the three configuration space dimensions, could be lowermagnitude and thus lead to smaller contributions to the total CFL frequency. Solong as we are careful to stay within the region of stability for our SSP-RK scheme,there is little reason not to take the largest possible time-step.An additional remark is to connect the maximum characteristic, for examplethe maximum acceleration or the maximum drag, to the numerical flux functionsdefined previously, Eqns. (2.61) and (2.128). In the global Lax-Friedrichs fluxesdefined for the Vlasov equation and drag component of the Fokker–Planck equation,we required the maximum of the flux, either collisionless or drag, sampled over thewhole phase space domain, T . This term, τ for example in Eq. (2.61), is exactly the117equired component of the CFL frequency in each dimension in Eqns. (2.175) and(2.176). Historically, the definition of the penalization term has also been done inthe opposite direction, with for example τ i = 12 p + 1 ∆ z i ∆ t , (2.180)as in [Lax, 1954]. Though this particular penalization term is a critical component ofsome stability bounds proved for the hyperbolic partial differential equations studiedin [Lax, 1954], such a large penalization can have unintended consequences for theaccuracy of the scheme, leading to a combination of overdiffusion and monotonicityerrors in the discrete solution. We will avoid such an extreme definition and insteadcontinue to use Eqns. (2.61) and (2.128) when we discuss the actual implementationof the method in the next chapter, Chapter 3.With both the CFL constraint for Maxwell’s equations and the CFL constraintfor the Vlasov–Fokker–Planck equation in hand, we have completed the mathemat-ical formulation of our discrete VM-FP system of equations. We evaluate the oper-ators defined in our semi-discrete scheme, Eqns. (2.22) and (2.133) for the Vlasov–Fokker–Planck equation, and Eqns. (2.29) and (2.30) for Maxwell’s equations, andthen determine from these evaluations which of the two CFL conditions, Eq. (2.179)or Eq. (2.172), is more restrictive. Having calculated both the linear operator L forthe complete semi-discrete VM-FP system of equations and the size of the time step∆ t , we can then plug the results into a forward Euler time step, Eq. (2.166), andrepeat the process as desired for a multi-stage SSP-RK method, e.g., SSP-RK3 inEq. (2.168). Before we move on from the mathematical foundation we have laid in118his chapter to the details of turning this mathematical foundation into algorithmsand code, we summarize the results of this chapter in the next section. We now summarize the contents of this chapter, and in doing so, foreshadowsome of the most important issues we will have to address in Chapter 3 when wemove from a mathematical formulation of the discrete scheme to an algorithmicformulation of the numerical method. In this regard, it is worth further drivingthe point of this chapter home: Eqns. (2.22) and (2.133) for the Vlasov–Fokker–Planck equation, and Eqns. (2.29) and (2.30) for Maxwell’s equations, followed by anappropriate ordinary differential equation integrator such as an SSP-RK3 method,Eq. (2.168), are a mathematically complete description of the discrete scheme. Tonow be a bit glib, mathematically, we are done.We have formulated a discrete scheme, which has provably retained proper-ties of the continuous system, with some flexibility in the choice of numerical fluxfunction, e.g., for Maxwell’s equations, central fluxes, Eqns. (2.33)-(2.34), or up-wind fluxes, Eqns. (2.48)-(2.51), both of which are perfectly acceptable numericalflux functions for Maxwell’s equations which have different, but potentially betterproperties depending on the problem being tackled. For example, we showed inLemma 2 that central fluxes for Maxwell’s equations conserves the electromagneticenergy, thus producing a completely conservative scheme in Proposition 12, whileupwind fluxes for Maxwell’s equations introduces numerical diffusion in the electro-119agnetic energy, thus leading to a monotonic decay of the energy. Central fluxesfor Maxwell’s equations is not free of numerical errors though, replacing diffusiveerrors with dispersive errors, errors in the phases of the solutions, e.g., when propa-gating an electromagnetic wave. These dispersive errors can be equally problematic[Hesthaven and Warburton, 2004], but regardless of the choice of numerical fluxfunction, the central point remains: the mathematical formulation of the discreteVlasov–Maxwell–Fokker–Planck (VM-FP) system of equations using a discontinu-ous Galerkin finite element method, with a polynomial basis, is completely specifiedby Eqns. (2.22) and (2.133) for the Vlasov–Fokker–Planck equation, and Eqns. (2.29)and (2.30) for Maxwell’s equations. Of course, to go from Eqns. (2.22) and (2.133)for the Vlasov–Fokker–Planck equation, and Eqns. (2.29) and (2.30) for Maxwell’sequations, to a numerical algorithm and code is its own non-trivial task, which weaddress in Chapter 3. So, to summarize:1. The discontinuous Galerkin finite element method (DG) is a spatial discretiza-tion scheme which combines aspects of finite element and finite volume meth-ods and leverages the benefits of both numerical methods to produce highorder accurate, robust, physically motivated spatial discretizations of a widespectrum of partial differential equations. The essential idea is an L mini-mization of the error after expanding the quantity of interest, for example thedistribution function, f ( z , t ) ≈ f h ( z , t ) = N (cid:88) k =1 f k ( t ) w ( z ) , in a basis set w = w ( z ), which we took to be the space of polynomials of120rder p, P p throughout this Chapter. The L minimization of the error can beformulated in the language of weak equality , ∂f h ∂t . = G [ f h ] , where G [ f h ] is a general operator acting on the quantity of interest, for examplethe Vlasov–Fokker–Planck spatial operator, and . = in the space spanned by w = w ( z ) denotes the operation (cid:90) I ∂f h ∂t w (cid:96) ( z ) d z = (cid:90) I G [ f h ] w (cid:96) ( z ) , ∀ (cid:96) = 1 , . . . , N. Note that weak equality, unlike strong equality where functions are everywhereequal, determines the solution up to an equivalence class, enforcing that theprojections of the left hand side and right hand side on the basis set spannedby w (cid:96) ( z ) , ∀ (cid:96) = 1 , . . . , N are equal.2. With the machinery of weak equality and an L minimization of the error, wecan formulate the DG discretization of our equation system of interest andderive the discrete-weak forms of the VM-FP system of equations, (cid:90) K j w ∂f h ∂t d z + (cid:73) ∂K j w − n · ˆ F dS − (cid:90) K j ∇ z w · α h f h d z = (cid:73) ∂K j ν w − n · ˆ G dS − (cid:73) ∂K j ν n · ∇ v w − T h m ˆ f dS − (cid:90) K j ν (cid:20) ∇ v w · ( v − u h ) f h − ∇ v w (cid:18) T h m f h (cid:19)(cid:21) d z , (cid:90) Ω j ϕ ∂ B h ∂t d x + (cid:73) ∂ Ω j d s × ( ϕ − ˆ E h ) − (cid:90) Ω j ∇ x ϕ × E h d x = 0 ,(cid:15) µ (cid:90) Ω j ϕ ∂ E h ∂t d x − (cid:73) ∂ Ω j d s × ( ϕ − ˆ B h ) + (cid:90) Ω j ∇ x ϕ × B h d x = − µ (cid:90) Ω j ϕ J h d x , two integration by parts on the diffusion operator to ultimately demonstrate thesemi-discrete scheme retains some of the properties of the continuous Fokker–Planck equation discussed in Section 1.6.122. There are many potential options for numerical flux functions, but a criticalproperty of the numerical flux function to prove our semi-discrete spatial dis-cretization retains properties of the continuous system is that the numericalflux function obeys the Godunov flux condition, (cid:73) ∂K j w − n · ˆ F dS = − (cid:73) ∂K j w + n · ˆ F dS, i.e., the flux into a cell is equal and opposite to the flux out of its neigh-bor cell along the shared interface. Example numerical flux functions for thecollisionless advection in phase space are n · ˆ F ( α h f + h , α h f − h ) = 12 n · α h (cid:0) f + h + f − h (cid:1) , n · ˆ F ( α h f − h , α h f + h ) = n · α h f − if sign( α h ) > , n · α h f + if sign( α h ) < , n · ˆ F ( α h f − h , α h f + h ) = 12 n · α h (cid:0) f + h + f − h (cid:1) − τ f + − f − ) , i.e., central fluxes, upwind fluxes, and global Lax-Friedrichs fluxes. Note thatthese forms of the numerical flux function exploit the fact that the discretephase space flow, α h , is continuous at the corresponding surface interfaces,Lemma 1. Likewise, similar flux functions can be defined for the numericalflux function for Maxwell’s equations,ˆ E h = (cid:74) E (cid:75) , ˆ B h = (cid:74) B (cid:75) , E = (cid:74) E (cid:75) − c { B } , ˆ E = (cid:74) E (cid:75) + c { B } , ˆ B = (cid:74) B (cid:75) + { E } /c, ˆ B = (cid:74) B (cid:75) − { E } /c, with (cid:74) g (cid:75) ≡ ( g + + g − ) / , { g } ≡ ( g + − g − ) / , and the drag component of the Fokker–Planck equation, n · ˆ F drag = 12 n · ( v − u h )( f + + f − ) , n · ˆ F drag = n · ( v − u h ) f − if sign( v − u h ) > , n · ( v − u h ) f + if sign( v − u h ) < , n · ˆ F drag = 12 n · ( v − u h )( f + + f − ) − max T | v − u h | f + − f − ) , where we have used the fact that v − u h is continuous across velocity spacesurfaces to simplify a central flux, upwind flux, and global Lax-Friedrichs fluxfor the drag component of the numerical flux function for the Fokker–Planckequation. The total numerical flux function for the Fokker–Planck equation is n · ˆ G = n · (cid:18) ˆ F drag + T h m ∇ v ˆ f (cid:19) , f is the distribution function at the interface using the recovery pro-cedure, and we require both the gradient, and the value, of the recovereddistribution function since we integrated the diffusion term by parts twice.4. The recovery procedure for computing the surface terms for the diffusion alsoleverages weak equality. We have the distribution function in two neighboringcells sharing an interface, ˆ f . = f L , ˆ f . = f R , where f L is the distribution function in the cell to the “left” of the interfaceand f R is the distribution function to the “right” of the interface. Definingthe recovery polynomial as ˆ f ( x ) = N − (cid:88) m =0 ˆ f m x m , (2.181)in one dimension, we can then uniquely compute a continuous polynomial(with continuous first derivatives, too). Importantly, the recovery procedure isfundamentally one-dimensional, since the discontinuity we are constructing thecontinuous representation along is a discontinuity at a surface. A continuousfunction, with continuous first derivatives, is “recovered” using the data thatis discontinuous at a given surface, i.e., the discontinuity is along the onedimension that is fixed at that surface. The reconstruction of the recoverypolynomial’s functional dependence along the surface in arbitrary dimensions125ill be addressed as part of our discussion of how to turn the mathematicalformulation of the discrete scheme into code in Chapter 3.5. We further utilize weak equality to determine the required velocity spacemoments for the coupling between the Vlasov–Fokker–Planck equation andMaxwell’s equations, as well as the moments required for the drag and dif-fusion coefficients in the Fokker–Planck equations. Weak equality allows usto define fundamental operators, e.g., division and multiplication, when thequantities being manipulated are themselves projections. The velocity spacemoments are M h . = (cid:88) j (cid:90) K j \ Ω k f h d v , M h . = (cid:88) j (cid:90) K j \ Ω k v f h d v ,M h . = (cid:88) j (cid:90) K j \ Ω k | v | f h d v , with the charge and current densities required for coupling to Maxwell’s equa-tions given by ρ c h = (cid:88) s q s M hs , J h = (cid:88) s q s M hs . Note that the charge and current density are strongly equal to the sum overspecies of the velocity space moments, since we have already projected downto the configuration space expansion. Likewise, for the flow and temperature126n the drag and diffusion coefficients, T h (cid:32)(cid:88) j (cid:73) ∂V maxj f h dS V max − (cid:88) j (cid:73) ∂V minj f h dS V min (cid:33) + m M h − mM h u h . = 0 ,T h (cid:34)(cid:88) j (cid:73) ∂V maxj ( n · v max ) f h dS V max − (cid:88) j (cid:73) ∂V minj ( n · v min ) f h dS V min (cid:35) + mM h − m M h · u h − M h T h . = 0 , which require weak multiplication and division, or weighted L projections, asdefined in Section 2.4. These expressions can be modified, for the discrete cur-rent density, temperature, and flow, in the case of running with only piecewiselinear polynomials, as discussed in Corollary 3 and Proposition 16 respectively.6. Using weak equality to construct consistent projections of velocity moments, aGodunov numerical flux function, and appropriate boundary conditions, i.e.,zero-flux in velocity space and a self-contained boundary condition in con-figuration space, like periodic boundary conditions, we can prove that thesemi-discrete scheme retains a number of the continuous VM-FP system ofequations’ properties. In particular, the whole system conserves mass and en-ergy, even when using piecewise linear polynomials and projecting | v | ontolinear polynomials, and we can show that even though the collisionless evolu-tion does not obey momentum conservation, the semi-discrete Fokker–Planckequation conserves momentum. Importantly, the lack of momentum conser-vation arises from our discretization of Maxwell’s equations, and thus onlydepends on configuration space resolution, a property we will numericallydemonstrate in Chapter 4. The collisionless component, the semi-discrete127lasov–Maxwell system of equations, is also L stable, either conserving ordecaying the L norm. This L stability leads to a discrete analogue of thesecond Law of Thermodynamics for the semi-discrete Vlasov–Maxwell systemof equations, with numerical diffusion arising as a production of entropy inour discrete system. Although we did not analytically prove a discrete secondLaw of Thermodynamics for the semi-discrete Fokker–Planck equation, wewill compare the entropy behavior between collisionless and collisional simula-tions in Chapter 4, and show that the collisionless entropy production is smallcompared to the collisional entropy production. Because many of these prop-erties, especially for the semi-discrete Vlasov–Fokker–Planck equation, onlydepended on the numerical flux function being Godunov and not a specificform of the numerical flux function, we can imagine further flexibility in termsof the mathematical formulation of the scheme. For example, we could extendthe recovery procedure to handle the collisionless and drag components of thediscretization and still retain the properties proved.7. Having specified a spatial discretization and constructed the semi-discrete VM-FP system of equations, we only require an ordinary differential equation in-tegrator for the time integration to complete the discretization and integratethe equation system in time. Example integrators include strong-stability pre-128erving Runge–Kutta methods, e.g., a three-stage third order method, f (1) = F ( f n , t n ) ,f (2) = 34 f n + 14 F (cid:0) f (1) , t n + ∆ t (cid:1) ,f n +1 = 13 f n + 23 F (cid:0) f (2) , t n + ∆ t/ (cid:1) , with F defining the complete evaluation of the semi-discrete VM-FP systemof equations, Eqns. (2.22) and (2.133) for the Vlasov–Fokker–Planck equation,and Eqns. (2.29) and (2.30) for Maxwell’s equations. These explicit time inte-grators have Courant-Friedrichs-Lewy constraints on the size of the time-step, CF L collisionless = CDIM (cid:88) i =1 max T (cid:12)(cid:12)(cid:12)(cid:12) v i ∆ x i (cid:12)(cid:12)(cid:12)(cid:12) + V DIM (cid:88) j =1 max T (cid:12)(cid:12)(cid:12)(cid:12) q s /m s ( E h + v × B h ) j ∆ v j (cid:12)(cid:12)(cid:12)(cid:12) ,CF L c = V DIM (cid:88) j =1 max T (cid:12)(cid:12)(cid:12)(cid:12) ν ( v − u h ) j ∆ v j (cid:12)(cid:12)(cid:12)(cid:12) + V DIM (cid:88) j =1 max T (cid:12)(cid:12)(cid:12)(cid:12) ν T h m s v j ) (cid:12)(cid:12)(cid:12)(cid:12) , ∆ t ( CF L collisionless + CF L c ) ≤ p + 1 , for the Vlasov–Fokker–Planck equation, and c ∆ t ∆ x ≤ /CDIM p + 1 , for Maxwell’s equations. Here, we have abbreviated the number of configura-tion space dimensions as CDIM and the number of velocity space dimensionsas
V DIM . The more restrictive of the two conditions tells us the maximumstable time-step, and completes the prescription for the numerical integrationof the VM-FP system of equations in space and time.Thus, we can now move to a discussion of how to evaluate the discrete scheme,129.e., how do we turn the math into code, an algorithmic formulation of the discretescheme that allows one to actually perform numerical experiments. Throughout thissummary, we have emphasized the requirements that components of the discretescheme be constructed consistently , e.g., computing velocity moments using weakequality. This emphasis is not without merit. When we first described plasmas asrich in their underlying physics in Chapter 1, we alluded to the fact that importantphysics properties are implicit to the underlying equation system. For example, weare discretizing the Vlasov–Fokker–Planck equation for the evolution of the particledistribution function, but just as important is that velocity moments such as thezeroth, mass, and second, energy, obey conservation equations. To actually retainthese properties that we painstakingly proved in this Chapter, we will find thatthe ultimate algorithmic formulation of the scheme requires a comparable amountof precision to the amount of mathematical care that was taken when deriving thediscrete scheme. 130 ome of the material in thischapter has been adapted fromJuno et al. [2018], Hakim,Francisquez, Juno, andHammett [2019], and Hakimand Juno [2020].
Chapter 3: From Math to Code: Efficient Implementation of DG forthe Vlasov–Maxwell–Fokker–Planck System of Equations
It is now time to undertake the task of translating the discrete scheme de-scribed in Chapter 2 into an algorithm which can be implemented in a code, in thiscase, the
Gkeyll simulation framework. As part of our derivation of the discretescheme, there were many components of the scheme we left deliberately abstractas they were unnecessary for describing the numerical method mathematically andproving properties of the discretization of the VM-FP system of equations. Wehave a long to-do list for converting Eqns. (2.22) and (2.133) for the Vlasov–Fokker–Planck equation, and Eqns. (2.29) and (2.30) for Maxwell’s equations, into code.We have restricted ourselves to basis sets of polynomials as part of the proofsof the various conservation properties that our discrete scheme retains from the131ontinuous system, such as conservation of mass and energy, but we have made nomention yet of what specific form this polynomial basis takes. We likewise mustnow evaluate these integrals in the discrete weak forms of the VM-FP system ofequations in some fashion, including a potential transformation from a more con-venient computational space to the physical domain on which the equations aredefined. Finally, in tandem with actually performing the integrals in the discreteweak forms, we must determine algorithmically how to compute the various compo-nents of the scheme, such as velocity moments for the coupling between Maxwell’sequations and the Vlasov–Fokker–Planck equation and the recovery of the distri-bution function for the Fokker–Planck equation. With a prescription for how toperform these operations, we will then be able to bring the whole algorithm to-gether and evaluate computationally the spatial discretization. Combined with thetime discretization described in Section 2.7, we will then have completed the conver-sion from the mathematical machinery described in Chapter 2 to the computationalmachinery required to perform the numerical integration of the VM-FP system ofequations in our simulation framework
Gkeyll . Even in one dimension, there is tremendous freedom in the definition of thepolynomial basis. The definition of the function space, P p , only restricts us topolynomials of, at most, order p . We could, for example, take our basis set to be132imply ψ k ( x ) = x k , k = 0 , . . . , p, x ∈ [ − , , (3.1)where we have defined the polynomials on the interval [ − ,
1] for convenience.We could define the polynomials with respect to the local grid cell immediately,as we did in the brief one dimensional DG example in Section 2.1 in Eq. (2.18)wherein the linear polynomial included the local grid cell volume and cell centercoordinate. However, as we will show in Section 3.3, we can always transform ourcomputational domain to the physical domain on which the equations are defined.We will find certain properties of the polynomial basis are ultimately more intuitiveby defining the polynomials on a reference element, in this case the element [ − , L stable. However, the basis defined in Eq. (3.1) is a very bad choice for our basisexpansion because the basis has serious computational issues.To see why Eq. (3.1) forms a bad basis computationally, consider the following133peration that will be required as part of our discretization, (cid:90) K j ∂f h ( z , t ) ∂t w (cid:96) ( z ) d z = (cid:88) k df k ( t ) dt (cid:90) K j w k ( z ) w (cid:96) ( z ) d z = M d f dt , (3.2)where the matrix M has entries M k(cid:96) = (cid:90) K j w k ( z ) w (cid:96) ( z ) d z , (3.3)and we have added back in the spatial dependence to the basis functions to makethe meaning of evaluation of entries of the matrix M more clear. In other words,each combination of basis functions, integrated over the cell K j , produces a matrixwith size N p × N p , where N p is the number of basis functions in the expansion withina cell. This matrix, Eq. (3.3), is often called the mass matrix in the DG and finiteelement literature [Hesthaven and Warburton, 2007]. Note that Eq. (3.2) impliesthat we will require the inverse of the mass matrix, M , to ultimately discretize thesystem of ordinary differential equations for f , the vector of expansion coefficientswithin a cell.Now, this mass matrix in one dimension on the reference cell is simply M k(cid:96) = (cid:90) − ψ k ( x ) ψ (cid:96) ( x ) dx. (3.4)To make this example concrete, for the basis defined in Eq. (3.1), consider the mass134atrix in one dimension for polynomial order four: M k(cid:96) = (cid:90) − x k x (cid:96) dx = . (3.5)Perhaps unremarkable, but let us examine the condition number for the matrix inEq. (3.5), κ ∞ ( M ) .. = || M − || ∞ || M || ∞ = 821116 , (3.6)where || · || ∞ is the L ∞ matrix norm , || A || = max ≤ k ≤ N N (cid:88) (cid:96) =1 | A kl | . (3.7)The condition number measures the sensitivity of the solution to small changesin the initial data. Because we require the inverse of the mass matrix, M , before wecan discretize the system of ordinary differential equations for the time evolution of f a large condition number for the mass matrix is very bad. A rough rule of thumbis that for κ ∞ ( A ) = 10 n , we expect to lose n digits of accuracy due to a loss ofprecision from the inversion of the matrix [Press et al., 2007]. So, for the matrix in Note the condition number can be defined with any suitable matrix norm, such as the Frobeniusnorm, || A || = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) k =1 N (cid:88) (cid:96) =1 | A kl | . ( κ ∞ ( M )) ∼ . υ ( ψ ) = ( ψ, υ ) L ( υ, υ ) L υ, (3.8)where the L inner product, ( · , · ) L , is the inner product we have been continuallyemploying, ( ψ, υ ) L = (cid:90) − ψ ( x ) υ ( x ) dx, with natural generalizations to higher dimensions. We then use this projectionoperator to transform the monomial basis in Eq. (3.1) into a set of orthogonal poly-136omials. Proceeding sequentially through the polynomial set, υ = ψ = 1 ,υ = ψ − proj υ ( ψ ) = x,υ = ψ − proj υ ( ψ ) − proj υ ( ψ ) = 3 x − ,υ = ψ − proj υ ( ψ ) − proj υ ( ψ ) − proj υ ( ψ ) = x (5 x − υ = ψ − proj υ ( ψ ) − proj υ ( ψ ) − proj υ ( ψ ) − proj υ ( ψ ) = 35 x − x + 335 . (3.9)This procedure generalizes to higher polynomial orders as we might expect, with υ k = ψ k − k − (cid:88) j =1 proj υ j − ( ψ k ) . (3.10)We can make these polynomials orthonormal usingˆ υ = υ (cid:112) ( υ, υ ) L , (3.11)i.e., dividing by the L norm of the polynomials. This procedure gives us thefollowing set of orthonormal polynomials for the one dimensional, p = 4, basis,ˆ υ = 1 √ , ˆ υ = (cid:114) x, ˆ υ = (cid:114)
58 (3 x − , ˆ υ = (cid:114)
78 (5 x − x ) , ˆ υ = 38 √ x − x + 3) . (3.12)137ecause these polynomials are orthonormal, (cid:90) − ˆ υ k ˆ υ (cid:96) dx = δ k(cid:96) , (3.13)Eq. (3.5) reduces to M = ←→ I , (3.14)the identity matrix, whose condition number is trivially κ ∞ ( M ) = 1.As an aside, we can gain intuition for why the conditioning of the mass matriximproves so dramatically when employing orthonormal polynomials by examiningthe behavior of our two choice of basis sets on the interval [ − , − , − ,
1] with an identical inner product to the inner product we havebeen employing, Eq. (2.1). Legendre polynomials are normalized to be equal to ± − ,
1] so that itis easy to imagine why higher order orthonormal polynomials do actually improvethe accuracy of the representation.at the edges of the interval. Although Legendre polynomials are orthogonal andvery similar to the orthogonal polynomials we first found with our Gram-Schmidtprocedure, they are not orthonormal, (cid:90) − P n ( x ) P m ( x ) dx = 22 n + 1 δ mn . (3.15)Importantly, we have our first instance justifying out choice to define the poly-nomials on a reference element [ − , modal basis for ourDG discretization. This terminology follows from the fact that in the projection ofa quantity of interest onto our basis set, we are projecting onto a set of modes. Analternative prescription is called a nodal basis, wherein the basis set is defined by aset of polynomials whose values are known at nodes. In other words, a basis suchas f ( x, t ) ≈ f h ( x, t ) .. = N p − (cid:88) k =0 f k ( ξ k , t ) (cid:96) k ( x ) , (3.16)where (cid:96) k are the Lagrange interpolating polynomials, (cid:96) k ( x ) = N p − (cid:89) j =0 ,j (cid:54) = k x − ξ j ξ k − ξ j , (3.17)and ξ k are the k nodes by which the polynomials are defined. In other words, in thisbasis set, the polynomials take the value of one at one node and zero at all othernodes, thus the coefficients f k in Eq. (3.16) are known at the nodes ξ k .Just as Eq. (3.1) was related mathematically to the orthonormal, modal basisby the Gram-Schmidt orthogonalization (orthonormalization) process, so too do theone dimensional modal and nodal bases have a mathematical connection. Using theVandermonde matrix, V k(cid:96) = ˆ υ (cid:96) ( ξ k ) , (3.18)i.e., the matrix whose entries are each of the (cid:96) orthonormal polynomials evaluated atthe nodes ξ k , we can transform the coefficients in the modal basis to the coefficients140n the nodal basis, V k(cid:96) f (cid:96) ( t ) = f k ( ξ k , t ) . (3.19)And just as with Eq. (3.1), both the modal and nodal bases are perfectly mathemat-ically acceptable basis sets for implementing the DG scheme for the VM-FP systemof equations described in Chapter 2, but they have quite different computationalproperties. Before we can explore the full extent of the computational consequencesfor a modal versus a nodal basis set, we should first discuss the generalization ofthese basis sets to higher dimensions. From the beginning, we have been interested in the numerical integration ofan equation system which is high-dimensional, up to six dimensions plus time. Thishigh dimensionality of the VM-FP system of equations presents a special set ofchallenges for the design and implementation of our numerical method. The “curseof dimensionality,” the exponential cost scaling of a numerical method with the di-mensionality of the problem, is not a “curse” to be taken lightly. This exponentiallyincreasing cost scaling with dimensionality is in fact one of the principal reasons forthe popularity of the particle-in-cell method discussed in Chapter 1, as it is arguedthat the integration of particles on a three-dimensional grid, instead of the inte-gration of the particle distribution function on a six-dimensional grid, is inevitably141ore cost effective. Of course, we have strong motivation for the direct discretiza-tion approach, so we instead want to focus on whether this burden of cost can beovercome.The standard higher dimensional generalization of the one dimensional basesdefined in Section 3.1 is a tensor basis constructed from a tensor product of theone dimensional basis sets for each dimension of interest. For example, in twodimensions, the generalization of the monomial basis is simply Q p = span ≤ m,n ≤ p { x m y n } . (3.20)Due to the nature of the tensor product, the number of basis functions within a cellscales like ( p + 1) d , exactly the exponential scaling we predicted at the beginning ofthis section. We seek reductions then of this tensor product basis.The first reduction we consider is known as the Serendipity basis set [Arnoldand Awanou, 2011]. The Serendipity basis set is obtained by dropping all monomialterms which have “super-linear” degree greater than the specified polynomial order p . For example, for the piecewise quadratic, two dimensional, Serendipity basisexpansion, we would have S = { , x, y, xy, x , y , x y, xy , x y } , (3.21)because the “super-linear” degree of x y is four, which is greater than the specifiedpolynomial order of two. We could then apply the appropriate higher dimensionalgeneralization of the Gram-Schmidt orthonormalization procedure described in theprevious section, Section 3.1. In two dimensions, this generalization of the inner142roduct would be ( υ, ψ ) L = (cid:90) − (cid:90) − υ ( x, y ) ψ ( x, y ) dxdy, (3.22)so that we would find the two dimensional, piecewise quadratic, orthonormal, modal,Serendipity basis to be ˆ υ ( x, y ) = 12 , ˆ υ ( x, y ) = √ x , ˆ υ ( x, y ) = √ y , ˆ υ ( x, y ) = 3 xy , ˆ υ ( x, y ) = √ x − , ˆ υ ( x, y ) = √ y − , ˆ υ ( x, y ) = √ x − y , ˆ υ ( x, y ) = √ y − x . (3.23)The general scaling of the Serendipity basis set is given by N p = min( d,p/ (cid:88) i =0 n − i (cid:18) di (cid:19)(cid:18) p − ii (cid:19) , (3.24)where N p is the number of polynomials, d is the dimensionality of the basis set, and p is the polynomial order. This particular reduced basis set has been extensivelystudied in the literature, and found to have the same formal convergence order asthe tensor basis, though the generalization of the Serendipity basis to unstructuredgrids requires care as arbitrary refinements of an unstructured grid will destroy theconvergence order of the Serendipity expansion [Arnold et al., 2002]. By convergence143rder, we mean the rate of convergence to the true solution of the continuous systemin the limit that the grid spacing goes to zero. So a second order method correspondsto a method where the errors decrease as (∆ x ) as ∆ x →
0. Although we have notsaid so explicitly up to this point, all the work of this thesis uses structured grids,specifically structured quadrilaterals.We can consider a further reduction on top of the Serendipity basis to dropall monomials of total degree greater than the polynomial order specified, which wecall the maximal order basis set. For this reduced basis set, we would only retainpolynomials zero through five in Eq. (3.23), since polynomials six and seven havetotal degree three. The general scaling of the maximal order basis set is N p = ( p + d )! p ! d ! . (3.25)Elsewhere in the finite element literature, these three basis sets, the tensor ba-sis, Serendipity, and what we are calling maximal order, are sometimes abbreviatedas the Q , S , and P spaces respectively. Like the Serendipity basis set, the maximalorder basis set has been the subject of a large number of studies to examine itsconvergence order and accuracy relative to the tensor basis. While maintaining thesame convergence order, the maximal order basis set is generally found to be lessaccurate, and this basis can have further detrimental consequences for the physical-ity of the solution. For example, Cheng et al. [2013b] found the maximal order basisto have more serious issues with artificial dissipation compared to the tensor basisin a Vlasov–Poisson study using the discontinuous Galerkin method.For reference, the number of degrees of freedom in a cell for a variety of poly-144omial orders and up to six dimensions for the three basis sets, tensor, Serendipity,and maximal order, is included in Tables 3.1, 3.2, and 3.3, respectively. We can Q Polynomial Order
Dimension ( p + 1) d S Polynomial Order
Dimension (cid:80) min( d,p/ i =0 (cid:0) di (cid:1)(cid:0) p − ii (cid:1) Polynomial Order
Dimension ( p + d )! p ! d ! , with the reference4D element being comprised of reference 3D elements for each of the 4D element’seight 3D faces, a reference 5D element consisting of a reference 4D element for allten 4D faces of a 5D element, and so on. This approach has the advantage of greatlysimplifying surface integral calculations. Since every higher dimensional element isrecursively generated from lower dimensional elements, every face of a higher di-mensional element, the 2D faces in 3D or the 4D faces in 5D, forms a unisolventexpansion for that surface. We thus only require the nodal information local to that This fact is true in general, but higher polynomial orders may modify the lower dimensionalreference elements such that the recursive algorithm is not quite as obvious as the one presentedhere. Just as polynomial order four introduces an interior node to a reference 2D element, so canhigher polynomial orders introduce interior nodes to higher dimensional reference elements whichwould have to be taken into account in the recursive generation of the reference element. For up topolynomial order four though, every higher dimensional object can be easily generated as described,with 2D reference elements making up the faces of a 3D reference element, 3D reference elementsmaking up the faces of a 4D reference element, and so on. Considering that the Serendipity basis infour, five, and six dimensions, with polynomial order four, involves the solution of a large numberof degrees of freedom per cell, we will not consider further extensions of this recursive algorithmdue to the same performance and cost considerations that motivated the use of the Serendipitybasis—we seek to avoid evolving thousands of degrees of freedom per cell.
Having defined suitable polynomial basis sets for the full spectrum of dimen-sionality of interest, for arbitrary polynomial order, we return to an issue discussedin Section 3.1. We require integrals over the physical domain, i.e., a physical cell K j in phase space, such as in Eq. (3.3), but we have defined the polynomials on the in-148erval [ − , d , where d is the dimensionality of the reference element. To transformEq. (3.3), we can make a change of variables, M k(cid:96) = (cid:90) K j w k ( z ) w (cid:96) ( z ) d z = (cid:90) I w k ( z ( η )) w (cid:96) ( z ( η )) (cid:12)(cid:12)(cid:12)(cid:12) d z d η (cid:12)(cid:12)(cid:12)(cid:12) d η = (cid:90) I ˆ υ k ( η )ˆ υ (cid:96) ( η ) (cid:12)(cid:12)(cid:12)(cid:12) d z d η (cid:12)(cid:12)(cid:12)(cid:12) d η , (3.26)where (cid:18) d z d η (cid:19) ij = dz i dη j (3.27)is the Jacobian matrix, and we require its determinant to perform the transforma-tion. In this procedure, we have transformed the basis functions w ( z ) defined onthe physical phase space mesh to ˆ υ ( η ), the orthonormal basis set defined on the ref-erence element I = [ − , d , where d is the dimensionality of the reference element.We could also just as easily transform the phase space basis functions w ( z ) to thenodal basis defined on the reference element I = [ − , d .To determine the Jacobian matrix and its determinant, we must know thefunctional form for the change of variables from the coordinate z to the coordinate η . To take a simple example, we could transform from a uniform, structured,Cartesian grid to the reference element with the formula z = η ∆ z z center , (3.28)where ∆ z is the grid spacing in each direction of phase space, and z center is the cell149enter. The entries of the Jacobian matrix would then be dz i dη j = ∆ z i δ ij , (3.29)and since this matrix is diagonal, the determinant is straightforwardly (cid:12)(cid:12)(cid:12)(cid:12) d z d η (cid:12)(cid:12)(cid:12)(cid:12) = 12 d d (cid:89) i =1 ∆ z i . (3.30)The change of variables need not be so simple. But, so long as the Jacobianfor the change of variables is known, we can map the reference element onto ascomplex a physical grid as we can imagine. For example, we can construct a non-orthogonal coordinate system which follows magnetic field lines, as is done with thesimulation framework the VM-FP solver is built in, Gkeyll , for other applications[Bernard et al., 2019, Shi et al., 2019, Mandell et al., 2020, Bernard et al., 2020,Francisquez et al., 2020]. Depending on the complexity of the Jacobian though, e.g.,if the transformation itself varies in space, further modification of the integrals maybe required, especially for the terms involving gradients.Let us now, in the lead up to the next section, return to the explicit expres-sion for the discrete weak form of the Vlasov equation, Eq. (2.22), and attempt toreveal exactly the integrals we need to compute. Substituting the expansions of thedistribution function, f h and the phase space flow, α h , into Eq. (2.22), we obtain (cid:88) k df k ( t ) dt (cid:90) K j w k ( z ) w (cid:96) ( z ) d z + (cid:88) m ˆ F m ( t ) · (cid:73) ∂K j n w − (cid:96) ( z ) w m ( z ) dS − (cid:88) m,n f m ( t ) α n ( t ) · (cid:90) K j ∇ z w (cid:96) ( z ) w m ( z ) w n ( z ) d z = 0 . (3.31)Assuming our grid is uniform, structured, and Cartesian we can rearrange this ex-150ression using the procedure in Eq. (3.26), as well as the determinant of the Jacobianmatrix in Eq. (3.30), to obtain (cid:88) k df k ( t ) dt d d (cid:89) i =1 ∆ z i (cid:90) I ˆ υ k ( η )ˆ υ (cid:96) ( η ) d η + (cid:32) d d (cid:89) i =1 ,i (cid:54) = j ∆ z i (cid:33) (cid:88) m ˆ F m ( t ) · (cid:73) ∂I j n ˆ υ − (cid:96) ( η )ˆ υ m ( η ) dS − (cid:32) d d (cid:89) i =1 ∆ z i (cid:33) (cid:88) m,n f m ( t ) α n ( t ) · (cid:90) I z ∇ η ˆ υ (cid:96) ( η )ˆ υ m ( η )ˆ υ n ( η ) d η = 0 . (3.32)Note the slight change in notation, where we are denoting the surface ∂I j as thesurface with constant j dimension, where j = x, y, z, v x , v y , v z , since the determinantof the Jacobian for the surface integral will not have the volume factor for that di-mension. In addition, we have obtained an additional factor of 2 / ∆ z in transformingthe gradient from ∇ z to ∇ η . Importantly, this term is still a vector, and one onlypicks up the factor of 2 / ∆ z for the particular gradient being transformed.Since Eq. (3.32) must be solved for every ˆ υ (cid:96) in our basis expansion, we canmake Eq. (3.32) more elegant by rearranging it to be a linear system, df k dt = ( M k(cid:96) ) − (cid:34)(cid:88) m U (cid:96)m · ˆ F m ( t ) + (cid:88) m,n C (cid:96)mn · α n ( t ) f m ( t ) (cid:35) , (3.33)where ( M k(cid:96) ) − is the inverse of the transformed mass matrix, M k(cid:96) = (cid:90) I ˆ υ k ( η )ˆ υ (cid:96) ( η ) d η , (3.34)and the tensors U (cid:96)m and C (cid:96)mn are U (cid:96)m = 2∆ z j (cid:73) ∂I j n ˆ υ − (cid:96) ( η )ˆ υ m ( η ) dS, (3.35) C (cid:96)mn = (cid:90) I z ∇ η ˆ υ (cid:96) ( η )ˆ υ m ( η )ˆ υ n ( η ) d η . (3.36)151 few remarks on these matrices and tensors are in order. The first remark is theimplicit sum in retaining the dot products in Eq. (3.33), i.e., we have to performthe surface integrals for each of the j surfaces and sum over the contribution, andlikewise we must sum over each contribution from the phase space flux, α h , in thevolume term. In addition, we remark that the contribution from the determinantof the Jacobian matrix has been cancelled when going from Eq. (3.32) to Eq. (3.33).The only coordinate transform contributions that survive are the factor from trans-forming the gradient ∇ z to ∇ η , and the remaining inverse volume factor, 2 / ∆ z j , inthe surface integral for the dimension which is constant at the corresponding surface, ∂I j . While we chose to illustrate the change of coordinates and construction of thelinear system with the orthonormal modal basis expansion, i.e., ˆ υ (cid:96) for each of the (cid:96) basis functions in the expansion, we could have just as easily illustrated thesetransformations with the nodal basis expansion. Importantly, a key operation wemust perform to be able to construct the linear system shown in Eq. (3.33) is toproject the numerical flux function onto our basis expansion. For example, if weemploy central fluxes, then using the machinery of weak equality from Section 2.4,we have ˆ F . = 12 α h ( f + h + f − h ) , (3.37)where the projection is done over the full basis expansion, but the phase space flux α h and the distribution function f ± h are evaluated at the corresponding surface.Similar manipulations which produced Eq. (3.33) can also be performed for152ur semi-discrete forms of Maxwell’s equations and the Fokker-Planck equation.The essential idea is always to construct the mass matrix which multiplies the timederivative, and the two tensors which encode the spatial discretization, one for thesurface integral contributions, for each surface on the reference element, and onefor the volume integral contribution. Note that in the construction of the tensorfor the surface integral contributions, we must project the flux functions onto thecorresponding basis set, i.e., we must project central, Eqns. (2.33)-(2.34), or upwindfluxes, Eqns. (2.48)-(2.51), for Maxwell’s equations onto configuration space basisfunctions. Likewise, we must project the two surface fluxes for the semi-discreteFokker–Planck equation onto phase space basis functions.The evaluation of all of these linear operations in each cell K j in phase spaceand Ω j in configuration space then completes the algorithm for the spatial discretiza-tion. To actually evaluate these linear operations though, we now need to constructthese tensors for the surface integrals and volume integral by specifying how tocompute the integrals in Eqns. (3.34–3.36). What may seem relatively straightfor-ward belies a subtlety that is of singular consequence for the construction of thealgorithm. At first glance, there is nothing remarkable about the integrals which must beperformed in the construction of Eqns. (3.34–3.36). They are products of polynomi-153ls; we could either use Gaussian quadrature of an appropriate degree, or even ex-actly integrate the combinations of polynomials and store the entries of the matricesand tensors defined in Eqns. (3.34–3.36) for the Vlasov equation and the analogousmatrices and tensors for the Fokker–Planck equation and Maxwell’s equations.Consider what the application of Gaussian quadrature to Eq. (3.36) wouldentail. In one dimension, the numerical integration of a function with Gaussianquadrature is done via (cid:90) − f ( x ) dx ≈ N q (cid:88) i =1 W i f ( x i ) , (3.38)where W i and x i are the i weights and abscissas for the Gaussian quadrature rule.The extension to higher dimensions is done using a tensor product of one dimensionalweights and abscissas, e.g., in two dimensions, (cid:90) − (cid:90) − f ( x, y ) dxdy ≈ N q (cid:88) i =1 N q (cid:88) j =1 W i W j f ( x i , y j ) . (3.39)An example Gaussian quadrature rule, Gauss-Legendre, is shown in Table 3.4. Toperform Gaussian quadrature on integrals such as Eq. (3.36), we require a tensorproduct of N q quadrature points in each direction for every dimension we wish tointegrate. This approach will integrate exactly monomials of a particular order,e.g., 2 N q − N q − q x i W i Order of Accuracy (2 N q − ± √ ± (cid:113)
35 59 ± (cid:114) − (cid:113)
65 18+ √ ± (cid:114) + (cid:113)
65 18 −√
75 0 ± (cid:114) − (cid:113)
107 322+13 √ ± (cid:114) (cid:113)
107 322 − √ P N q ( x ) and the weights W i = 2 / [(1 − x i )( P (cid:48) N q ( x i )) ] [Abramowitz and Stegun, 1985].For example, consider integrating the volume term in five dimensions with secondorder polynomials. Naively, one expects this to require the integration of monomialswith degree 3 p = 6 in each dimension, because both α h and f h have polynomialexpansions, thus requiring at least 4 quadrature points in each dimension, or a to-tal of 4 = 1024 quadrature points, to avoid under-integrating the volume term inEq. (3.36). Given that the scaling of the computation of the volume integral in acell is O ( N totq N p ), where N totq is the total number of quadrature points, the numberof operations per phase space cell becomes quite large for modest polynomial ordersin high dimensions.Leveraging the fact that Eq. (3.36) is just a triple product of polynomials andexactly integrating each term in the tensor to some specified precision, e.g., double155recision, is not guaranteed to produce a more favorable computational complex-ity. If every degree of freedom within a phase space cell is coupled, the resultingtensor would be dense and the computational complexity of evaluating this tensorconvolution would then be O ( N p ), where N p is the number of basis functions inour phase space expansion. It is perhaps the case that the scaling would not beas dire as O ( N p ), since the phase space flux, α h , requires the expansions of theelectromagnetic fields, which live in the configuration space subspace of our phasespace expansion, but α does vary linearly in velocity space via the v × B compo-nent of the Lorentz force. Thus, we expect the computational complexity would bebetween O ( N p ) and O ( N c N p ), where N c is the number of configuration space basisfunctions, and not the full reduction to the more favorable O ( N c N p ) scaling.An approach that is standard with nodal bases is to reduce the cost of thescheme by only evaluating the terms in these integrals, such as Eq. (3.35) andEq. (3.36), at the specified nodes that define the polynomials [Hesthaven and War-burton, 2007, Hindenlang et al., 2012]. In doing so, the required number of op-erations would be significantly decreased, as the values of the coefficients at thenodes are known by the definition of the nodal basis, reducing the computationalcomplexity to O ( N p ). But, this approach incurs the very same aliasing errors wewarned about in Section 2.4. Even if the values of the various quantities such as α h and f h are known at the nodes, the product of the two quantities required for thevolume term is not known at the nodes because the product of the two quantitiesis higher order. Thus, we will be unable to determine the nonlinear term uniquelyif we evaluate α h and f h at the nodes and multiply the result.156e now make concrete one of the principal advancements of this thesis: theintolerable consequences of aliasing errors in a DG discretization of an equationsystem such as the VM-FP system of equations. We emphasized in Section 1.6 andAppendix A for the continuous system, and again when we discussed the propertiesof the discrete system in Sections 2.3 and 2.6, that many properties of the VM-FPsystem of equations are implicit to the equation system. The Vlasov–Fokker–Planckequation is a conservation equation for the particle distribution function, and thefact that it is a conservation equation makes certain properties explicit, such as phasespace incompressibility for the collisionless component of the equation system. How-ever, other properties are contained in velocity moments of the equation system. Forexample, it is the second velocity moment of the Vlasov–Fokker–Planck equation,combined with Maxwell’s equations, that gives us total energy conservation.When proving that the discrete scheme maintains properties of the continu-ous system such as conservation of mass and energy, we substituted for the testfunctions, w , expressions we presumed we would be able to integrate. In one case,we substituted w = 1 / m | v | and evaluated the integrals to massage the volumeterm into forms which determined the conditions for which energy would be con-served. While at first glance this may seem like an obvious assertion: we have toevaluate the integrals correctly to actually retain properties such as conservation ofmass and energy, it is important to realize why this is the case. Were we evalu-ating explicit conservation relations, such as the conservation of mass, momentum,and energy equations in the Euler equations, the Navier-Stokes equations, or theequations of magnetohydrodynamics, aliasing errors could be problematic, but they157ould not destroy conservation relations. The aliasing errors arising from not ex-actly representing the fluid equation solution in a DG algorithm exactly might causeanomalous energy transport, but the aliasing induced transport would not destroyenergy conservation of the equation system.We do not wish to be overly uncharitable on this point. It is well knownwithin the DG computational fluid dynamics community that aliasing errors canlead to stability issues [Kirby and Karniadakis, 2003]; however, because the aliasingerrors manifest in the smallest scales and highest wavenumbers, techniques such asfiltering and artificial dissipation are commonly employed to ameliorate these errors[Fischer and Mullen, 2001, Gassner and Beck, 2013, Flad et al., 2016, Moura et al.,2017]. And because fluids equations such as the Euler equations, the Navier-Stokesequations, or the equations of magnetohydrodynamics involve the discretization ofexplicit conservation relations for mass, momentum, and energy, there is far lessconcern that such filtering or artificial dissipation will destroy the quality of thesolution, at least at scales above the resolution of the simulation. There are othermeans of alleviating or eliminating aliasing errors using split-form formulations ofthe DG method [Gassner, 2013, 2014, Gassner et al., 2016a,b, Flad and Gassner,2017], and overintegration, essentially the idea we already discussed of adding suf-ficient quadrature points to exactly integrate the nonlinear term [Mengaldo et al.,2015, Kopriva, 2018, Fehn et al., 2019]. For a comparison of these two approaches,see Winters et al. [2018]. Importantly, with the exception of overintegration , tech- In the split-form forumulation, conservative and non-conservative forms of the equation atthe continuous level are averaged to produce a different, but ultimately more computationallyfavorable, equation to discretize. And only overintegration in specific circumstances, as overintegration of expressions such as
Eq. (2.110) for computing the discrete flow, u h will always incur aliasing errors unless you applyoverintegration to the linear operation defined in Eq. (2.111), because Eq. (2.110) involves integra-tion of a rational function, which Gaussian quadrature cannot integrate exactly. O ( N totq N p ), while exact analytic in-tegration will produce an algorithm we expect will lie between O ( N c N p ) and O ( N p ),at least if one assumes that every degree of freedom couples to every other degreeof freedom in the expansion. We can ask the question if there is any way to reducethis cost, and indeed for numerical integration, some savings can be obtained by useof an anisotropic quadrature scheme. For example, if we consider the advection invelocity space, (cid:90) K j ∇ v w (cid:96) · α vh f h d z = (cid:90) K j ∇ v w (cid:96) · qm ( E h + v × B h ) f h d z , (3.40)for each of the (cid:96) basis functions in our phase space expansion, we can see that,while we require integrating monomials of degree 3 p in configuration space, in ve-locity space we require at most integrating monomials with degree 2 p + 1. Table3.5 considers the impact anisotropic quadrature, using only the minimum number160f quadrature points required along each direction of integration, has on a few com-binations of velocity space and configuration space dimensions. While there is no Polynomial Order
Dimension ((3 p + 1) / CDIM × ((2 p + 2) / Cost(Original/New) Polynomial Order
Dimension ∼ . ∼ . ∼ . ∼ . ∼ . ∼ . ∼ . ∼ . ∼ . O ( N totq N p ). There are someexceptions: for example, we can rewrite the phase space flux in configuration space161o exploit the fact that the we are employing structured, Cartesian grids, (cid:90) K j ∇ x w (cid:96) · v f h d z = (cid:90) K j ∇ x w (cid:96) · ( v − v center ) f h d z + (cid:90) K j ∇ x w (cid:96) · v center f h d z , (3.41)for each of the (cid:96) basis functions in our phase space expansion, where v center = v isthe cell center velocity. These integrals can be pre-computed on the phase space ref-erence elements because they are only coordinate weighted matrices, independent ofone’s exact position in velocity space, thus reducing their computational complexityto O ( N p ).However, the rearrangement of the phase space flux in configuration spaceto reduce the cost is the exception and not the norm. The individual pieces ofthe semi-discrete Fokker–Planck equation will be limited in cost by the number ofquadrature points required to integrate exactly the semi-discrete form because theFokker–Planck equation is nonlinear, just like the advection in velocity space dueto the electromagnetic fields.So, numerical quadrature will be inescapably expensive if we are to satisfy ourconstraint that we must integrate the semi-discrete VM-FP system of equations ex-actly to prevent aliasing errors from destroying the quality of our solution. As statedabove, at first glance, the analytical integration to pre-compute and construct thetensors, for example Eq. (3.36), for convolution as part of the update, are very dense.The convolution of these dense tensors will lead to an unfavorable computationalcomplexity, similar to the numerical quadrature approach, between O ( N c N p ) and O ( N p ). However, if we could sparsify these tensors in some way, thereby reducingthe couplings between all of the polynomials in our basis expansion, we may dra-162atically improve the computational complexity, and thus reduce the cost, of ournumerical method for the VM-FP system of equations.It is no coincidence we have drawn continual attention to the modal, orthonor-mal basis in our discussion of the specific forms our polynomial bases might take.We now emphasize the second of our most important algorithmic advances in ourimplementation of our DG discretization of the VM-FP system of equations: em-ploying a modal, orthonormal basis set for our polynomial basis expansion. Thisjudicious choice of basis functions allows us to significantly sparsify the requisite ten-sors needed to evaluate the spatial discretization of the VM-FP system of equations,while still respecting the requirement that our algorithm be alias-free for stabilityand accuracy.To get a sense for just how sparse the update with a modal, orthonormal basisis, we consider again the collisionless update, the Vlasov equation, and the volumeterm defined in Eq. (3.36). Now, we will project the phase space flux, α h , onto thismodal, orthonormal basis, α xj ( t ) = (cid:90) I ( v − v center )ˆ υ j ( η ) d η + (cid:90) I v center ˆ υ j ( η ) d η (3.42) α vj ( t ) = (cid:88) i (cid:90) I qm [ E i ( t ) + v center × B i ( t )] ˆ ϑ i ( ζ )ˆ υ j ( η ) d η + (cid:88) i (cid:90) I qm ( v − v center ) × B i ( t ) ˆ ϑ i ( ζ )ˆ υ j ( η ) d η , (3.43)where we have denoted the orthonormal basis expansion in phase space as ˆ υ ( η )and the orthonormal expansion in configuration space as ˆ ϑ ( ζ ). Importantly, theseexpressions have already leveraged the fact that the mass matrix is the identity163atrix, up to the volume factor in a cell, to simplify the resulting expressions so thatthe index α j maps to the j th basis function on the right hand side. By separating v → ( v − v center ) + v center , we can cleanly separate the velocity dependence intothe piecewise constant basis function and a piecewise linear basis function. In otherwords, we can clearly see that we require only a small fraction of the full basisexpansion’s dependence in velocity space to represent both the configuration spaceand velocity space phase space flux α x,vh .These expressions for the phase space flux can be plugged in for the coefficientsin Eq. (3.33), and the whole update evaluated, after exploiting a similar sparsity inthe collisionless numerical flux function and the other components of the discreteweak forms of the VM-FP system of equations. To actually evaluate matrices suchas Eq. (3.36), we can use a computer algebra system, for example Maxima [Maxima,2019], and compute the explicit form of the sums in Eq. (3.33). In other words, byevaluating out k = (cid:88) m,n C kmn · α n f m , (3.44)where out k is a component of the update for df k /dt , and using the fact that themass matrix is the identity matrix to change variables (cid:96) → k , we obtain the updateshown in Figure 3.3 for the piecewise linear tensor product basis in one spatial andtwo velocity dimensions (1X2V).Figure 3.3 shows a C++ computational kernel that can be called for every cell K j of a structured, Cartesian grid in phase space, as we are passing all the informa-tion required to the kernel to determine where we are physically in phase space, i.e.,164igure 3.3: The computational kernel for the volume integral, Eq. (3.36), for thecollisionless advection in phase space of the particle distribution function in onespatial dimension and two velocity dimensions (1X2V) for the piecewise linear tensorproduct basis. Note that this computational kernel takes the form of a C++ kernelthat can be called repeatedly for each grid cell K j depending on the local cell centercoordinate and the local grid spacing. Here, the local cell coordinate is the input“const double w” and the local grid spacing is the input “const double dxv”. The outarray is the increment to the right hand side due this volume integral contribution ina forward Euler time-step, i.e., a piece of Eq. (2.165) for the Vlasov–Fokker–Planckequation. To complete the right hand side of Eq. (2.165) for the evolution of theparticle distribution function, for a given phase space cell, we require the surfacecontributions for the collisionless advection, as well as the computational kernels forthe corresponding tensors encoding the spatial discretization of the Fokker–Planckequation.the local cell center coordinate and grid cell size. The output of this computationalkernel, the out array, is a piece of Eq. (2.165) for the Vlasov–Fokker–Planck equa-tion, the volume integral of the collisionless advection in phase space. To completethe right hand side of Eq. (2.165) for a given phase space cell, we require the surfacecontributions for the collisionless advection, as well as the computational kernels forthe corresponding tensors encoding the spatial discretization of the Fokker–Planck165quation. We will likewise have computational kernels for Maxwell’s equations whichcompletely specify the volume and surface contributions, and allow for the incre-menting of the solution in a forward Euler time-step.Notably, the computational kernel in Figure 3.3 has no matrix data structure,much less the requirement to perform quadrature since we have already analyticallyevaluated the integrals in Eq. (3.36) with a computer algebra system and written outthe results to double precision. We refer to this as a “quadrature- and matrix-free”implementation of the DG method. Such quadrature-free methods using orthogonal(orthonormal) polynomials were studied in the early days of the DG method [Atkinsand Shu, 1998, Lockard and Atkins, 1999] and are still applied to a variety of linearhyperbolic equations, such as the acoustic wave equation for studies of seismic ac-tivity, the level set equation, and Maxwell’s equations [K¨aser and Dumbser, 2006,Marchandise et al., 2006, Koutschan et al., 2012, Kapidani and Sch¨oberl, 2020].Even for alternative formulations of DG which do not seek to eliminate aliasingerrors by exactly integrating the components of the discrete weak form, matrix-freeimplementations are desirable to reduce the memory footprint of the scheme [Fehnet al., 2019]. Minimizing the memory footprint can lead to performance gains evenbeyond the reduction in the number of operations required to take a time-step.We emphasize again the novelty of our approach. Using a modal, orthonormalbasis, we produce a “quadrature- and matrix-free” method that respects our require-ment that our algorithm be alias-free by analytically evaluating the integrals in thediscrete weak forms of the VM-FP system of equations, thus the quadrature-freecomponent. And the matrix-free component follows from the fact that the result-166ng integrals produce sparse tensors whose convolutions can be unfolded in theirentirety, eliminating the need for a matrix data structure to actually evaluate thetensor-tensor convolutions. All that is required is entry-by-entry evaluation of theresults of these convolutions, as demonstrated in Figure 3.3 by the out array.As a frame of reference the sparseness of our “quadrature- and matrix-free”method, the computational kernel in Figure 3.3 has ∼
70 multiplications; whereas,the update for numerical quadrature applied to a nodal basis has ∼
250 multiplica-tions. The potential gains from a nodal basis by only requiring the expansion localto a surface in the surface integrals do not provide enough computational savingsto compete with the sparsity of the orthonormal, modal expansion. We will do athorough computational complexity experiment in Section 3.7 to determine bothexactly what the computational complexity of the sparse, orthonormal, modal basisexpansion is, as well as compare in totality the performance of a sparse, orthonor-mal, modal basis expansion to an optimized nodal basis expansion using anisotropicquadrature with high performance linear algebra libraries. Before we do this com-parison though, it is worth going through the final details of the algorithm. Wemust now discuss how we compute the recovery polynomial in generality, and howwe compute velocity moments, to complete the implementation of our numericalmethod for the VM-FP system of equations.167 .5 Extending the Recovery Scheme to Higher Dimensions
As stated above in Section 3.3, many of the components of the surface inte-grals, for example the numerical flux functions for the collisionless advection anddrag term, are simple enough to project onto our phase space basis expansion,compute the coefficients in our modal, orthonormal basis expansion, and then con-volve tensors such as Eq. (3.35) to evaluate the surface integral contributions in ourdiscretization of the VM-FP system of equations. However, we require a prescrip-tion for computing the recovery polynomial in generality so we can evaluate thecorresponding surface integrals in the discrete Fokker–Planck equation. Whereasprojections such as Eq. (3.37) for central fluxes applied to the collisionless advec-tion naturally retain the spatial dependence at the surface, and thus the high ordernature of our scheme, we have not yet described a procedure for the non-recoveredspatial dependence in our computation of the recovery polynomial.We said in the summary of Chapter 2, Section 2.8, that the recovery procedureis fundamentally one dimensional: we are only generating a recovery polynomialacross the surface where the function has a discontinuity. So, let us consider theoperation of projecting a two dimensional function, f ( x, y ), onto a one-dimensionalbasis, (cid:90) − g ( x, y ) ψ k ( x ) dx = (cid:90) − f ( x, y ) ψ k ( x ) dx, (3.45) g k ( y ) = (cid:90) − f ( x, y ) ψ k ( x ) dx, (3.46)i.e., each of the k coefficients for the component expansion in the x dimension retain168heir y variation. Note that the simplified form of Eq. (3.46) assumes the basis ψ k is our modal, orthonormal basis expansion to simplify the left hand side, and thatas part of this operation f ( x, y ) has a two dimensional basis expansion in x and y .Although we characterized the recovery procedure mathematically in Sec-tion 2.4, we should now explicitly compute the recovery polynomial in a specific testcase to make apparent how to use Eq. (3.46) to compute the recovery polynomialin generality. Let us use the piecewise linear, one dimensional, modal, orthonormalbasis for this demonstration, ˆ υ ( x ) = 1 √ , ˆ υ ( x ) = (cid:114) x, (3.47)but on a slightly different reference element, K L = [ − ,
0] on the left, and K R = [0 , − , x = 0. These shiftedbasis functions are then ˆ υ L ( x ) = 1 √ , ˆ υ R ( x ) = 1 √ , ˆ υ L ( x ) = (cid:114)
32 ( x + 1) , ˆ υ R ( x ) = (cid:114)
32 ( x − , (3.48)169o that the full basis expansions in each cell are, f L ( x ) = 1 √ f L + (cid:114)
32 ( x + 1) f L ,f R ( x ) = 1 √ f R + (cid:114)
32 ( x − f R . (3.49)Since we are using piecewise linear polynomials in the left and right cells, two basisfunctions in each cell, four basis functions total, we can represent a cubic functionacross the interface, h ( x ) = h + h x + h x + h x . (3.50)We then solve the following set of equations (cid:90) − [ h ( x ) − f L ( x )]ˆ υ L ( x ) dx = 0 , (cid:90) − [ h ( x ) − f L ( x )]ˆ υ L ( x ) dx = 0 , (cid:90) [ h ( x ) − f R ( x )]ˆ υ R ( x ) dx = 0 , (cid:90) [ h ( x ) − f R ( x )]ˆ υ R ( x ) dx = 0 , (3.51)using a computer algebra system to analytically evaluate each integral and invertthe matrix equation for the coefficients, h = √ (cid:0) − √ f R + 2 √ f L + 3 f R + 3 f L (cid:1) ,h = − √ (cid:0) √ f R + 5 √ f L − f R + 9 f L (cid:1) ,h = − √ f L − f R ) √ ,h = √ (cid:0) √ f R + 5 √ f L − f R + 5 f L (cid:1) . (3.52)Now we can use Eq. (3.46) to modify the individual pieces of Eq. (3.52). For example,170f the original function f = f ( x, y ), we can compute in the right cell f R ( y ) = (cid:90) f ( x, y )ˆ υ R ( x ) dx,f R ( y ) = (cid:90) f ( x, y )ˆ υ R ( x ) dx, (3.53)and likewise for the left cell.This procedure, combining the one dimensional recovery in Eq. (3.52) withthe projection from the higher dimensional space onto the one dimensional basis,Eq. (3.46), to determine how the coefficients vary in the other dimensions, is generaland can be extended to as high dimensionality and as high polynomial order aswe choose. Notably, regardless of the specific form of the recovery polynomial,we emphasize that we only require the first and second coefficients, h and h inEq. (3.52), because we are evaluating the recovery polynomial and its first derivativeat the x = 0 surface. In other words, the value of the recovery polynomial at thesurface of the reference element is h , and the value of the gradient of the recoverypolynomial at the surface of the reference element is h , at least for piecewise linearpolynomials. We have thus completely specified the required recovered function,e.g., the recovered distribution function in the discrete Fokker–Planck equation,the value and the gradient of the recovered function, and the recovered function’svariation along the surface across which we are constructing the recovered function.We can then project the results of this recovery process onto phase space basisfunctions, and construct a similar tensor to Eq. (3.35) to convolve and evaluate thesurface contributions in the discrete Fokker–Planck equation.171 .6 Computing the Coupling Moments The final component of our implementation is a means of computing the veloc-ity moments which close our equation system, such as J h for coupling to Maxwell’sequations. In the same way we demonstrated how one leverages weak equality to ac-tually calculate the recovery polynomial in arbitrary dimensions in Section 3.5, thegoal of this section is to illustrate the use of weak equality to compute the couplingmoments, and the form these computational kernels take. Recall the operations wedefined in Eqns. (2.105–2.106), which we here write out explicitly transformed tothe reference element on which the modal, orthonormal basis sets are defined, (cid:88) m M m (cid:90) I Ω ˆ ϑ (cid:96) ( ζ ) ˆ ϑ m ( ζ ) d ζ = (cid:32) V DIM V DIM (cid:89) i =1 ∆ v i (cid:33) (cid:88) n (cid:88) j (cid:90) I j \ I Ω f n ( t )ˆ υ n ( η ) ˆ ϑ (cid:96) ( ζ ) d η , (3.54) (cid:88) m M m (cid:90) I Ω ˆ ϑ (cid:96) ( ζ ) ˆ ϑ m ( ζ ) d ζ = (cid:32) V DIM V DIM (cid:89) i =1 ∆ v i (cid:33) (cid:88) n (cid:88) j (cid:90) I j \ I Ω v f n ( t )ˆ υ n ( η ) ˆ ϑ (cid:96) ( ζ ) d η , (3.55) (cid:88) m M m (cid:90) I Ω ˆ ϑ (cid:96) ( ζ ) ˆ ϑ m ( ζ ) d ζ = (cid:32) V DIM V DIM (cid:89) i =1 ∆ v i (cid:33) (cid:88) n (cid:88) j (cid:90) I j \ I Ω | v | f n ( t )ˆ υ n ( η ) ˆ ϑ (cid:96) ( ζ ) d η . (3.56)We note that the matrix on the left hand side is simply the mass matrix in config-uration space, and since we have already canceled the configuration space volumefactor, the matrix is simply the identity matrix. However, we require a means tomake the integrals on the reference element independent of our location in phase172pace, and so we perform a similar transform as done in Eqns. (3.42) and (3.43), M (cid:96) = (cid:32) V DIM V DIM (cid:89) i =1 ∆ v i (cid:33) (cid:88) n (cid:88) j (cid:90) I j \ I Ω f n ( t )ˆ υ n ( η ) ˆ ϑ (cid:96) ( ζ ) d η , (3.57) M (cid:96) = (cid:32) V DIM V DIM (cid:89) i =1 ∆ v i (cid:33) (cid:88) n (cid:88) j (cid:90) I j \ I Ω ( v − v center ) f n ( t )ˆ υ n ( η ) ˆ ϑ (cid:96) ( ζ ) d η + (cid:32) V DIM V DIM (cid:89) i =1 ∆ v i (cid:33) (cid:88) n (cid:88) j (cid:90) I j \ I Ω v center f n ( t )ˆ υ n ( η ) ˆ ϑ (cid:96) ( ζ ) d η (3.58) M (cid:96) = (cid:32) V DIM V DIM (cid:89) i =1 ∆ v i (cid:33) (cid:88) n (cid:88) j (cid:90) I j \ I Ω | v − v center | f n ( t )ˆ υ n ( η ) ˆ ϑ (cid:96) ( ζ ) d η + (cid:32) V DIM V DIM (cid:89) i =1 ∆ v i (cid:33) (cid:88) n (cid:88) j (cid:90) I j \ I Ω v center · ( v − v center ) f n ( t )ˆ υ n ( η ) ˆ ϑ (cid:96) ( ζ ) d η + (cid:32) V DIM V DIM (cid:89) i =1 ∆ v i (cid:33) (cid:88) n (cid:88) j (cid:90) I j \ I Ω | v center | f n ( t )ˆ υ n ( η ) ˆ ϑ (cid:96) ( ζ ) d η , (3.59)which can be further simplified to, M (cid:96) = v center M (cid:96) + (cid:32) V DIM V DIM (cid:89) i =1 ∆ v i (cid:33) (cid:88) n (cid:88) j (cid:90) I j \ I Ω ( v − v center ) f n ( t )ˆ υ n ( η ) ˆ ϑ (cid:96) ( ζ ) d η , (3.60) M (cid:96) = 2 M (cid:96) · v center − | v center | M (cid:96) + (cid:32) V DIM V DIM (cid:89) i =1 ∆ v i (cid:33) (cid:88) n (cid:88) j (cid:90) I j \ I Ω | v − v center | f n ( t )ˆ υ n ( η ) ˆ ϑ (cid:96) ( ζ ) d η . (3.61)We can then generate a computational kernel to compute these coupling momentssequentially, and the needed quantities such as the current density can be computedfrom the results, e.g., via Eq. (2.109). Using a 1X2V, one configuration space dimen-sion and two velocity space dimensions, piecewise linear, tensor product basis againas an example, we show the results of a computer algebra system evaluating theintegrals in Eqns. (3.57–3.59), with the simplifications outlined in Eqns. (3.60) and173igure 3.4: Example computational kernel for the calculation of the zeroth throughsecond moments using weak equality in one spatial dimension and two velocitydimensions (1X2V) with piecewise linear, tensor product, modal, orthonormal poly-nomials. Note that this computational kernel is called inside a loop over velocityspace for a given configuration space cell, as we are integrating over velocity space.(3.61), in Figure 3.4. It is critical to note that the computational kernel in Figure 3.4is called for every velocity space cell associated with a given configuration space cell,i.e., these kernels form a reduction operation across velocity space, as expected sincewe are integrating over velocity space at a given configuration space cell. The beautyof Eqns. (3.57–3.59), with the simplifications outlined in Eqns. (3.60) and (3.61), isthat this same computational kernel can be called irrespective of our location inphase space, so long as we pass the correct cell center coordinate and local gridcell size. Analogous to the updates for the Vlasov–Fokker–Planck equation andMaxwell’s equations, the computation of the coupling moments is also free of bothquadrature and matrix data structures.We note in concluding this section that these procedures can be, and within Gkeyll are, extended to other diagnostic moments, for example the stress tensor174nd heat flux, ←→ S h . = (cid:88) j (cid:90) K j \ Ω k vv f h d v , (3.62) Q h . = 12 (cid:88) j (cid:90) K j \ Ω k | v | v f h d v , (3.63)which can be rearranged similarly with the same variable manipulation as before, v → ( v − v center ) + v center . In general, the mathematical machinery of weak equalitycan be straightforwardly converted to linear equations which can be computed todetermine the desired projection of some quantity, whether it is a velocity moment,a numerical flux function, or a more complicated constraint equation for quantitiessuch as u h and T h . The components of the linear equation, the integrals overcomplex combinations of basis functions, can then be analytically evaluated using acomputer algebra system such as Maxima [Maxima, 2019], and with the help of themodal, orthonormal polynomial basis, significantly sparsified, reducing the numberof operations required to evaluate and solve the linear equations.Although we have focused on the components of the discretization which areboth quadrature- and matrix-free, we should briefly discuss the parts of the dis-cretization which are not necessarily matrix-free. For example, the solution to the setof linear equations for the discrete flow and temperature, u h and T h , e.g., Eqns. 2.141and 2.147 when using at least piecewise quadratic polynomials, is not matrix-freebecause of the coupling between the projections of u h and T h due to the boundarycorrections from finite velocity space extents. All the computational machinery wehave outlined, i.e., the analytic evaluation of the integrals using a computer alge-175ra system, is still the procedure for evaluating u h and T h . Now though, insteadof completely unrolling the evaluation of the matrix equations and eliminating theneed for a matrix data structure by evaluating every individual term in the linearequation, we construct the relevant matrix and invert the linear system to obtain oursolution for u h and T h . In Figure 3.5 we show an example computational kernel tosolve the coupled linear system for u h and T h , using the Eigen linear algebra library[Guennebaud, Jacob, et al., 2010], in one configuration space and one velocity spacedimension (1X1V) with piecewise quadratic Serendipity polynomials. Importantly,the fact that our basis is modal and orthonormal reduces the number of terms inthe matrix we have to invert. These computational kernels can then be called inevery configuration space cell to calculate the local expansion of u h and T h requiredfor the discretization of the Fokker–Planck equation.Now that all the pieces of our discrete scheme are complete, including themeans of computing the coupling moments between Maxwell’s equations and theVlasov–Fokker–Planck equation, the implementation of our discrete scheme is fin-ished. We turn now to the question of the computational complexity of our discretescheme. Although we expect the modal, orthonormal basis to have significantlydecreased the cost of numerically integrating our DG discretization of the VM-FPsystem of equations, we require quantitative proof of this cost reduction.176igure 3.5: A C++ computational kernel for the construction and inversion of thematrix to solve the coupled linear system for the discrete flow and temperature, u h and T h . Here, we show the form of the matrix in one spatial and one velocitydimension (1X1V) using piecewise quadratic Serendipity polynomials. Since both u h and T h have three degrees of freedom, i.e., three basis functions, which describetheir projection, the coupled linear system is six by six. We construct the individualterms in the matrix using a combination of weak multiplication, weak division, andthe corrections at the boundary due to our finite velocity space extents. We canthen use a linear algebra library, in this case Eigen, to solve the linear system anddetermine the discrete flow and temperature required in the evaluation of the dragand diffusion coefficients in the discrete Fokker–Planck equation. We know the choice of a modal, orthonormal polynomial basis leads to thetensors over which we need to sum, such as Eq. (3.36), being sparse, and we haveevidence from the computational kernel presented in Figure 3.3 that the number ofoperations is indeed reduced compared to the use of numerical quadrature. We would177ike to determine generally how sparse the tensors required to update our discreteVM-FP system of equations are. In Figure 3.6, we plot the results of a numericalexperiment using the computational kernels for updating the collisionless componentof the VM-FP system of equations. We show the time to evaluate the computationalFigure 3.6: Scaling, i.e., the time to evaluate the update versus the number de-grees of freedom, N p , in a cell, of just the streaming term, α xh = v , (left) and thetotal, streaming and acceleration, update (right) for the Vlasov solver. The dimen-sionality of the solve is denoted by the relevant marker, and the three colors cor-respond to three different basis expansions: black:maximal-order, blue:Serendipity,and red:tensor. Importantly, this is the scaling of the full update, for every dimen-sion, i.e., the 3x3v points include the six dimensional volume integral and all twelvefive dimensional surface integrals.kernels for just the streaming term, α xh = v , in the left plot of Figure 3.6, and theevaluation of the full phase space update, streaming and acceleration, in the rightplot. From the scaling of the cost to evaluate these computational kernels we candetermine the computational complexity of the algorithm with respect to the numberof degrees of freedom per cell, i.e., the number of basis functions in our expansion, N p . It is immediately apparent that even with the steepening of the scaling as thenumber of degrees of freedom increases there is at least some gain over the use of di-178ect quadrature to evaluate the integrals in the discrete weak form because, at worst,the total, streaming plus acceleration, update scales roughly as O ( N p ). In fact, thisscaling of, at worst O ( N p ), is exactly the scaling obtained by under-integratingthe nonlinear term in a nodal basis, as mentioned in Section 3.4 [Hesthaven andWarburton, 2007, Hindenlang et al., 2012]. But critically, we have obtained thiscomputational complexity while eliminating aliasing errors from our scheme, as werequire for stability and accuracy! We can explicitly evaluate the gain compared tothe anisotropic quadrature shown in Table 3.5. For example, for piecewise quadraticbasis functions in six dimensions, the Serendipity space has 256 degrees of freedomin a cell but requires 1728 quadrature points to evaluate the nonlinear termHowever, the improvement in the scaling is actually better than it first ap-pears. The scaling shown in Fig. 3.6 is the cost scaling of the full update to performa forward Euler step in a phase space cell, i.e., in six dimensions, three spatial andthree velocity, the total update time in the right plot of Fig. 3.6 is the time to com-pute the six dimensional volume integral plus the twelve required five dimensionalsurface integrals. This means the scaling we are quoting is irrespective of the dimen-sionality of the problem, unlike in the case of the nodal basis, where the quadraturemust be performed for every integral and there is a hidden dimensionality factor inthe scaling. In other words, in six dimensions, what at first may only seem like afactor of ∼ ∼
40 improvement in the scaling once one includesthe dimensionality factor, up to the constant of proportionality of the scaling. Ofcourse, one must also compare the size of the constant of proportionality multiplying179oth scalings to accurately compare the reduction in the number of operations andimprovement in the overall performance, since said constant of proportionality caneither tell us the picture is much rosier, that in fact the improvement in performanceis larger than we expected, or much more dire, that the improvement in the scalingis offset by a larger constant of proportionality.To determine the constant of proportionality, we will perform a more thor-ough numerical experiment and compare the cost of the alias-free nodal scheme andalias-free modal scheme for a complete collisionless Vlasov–Maxwell simulation. Weconsider the following test: a 2X3V computation done with both the nodal and themodal algorithms, with a detailed timing breakdown of the most important step ofthe algorithm, the Vlasov time step. The reader is referred Table 3.7 for a summaryof the following two paragraphs if they wish to skip the details of the computer ar-chitecture and optimizations employed. Both computations are performed in serialon a Macbook Pro with an
Intel Core i7-4850HQ (“Crystal Well”) chip, thesame architecture on which the scaling analysis was performed. The only optimiza-tion in the compilation of both algorithms is “O3” and both versions of the codeare compiled with the C++
Clang 9.1 compiler.Specific details of the computations are as follows: a 16 × grid, withpolynomial order two, and the Serendipity basis, 112 degrees of freedom per cell. Thetwo simulations were run for a number of time-steps to allow us to more accuratelycompute the time per step of just the Vlasov solver, as well as the time per step ofthe complete simulation. The time-stepper of choice for this numerical experiment isthe three-stage, third order, SSP-RK method, Eq. (2.168). To make the simulations180s realistic as possible in terms of memory movement, we also evolve a “proton”and “electron” distribution function, i.e., we evolve the Vlasov–Maxwell system ofequations for two plasma species.To make the comparison as favorable as possible for the nodal algorithm, wealso employ the Eigen linear algebra library, Eigen 3.3.4 [Guennebaud, Jacob,et al., 2010], to perform the dense matrix-vector multiplies required to evaluatethe higher order quadrature needed to eliminate aliasing errors in the nodal DGdiscretization. And we note that the nodal algorithm is optimized to use only thesurface basis functions in the surface integral evaluations, so we are doing as muchas possible to reduce the cost of the alias-free nodal scheme.The results are as follows: for the nodal basis, the computation required seconds per time step, of which seconds were spent solving theVlasov equation. The remaining time is split between the computation of Maxwell’sequations, the computation of the current from the first velocity moment of thedistribution function to couple the particles and the fields, and the accumulation ofeach Runge-Kutta stage from our three stage Runge-Kutta method. For the modal basis, the computation required seconds per time step, of which seconds were spent solving the Vlasov equation.In the nodal case, we emphasize that we achieve a reasonable CPU efficiency,and the nodal timings are not a matter of poor implementation. We estimate thenumber of multiplications in the alias-free nodal algorithm required to perform a fulltime-step is ∼ e
12, three trillion, once one considers the fact that we are evolvingtwo distribution functions with a three-stage Runge–Kutta method. One thousand181econds to perform three trillion multiplications corresponds to an efficiency of ∼ e Eigen 3.3.4 on asimilar CPU architecture to the one employed for this test [Guennebaud, Jacob,et al., 2010], so we argue that the cost of the alias-free nodal algorithm is dueto the number of operations required and not an inefficient implementation of thealgorithm.It is then worth discussing how this improvement in the timings using themodal algorithm compares with our expectations. Given the scaling of the modalbasis, we would anticipate the gain in efficiency in five dimensions would be arounda factor of twenty, a factor of four from the reduction in the scaling from O ( N q N p )to O ( N p ), and a factor of five from the latter scaling containing all of the fivedimensional volume integrals and the ten four dimensional surface integrals. Wecan see that the gain in just the Vlasov solver is ∼
17, while the gain in the overalltime per step is ∼
16, not quite as much as we would naively expect, but still asizable increase in the speed of the Vlasov solver. The reduction in the overall timeis due to the fact that, while the time to solve Maxwell’s equations and compute thecurrents to couple the Vlasov equation and Maxwell’s equations is reduced, theseother two costs, in addition to the cost to accumulate each Runge-Kutta stage, isnot reduced as dramatically as the time to solve the Vlasov equation is. Again, thedetails of this comparison are summarized in Table 3.7.So, we have achieved our goal of respecting the requirement that our DGmethod for the VM-FP system of equations be alias-free, while measurably reducing182 omputer Architecture Compiler
MacBook Pro Intel Core i7-4850HQ Clang 9.1 C++(High Sierra OS) (“Crystal Well”)
Optimization Flags Grid Size Polynomial Order “O3,” 16 × Serendipity quadratic,Eigen 3.3.4 for nodal 112 degrees of freedom
Nodal Total Time Modal Total Time Total Time Reduction1079.63 secondstime-step secondstime-step ∼ Nodal Vlasov Time Modal Vlasov Time Vlasov Time Reduction1033.89 secondstime-step secondstime-step ∼ The focus of this chapter has been principally on the evaluation of the linearoperator in Eq. (2.165) which goes into a forward Euler time-step, Eq. (2.166). Wesummarize now all the steps in the evaluation of this linear operator, for the discreteVlasov–Fokker–Planck equation and Maxwell’s equations, so that we can perform aforward Euler time-step.1. Loop over configuration space cells, and for each configuration space cell, com-183ute the needed coupling moments from the distribution functions for eachspecies at the old time-step, f nh , where superscript n denotes the known time-step. • Within each configuration space cell, loop over velocity space to com-pute velocity moments using computational kernels, such as the 1X2Vkernel shown in Figure 3.4. These kernels will give M n h , M n h and M n h ,Eqns. (3.57–3.59). • Calculate the current density from M n h for each plasma species, J nh = (cid:88) s q s M n hs . • Calculate the discrete flow and temperature, u nh and T nh , from M n h , M n h and M n h , as well as the boundary corrections in velocity space, usingcomputational kernels such as the one shown in Figure 3.5 for a 1X1V,polynomial order two, simulation. Note that if using piecewise linearpolynomials, we require the additional “star moments” in the computa-tion of u nh and T nh , Eqns. (2.158–2.160).2. Loop over configuration space cells and update the electromagnetic fields, E nh , B nh , forward in time. • Project the chosen numerical flux function for the electric and magneticfields, central fluxes, Eqns. (2.33)-(2.34), or upwind fluxes, Eqns. (2.48)-(2.51), onto the modal, orthonormal configuration space basis expansion.184
Evaluate the volume and surface integrals using the corresponding com-putational kernels, analogous to the volume and surface tensors for thecollisionless Vlasov equation, Eqns. (3.35) and (3.36), but note that thesecomputational kernels only involve the configuration space basis expan-sion. After evaluation of the volume and surface integrals, increment theelectromagnetic fields with this contribution multiplied by the size of thetime-step ∆ t , E n +1 h = E nh + ∆ t L EM ( E nh , B nh ) , and likewise for the magnetic field. • Increment the current density at the known time-step onto the electricfield, E n +1 h = E nh + ∆ t(cid:15) J nh . (3.64)3. Loop over phase space cells and update the particle distribution function foreach species, f nh , forward in time. • Project the chosen numerical flux functions for both the collisionless ad-vection and the drag term in the Fokker–Planck equation, e.g., centralfluxes, Eq. (2.59), or global Lax-Friedrichs fluxes, Eq. (2.61), for the colli-sionless advection and central fluxes, Eq. (2.126), or global Lax-Friedrichsfluxes, Eq. (2.128), for the drag term in the Fokker–Planck equation. • Determine the recovered distribution function from the general recovery185rocedure described in Section 3.5, i.e., recover a continuous functionacross the interface from the distribution function in the two neighbor-ing cells, while retaining the phase-space dependence of the distributionfunction representation on the surface. Compute the value and gradi-ent of the recovered distribution function at the surface, and add thesecontributions to the numerical flux functions for computing the surfaceintegral contributions to the discrete Fokker–Planck equation. • Evaluate the volume and surface integrals in the DG discretization of theVlasov–Fokker–Planck equation, e.g., the volume kernel in Figure 3.3 fora piecewise linear, 1X2V, simulation, and increment these contributionsmultiplied by the size of the time-step ∆ t onto the old values of theparticle distribution function, f n +1 h = f nh + ∆ t L V F P ( f nh , E nh , B nh , u nh , T nh ) . • Repeat each calculation, the flux function project, the recovery proce-dure, and the evaluation of volume and surface integrals, for each speciesin the plasma.The above steps form the core of forward Euler time-step, which can then becombined into a multi-stage Runge–Kutta method, such as our preferred three-stage,third order, SSP-RK3 scheme, Eq. (2.168). Note that for computing the size of thetime-step, while the CFL condition for Maxwell’s equation at each stage will remainfixed since the speed of light is a constant, we can evaluate the CFL constraint for186he Vlasov–Fokker–Planck equation at each stage. The general structure of thisforward Euler method is unchanged, even if we modify components of the update,for example applying the recovery procedure for the update of the advective termssuch as the collisionless update of the Vlasov equation. However, we could modifythis update to separate the collisionless and collision operators if an operator splitwould provide a more favorable time-stepping scheme. For example, as the collision-ality increases and the collision operator becomes the more restrictive componentof taking a time-step, standard operator splits that employ Runge–Kutta-Legendremulti-stage methods for advection-diffusion equations are an option [Meyer et al.,2014].So, we have formulated and implemented a Runge–Kutta discontinuous Galerkindiscretization of the Vlasov–Maxwell–Fokker–Planck system of equations—a sizableeffort! But now we turn to the equally important question: does the code give theright answer? In the next chapter, Chapter 4, we will pursue an extensive bench-marking endeavor to determine the validity of our numerical method.187 ome of the material in thischapter has been adapted fromJuno et al. [2018], Hakim,Francisquez, Juno, andHammett [2019], and Hakimand Juno [2020].
Chapter 4: Benchmarking our DG Vlasov–Maxwell–Fokker–PlanckSolver in
Gkeyll
We will proceed on three different fronts to determine the validity of our imple-mented DG scheme for the VM-FP system of equations. First, we will examine justthe Vlasov–Fokker–Planck equation, in the absence of electromagnetic fields. Then,we will benchmark the collisionless Vlasov–Maxwell system of equations, with spe-cial focus on self-consistent simulations including the feedback between the plasmaand the electromagnetic fields. Finally, we will bring it all together for a benchmarkof the complete equation system, a validation of the VM-FP system of equations intheir entirety.We reiterate a few definitions for convenience here. We will make use of the188axwellian velocity distribution as a common initial condition, f s ( x , v , t = 0) = n s ( x ) (cid:18) m s πT s ( x ) (cid:19) V DIM exp (cid:18) − m s | v − u s ( x ) | T s ( x ) (cid:19) , (4.1)where V DIM is the number of velocity dimensions. We note that we will have toproject Eq. (4.1) onto our basis expansion at the start of any simulation. Althoughthis distribution function defines local thermodynamic equilibrium, as we discussedin Corollary 1 in Chapter 1 and in Appendix A, the Maxwellian velocity distributionmight have some configuration space dependence that is unstable to perturbations.The system will then rearrange itself to a different energy state in a collisionlesssystem, and to a higher entropy state in the presence of collisions. Eq. (4.1) is thusoften a convenient initial condition, though we will make clear when we employdifferent initial plasma distributions. We will also use consistently the definitionof the thermal velocity v th s = (cid:112) T s /m s , especially to define the extents in velocityspace.Although we will reiterate many of the specifics for every benchmark, wenote here a few details which will be unchanged throughout our benchmarks. Wewill consistently use the Serendipity element space for our polynomial basis as anoptimal middle ground of cost and accuracy between the tensor product basis andthe maximal order basis. As an optimization of the computation and memoryrequired in a multi-stage method, and accuracy of the time integration, we will alsoemploy the three stage, third order, SSP-RK3 method for the time integration of allbenchmarks presented. Importantly, we will use the same numerical flux functionsfor all the presented benchmarks, upwinding, Eq. (2.60) for α x = v , the streaming189erm, and global Lax-Friedrichs for both the acceleration α v = q/m ( E h + v × B h ),Eq. (2.61), and the drag term, Eq. (2.128). Finally, we will uniformly use zero-flux boundary conditions in velocity space, with the additional boundary term wemust evaluate in the Fokker–Planck operator due to integrating by parts twice,Eq. (2.134), so as to retain the proved conservation properties in Chapter 2. Whenwe refer to zero flux boundary conditions in velocity space in all of the forthcomingboundary conditions, and when numerically integrating the discrete Fokker–Planckequation in Sections 4.1 and 4.3, we are implicitly also taking into account thisadditional boundary condition in Eq. (2.134). In the absence of streaming and body forces, any initial distribution functionshould relax to a Maxwellian. Although we did not demonstrate this to be thecase via analytic examination of our discretization of the Fokker–Planck equation,we now consider a numerical demonstration of a discrete analog to the H-theoremproved in Corollary 1 in Chapter 1. Importantly, a proper implementation of thediscrete Fokker–Planck equation has a maximum entropy state, which by definition is the discrete Maxwellian. However, such a discrete Maxwellian is not necessarilythe projection of Eq. (4.1) onto basis functions, as Eq. (4.1) is a continuous functiondefined on all of velocity space, v ∈ ( −∞ , ∞ ), and we are employing finite velocityspace extents. Nevertheless, these two quantities, the projection of Eq. (4.1) and190he maximum entropy state of our discrete Fokker–Planck operator will convergetowards each other as the grid is refined.In this first test, the relaxation of an initial non-Maxwellian distribution func-tion to a discrete Maxwellian, due to collisions, is studied. We will avoid the use ofa species index in this test since the electromagnetic fields are zero, and we are onlystudying the effects of the collision operator. The initial distribution function is astep-function in velocity space, f ( x, v, t = 0) = / (2 v ) | v | < v | v | ≥ v , (4.2)where v = √ v th . Piecewise linear and quadratic Serendipity basis sets on 16 and8 velocity space cells, respectively, are used. Note that there is no variation inconfiguration space in this problem, so only one configuration space cell is required.Velocity space extents, ( v min , v max ), are placed at ± v th the simulation is run to νt = 5, five collisional periods, and zero flux boundary conditions are used in velocityspace.In each case, the relative change in density and energy are close to machineprecision, demonstrating excellent conservation properties of the scheme. In Fig-ure 4.1, the time-history of the error in normalized energy change is plotted. Theerrors per time-step of the conservative scheme are machine precision, and the smallchange in energy is due to the numerical diffusion inherent to the SPP-RK3 scheme.For fixed time-step size, changing resolution or polynomial order has little impact onthe magnitude of energy errors, and they always remain close to machine precision.191igure 4.1: (a) Relative change in energy, ∆ M /M ( t = 0) = [ M ( t ) − M ( t =0)] /M ( t = 0), for p = 1, N = 16 (solid and dashed blue) and p = 2, N = 8 (dot-ted and dash-dot orange) cases for relaxation of a square distribution to a discreteMaxwellian. The decrease in energy in our conservative scheme is close to machineprecision. The curves labeled ‘no conservation’ omit the boundary correction termsand use regular moments instead of “star moments” (for p = 1) needed for momen-tum and energy conservation. (b) Time-history of relative change in entropy. Whenusing the conservative scheme, the entropy rapidly increases and remains constantonce the distribution function becomes a discrete Maxwellian.Figure 4.1 also shows that as the distribution function relaxes, the entropyrapidly increases and then remains constant once the discrete Maxwellian state isobtained. The change in entropy between p = 1 and p = 2 is indicative that differentdiscrete Maxwellians will be obtained depending on grid resolution and polynomialorder. The same figure shows that neglecting the boundary corrections and “starmoments” (for p = 1) needed for conservation degrade energy conservation by manyorders of magnitude, and in the p = 1 case, can even lead to decreasing entropy.In fact, the violation of the second law of thermodynamics when neglecting theboundary corrections to the drag and diffusion coefficients provides solid evidencethat the care taken in accounting for the finite velocity space extents in formulatingthe scheme in Chapter 2 produces a more reliable scheme for the physics content ofthe equation system. Note that this is not a good test for momentum conservation,192ecause the initial momentum is zero.We now consider relaxation in a 1X2V setting. For this test, the initial condi-tion is selected as a sum of two Maxwellians, the first with drift velocity u = (3 v th , u = (0 , v th ). Both Maxwellians have a thermalspeed of v th = 1 /
2. A 16 grid in velocity space with p = 2 Serendipity basis func-tions is used. Again, there is no variation in configuration space in this problem, soonly one configuration space cell is required.As the particles collide, the distribution function will relax to a new Maxwellianwith non-zero drift and different temperature, thus allowing us to test momentumconservation. The simulation is run to νt = 5, five collisional periods. Figure 4.2shows the initial and final distribution function demonstrating the relaxation tothe discrete Maxwellian. The errors in the energy and the x - and y -componentsof momentum are close to machine precision for our conservative scheme, as shownin panel (c). Neglecting boundary correction terms degrades conservation by manyorders of magnitude. Also, panel (d) demonstrates that the entropy increases mono-tonically, reaching its steady-state value once the discrete Maxwellian is obtained.These tests demonstrate the high accuracy with which the moments are conservedas well as providing empirical evidence that entropy is a non-decreasing functionof time, so long as we are careful to include the corrections in the computationof the moments and additional boundary condition which arise from solving theVlasov–Fokker–Planck equation on a finite velocity grid.193igure 4.2: The initial (a), relaxed (b) distribution function in a 1X2V relaxationtest. Conservation (c) of energy (orange) and momentum (green, purple) is atmachine precision for our conservative scheme. Neglecting boundary correctionsbreaks conservation by more than 8 orders of magnitude. Purple and green curvesoverlay each other on this scale. (d) The entropy increases rapidly and then remainsconstant once the discrete Maxwellian is obtained. We now add in the streaming of particles in configuration space, α xh = v , whilekeeping the electromagnetic fields zero, to test the accuracy of our DG Vlasov–Fokker–Planck equation in the presence of spatial gradients. In this benchmark, westudy shock structure in the kinetic regime with the classic Sod-shock [Sod, 1978]194nitial conditions in one spatial dimension and one velocity dimension (1X1V), ρ l u l p l = . . , ρ r u r p r = . . . , (4.3)where this mass density, flow, and pressure are used to initialize the Maxwellianvelocity distribution defined in Eq. (4.1) on the left, l , and right, r , sides of thedomain. The phase space domain is [0 , L ] in configuration space and [ − v th,l , v th,l ]in velocity space, with v th,l = p l /ρ l = 1 since p l = n l T l = 1 and ρ l = mn l = 1,and we initialize the discontinuity to be at x = L/
2. Note that for this 1X1Vsystem, the gas adiabatic constant is γ = 3 because the internal energy is defined as p/ ( γ −
1) =
N ρv th / N = 1 in one dimension, and upon rearranging, we find γ = 3.The simulations were run on a 64 ×
16 grid, with piecewise quadratic Serendipityelements, L = 1, and t end = 0 .
1. Zero flux boundary conditions are used in velocityspace and copy boundary conditions are used for configuration space, where thevalue of the distribution function at x = 0 and x = L is copied into the ghost layerfor the computation of the fluxes at the configuration space boundary. Note thatthis copy boundary condition copies the full expansion of the distribution functionfrom the skin cells at x = 0 and x = L into the ghost layer, and so is not thesame as a homogeneous Neumann boundary condition, but more akin to a perfectlymatched layer, i.e., an open boundary condition.The Knudsen number (Kn = λ mfp /L , where ν = v th /λ mfp ) is varied between1 /
10, 1 /
100 and 1 / L (cid:29) λ mfp .Hence, in the last case the solution should match, approximately, the solution fromthe Euler equations for the evolution of a fluid .Figure 4.3 shows the density, velocity, temperature and gas frame, or kinetic,heat-flux, q h ( x, t ) . = (cid:88) j (cid:90) K j \ Ω k ( v − u h ( x, t )) f h ( x, v, t ) dv, (4.4)obtained from the kinetic simulations. For comparison, the exact solution to thecorresponding inviscid Euler Riemann problem is also shown. It is observed, asexpected, that as the gas becomes more collisional, the moments tend to the Eulersolution. An interesting aspect of the kinetic results, though, are the viscosity, heat-conductivity and other transport effects which smooth the shock structures that aresharp in the Euler solution. In particular, the lower-right plot of Figure 4.3 showsthat the heat-flux is completely absent in the inviscid equations. There is significantheat-flux in the low collisionality case, but this heat flux vanishes as the collisionalityincreases. It is a testament to the accuracy of our discrete Vlasov–Fokker–Planckimplementation that we can transition from the low to high collisionality limit,comparing favorably with the Euler equation solution in the high collisionality limit. We note that the Euler equations are formally derived with the full Boltzmann collision op-erator accounting for hard sphere collisions of gas particles, and then taking viscosity and heatconduction to be zero. In this case, even the simplified Fokker–Planck operator leads to a highcollisionality limit. However, the transport coefficients for matching a Navier-Stokes solution withfinite viscosity and heat conduction, i.e., finite momentum and heat transport, would need to bemodified to account for this particular collision operator. /
10 (red), 1 /
100 (magenta), and 1 /
500 (blue), with the inviscid Euler results(black dashed) shown for comparison. As the gas becomes more collisional, i.e.,decreasing Knudsen number, the solutions tend to the Euler result. Note that thereis no heat-flux in the inviscid limit.We next consider a Sod-shock with a sonic point in the rarefaction wave. Theinitial conditions are selected as ρ l u l p l = . . , ρ r u r p r = . . . , (4.5)and this mass density, flow, and pressure are again used to construct an initialMaxwellian velocity distribution, Eq. (4.1). We employ the same 64 ×
16 grid with197iecewise quadratic Serendipity elements, [ − v th,l , v th,l ] velocity space extents, andzero-flux boundary conditions in velocity space. In contrast to the standard Sod-shock, this problem is run on a periodic domain [ − , | x | < .
3. The Knudsen number is 1 /
200 and the simulation is run to t = 0 . t = 0 . p = 1 and p = 2 cases. In each case, the errors are close to machine precisionwhen using our conservative scheme, but neglecting boundary corrections and usingregular moments instead of ‘star moments’ (for p = 1) leads to errors many orders ofmagnitude greater. We note that, even in the presence of spatial gradients, the errorsare independent of polynomial order and only depend on the number of time-stepstaken in the simulations, as we expect from our mathematical formulation in thealgorithm in Chapter 2. So, not only do we converge to the inviscid Euler solution inthe limit of high collisionality as we expect, but momentum and energy are conservedto a high precision by the scheme. Importantly, while we did not discuss the limit ofno electromagnetic fields in Chapter 2 when we discussed momentum conservationin the discrete scheme, we did note that the errors in momentum conservation arose198igure 4.4: The density (a), velocity (b), and distribution function (c) for the Sod-shock problem with a sonic point in the rarefaction. Complicated shock structuresare formed and are visible both in the moments as well as the distribution function.from our discretization of Maxwell’s equations. Thus, we find here by numericaldemonstration that our DG discretization of the Vlasov–Fokker–Planck equation inthe limit of E = B = 0 exactly conserves the momentum, in addition to the energy,as the momentum conservation errors in the relaxation test and kinetic Sod-shockbenchmark are only a function of the size of the time-step.199igure 4.5: The relative change in the momentum (a) and energy (b) for p = 1(blue) and p = 2 (orange) cases for the Sod-shock problem with a sonic point in therarefaction. Our conservative scheme gives us machine precision errors in momentumand energy errors that are nearly independent of polynomial order and only dependon the number of time-steps taken in each simulation. However, neglecting theboundary corrections needed for conservation leads to errors orders of magnitudegreater. To test the conservation properties of the discrete Vlasov–Maxwell system ofequations, we set up a drifting electron-proton plasma with a large density gradientin both species to drive strong asymmetric flows. We initialize a Maxwellian velocitydistribution, Eq. (4.1), for both protons and electrons with a density gradient, n ( x, t = 0) = n (1 + 4 exp( − β l ( x − x m ) )) x < x m , = n (1 + 4 exp( − β r ( x − x m ) )) x > x m , , (4.6)in a 1X1V box. The phase space domain is L x = 96 λ D with velocity space extents[ − . v th e , . v th e ] and [ − . v th p + v th e , . v th p + v th e ] for the electrons and protons200espectively. Here, λ D is the Debye length, Eq. (1.5), and v th e and v th p are theelectron and proton thermal velocities.We set β l = 0 . λ − D , β r = 0 . λ − D , x m = L x / λ D , and n = 1 inEq. (4.6). There is a constant drift in both the protons and electrons, u ( x, t = 0) = v th e , and the following parameters are chosen: m p /m e = 1836, T p /T e = 1 .
0, and v th e = 1 .
0. The latter is a normalization such that the velocity normalization in thesystem is the electron thermal velocity, a reasonable choice in 1X1V when Maxwell’sequations reduce to just Ampere’s Law, ∂ E ∂t = J (cid:15) , (4.7)and thus there are no light waves in the system.We employ periodic boundary conditions in x and zero-flux boundary condi-tions in v x , though we note that this density gradient is not periodic. However, thevalue of the gradient at the edge of configuration space is small, far below machineprecision. To demonstrate energy conservation, irrespective of configuration spaceresolution or polynomial order, we perform a number of simulations with N x = 4,∆ x = 24 λ D , and N v = 12, ∆ v = 1 v th s . Simulations are run for 1000 ω − pe , where ω pe is the electron plasma frequency, Eq. (1.23). Results are plotted in Figure 4.6,where the change in the total energy is defined as∆ E = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:82) L x E ( t ) − E ( t = 0) d x (cid:82) L x E ( t = 0) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.8)201
200 400 600 800 1000 t(ω −1pe ) −9 −8 −7 Δ PolynomialΔOrderΔComparison p=2,Δt=0.4ω −1pe p=2,Δt=0.2ω −1pe p=2,Δt=0.1ω −1pe p=3,Δt=0.4ω −1pe p=3,Δt=0.2ω −1pe p=3,Δt=0.1ω −1pe t(ω −1pe ) −10 −9 −8 −7 −6 −5 Δ Polynomial Order Comparison p=1, v=1v t , t=0.2ω −1pe p=1, v=1/2v t , t=0.2ω −1pe p=1, v=1/4v t , t=0.2ω −1pe p=2, t=0.2ω −1pe p=3, t=0.2ω −1pe t(ω −1pe ) −9 −8 −7 Δ PolynomialΔOrderΔ2
Δx=24λ D ,Δt=0.4ω −1pe Δx=24λ D ,Δt=0.2ω −1pe Δx=24λ D ,Δt=0.1ω −1pe Δx=12λ D ,Δt=0.2ω −1pe Δx=6λ D ,Δt=0.1ω −1pe t(ω −1pe ) −9 −8 −7 Δ PolynomialΔOrderΔ3
Δx=24λ D ,Δt=0.4ω −1pe Δx=24λ D ,Δt=0.2ω −1pe Δx=24λ D ,Δt=0.1ω −1pe Δx=12λ D ,Δt=0.2ω −1pe Δx=6λ D ,Δt=0.1ω −1pe Figure 4.6: The change in the total, electron plus proton and electromagnetic, energyfor a number of simulations to demonstrate the robustness of our energy conservingscheme. The scheme’s energy conservation is independent of the polynomial order(top left/right), with the caveat that the choice of polynomial order 1 requiressufficient velocity resolution to reduce the projection errors in projecting | v | . Thelatter caveat of projection errors in the polynomial order 1 simulations is also thereason for the dip in the most resolved polynomial order 1 calculation, where thecomputation of errors is the most sensitive and we must be careful about finiteprecision effects. We note though that for fixed time-step we recover the energyconservation result of p = 2 and p = 3 if we use enough velocity space resolutionwith the p = 1 simulations. Likewise, the scheme’s energy conservation depends onlythe size of the time-step, not the configuration space resolution (bottom left/right).The convergence of the energy errors in the top left plot match our expectations fora third order time-stepping method, 2.5 and 2.9 for p = 2, and 2.0 and 2.9 for p = 3.with E = 12 m e (cid:90) v max v min | v | f e d v + 12 m p (cid:90) v max v min | v | f p d v + 12 (cid:15) | E | . (4.9)Note that the absolute value in the definition of the relative energy change is due202o the fact that the total energy decreases with time.We emphasize a number of results. Defining the convergence order as, C ( E , E ) = log (cid:18) E E (cid:19) = log( E ) − log( E )log(2) , (4.10)we find the order of convergence with decreasing time-step to match our expectationsfor a third-order Runge-Kutta method, 2.5 and 2.9 for p = 2, and 2.0 and 2.9for p = 3. In addition, the energy conservation errors are independent of choiceof polynomial order. We note in particular that energy can be conserved withpolynomial order 1, but depending on the size of the time-step, one may require morevelocity resolution so that projection errors from projecting | v | onto linear basisfunctions do not dominate the error in the computation of the energy. Finally, asexpected, the conservation of energy is determined by the error in the time-steppingscheme, and refining the grid and increasing the configuration space resolution from N x = 4 to N x = 8 ,
16 does not improve the energy conservation compared todecreasing the size of the time-step.We can likewise examine the extent to which momentum is conserved, eventhough our algorithm does not formally conserve the total momentum. In Figure 4.7,we plot the integrated total momentum, relative to the total momentum at thebeginning of the simulation,∆ M = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:82) L x M ( t ) − M ( t = 0) d x (cid:82) L x M ( t = 0) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.11)203
200 400 600 800 1000 t(ω −1pe ) −12 −11 −10 −9 −8 −7 −6 Δ PolynomialΔOrderΔ2
Δx=6λ D ,Δv=1/4v t Δx=3λ D ,Δv=1/4v t Δx=3λ D ,Δv=1/8v t Δx=1.5λ D ,Δv=1/4v t Δx=1.5λ D ,Δv=1/8v t Δx=0.75λ D ,Δv=1/4v t Δx=0.75λ D ,Δv=1/8v t Δx=0.375λ D ,Δv=1/4v t Δx=0.375λ D ,Δv=1/8v t t(ω −1pe ) −13 −12 −11 −10 −9 −8 −7 −6 Δ PolynomialΔOrderΔ3
Δx=6λ D ,Δv=1/4v t Δx=3λ D ,Δv=1/4v t Δx=3λ D ,Δv=1/8v t Δx=1.5λ D ,Δv=1/4v t Δx=1.5λ D ,Δv=1/8v t Δx=0.75λ D ,Δv=1/4v t Δx=0.75λ D ,Δv=1/8v t Δx=0.375λ D ,Δv=1/4v t Δx=0.375λ D ,Δv=1/8v t Figure 4.7: The change in the total, electron plus proton, momentum in a numberof simulations. Simulations with polynomial order 2 (left) and polynomial order 3(right) are performed with increasing configuration space and velocity space reso-lution to demonstrate that errors in the total momentum decrease with increasingconfiguration space resolution, while only weakly depending on velocity space reso-lution. The convergence orders of the polynomial order 2 simulations are 1.35, 2.55,2.93, and 3.14, and the convergence orders of the polynomial order 3 simulationsare 2.83, 3.32, 3.38, and 4.76, and these convergence orders are calculated usingthe higher velocity resolution results. We note the convergence orders are largelyunaffected by using the lower velocity resolution simulations to compute them.where M = m e (cid:90) v max v min | v | f e d v + m p (cid:90) v max v min | v | f p d v , (4.12)is the total, electron plus proton, momentum. We note again the absolute value inEq. (4.11) is due to the fact that the total momentum decreases with time. Whilewe cannot show that our scheme conserves the total momentum, the errors in thetotal momentum converge rapidly with increasing configuration space resolution,and depend only weakly on resolution in velocity space. The convergence order asdefined by Eq. (4.10) are 1.35, 2.55, 2.93, and 3.14 for p = 2, and 2.83, 3.32, 3.38, and4.76 for p = 3, calculated using the higher velocity resolution results, though one canuse the lower velocity resolution results and obtain virtually identical convergence204ates. We have thus demonstrated one aspect of the scheme that is high-order: theconvergence of the errors in the total momentum with our orthonormal, modal, DGalgorithm are super-linear in polynomial order.Finally, we examine two additional convergence metrics for our discretization ofthe Vlasov–Maxwell system with this initial condition: the behavior of the L normof the distribution function and the divergence errors in Gauss’ law for the electricfield. We expect with our choice of numerical flux function, upwinding, Eq. (2.60)for α x , the streaming term, and global Lax-Friedrichs for the acceleration α v , thatthe L norm of the distribution function is a monotonically decaying function. We t(ω −1pe ) −3 −2 −1 L e l e c t r o n Polynomial Order Comparison Electrons p=2,Δx=24λ D ,Δv=1v t p=2,Δx=12λ D ,Δv=1/2v t p=2,Δx=6λ D ,Δv=1/4v t p=2,Δx=3λ D ,Δv=1/8v t p=3,Δx=24λ D ,Δv=1v t p=3,Δx=12λ D ,Δv=1/2v t p=3,Δx=6λ D ,Δv=1/4v t p=3,Δx=3λ D ,Δv=1/8v t t(ω −1pe ) −3 −2 −1 L p r o t o n Polynomial Order Comparison Protons p=2,Δx=24λ D ,Δv=1v t p=2,Δx=12λ D ,Δv=1Δ2v t p=2,Δx=6λ D ,Δv=1Δ4v t p=2,Δx=3λ D ,Δv=1Δ8v t p=3,Δx=24λ D ,Δv=1v t p=3,Δx=12λ D ,Δv=1Δ2v t p=3,Δx=6λ D ,Δv=1Δ4v t p=3,Δx=3λ D ,Δv=1Δ8v t Figure 4.8: The change in the L norm of the electron (left) and proton (right)distribution function with increasing resolution and polynomial order. As expected,the behavior of the L norm of the distribution function is monotonic and decays intime. We note as well that increasing the polynomial order from 2 to 3 correspondsextremely well with a doubling of the resolution, providing direct evidence for theoften assumed benefit of a high order method.present numerical evidence for this proof in Figure 4.8 for both the protons andelectrons by plotting the relative change in the L norm, L s = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:82) L x (cid:82) v max v min f s ( t ) − f s ( t = 0) d x d v (cid:82) L x (cid:82) v max v min f s ( t = 0) d x d v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4.13)205t is interesting to note the behavior of polynomial order 3 compared to polynomialorder 2, which provides anecdotal evidence that increasing the polynomial order ofthe simulation is analogous to increasing the resolution in configuration and velocityspace. Although this behavior is often touted as prima facie for employing highorder methods, such behavior is difficult to demonstrate analytically for nonlinearequation systems, if it is demonstrable at all.Likewise, we consider how well Gauss’ law for the electric field is satisfied in adiscrete sense. In one dimension, Eq. (1.61) becomes ∂E x ( x ) ∂x = | e | n p ( x ) − n e ( x ) (cid:15) , (4.14)where we have already substituted in for the charge density, ρ c = | e | ( n p − n e ). Weplot the results for the suite of simulations considered above, polynomial order 2and 3, in Figure 4.9 at the end of the simulations, t = 1000 ω − pe . We note that,while the agreement is not perfect, the two quantities track remarkably well, evenas larger amplitude, smaller scale, electric fields are formed with increasing resolu-tion. Especially for the finest resolution, polynomial order 3, when very fine scalestructure forms in the electric field as the resolution approaches the Debye length,the characteristic length scale of these simulations, the charge density and diver-gence of the electric field agree very well. We reiterate that we currently do notenforce this condition, as the charge density ρ c does not appear anywhere in theVlasov equation or Ampere’s law, and thus it is a testament to the robustness ofour numerical method that we do not observe large divergence errors in Gauss’s lawfor the electric field. 206
20 40 60 80
X(λ D ) −0.00050.00000.00050.00100.0015 Polynomial Order 2, t = 1000ω −1pe ∂E x ∂x ,Δx=12λ D ,Δv=1Δ2v tρ c ε ,Δx=12λ D ,Δv=1Δ2v t∂E x ∂x ,Δx=6λ D ,Δv=1Δ4v tρ c ε ,Δx=6λ D ,Δv=1Δ4v t∂E x ∂x ,Δx=3λ D ,Δv=1Δ8v tρ c ε ,Δx=3λ D ,Δv=1Δ8v t X(λ D ) −0.006−0.004−0.0020.0000.0020.0040.0060.0080.010 Polynomial Order 3, t = 1000ω −1pe ∂E x ∂x ,Δx=12λ D ,Δv=1Δ2v tρ c ε ,Δx=12λ D ,Δv=1Δ2v t∂E x ∂x ,Δx=6λ D ,Δv=1Δ4v t ρ c ε ,Δx=6λ D ,Δv=1Δ4v t∂E x ∂x ,Δx=3λ D ,Δv=1Δ8v tρ c ε ,Δx=3λ D ,Δv=1Δ8v t Figure 4.9: Comparison of the divergence of the electric field (dashed line) andthe charge density (stars) for polynomial order 2 (left) and polynomial order 3(right) simulations at the end of the simulation, t = 1000 ω − pe . We can see thatthe two quantities agree reasonably well, especially as we refine the grid. Even ashigher amplitude, smaller scale, electric fields are excited in the higher resolutionsimulations, the two quantities track each other well, despite the fact that we do notenforce this condition, and the charge density does not appear anywhere in evolvedsystem of equations. We now turn our attention to another simple, yet subtle, test of the Vlasov–Maxwell solver: advection in specified electromagnetic fields. Since charged particlescirculate around magnetic fields, and we are employing a Cartesian mesh, we checkthat our numerical method can handle the advection of the distribution functionin phase space. In other words, we are checking that our algorithm can handlecorner transport across cells. Consider a constant magnetic field in the z direction, B = B e z and an oscillating electric field of the form, E ( t ) = E cos( ωt ) e x . (4.15)207he evolution of charged particles in such a system can be solved analytically. As-suming no spatial variation of the electric and magnetic fields, we have two ordinarydifferential equations for the evolution of the particles’ velocities, dv x dt = q s m s E cos( ωt ) + Ω c v y , (4.16) dv y dt = − Ω c v x , (4.17)where Ω c = q s B /m s is the cyclotron frequency of the particles in this particularmagnetic field. For simplicity, let us normalize the time and frequency to the inversecyclotron frequency and cyclotron frequency respectively so that our two ordinarydifferential equations become, dv x d ˜ t = E B cos(˜ ω ˜ t ) + v y , (4.18) dv y d ˜ t = − v x , (4.19)where tildes indicate normalized quantities.We can convert this system of coupled first-order ordinary differential equa-tions into a set of uncoupled second order ordinary differential equations and solvefor the particular solutions of each to obtain, v x (˜ t ) = w x (˜ t ) + v x (0) cos(˜ t ) + v y (0) sin(˜ t ) , (4.20) v y (˜ t ) = w y (˜ t ) − v x (0) sin(˜ t ) + v y (0) cos(˜ t ) , (4.21)208here, w x (˜ t ) = E B (1 − ˜ ω ) [sin(˜ t ) − ˜ ω sin(˜ ω ˜ t )] ˜ ω (cid:54) = 1 , E B [˜ t cos(˜ t ) + sin(˜ t )] ˜ ω = 1 , (4.22) w y (˜ t ) = E B (1 − ˜ ω ) [cos(˜ t ) − cos(˜ ω ˜ t )] ˜ ω (cid:54) = 1 , − E B ˜ t sin(˜ t ) ˜ ω = 1 . (4.23)Note that ˜ ω = 1 means that the denormalized frequency is equal to the cyclotronfrequency, i.e., when ˜ ω = 1, that is the resonant case for the particles. Since themotion of a distribution of particles is constant along characteristics, we know that,given an initial distribution f ( v x , v y ), the distribution of particles at any later timeis f ( v x ( t ) , v y ( t ) , t ) = f ( v x (0) , v y (0) , . (4.24)Consider an initial Maxwellian distribution of electrons in one spacial dimensionand two velocity dimensions, 1X2V, Eq. (4.1). Using our solution for the particles’velocities, we can see that,[ v x (˜ t ) − w x (˜ t )] + [ v y (˜ t ) − w y (˜ t )] = v x (0) + v y (0) . (4.25)So, the exact solution for an initial Maxwellian distribution of particles is just aMaxwellian with drift velocities w x (˜ t ) , w y (˜ t ) for all future times.We simulate the evolution of an initially Maxwellian distribution functionof electrons under the influence of a constant magnetic field in the z direction, B = B e z , and a time-varying electric field given by Eq. (4.15), one simulation209ith ˜ ω = 0 . , E /B = 1 .
0, a non-resonant case, and one simulation with ˜ ω =1 . , E /B = 0 .
5, a resonant case. We compare the analytic solution from Eqns.(4.22)–(4.25) to simulations using our Vlasov–Maxwell solver in Figures 4.10 and4.11. Both simulations are performed on a 1X2V grid with L x = 2 π , and velocityspace extents [ − v th e , v th e ] in both the v x and v y dimensions. We use polynomialorder 2, N x = 2, and N v x = N v y = 16, so ∆ v x = ∆ v y = 1 v th e . Periodic boundaryconditions are employed in configuration space, and zero flux boundary conditionsare employed in velocity space. Even on a coarse velocity space mesh, the evolutionof the distribution function is well-described by our analytic solution, with very lit-tle diffusion as electrons circulate around the magnetic field. Additionally, we runthe non-resonant case, ˜ ω = 0 . , E /B = 1 .
0, to t = 1000Ω − c and plot the finaldistribution function in Figure 4.12. While we note some noticeable diffusion in thepolynomial order 2 simulation, by increasing to polynomial order 3 on the same grid,we virtually eliminate this diffusion, again illustrating the virtues of a high-ordermethod applied to the discretization of the Vlasov–Maxwell system.It is worth emphasizing an inherent flexibility we have in our Vlasov–Maxwellsolver in Gkeyll : we can choose whatever polynomial order is ultimately necessaryfor the required dynamics. While the polynomial order 3 simulation of the non-resonant case is slightly more expensive, an 80 percent increase in cost for a t =1000Ω − c simulation for the specified grid resolution of N x = 2, N v x = N v y = 16, thisfreedom to increase the polynomial order as needed ultimately allows us to tacklea wider range of problems. And, we wish to point out that an 80 percent increasein cost is actually better than we would naively expect, as there are 60 percent210 v y ( v t h ) t = 10 c t = 40 c v x ( v th ) v y ( v t h ) t = 70 c v x ( v th ) t = 100 c v y ( v t h ) t = 4 c t = 8 c v x ( v th ) v y ( v t h ) t = 12 c v x ( v th ) t = 16 c Figure 4.10: The non-resonant (top) and resonant (bottom) advection of a distri-bution of electrons in phase space, over-plotted with the analytical solution. Theelectron distribution function is plotted at f ( x = π, v x , v y ). We can see that inboth cases the distribution function’s evolution is well described by our derived an-alytical solution, and that in the non-resonant case, where the distribution functionis advected for a large number of inverse cyclotron periods, there is no noticeablediffusion of the distribution function in phase space. We emphasize that these sim-ulations are performed with polynomial order 2 on a relatively coarse velocity spacemesh, N v x = N v y = 16 with velocity space extents [ − v th e , v th e ] in both the v x and v y dimensions, so ∆ v x = ∆ v y = 1 v th e . 211
20 40 60 80 100−2−1012 u x Time (Ω −1c ) −1012 u y u x Time (Ω −1c ) −4−2024 u y Figure 4.11: The value of the flow computed from the simulations (red dots) over-plotted with the analytic solution (black line) for non-resonant (top) and resonant(bottom) cases. The values of the flow are plotted at u x ( x = π ) , u y ( x = π ).212 v x ( v th ) v y ( v t h ) Polynomial Order 2 v x ( v th ) Polynomial Order 3 t = 1000 c Figure 4.12: Comparison of a polynomial order 2 (left) and polynomial order 3(right) simulation of the non-resonant case at t = 1000Ω − c . The electron distributionfunction is plotted at f ( x = π, v x , v y ). On this coarse mesh, N v x = N v y = 16with velocity space extents [ − v th e , v th e ] in both the v x and v y dimensions, so∆ v x = ∆ v y = 1 v th e , the diffusion of the distribution function in phase space startsto become noticeable for the polynomial order 2 case after running the simulationfor a long enough time. But, we note that for the same coarse mesh, the distributionfunction in the polynomial order 3 simulation remains pristine at this late time.more basis functions, 32 /
20 = 1 .
6, going from polynomial order 2 to 3, and werequire 50 percent more time-steps for the high polynomial order simulation from amore restrictive CFL condition. This back-of-the-envelope calculation suggests thatpolynomial order 3 should be 2.5 times more expensive for the same grid resolutionand end time. The improvement over the naive cost scaling occurs because thehigher polynomial order computational kernels obtain better efficiency in terms ofarithmetic intensity, i.e., the number of floating point operations per byte of memorymoved. 213 .2.3 Landau Damping of Langmuir Waves
Consider a plasma, or Langmuir, wave propagating in a plasma of protons andelectrons whose distribution functions are given by Maxwellians, Eq. (4.1). Langmuirwaves are dispersive waves, with a dispersion relation given by1 − k λ De Z (cid:48) (cid:18) ω √ v th e k (cid:19) = 0 , (4.26)in the limit that the proton mass is much larger than the electron mass and theprotons can thus be considered immobile. Z ( ζ ) is the plasma dispersion function,defined as Z ( ζ ) = 1 √ π (cid:90) ∞−∞ e − x x − ζ dx, (4.27)with the derivative of the plasma dispersion function given by Z (cid:48) ( ζ ) = − ζZ ( ζ )] . (4.28)An application of complex integration techniques shows that depending on the signof the largest imaginary component of the frequency ω = ω r + iγ , the wave is eitherunstable and will grow with time, or will damp away, a phenomenon known asLandau damping.For Langmuir waves propagating in a Maxwellian plasma of protons and elec-trons, the waves quickly damp. Using a 1X1V setup, we can initialize Langmuirwaves in the Vlasov–Maxwell system with a small density perturbation and the214orresponding electric field to support this density perturbation, n e ( x ) = n [1 + α cos( kx )] (4.29) n p ( x ) = n (4.30) E x ( x ) = −| e | α sin( kx ) (cid:15) k , (4.31)where n = 1 . α is the size of the perturbation, and k is the wavenumber of thewave. The electric charge e and permittivity of free space (cid:15) are included in theelectric field to satisfy Eq. (1.61). Choosing α (cid:28) L x = 2 π/k so exactlyone wavelength fits in the domain. Specific parameters for these runs are: α = 10 − , m p /m e = 1836, T p /T e = 1 .
0, and v th e /c = 0 .
1. For the proton species, the velocityspace extents are ± v th p , and for the electrons, the velocity space extents are ± v th e .The boundary conditions in configuration space are periodic, while the boundaryconditions in velocity space are zero flux.The resolution is chosen for each simulation to adequately resolve the De-bye length in configuration space and to mitigate numerical recurrence in velocityspace. By numerical recurrence, we refer to the process by which the collisionlesssystem artificially “un-mixes” if the distribution function forms structure at the ve-locity space grid scale, see, e.g., Cheng et al. [2013b] for a discussion of numericalrecurrence in DG schemes. Numerical recurrence is inevitable with finite velocityresolution for this particular problem, because the Landau damping of the wave willcreate smaller and smaller velocity space structure through the phase-mixing of the215ave. We could completely eliminate this issue with a diffusive process in velocityspace, such as a collision operator, and we will explore the effects of collisions on theLangmuir wave in Section 4.3.1. Here, we choose ample velocity resolution so thatthe wave damps enough for us to extract a clean damping rate and frequency forthe initialized wave. We find for the longest wavelengths, using polynomial order2, a resolution of 64 points in configuration space adequately resolves the Debyelength, and 128 points in velocity space permits the wave to phase-mix sufficientlyto extract damping rates.The evolution of the electromagnetic energy, as well as the other componentsof the energy, in a prototypical simulation is given in Figure 4.13. Comparisons of -17 -16 -15 -14 -13 -12 -11 F i e l d E n e r g y ∫ | E | d x T o t a l C h a n g e i n E l e c t r o n P a r t i c l e E n e r g y Electromagnetic EnergyElectron Particle EnergyTotal Energy
Figure 4.13: Prototypical evolution of the electromagnetic energy (blue), (cid:15) (cid:82) | E | dx ,for the damping of a Langmuir wave, in this case kλ D = 0 .
5, for a number of plasmaperiods (left), and the evolution of various components of the energy for the fulllength of the simulation (right). The right plot is the relative change in the energycomponent compared to the total energy at t = 0, i.e, ∆ E comp /E . The localmaxima (red circles) of the evolution in the left plot are used to determine both thedamping rate and frequency of the excited wave via linear regression, with the blackline being our reference fit for the damping rate. We note that energy is very wellconserved, and, as expected, the plasma waves damp on the electrons, convertingelectromagnetic energy to electron thermal energy.a number of Vlasov–Maxwell simulations with theory for both the damping rates216nd the frequencies of the waves are given in Figure 4.14. For the theoretical result, D -8 -7 -6 -5 -4 -3 -2 -1 D a m p i n g R a t e D F r e q u e n c y Figure 4.14: Damping rates (left) and frequencies (right) of Langmuir waves fromtheory (solid line) and for a number of Vlasov–Maxwell simulations (red circles).The solid lines are obtained using a root finding technique applied to Eq. (4.26).The x-axis of both figures is normalized to the Debye length, λ D , and the y-axis ofboth figures is normalized to the plasma frequency, ω pe .we solve Eq. (4.26) using a root-finding technique. We emphasize that we solve theVlasov–Maxwell system in its entirety, including the nonlinear term, for both theprotons and electrons. With the above simulation parameters, the plasma wavesdamp entirely on the electron species, so the approximation that the protons areessentially immobile in our dispersion relation holds to high precision. We also wishto note that the resolution of 64 points in configuration space is not required forevery simulation. For example, the prototypical simulation presented in Figure 4.13uses only 16 points in configuration space, or approximately one grid cell per De-bye length. As long as the gradients are properly resolved, the Vlasov–Maxwelldiscretization is extremely robust. 217 .2.4 Three-Species Collisionless Electrostatic Shock We turn now to benchmarking the flexibility of our Vlasov–Maxwell solve in
Gkeyll by considering the evolution of a plasma with more than two species. InPusztai et al. [2018], a semi-analytic model for electrostatic collisionless shocks wasderived and then checked against the results of a number of fully nonlinear Vlasov–Maxwell calculations. The Vlasov–Maxwell simulations performed in Pusztai et al.[2018] were done with an initially alias-free nodal scheme implemented and describedin Juno et al. [2018], before the algorithm was improved with an orthonormal, modalbasis—see Chapter 3 for details on the othornormal, modal basis compared to thenodal basis. In the following test, we employ the orthonormal, modal basis algorithmfor the three-species shock problem and reproduce the results of Pusztai et al. [2018]with our new and improved implementation of the DG scheme for the VM-FP systemof equations.The three-species collisionless shock setup described in Pusztai et al. [2018] isrepeated here for clarity. A Maxwellian, Eq. (4.1), with a density gradient in 1X1Vin all three species is initialized and allowed to evolve freely, as in Section 4.1.2, butnow allowing the electromagnetic fields to evolve as well. This density gradient is astep function, with n L = n , and n R = 2 n , where n is the density normalization,and the subscripts L and R denote the left and right values of the density in the 1Dconfiguration space domain.The three species in the plasma are electrons, fully ionized aluminum, and aproton impurity species. The real mass ratios of the various species are employed218o that m p /m e = 1836 , m i /m p = 27, where the subscript i denotes the mass ofthe aluminum ion species. Note that Z i = 13 for fully ionized aluminum. Sincethe proton species is an impurity, we choose n p /n i = 0 .
01. The electrons are muchhotter than either ion species, T e /T p = 45 , T p = T i . The configuration space domainhas length L x = 100 λ D . Note that the jump in the density is initialized at x = 50 λ D ,the middle of the domain. The velocity space extents of the electrons, aluminumions, and proton impurity are [ − v th e , v th e ] , [ − v th i , v th i ], and [ − v th p , v th p ]respectively, with v th s denoting the thermal velocity of the specified species. We usethe same resolution as Pusztai et al. [2018], N x = 256 and N v = 96 for all threespecies, and p = 2 Serendipity elements. Copy boundary conditions are employedin the x dimension as in Section 4.1.2, i.e., we employ a perfectly matched layerin configuration space to allow the electromagnetic fields and distribution functionto evolve freely at x = 0 and x = 100 λ D , and zero flux boundary conditions areemployed in velocity space.We plot the aluminum and proton distribution functions in the vicinity of theshock in Figure 4.15. We note that this figure is similar to Figure 9 in Pusztai et al.[2018]. These distribution functions are plotted at t = 35 (cid:112) m e /m p ω − pe ∼ ω − pe and over-plotted in white are contours of constant H ( x, v ) = m s v + q s φ ( x ), theHamiltonian. We note that the Hamiltonian has been transformed to the rest frameof the shock, ˆ v = v − V shock , and there is some freedom in computing φ ( x ) from theelectric field in our simulations. We choose φ ( x = 0) = 0 on the left edge of thedomain, and then integrate E x along the 1D domain to determine the electrostaticpotential. 219 X ( D ) V x ( v t h p ) V s h o c k Aluminum f i m a x ( f i )
65 70 75 80 85 90 X ( D ) V x ( v t h p ) V s h o c k Proton impurity f p m a x ( f p ) Figure 4.15: The aluminum (left) and proton impurity (right) distribution functionsin the vicinity of the shock at t = 35 (cid:112) m e /m p ω − pe ∼ ω − pe . Over-plotted inwhite are contours of constant H ( x, v ) = m s v + q s φ ( x ), the Hamiltonian. Wenote that the Hamiltonian has been transformed to the rest frame of the shock,ˆ v = v − V shock , and there is some freedom in computing φ ( x ) from the electric fieldin our simulations. We choose φ ( x = 0) = 0 on the left edge of the domain, andthen integrate E x along the 1D domain to determine the electrostatic potential. Wedraw attention to the trapped particle regions in the proton distribution functionjust down-stream of the shock, which amplify the cross-shock potential and leadto a large reflected population of protons. Note that we are plotting a normalizedvalue for the distribution function, as in Pusztai et al. [2018], and that the v-axesare different for the two species.We find similar results to Pusztai et al. [2018] for the value of the shockvelocity, V shock = 5 . v th p , M = 1 . M = V shock / (cid:112) Z i T e /m i is the machnumber, the value of the maximum normalized electrostatic potential, ˆ φ max = 23 . φ = eφ/T p , and the measured ratio of the reflected population of the protonimpurity species, α p = 0 . x = 85 λ D . Theseresults are in good agreement with the semi-analytic model derived in Pusztai et al.2202018], especially for the reflected proton ratio, α p ∼ . The Vlasov–Maxwell system of equations supports a large zoo of instabilities.Many of these instabilities are fundamentally “kinetic” in nature, meaning theirultimate evolution is challenging to model with fluid systems of equations. In otherwords, the actual collisionless dynamics of the plasma is a critical component to theevolution of the instability, and equations that evolve a truncated set of of velocitymoments of the Vlasov–Maxwell system of equations will have difficulty modelingthese instabilities.Determining whether an extended two-fluid model could capture the dynamicsof current sheets unstable to modes such as the lower-hybrid drift instability (LHDI)[Hirose and Alexeff, 1972, Davidson et al., 1977, Yoon et al., 2002] was the focus ofa recent paper, Ng et al. [2019] (see also Ng [2019]). Due to the inhomogeneities inthe magnetic field and density in the vicinity of the current, diamagnetic effects may221ecome important and drive instabilities such as the LHDI. As part of this study,Vlasov–Maxwell simulations of the LHDI were performed with
Gkeyll to compareboth the linear and nonlinear stages of the evolution of the unstable current sheet in afully kinetic model and the aforementioned extended two-fluid models. A simulationof a current sheet unstable to the LHDI is reproduced here as evidence our modal,orthonormal DG discretization of the Vlasov–Maxwell system of equations providesa fiducial representation of the dynamics of this kinetic instability.We use the same parameters as Ng et al. [2019]. In 2X2V, two spatial, ( x, y ),and two velocity, ( v x , v y ), dimensions, we initialize a gradient in an out-of-planemagnetic field, B z ( x, y ) = B ( y ) + δB ( x, y ) , (4.32) B ( y ) = − C tanh (cid:16) y(cid:96) (cid:17) , (4.33) δB ( x, y ) = C cos (cid:18) πyL y (cid:19) sin (cid:18) πmxL x (cid:19) , (4.34)where (cid:96) = ρ p and m = 8, i.e., a current sheet of width ρ p and an m = 8 perturba-tion to the current sheet. Here, ρ p is the proton Larmor radius, ρ p = v th p / Ω cp . Thebox size is L x × L y = 6 . ρ p × . ρ p . The velocity space extents for electrons are[ − v th e , v th e ] , and the velocity space extents for the protons are [ − v th e , v th e ] .Zero flux boundary conditions are used in velocity space, periodic boundary condi-tions are used in x , and reflecting boundary conditions are used in y . By reflecting,we mean that the particles reflect off the y -boundary, and the boundary conditionfor Maxwell’s equations is that of a perfect conductor, zero tangent electric field222nd zero normal magnetic field. The grid resolution is N x × N y = 128 × N v = 32 grid points in velocity space for the electrons, and N v = 24 for theprotons, with piecewise quadratic Serendipity elements.Additional parameters are v th e /c = 0 . , m p /m e = 36 , T p /T e = 10, and β tot =1 .
0. Since β tot = 1 . β p = 10 . / . β e = 1 . / .
0. The system is normalized such that the constantsare C = v th e / √ β e = v Ae , the electron Alfv´en velocity, and C = 10 − /m where m is the mode number being initialized. Note that with the chosen parameters, theresolution is such that ∆ x ≈ ρ e , where ρ e is the electron gyroradius, ρ e = v th e / Ω ce .Finally, we note two critical components to initializing the system. First, theastute reader will notice that the the initial magnetic field has non-zero curl, andtherefore there must be a supporting current in the plasma, thus we refer to thisinitial condition as a current sheet, J x = − C (cid:96) sech (cid:16) y(cid:96) (cid:17) − C πL y sin (cid:18) πyL y (cid:19) sin (cid:18) πmxL x (cid:19) , (4.35) J y = − C πmL x cos (cid:18) πyL y (cid:19) cos (cid:18) πmxL x (cid:19) . (4.36)Since the protons are 10 times hotter than the electrons, we give the appropriatefraction of the current to the protons and electrons, 10.0/11.0 to the protons and1.0/11.0 to the electrons. Second, to initialize the particle distribution functions,we initialize both a current carrying and background Maxwellian, the sum of two223nstances of Eq. (4.1), for each species, f s ( x, y, v x , v y ) = m s n sech (cid:0) y(cid:96) (cid:1) πT s exp (cid:18) − m s ( v x − u x s ) + ( v y − u y s ) T s (cid:19) + m s n B πT s exp (cid:18) − m s v x + v y T s (cid:19) , (4.37)where, u x s = T frac J x q s sech (cid:0) y(cid:96) (cid:1) , (4.38) u y s = T frac J y q s sech (cid:0) y(cid:96) (cid:1) , (4.39)and n = 1 . n B = 10 − . Note that T frac is the aforementioned fraction ofthe current given to the protons and electrons, 10.0/11.0 and 1.0/11.0 respectively.This background density is for numerical stability, so that the density does not goto zero away from the current sheet.We plot the results of this simulation in Figures 4.16 and 4.17, focusing on thelate linear stage when the traditional mode structure of the LHDI is most visuallyevident. In Figure 4.16, we see the logarithmic growth of the electric field associatedwith the LHDI , with a growth rate found γ ∼ . ci , in agreement with lineartheory and Ng et al. [2019]’s computation, as well as the mode structure expectedfor an m = 8 perturbation. Likewise the structure is concentrated away from thecurrent sheet centered at y = 0, as expected since it is the edge of the current sheetwhere the density gradient is largest and thus most unstable to the LHDI. Note that we use a slightly different coordinate system from Ng et al. [2019], who insteaddefine the 2X2V domain as ( y, z, v y , v z ). This is why the equivalent mode structure found in Nget al. [2019] is in the y-electric field, as opposed to here, where the LHDI mode structure is foundin the x-electric field. t ( ci ) E x X ( i ) Y ( i ) t = 6 ci E x Figure 4.16: The exponential growth of the LHDI electric field (left) and the LHDIelectric field visualized in configuration space late in the linear stage at t = 6Ω − ci (right). The growth rate, γ ∼ . ci , compares well with linear theory and theresults presented in Ng et al. [2019]. Likewise, the mode structure in a snapshot ofthe LHDI electric field corresponds to the typical LHDI electric field for an m = 8perturbation, with the electric field localized to the edge of the current sheet wherethe density gradient is largest. The LHDI electric field magnitude is normalizedto B v A = B / √ µ n m p where B is the asymptotic magnetic field and n is thedensity in the current layer.In Figure 4.17, we present the proton distribution function at the edge of thecurrent sheet and confirm the presence of the proton resonance expected for theLHDI. Both the initial drift and the phase velocity for the ion resonance conditionare over-plotted with a cut of the distribution function through x = 2 . ρ p , y = − . ρ p , v y = 0 . v th p . The resonant velocity is computed by solving Eq. (18) in Nget al. [2019]. The clear resonance structure in the ion distribution function, used as Note that Ng et al. [2019] contains a sign difference in the initial magnetic field profile, whichmanifests as a difference in the sign of the proton flow. The growth rate, mode structure, andresonant velocity are manifestly unaffected, because in 2D a change in sign of the initial flow profileis analogous to a rotation of the whole system by 180 degrees, and the Vlasov-Maxwell system hasrotational symmetry. X ( i ) V X ( v t h p ) t = 6 ci V X ( v th p ) f p ( v x ) t = 6 ci Initial drift velocityPhase velocity
Figure 4.17: The distribution function for the protons plotted at f ( x, y = − . ρ p , v x , v y = 0 . v th p ), at the edge of the current sheet (left), and a further cut ofthe 2D distribution function, f ( x = 2 . ρ p , y = − . ρ p , v x , v y = 0 . v th p ) (right). Themode structure for an m = 8 perturbation is again easily seen in the 2D visualizationof the proton distribution function, as the protons at the edge of the current sheetare resonant with the growing electric field from the LHDI. We have over-plotted theinitial drift velocity (red solid) and the phase velocity for the resonance condition(green dashed) on top of the 1D cut of the distribution function at x = 2 . ρ p .proof of the importance of ion kinetics in the dynamics of the instability in Ng et al.[2019], is again a prominent aspect of the algorithm presented here in this thesis.While there have been numerous particle-in-cell studies of the LHDI [Lapenta andBrackbill, 2002, Lapenta et al., 2003, Daughton, 2003, Roytershteyn et al., 2012],the phase space structure lucidly provided by a continuum approach presents analternative means of understanding the plasma physics of these small scale, kinetic,instabilities. 226 .2.6 Hybrid Two-stream/Filamentation Instability Our final benchmark of our collisionless Vlasov–Maxwell solver is in the samevein as the previous section and concerns the modeling of small scale, kinetic insta-bilities. In astrophysical settings, interpenetrating beams, or flows, of plasma arequite common, as they can serve as a free energy source for a myriad of instabilities.In particular, in the unmagnetized case, the two-stream instability, filamentationinstability [Fried, 1959], and a hybrid mode of the two-stream and filamentation re-ferred to as the electromagnetic oblique mode [Bret, 2009] are of interest for a varietyof astrophysical systems from gamma ray bursts [Medvedev and Loeb, 1999] to pul-sar wind outflows [Kazimura et al., 1998] to cosmological scenarios [Schlickeiser andShukla, 2003, Lazar et al., 2009]. It is of particular interest in these astrophysicalcontexts if the filamentation instability, or filamentation-like instabilities, are effi-cient enough to produce dynamically important magnetic fields, and, for example,explain the observed emission or the presence of a magnetic field in the system.The dynamics of these instabilities, especially their competition, served as themotivation for a recent study using the Vlasov–Maxwell solver in
Gkeyll [Skoutnevet al., 2019]. Skoutnev et al. [2019] found that in a certain parameter regime, asthe beams internal temperature was decreased and v th /u d , the ratio of the thermalvelocity to the drift speed of the beam, became smaller, the electromagnetic obliquemodes had comparable growth rates to the two-stream instability. These modes thussaturated on similar time scales, leading to the dynamics of a single mode havinga manifestly different final nonlinear state in comparison to an initialization of a227pectrum of modes.We will consider the results of these nonlinear simulations from Skoutnev et al.[2019] in Chapter 5, but here we focus on the ability of the DG Vlasov–Maxwellsolver to accurately capture the linear growth of these modes, two-stream, filamen-tation, and electromagnetic oblique. For the purposes of demonstrating that thealgorithm adequately captures the growth of these modes, we will focus on singlemode simulations, in contrast to the simulations presented in Skoutnev et al. [2019],which were initialized from a bath of random fluctuations. We will focus particularattention on the electromagnetic oblique modes in anticipation of how their uniquephysics will prove a critical component of the nonlinear evolution of a spectrum ofmodes discussed in Chapter 5.To initialize these single mode simulations, we consider an electron-protonplasma in 2X2V, but with the protons forming a stationary, charge-neutralizingbackground . The electrons are initialized as two drifting Maxwellians, Eq. (4.1), f e ( x, y, v x , v y ) = m e n πT e exp (cid:18) − m e ( v x ) + ( v y − u d ) T e (cid:19) + m e n πT e exp (cid:18) − m e ( v x ) + ( v y + u d ) T e (cid:19) , (4.40)where n = 0 . u y = 0 . c , with c being thespeed of light. The electron temperature is chosen so that v th e /u d = 1 / v th e = 0 . c .The simulations are performed with N x × N y × N v = 8 × × configuration andvelocity space resolution, with polynomial order 3 and the Serendipity element basis. For the purposes of the simulation, this limit is achieved by not adding a proton contributionto the current in Maxwell’s equations so that the only contribution to the current comes from thedynamic electron species. L x × L y = 2 π/k x × π/k y , and the velocity space extents are [ − u y , u y ] ,with periodic boundary conditions in configuration space and zero-flux boundaryconditions in velocity space. A small perturbation is seeded in the electric andmagnetic fields of the form E x = − δ sin( k x x + k y y ) k x + k y α , (4.41) E y = αE x , (4.42) B z = k x E y − k y E x , (4.43)where δ is the size of the perturbation and α is a coefficient determined by theeigenfunctions of the linear theory and corresponds to the ratio of the y-electricfield to the x-electric field.In the notation of Skoutnev et al. [2019], we define an angle θ with respect tox-axis so that the wave vector, k = ( k x ˆx , k y ˆy ), corresponds to a pure filamentationmode when θ = 0 degrees, and a pure two-stream mode when θ = 90 degrees. Inother words, a pure k x mode is a filamentation mode, and a pure k y mode is a two-stream mode, with all the intermediate angles defining the aforementioned obliquemodes. We note in both cases the initial condition simplifies, as a filamentation modereduces to a perturbation in B z , and a two-stream mode reduces to a perturbationin E y . For all of the simulations, δ is chosen to be sufficiently small to maximizethe linear regime of the simulation and insure a reasonable fit of the growth rate.In Figure 4.18, we compare the results of the linear theory with a sequence229f Vlasov-Maxwell simulations using Gkeyll for a variety of initial perturbations.The linear theory solution is found by linearizing the Vlasov–Maxwell system ofFigure 4.18: Comparison of linear theory (solid line) calculated from the disper-sion relation in Eq. (4.44) after rotation to the coordinate system aligned with k ,Eqns. (4.47–4.49), with a number of Gkeyll simulations (stars) for the filamenta-tion limit, θ = 0 ◦ , an oblique mode at θ = 45 ◦ , and the two-stream limit, θ = 90 ◦ .We observe good agreement between the linear theory and our DG Vlasov-Maxwellsolver.equations to obtain the dispersion matrix, D ij = ω c (cid:0) k i k j − k δ ij (cid:1) + (cid:15) ij , (4.44)where, (cid:15) ij = (cid:32) − (cid:88) s ω p s ω (cid:33) δ ij + (cid:88) s ω p s ω (cid:90) ∞−∞ v i v j k · ∇ v f s ω − k · v d v . (4.45)It is most convenient to rotate the dispersion matrix to the coordinate system aligned230ith the wave vector k , i.e., a rotation by the angle θ previously defined, D = D D D D , (4.46)where D = 1 − ω pe k v th [ Z (cid:48) ( ξ + ) + Z (cid:48) ( ξ − )] , (4.47) D = D = ω pe u d cos θ ωkv th [ Z (cid:48) ( ξ + ) − Z (cid:48) ( ξ − )] , (4.48) D = 1 − ω pe ω − k c ω − ω pe ( u d cos θ + v th )4 ω v th [ Z (cid:48) ( ξ + ) + Z (cid:48) ( ξ − )] . (4.49)Here, Z ( ξ ± ) is the plasma dispersion function previously employed in Section 4.2.3,Eq. (4.27), but now with ξ ± = ω ± ku d sin θ √ kv th . The linear solution, the solid lines inFigure 4.18, are eigenmodes of the system found by solving det( D ) = 0 for ω withthe corresponding eigenvectors satisfying R T DR E = 0, where R is the rotationmatrix for the angle − θ .We now turn to the evolution of an electromagnetic oblique mode in the nonlin-ear regime. We repeat the oblique mode calculation with θ = 45 ◦ with an increasedresolution, N x × N y × N v = 48 × × , and slightly larger velocity extents,[ − v th e , v th e ] , running the simulation for t = 500 ω − pe , deep into the nonlinearevolution of the mode, with wave-vector k x = k y = 2 .
0. In Figure 4.19, we plot thethree field components, E x , E y , and B z , as well as the particle distribution functionat ( y = L y / , v y = 0) , ( x = L x / , v x = 0), and ( x = L x / , y = L y /
2) at t = 125 ω − pe at the initial nonlinear phase. We can see that the oblique mode grows all threecomponents of the field that are initialized, as well as the standard signatures of the231igure 4.19: The evolution of the electromagnetic fields, E x (top left), E y (topmiddle), and B z (top right), as well as the electron distribution function at( y = L y / , v y = 0) (bottom left), ( x = L x / , v x = 0) (bottom middle), and( x = L x / , y = L y /
2) (bottom right) at t = 125 ω − pe as the oblique mode, θ = 45 ◦ ,instability is going nonlinear. We observe the growth of all three components of theinitial electromagnetic fields, with standard signatures of both two-stream- and fila-mentation modes in the distribution function: the phase space vortices in the x − v x and y − v y plane, and the deflection of the beams in the v x − v y plane respectively.the two-stream and filamentation instability, the phase space vortices in the x − v x and y − v y plane, and the deflection of the beams in the v x − v y plane respectively.Late in time at t = 500 ω − pe in Figure 4.20, we see that the saturated state has littleif any magnetic field, as potential wells have formed in the electric fields that havescattered the particles to a fairly isotropic state in the v x − v y plane and depleted thephase space structure required to support a magnetic field. This particle scatteringwill prove to be an important component of the nonlinear evolution of a spectrumof unstable modes in Chapter 5. 232igure 4.20: The evolution of the electromagnetic fields, E x (top left), E y (topmiddle), and B z (top right), as well as the electron distribution function at ( y = L y / , v y = 0) (bottom left), ( x = L x / , v x = 0) (bottom middle), and ( x = L x / , y = L y /
2) (bottom right) at t = 500 ω − pe of the oblique mode, θ = 45 ◦ , instability deep inthe nonlinear phase of the dynamics. Here, we observe little, if any, magnetic field,as the electrostatic wells forming in the electric field components scatter particles toa nearly isotropic state in the v x − v y plane and deplete the phase space structurerequired to support the magnetic field. We return now to the Landau damping of Langmuir waves discussed in Sec-tion 4.2.3, but now including the effects of collisions with our discretization of theFokker–Planck equation. Collisions can significantly change the damping rate, andin the limit of high collisionality, the damping can be “shut off.” This shut off233appens when the mean free path becomes shorter than the wavelength, preventingthe particles from resonating with the wave and gaining energy before being scat-tered via collisions. We are interested in demonstrating that the discrete VM-FPsystem of equations in
Gkeyll can smoothly transition from the collisionless to col-lisional regimes, similar to our benchmarks in Section 4.1.2, but now including theself-consistent plasma-electromagnetic field feedback.We again initialize Maxwellian, Eq. (4.1), proton and electron distributionfunctions with the initial density and electric field again given by Eqns. (4.29–4.31).We choose a fixed k for this study scanning collisionality, kλ D = 0 .
5, and still set L x = 2 π/k so exactly one wavelength fits in the domain. The proton and electronvelocity space limits are again set to ± v th s , with periodic boundary conditions inconfiguration space and zero flux boundary conditions in velocity space.Figure 4.21 shows the electric field energy as a function of time for ν =0 . ω pe , . ω pe , and 1 . ω pe . As the collision frequency increases, we find a rapidly de-creasing damping rate in the moderate collisionality regime, as seen in Figure 4.22.We compare this damping rate cut off with the results of Anderson and O’Neil[2007b], who employ a similar simplified collision operator, though they considerthe 1X3V case, and here we are examining the 1X1V case, so the results are notexpected to match exactly. Nevertheless, a fit (black dashed line in Figure 4.22)to the slope in the intermediate collisionality transition regime from the theory inAnderson and O’Neil [2007b] shows reasonable agreement with the numerical results.234igure 4.21: Field energy as a function of time for the linear collisional Landaudamping problem with varying collisionality. Similar to Figure 4.13, we computethe damping rate of each simulation by fitting to the peaks of the field energy. Thecollision frequency ν is normalized to the electron plasma frequency.Figure 4.22: Damping rate versus collisionality computed from simulations such asthose shown in Figure 4.21. As expected, the damping rate shuts off with increas-ing collisionality due to the particles being scattered by collisions before they canresonate with the wave. The black dashed line shows an analytical estimate of thedamping rate computed from expressions found in Anderson and O’Neil [2007b] andagrees well with the results computed here.235 .3.2 Heating via Magnetic Pumping Our final benchmark provides an opportunity to perform our most exactingtest yet of the VM-FP system of equations. We will examine heating via magneticpumping, a process by which oscillations of the magnetic field are converted toparticle energy. Magnetic pumping relies on the approximate conservation of themagnetic moment, µ = mv ⊥ / B , in a magnetized plasma. As the magnetic fieldincreases, to maintain magnetic moment conservation, v ⊥ should also increase. Ina collisionless system, if the magnetic field is oscillating slowly compared to thegyro period, then v ⊥ oscillates up and down in a reversible way, and there is no netheating of the plasma. However, collisions can provide a route to pitch angle scatterthe energy into the parallel direction, leading to an overall irreversible heating ofthe plasma.This mechanism was originally proposed as a heating mechanism in the earlydays of fusion research and investigated extensively [Berger et al., 1958, Laroussiand Roth, 1989]. Recently, this same mechanism has been studied as a potentialsource of particle heating in the solar wind [Lichko et al., 2017]. We use a similarsetup as Lichko et al. [2017], with a small modification to the parameters and adifferent collision operator . Note that the collision operator employed by Lichkoet al. [2017] retains the velocity dependence of the collision frequency, and is thus Both our collision operator and the collision operator employed by Lichko et al. [2017] areFokker–Planck collision operators, but Lichko et al. [2017] discretizes the full, unsimplified Fokker–Planck equation written in Landau form, ∂f cs ∂t = (cid:88) s (cid:48) ν s,s (cid:48) ∇ v · (cid:90) d v (cid:48) ←→ U ( v , v (cid:48) ) · (cid:18) f s (cid:48) ( v (cid:48) ) ∇ v f s ( v ) − m s m s (cid:48) f s ( v ) ∇ v (cid:48) f s (cid:48) ( v (cid:48) ) (cid:19) , (4.50)
236 more accurate description of collisions in a plasma. Nevertheless, our simplifiedFokker–Planck operator contains pitch-angle scattering due to the isotropic diffusionterm, and thus can be used to test whether our discretization of the VM-FP systemof equations contains an accurate representation of magnetic pumping.We set up a 1X3V domain which has extents [0 , πρ e ] × [ − v th,s , v th,s ] ona 256 × grid. Here, ρ s = v th,s / Ω cs is the gyroradius of species s . A perturbationis driven on a background magnetic field B = B ˆ z using an antenna that drivescurrents given by J =ˆ y J sin (cid:104) π , ω ramp t ) (cid:105) sin( ω pump t ) × (cid:20) exp (cid:18) − ( x − x ) σ J (cid:19) − exp (cid:18) − ( x − x ) σ J (cid:19)(cid:21) . (4.53)The current is turned on slowly over one pumping period using ω ramp = ω pump .This ramping phase ensures that the antenna is “turned on” slowly and hence doesnot excite unwanted waves in the plasma. Further, we need to ensure that theplasma density is low enough that the electromagnetic waves are not “trapped” inthe density holes that are created around the antenna.The tests shown here use ω pump = 0 . ce , x = 50 πρ e , x = 150 πρ e , σ J =200 πρ e / ce = 2 . ω pe . We employ a proton mass ratio m p /m e = 1836 and where ν s,s (cid:48) = q s q s (cid:48) ln(Λ)8 πm s (cid:15) (4.51)is the collision frequency of species s colliding with species s (cid:48) , and ←→ U ( v , v (cid:48) ) is the Landau tensor, ←→ U ( v , v (cid:48) ) = 1 | v − v (cid:48) | (cid:18) ←→ I − ( v − v (cid:48) )( v − v (cid:48) ) | v − v (cid:48) | (cid:19) . (4.52) n ρ e = 2 . × , and thermal speed v th e /c = β Ω ce / [2 ω pe (1 + τ )]. Thetemperature ratio is τ = T p /T e = 1, and the ratio between plasma and magneticpressures is β = 2 × − . With these quantities, the normalized background mag-netic field amplitude is (cid:15) ω pe B / ( en ) = Ω ce /ω pe , and we use the normalized drivingcurrent density amplitude J / ( enc ) = Ω ce / (2 ω pe ).Figure 4.23 shows the evolution of the magnetic field and thermal energy in themiddle of the domain, x = 100 πρ e . As the antenna current ramps up, an oscillatingFigure 4.23: Time evolution of the magnetic field (top) in the middle of the do-main from the magnetic pumping problem. As the antenna currents ramp up, anoscillating field is created that then transfers energy, via pitch-angle scattering, tothe plasma, leading to an increase in the thermal energy (bottom). With zero colli-sionality (bottom, green), the energy exchange is completely reversible, and no netheating is observed, but as the collision frequency is made finite, magnetic pumpingbegins to heat the plasma. 238agnetic field structure is created. The amplitude of oscillations are about 15%of the background. This oscillating energy is then transferred to parallel heatingvia pitch angle scattering. This heating is shown in the bottom panel of the figure,which shows that as the collision frequency becomes finite, the plasma gains thermalenergy through the simulation. Importantly, these simulations show how taxing thistest problem is, as it relies on every part of the discretization of the VM-FP systemof equations, and that the scheme must be able to preserve the adiabatic invariants.Were the magnetic moment, µ , not conserved in the zero collisionality case, andthe overall scheme not conservative, we would not be able to confidently arguethe heating demonstrated is a consequence of the physics contained in the collisionoperator.As a comprehensive test of the algorithm’s ability to model heating via mag-netic pumping, we next turn to the heating rate versus the ratio of the collisionalityto the pump frequency, ν/ω pump . In Figure 4.24, we plot the heating rate computedfrom the code, γ H = 1 E ∂ E ∂t , (4.54)where E is the second velocity moment, Eq. (1.72), the particle energy. This quantityis computed in the middle of the domain, x = 100 πρ e . We compare the results of Gkeyll simulations with our DG VM-FP solver to the heating rate predicted by thetheory of magnetic pumping [Lichko et al., 2017], γ mp = ω pump (cid:18) δnn (cid:19) ν ( ω + 4 ν ) , (4.55)239 hea t i ng r a t e / ω pu m p ν / ω pump Braginskiimagnetic pumpingcode
Figure 4.24: Heating rate via magnetic pumping, plus an additional viscous heatingmechanism, as a function of normalized collision frequency. The code agrees wellwith the theoretical prediction (black line) magnetic pumping at lower collisionfrequency, but shows an additional heating mechanism at higher collisionalty due tothe viscous damping of out-of-plane flows, which are included in the Braginskii-basedtheory (red line).where δn and n are computed from the central density n ( t ) = n + δn sin( ω pump t )after the initial transients. Note that this heating rate is derived in terms of themagnetic fluctuations, δB/B , but if the plasma is frozen-in to the magnetic field, theratios δn/n and δB/B are equal. To correctly match the heating rates computedfrom the time evolution of the temperature, the density compression is also measuredin the middle of the domain, x = 100 πρ e .Our discretization of the VM-FP system of equations agrees with magneticpumping theory for small ν/ω pump (cid:46)
1, but indicates an additional heating mecha-nism for larger collisionality. This trend was also observed, but in a different param-eter regime and using the Landau form of the collision operator, Eq. (4.50), in Lichkoet al. [2017]. Because the pump frequency is larger than the proton cyclotron fre-240uency, ω pump > Ω cp , the protons are unmagnetized and unable to respond to thecompression of the magnetic field. Thus, when the electrons undergo compression,the protons are effectively stationary, leading to an electric field to maintain chargeneutrality, but this electric field drives an E × B flow. In our chosen geometry,the electric field to maintain quasi-neutrality develops in the x direction, so the z magnetic field drives a flow in the y direction. This flow is then viscously damped,leading to additional heating.This additional heating can be derived by considering a Braginskii calculation[Braginskii, 1965] in which flows are viscously damped in the limit ν (cid:29) ω pump . Wecan use the Braginskii stress tensor to compute the heating rate for the viscousdamping of the electron flows, γ B = 23 nT (cid:34)(cid:16) η η (cid:17) (cid:18) ∂u x ∂x (cid:19) + η (cid:18) ∂u y ∂x (cid:19) (cid:35) , (4.56)where η = 0 . nT τ c and η = 0 . nT / ( τ c Ω c ) are two of Braginskii’s viscosity co-efficients and τ c is the collision time for the species. These expressions are for ω pump (cid:28) ν (cid:28) Ω c , but are generalized for arbitrary ν/ Ω c in Braginskii [1965].The η term gives rise to magnetic pumping in the collisional limit [Kulsrud, 2005,Schekochihin et al., 2005], and asymptotic matching can be done to extend thedefinition of η into the low collisionality regime. We can then relate Braginskii’scollision time, τ c , to the collision rate for our simplified Fokker–Planck collision op-erator by τ c = 0 . /ν . The η term represents additional viscous heating due toclassical cross-field momentum transport. Note that the Braginskii calculation is performed as an asymptotic expansion of the full Fokker–Planck collision operator, written in the Landau form, Eq. (4.50) (cid:18) ∂u x ∂x (cid:19) = (1 / ω (cid:18) δnn (cid:19) , (4.57)and (cid:18) ∂u y ∂x (cid:19) = 12 ω pe Ω ce (cid:18) δnn (cid:19) , (4.58)to find that the out-of-plane flows are actually larger, with u y ≈ . u x for ourparameters. Viscous heating from damping these flows dominates at high collision-ality for these parameters. Note that because we are using δn/n in the formulas,we obtain a slightly smaller heating rate since δn/n = 0 . δB/B = 0 . Gkeyll .Although this benchmarking section has been by no means exhaustive, we havecovered a wide spectrum of functionality within our algorithm for the VM-FP sys-tems of equations. We have demonstrated numerically the conservation propertiesproved analytically in Chapter 2, and further shown numerically that our schemesatisfies discrete analogs of the Second Law of Thermodynamics and an H-theorem.We have shown the code obtains theoretical estimates for damping rates, growthrates, and heating rates in a variety of non-trivial test cases of both the collisionlessVlasov–Maxwell implementation and the full Vlasov–Maxwell–Fokker–Planck nu-242erical method. Further, we have shown that a continuum VM-FP solver providesa high fidelity representation of the particle distribution function which can be lever-aged to clearly identify everything from particle trapping to resonant wave-particleinteractions. We turn now to the question of critical importance: what science canbe done with this novel, well-tested tool that provides such high quality particledistribution function data? 243 ome of the material in thischapter has been adapted fromJuno et al. [2020] and Skoutnev,Hakim, Juno, and TenBarge[2019]
Chapter 5: Leveraging the Uncontaminated Phase Space
We turn now to a question of the utmost importance after the meticulous workto derive, implement, and test a novel numerical method for the VM-FP systemof equations: what new science can be done with this tool? As we discussed inChapter 4, the continuum representation of the particle distribution function, freeof the counting noise which normally pollutes a particle-based discretization, allowsfor the clear identification of plasma processes in phase space. We would like nowto leverage this high fidelity representation for the particle distribution functionin a variety of numerical experiments to provide new perspective on energizationprocesses and nonlinear saturation mechanisms in a number of plasma environments.This chapter will not be an exhaustive discussion of every ongoing project withthe VM-FP solver in
Gkeyll . It is merely our goal to demonstrate the versatility ofthis approach of a continuum discretization and to justify our effort in the previous244hapters deriving and implementing the DG algorithm for the VM-FP system ofequations. We refer the reader to a number of publications for the breadth ofapplicability of the VM-FP solver, including bounded plasma and plasma sheathstudies [Cagas et al., 2017a, Cagas, 2018, Cagas et al., 2020], electrostatic shocks[Pusztai et al., 2018, Sundstr¨om et al., 2019], instability calculations [Cagas et al.,2017b, Ng et al., 2019], and simulations of the plasma dynamo [Pusztai et al., 2020].We will focus on the ability to directly diagnose the energy transfer betweenthe electromagnetic fields and the plasma in phase space, and the nonlinear satura-tion of instabilities driven by counter-streaming beams of plasma. Using the clean,uncontaminated phase space, we will be able to identify phase space energization sig-natures as a complement to other methods of determining the mechanisms of energyexchange within a plasma. Likewise, we will leverage the high fidelity representa-tion of the distribution function to completely characterize the nonlinear dynamicsof the beam-driven instabilities discussed in Section 4.2.6, and in doing so, showcasea situation where the particle noise inherent to particle-based methods can lead todeceptive dynamics.
Before we dive into the distribution function, we require a means of interpretingthe structure in the distribution function and how this structure can be translatedto study the energy transfer between the electromagnetic fields and the plasma.245o probe the energy exchange between electromagnetic fields and the plasma inphase space, we will utilize a technique called the field-particle correlation [Kleinand Howes, 2016, Klein, 2017, Klein et al., 2017, Klein et al., 2020, Howes et al.,2017, Howes et al., 2018, Li et al., 2019]. The essential idea behind the field-particlecorrelation is to determine where in phase space the plasma is gaining or losingenergy, and thereby ascertain the specifics of the energization process, or processes,that may be occurring.To derive the field-particle correlation diagnostic, we examine the collisionlessVlasov equation weighted by 1 / m s | v | , ∂w s ∂t = − v · ∇ x w s − q s | v | E · ∇ v f s − q s | v | ( v × B ) · ∇ v f s , (5.1)where we have separated out each component of the phase space flux: the con-figuration space streaming term, the electric field, and the magnetic field. Here, w s ( x , v , t ) = m s | v | f s ( x , v , t ) / w s evolves by integrating over phasespace, ∂W s ∂t = − (cid:90) (cid:90) q s | v | E · ∇ v f s d x d v = − (cid:90) (cid:18)(cid:90) q s v f s d v (cid:19) · E d x = − (cid:90) J s · E d x , (5.2)where we have split the integral over phase space into an integral over configuration246pace and velocity space. Here, W s = (cid:90) E s d x , (5.3)the integral of the particle energy over all of configuration space. Note that we haveperformed similar operations to the proof of Proposition 5 in Appendix A, i.e., wehave integrated the velocity gradient by parts which eliminates the contribution fromthe magnetic field by properties of the cross product, and we have used a suitableboundary condition, such as periodic boundary conditions in configuration spaceand the distribution function vanishing at the edge of velocity space, to eliminatethe boundary terms. In other words, the exchange of energy between the plasmaand the electromagnetic fields is governed entirely by the electric field since only theelectric field can do work on the plasma, and vice versa. Both the magnetic fieldand streaming term can move energy around in phase space, but neither componentof the Vlasov equation corresponds to a net energization or de-energization of theplasma.We could stop here and only use J s · E as a proxy for the bulk energization ofthe plasma, but this would be restrictive, as J s · E gives us no information aboutwhat is happening to the particles as a function of their particular velocities. Fromthis formulation of the energy exchange, we would be unable to distinguish be-tween energization processes such as resonant wave-particle interactions and directacceleration via electric fields. In this vein, we have no way to distinguish betweena transfer of energy which is oscillatory, such as a wave propagating through theplasma, and a transfer of energy which is secular, such as the wave damping on the247lasma via a resonant process like Landau damping.So, we step back from performing the integration over phase space and focuson Eq. (5.2). Since we expect the electric field to be the only participant in thedirect energization and de-energization of the plasma, we will define the field-particlecorrelation C ( x , v , t, τ ) = − q s τ (cid:90) t + τt | v | E ( x , t (cid:48) ) · ∇ v f s ( x , v , t (cid:48) ) dt (cid:48) . (5.4)Here, τ defines a correlation time over which to average so we can address our pre-vious concern about distinguishing between oscillatory and secular energy transferby averaging over the oscillatory energy exchange. In the limit of τ →
0, we obtainthe instantaneous energy exchange, C ( x , v , t,
0) = ∂w s ∂t = − q s | v | E ( x , t ) · ∇ v f s ( x , v , t ) . (5.5)Importantly, because this diagnostic does not require integrations over config-uration space, it can be used as a single-point diagnostic. This feature has alreadybeen leveraged to discover the presence of electron Landau damping in observa-tions of the Earth’s turbulent magnetosheath using spacecraft measurements [Chenet al., 2019]. The result in Chen et al. [2019] provides sizable motivation to applythe field-particle correlation to other plasma systems beyond the Alfv´enic turbu-lence studied with the field-particle correlation in, e.g., Klein et al. [2017], that gavea frame of reference for the signature of Landau damping observed in Chen et al.[2019]. By applying the field-particle correlation to other plasma systems, we canbuild a Rosetta stone that can be used to translate the signatures observed in other248pacecraft observations. We undertake such a study in the next section. We now examine in greater detail the results of the simulation shown in Fig-ure 1.1 in Section 1.7. The particular simulation is a perpendicular collisionlessshock. Here, a collisionless shock refers to a shock-wave, a disturbance propagat-ing faster than the local (magneto)sonic speed, which inevitably dissipates its bulkkinetic energy as other forms of energy, e.g., thermal energy, by means other thanparticle collisions, because the shock wave forms on scales smaller than the inter-particle mean-free path. For a survey of studies of collisionless shocks relevant forthe heliosphere and Earth’s bow shock, we refer the reader to Wilson III et al. [2010,2012, 2014a,b] and references therein.Since these shock-waves are collisionless, we know that the energy transferfrom the kinetic energy of the incoming supersonic flow into thermal and electro-magnetic energy occur due to kinetic processes such as wave-particle interactionsand small-scale instabilities. And, since this energy conversion is collisionless, it canbe diagnosed directly in phase space with the aforementioned field-particle correla-tion technique, Eq. (5.4). We will use a perpendicular collisionless shock set-up in1X2V to determine how the upstream kinetic energy from the supersonic plasmaflows is converted to other forms of energy. Here, perpendicular refers to the orien-tation of the magnetic field with respect to the shock normal, the direction of theincoming supersonic flow. We now describe in detail the simulation parameters.249he particular geometry we choose is the one spatial coordinate is in the x direction, with the initial magnetic field in the z direction, B ( t = 0) = B ˆz . In thisgeometry, we can see why we only require the two velocity dimensions perpendicularto the magnetic field to describe the dynamics because of how Maxwell’s equationssimplify, ∂B z ∂t = − ∂E y ∂x , (5.6) ∂E y ∂t = − c ∂B z ∂x − J y (cid:15) , (5.7) ∂E x ∂t = − J x (cid:15) . (5.8)The electrons and protons are initialized with the same supersonic flow into a re-flecting wall, which leads to a shock wave that propagates from left to right in oursimulation. Note that the particles reflect from the wall, but the “reflecting wall”boundary condition for the electromagnetic fields is a conducting wall boundarycondition in the traditional sense, with zero normal magnetic field and zero tangen-tial electric field. This method of initialization is often called the “injection” setup,and this setup has been previously employed in numerous particle-in-cell studies ofcollisionless shocks [e.g., Caprioli and Spitkovsky, 2014a,b,c, and references therein].Detailed parameters are as follows: the reflecting wall for the particles andconducting wall for the electromagnetic fields are at x = 0, and plasma is injectedwith a copy boundary condition at x = 25 d p , where d p is the proton collisionless We previously employed this boundary condition in Sections 4.1.2 and 4.2.4, but we repeatthe definition of this boundary condition here for completeness. A copy boundary condition meansthat the value in the ghost layer at the rightmost grid cell is exactly equal to the value in therightmost grid cell, for all the quantities being evolved, including the distribution functions for theelectrons and protons, and the electromagnetic fields. Because the plasma is initialized with a flowpropagating from right to left, this boundary condition leads to a continuous injection of plasma d p = c/ω pp . Here, c is the speed of light, and ω pp is proton plasmafrequency, ω pp = (cid:112) e n /(cid:15) m p . We use a reduced mass ratio between the protonsand electrons, m p /m e = 100. The total plasma beta, β = 2 µ n ( T e + T p ) /B = 2,with the proton beta, β p = 1 .
3, and electron beta, β e = 0 . v th e /c = 1 / (16 √ v th s = (cid:112) T s /m s . The in-flow velocity to initialize the perpendicular, electromagnetic shock is U x = − v A ( U x < v A is the proton Alfv´enspeed, v A = B / √ µ n m p . Since the plasma is initialized with a flow transverse toa background magnetic field, we initialize the corresponding electric field necessaryto support this flow, E = − u × B = U x B ˆy . With these specified parameters andinitial flow, we can initialize Maxwellian velocity distribution, Eq. (4.1), functionsfor the protons and electrons.For the grid in configuration space, we use N x = 1536, ∆ x ∼ d e /
6, withpiecewise quadratic Serendipity elements for the discontinuous Galerkin basis ex-pansion. In velocity space, the electron extents are ± v th e , and the proton extentsare ± v th p , with zero-flux boundary conditions at the edges of velocity space, and N v x = N v y = 64 for both species, corresponding to ∆ v = v th e / v = v th p / from the right wall, with the corresponding electric field and magnetic field to support the E × B flow. ν ee = 1 . e − ce = 0 . cp ,much less than the proton cyclotron frequency, Ω cp = eB /m p , with the proton-proton collision frequency correspondingly smaller based on the square root of themass ratio, ν pp = 0 . cp .We will begin with a discussion of the overall structure of the collisionlessshock. In Figure 5.1, we show the electromagnetic fields and reduced particle dis-tribution functions in x − v x phase space, integrated over v y , for the electrons andprotons, after the perpendicular shock has formed and propagated through the sim-ulation domain, t end = 11Ω − cp . Although the downstream region after the shock haspassed through the plasma is fairly oscillatory, because the energy injected into theplasma by the shock sloshes back and forth between the electromagnetic fields andparticles, we can estimate the compression ratio of this low Mach number shockbased on the magnetic field to be roughly, r ∼ .
5. This estimate is based on themean value of the magnetic field, B z , in the downstream region (solid black line inFigure 5.1). With this estimate for the compression ratio, we calculate the shockvelocity to be U shock = U x / ( r −
1) = 2 v A .We have marked an approximate transition from the upstream of the shock tothe shock ramp (dashed-dotted lines) and likewise an approximate transition fromthe shock to the downstream region (dashed lines) in Figure 5.1. The full extent ofthe shock includes the foot, where the initial field variation begins, the ramp, wheremost of the reflected proton population can be found, and the overshoot. It is worth252mphasizing a striking feature of the electromagnetic fields through the shock: weexpect the y-electric field to be the dominant component of the energization ofthe protons and electrons through the shock, because the x-electric field is roughlybimodal through the shock and oscillates about 0 in the downstream. This featureis perhaps intuitive, as in this reduced dimensionality, the x-electric field is theelectrostatic component of the dynamics, and so we might naively expect that thedominant energy exchange will happen through the electromagnetic component ofthe fields, i.e., the component of the electric field which supports the compressionof the magnetic field. Still, these features fittingly foreshadow our ultimate analysisof the phase space signature of the energization mechanism.The particle distribution functions in x − v x phase space in Figure 5.1 areillustrative of the dynamics through the shock, showing a clear compression of theelectrons and a reflected population of protons. We can gain further insights intothe dynamics of this shock by looking at the distribution function in v x − v y atfixed points in configuration space through the shock. In Figure 5.2, we plot theproton and electron distribution functions in velocity space through the shock, fromupstream through the ramp to downstream. We draw special attention to the protondistribution function in the shock ramp, where we can identify a higher energy tailin v x − v y . 253igure 5.1: The x-electric field (top), y-electric field (second from top), z-magneticfield (middle), reduced proton distribution function (second from bottom), and re-duced electron distribution function (bottom), both integrated in v y , after the per-pendicular shock has formed and propagated through the simulation domain. Wehave marked an approximate transition from upstream of the shock to the shockedplasma (dashed-dotted lines), and likewise an approximate transition from the shockto the downstream region (dashed lines). To mark the mean values of the oscillat-ing downstream electromagnetic fields, we have used a solid black line to mark theapproximate compression of the magnetic field, along with E = 0.254igure 5.2: The proton (top two rows) and electron (bottom row) distributionfunctions plotted through the shock at t = 11Ω − cp . As we move from upstream, x = 24 . d p , through the shock ramp centered at x = 21 . d p , we can identify thereflected proton population as well as a broadening of the electron distribution func-tion. We would like to identify the energization mechanism for this high energytail of protons, along with the cause of the broadening of the electron distribution.We thus turn to Eq. (5.4), but instead of performing a time average, we use theinstantaneous limit, Eq. (5.5), since we expect the energization through this shock255o be impulsive and not require any averaging over an oscillatory component ofthe energy exchange. Further, we separate the field-particle correlation into theenergization in each of the two velocity directions and transform the fields andvelocities to the shock rest-frame, C v x ( x, v (cid:48) x , v (cid:48) y , t ) = − q s ( v (cid:48) x − U shock ) E x ( x, t ) ∂f s ( x, v (cid:48) x − U shock , v (cid:48) y , t ) ∂v (cid:48) x , (5.9) C v y ( x, v (cid:48) x , v (cid:48) y , t ) = − q s v (cid:48) y E y ( x, t ) − U shock B z ( x, t )] ∂f s ( x, v (cid:48) x − U shock , v (cid:48) y , t ) ∂v (cid:48) y , (5.10)where we have performed a Lorentz transformation of the the y electric field, E (cid:48) = E − u × B . (5.11)Here, primed coordinates denote the simulation frame and unprimed coordinatesdenote the shock rest-frame, so that, for example, the velocity in the shock rest-frame is v x = v (cid:48) x − U shock . (5.12)Note that we are multiplying by the velocity squared in the particular direction ofinterest, as we expect the orthogonal velocity coordinates, e.g., v y for the E x corre-lation, will integrate to zero as the x electric field can only provide net energizationin the v x direction.We first investigate the proton energization in the shock foot through thedownstream transition, x = 22 . d p → . d p in Figure 5.2. We plot in Figures 5.3and 5.4 the field-particle correlation separated into the v x and v y components, Eqns.(5.9) and (5.10), as well as the corresponding proton distribution function, through256he shock. We focus in Figure 5.3 on the shock foot and ramp, around x = 22 . d p and x = 21 . d p respectively, at the specified time of t = 11Ω − cp . The blue-red signatureidentifies the region in phase space in which particles are being accelerated to highervelocities. Blue regions correspond to a loss of phase space energy density, while redregions correspond to an increase, so a blue-red region means phase space energydensity is being transported from the blue to the red region. We note that in boththe shock foot and ramp, the energization is dominantly in v y and concentrated inthe vicinity of the high energy tail.In Figure 5.4, we examine the overshoot and transition to the downstream re-gion of the shock, where all of the secular energization is complete and the remainingenergy exchange is governed by a sloshing back and forth between the electromag-netic fields and plasma. In the overshoot and transition region, we note that theenergization has decreased in magnitude in the units of the field-particle correlationand become much more unstructured. The progression from the region of directenergization to the downstream region where no further secular energization occursand energy is merely exchanged back and forth between the fields and the particlesis then nearly complete. By this point, the shock is “done” in the sense of convertingthe incoming bulk kinetic energy of the supersonic flows to other forms of energy,though it remains for the downstream region to further partition the energy betweenthe thermal energy of the plasma and electromagnetic energy via other collisionlessprocesses.Although we can make some sense of the energy exchange occurring by therelative magnitudes of the field-particle correlation and the overall structure, we257igure 5.3: Proton distribution functions (top row), C v x field-particle correlations(middle row), and C v y field-particle correlations (bottom row) in the shock footand ramp region, where the shock has begun energizing the plasma. We see clearevidence in the proton distribution function of a high energy tail in v x − v y . Further,we note that the energization of the plasma is localized to this high energy tail.This energization is due to the component of the proton distribution function whichreturns upstream via its gyromotion, and is thus able to gain energy along themotional electric field, E y , which supports the E × B drift.258igure 5.4: Proton distribution functions (top row), C v x field-particle correlations(middle row), and C v y field-particle correlations (bottom row) in the overshoot andtransition regions of the shock, after much of the secular energization has beencompleted by the shock. We see that the magnitude of the field-particle correlationhas decreased in comparison to Figure 5.3, and that the correlation has becomemore unstructured. By this point in the shock, protons in the plasma are almostdownstream, and thus no long experience the gradient in the magnetic field offwhich the protons reflected, preventing them from gaining further energy along themotional electric field. What remains is oscillatory energy exchange between theplasma and the electromagnetic fields. 259ould like to understand what particular processes are present in the energy ex-change. We wish to further scrutinize the high energy tail in the shock ramp inthe proton distribution function which is prominent in Figure 5.3 and a “hot spot”for the energization of the protons. This higher energy tail in the proton distribu-tion function arises from the component of the proton distribution function whichreturns upstream via its gyromotion, and is thus able to gain energy along themotional electric field, E y , which supports the E × B drift.To understand this process of protons returning upstream and gaining energyalong the motional electric field, we consider a single-particle picture. In this single-particle picture, we approximate the shock as a discontinuity in the magnetic field,since the proton gyro-orbit, or Larmor orbit, is as large or larger than the shockscale length, ρ p (cid:38) L shock ∼ d p . In Figure 5.5, we plot (a) the trajectory of a protonin the ( x, y ) plane and (b) its corresponding trajectory in ( v x , v y ) velocity spacein the shock frame, where the colors indicate the corresponding segments of thetrajectory. The proton velocity is normalized to the proton thermal velocity, v th p .In the upstream region, x > E × B velocity (black star) corresponds to the gyro-orbit of the proton about theupstream inflow velocity in the ( v x , v y ) plane.Upon first crossing the magnetic discontinuity to x <
0, the particle changesto a Larmor gyration in the ( v x , v y ) plane (blue) about the downstream E × B velocity (green star). In the larger amplitude downstream perpendicular magneticfield, the radius of the Larmor motion in the ( x, y ) plane is reduced (blue), andunder appropriate conditions, it can lead to the particle crossing back upstream to260 -10 -8 -6 -4 -2 0 2 4 6 8 10-10-8-6-4-20246810 Figure 5.5: (a) Real space trajectory of a proton as it traverses the shock front and(b) the corresponding velocity space trajectory. Note that the magnetic gradientis assumed to be a discontinuity in this simple picture of the perpendicular shock.The colors of the particle trajectories in real space (a) correspond to the particle’slocation in phase space (b). Black is upstream, blue corresponds to a proton crossingthe magnetic discontinuity before returning upstream, gaining energy along the redtrajectory, and then returning downstream and following the green trajectory.261 > x >
0, it will once again undergo aLarmor orbit in the ( v x , v y ) plane (red) about the upstream E × B velocity (blackstar). In this segment of the trajectory (red), the proton gains perpendicular energyin the shock frame, given by the distance in velocity space of the proton from theorigin of the ( v x , v y ) plane. This picture is exactly what we observe in phase spacein Figure 5.3, and it is no coincidence that the segment of the trajectory in redroughly corresponds to the location in phase space of the high energy tail which isgaining energy in our self-consistent perpendicular shock simulation.Finally, the particle will eventually cross back into the downstream region to x < v x , v y ) plane (green) about thedownstream E × B velocity (green star). Without any additional crossings of themagnetic discontinuity, the proton will simply E × B drift downstream, periodicallygaining and losing energy, in the shock frame, due to work on the proton by themotional electric field E y <
0, but the proton will experience no net energizationover a complete Larmor orbit. This energy exchange, without any overall gain inenergy, is present in Figure 5.4, wherein the field-particle correlation becomes morestructured and lower amplitude. In the transition to the downstream region, weonly observe the oscillatory exchange of energy between the electromagnetic fieldsand plasma because the protons are drifting past the magnetic gradient. Once theprotons have drifted past the magnetic gradient, they no longer have the means toreturn upstream and gain energy off the motional electric field.262hether a given proton will be “reflected” by the increased magnetic fieldmagnitude beyond the discontinuity and return to the upstream region ( x >
0) fromdownstream ( x <
0) depends on three conditions in this idealized shock model: (i)the jump in the magnetic field magnitude B d /B u ; (ii) the perpendicular velocity inthe frame of the upstream E × B velocity relative to that inflow velocity, v ⊥ ,u /U u ;and (iii) the gyrophase θ of the proton’s gyro-orbit when it first reaches the magneticdiscontinuity at x = 0. For given values of B d /B u and v ⊥ ,u reflection may occur overa range of values of gyrophase θ . For the self-consistent perpendicular shock studiedhere, a portion of the the distribution of protons have the required gyrophase toreflect off the magnetic gradient and gain energy in Figure 5.3.The energization mechanism we have identified in Figure 5.3 is called shock-drift acceleration and has been studied previously in the literature [Paschmann et al.,1982, Sckopke et al., 1983, Anagnostopoulos and Kaliabetsos, 1994, Anagnostopou-los et al., 2009, Ball and Melrose, 2001]. We have identified, for the first time, thephase space signature of this energization process using the field-particle correlationand a continuum method for the solution of the VM-FP system of equations. Phasespace energization signatures, such as those shown in Figure 5.3 for shock-drift ac-celeration, are useful not just for the study of direct numerical simulations, but alsoas a means of interpreting observational results from in situ spacecraft—see Chenet al. [2019] and the motivating theoretical studies by Howes et al. [2017] and Kleinet al. [2017]. Note that, unlike many early simple models of collisionless shocks [eg., Sckopke et al., 1983],this is not a specular reflection at the magnetic discontinuity at x = 0, but rather the result of theLorentz force leading to a return of the proton upstream to x > .1.2.2 Electron Energization in a Perpendicular Shock Having identified the proton energization mechanism, we turn now to theelectron dynamics in the shock. We again examine the field-particle correlation in v x and v y through the shock foot to the transition to the downstream in Figures 5.6and 5.7. At first glance, the phase space signature appears to roughly cancel on eachside of v x,y = 0 for all of the correlations, each correlation has a slight asymmetrywhich leads to either net energization or net de-energization. In the shock foot andramp, Figure 5.6, these slight asymmetries correspond to a gain of energy due to E x ,and a loss of energy due to E y , and we note by their magnitudes that more energyis gained due to E x than lost due to E y . Thus, the electrons overall gain energy. Wesee the opposite trend in the overshoot and transition to the downstream, Figure 5.7,wherein the electrons gain energy due to E y and lose energy due to E x . Again, thegain in energy due to E y is larger than the loss of energy due to E x , so the electronsoverall continue to gain energy.The energy gain and loss due to E x can be thought of simply as electronsresponding to an electrostatic potential, E x = − ∂φ/∂x , as E x is the electrostaticcomponent of the electromagnetic fields. We are especially interested, though, inthe energy gain (and loss) due to E y , the electromagnetic component of the electricfield, since this component of the field supports the compression of the magneticfield. To understand the energy exchange between the electrons and E y , we againturn to a single-particle picture for intuition.Because the electron gyro-orbit is much smaller than the length scale of the264igure 5.6: Electron distribution functions (top row), C v x field-particle correlations(middle row), and C v y field-particle correlations (bottom row) in the shock foot andramp region. The field-particle correlation has a slight asymmetry that correspondsto an energy gain to the x field-particle correlation and an energy loss due to the y field-particle correlation. The gain in energy due to E x exceeds the loss in energydue to E y , corresponding to a net energization of the electrons.265igure 5.7: Electron distribution functions (top row), C v x field-particle correlations(middle row), and C v y field-particle correlations (bottom row) in the overshoot andtransition regions of the shock. Here, we observe the opposite behavior to Figure 5.6,where now the asymmetry in the field particle correlation is such that the particlesgain energy due to E y and lose energy due to E x . The gain in energy due to E y stillexceeds the loss in energy due to E x , so the electrons continue to gain energy in thisregion of the shock. This particular energization signature in the y field particlecorrelation arises from alignment of the ∇ x B drift and the motional electric field, E y , and relies on conservation of the electron’s magnetic moment, the first adiabaticinvariant. Because of the relationship between this energization mechanism and theelectron’s first adiabatic invariant, we call this adiabatic heating.266ollsionless shock, ρ e (cid:28) L shock ∼ d p , we approximate the shock in an idealizedmodel as a linear ramp in the magnetic field. In Figure 5.8, we plot in the toppanel the profile of the perpendicular magnetic field B z ( x ) (blue) and the motionalelectric field E y ( x ) (red) along the shock normal direction, and in the middle panelthe trajectory of an electron in the ( x, y ) plane as it flows through the shock ramp,0 ≤ x/d p ≤
2. The trajectory plot shows clearly the ∇ x B drift in the + y direction.A salient difference between the idealized single particle motion for electrons andprotons is that the electron thermal velocity is larger than the inflow velocity, soelectrons can move in the + x direction, even upstream of the shock. This conditionis also satisfied for the shock parameters in our self-consistent perpendicular shocksimulation, U shock ∼ v A (cid:28) v th e .Although the electron constantly gains and loses energy as part of its E × B drift due to the motional electric field E y , the net effect on the particle energy overa Larmor orbit is zero, because the drift in the − x direction is perpendicular to theelectric field, U E × B · E y = 0. But, in the region where the perpendicular magneticfield changes magnitude, 0 ≤ x/d p ≤
2, a ∇ x B drift arises in the + y direction, whichleads to a net energization of the electrons by E y . This alignment of the motionalelectric field, E y , with a drift, in this case the ∇ x B drift, allows the electrons togain energy, as shown in the bottom panel of Figure 5.8.As an aside, the rate of energization of the electrons by the ∇ x B drift in themotional electric field is precisely the rate required to satisfy the conservation ofthe first adiabatic invariant of the electron, the electron magnetic moment, µ = m e v ⊥ / B z . This connection can be shown by calculating the net rate of work done267 Figure 5.8: (Top panel) Profiles along the shock normal direction of the perpendicu-lar magnetic field B z (blue) and the motional electric field E y (red), (Middle panel)trajectory of an electron in the ( x, y ) plane, and (Bottom panel) the rate of workdone by the electric field on the electron j y E y .268y E y due to the ∇ x B drift, which contributes to the perpendicular kinetic energyof the electrons, dm e v ⊥ / dt = q e u ∇ x B E y , (5.13)where the magnitude of the ∇ x B drift in the + y direction is given by u ∇ B = m e v ⊥ q e B z (cid:18) B z ∂B z ∂x (cid:19) . (5.14)For the static fields in this idealized model, the total time derivative is determinedby the E × B velocity, ddt = ∂∂t + u x ∂∂x = u E × B ∂∂x . (5.15)Substituting u E × B = E y /B z , we can manipulate (5.13) to obtain ∂∂x m e v ⊥ B z = ∂µ∂x = 0 , (5.16)proving that the electron’s first adiabatic invariant µ is conserved. Because thisenergization process relies on the electron’s first adiabatic invariant being conserved,we call this energization adiabatic heating .This simple model for the electron energization presumes that the only electricfield participating is E y , but we can see from Figure 5.1 that this is not the case.Even if the electrostatic field is roughly bi-modal across the shock so that muchof the energy exchange between the electrostatic field and the electrons is reversedwhen the electrons cross downstream, the presence of this electrostatic field stillcomplicates the picture. The electrostatic electric field leads to an E × B flow in the269 y direction which counters the ∇ x B drift in the + y direction. Still, for at least acomponent of the energization through the shock, especially in the overshoot andtransition region in Figure 5.7, we see a signature in the field-particle correlation ofenergy gain in y , which is characteristic of this alignment between the ∇ x B driftand the motional electic field, E y . Because of the finite ∇ x B drift, there are moreelectrons with velocities aligned with the motional electric field, E y , leading to theasymmetry in the field-particle correlation in Figure 5.7, and thus a net gain ofenergy for the electrons.We conclude this study of a self-consistent perpendicular shock with our DGVM-FP solver noting that, with the combination of diagnostics such as the field-particle correlation and our continuum representation of the particle distributionfunction, we can directly diagnose the energy exchange of kinetic plasma processesin phase space. We have shown, for the first time, the phase space signature ofshock-drift acceleration of the protons and adiabatic heating of the electrons in acollisionless shock. Although these energization mechanisms have been studied pre-viously, especially using the same single particle, and more generally Lagrangian,picture we used to model the particulars of the energization processes, the Eulerianphase space picture presented here is of considerable value. Especially when inter-preting spacecraft observations of particle distribution functions, which must usuallybe done in the Eulerian frame to obtain good enough sampling statistics, having ameans of interpreting the specific energization mechanisms opens new possibilitiesfor diagnosing the details of the phase space dynamics.There is more that can be learned from this perpendicular shock simulation.270or example, we have only noted and not examined the competition between theelectrostatic and electromagnetic electric fields in energizing electrons. Given therequirements for adiabatic heating, ρ e (cid:28) L shock , we might expect more realisticmass ratios to yield different results for this competition as well.Finally, the distribution function structure we resolve in the downstream re-gion, where the plasma and electromagnetic fields continually exchange energy, isa rich problem for understanding the ultimate “mixing” of the plasma. Collision-less shocks are often discussed interchangeably with irreversible heating and entropyincrease, though we note that the energy exchange happens on length scales muchsmaller than the collisional mean-free path. Thus, despite the total energy exchangebeing “done” once the shock has passed through the plasma, we expect additionalkinetic mechanisms are at play which transfer energy to smaller velocity space scales,where collisions ultimately dissipate this energy. Given the structure we can rep-resent in phase space with the continuum VM-FP solver presented in this thesis,we expect the ultimate diagnosis of this collisionless mixing is ideally studied bythe approach taken here, as the details of the collisionless mixing may be obscuredin particle-based method with the artificial collisionality introduced by finite sizedparticles [Birdsall and Langdon, 1990].The focus of this section has been on how we can use the high fidelity repre-sentation of the distribution function to more carefully analyze plasma processes inphase space. Because diagnostics such as the field-particle correlation, Eq. (5.4) andEq. (5.5), involve gradients of the velocity distribution function, traditional particle-based methods may have difficulty leveraging these diagnostic to examine the pre-271ise processes present. Counting noise can add sizable errors to the computation ofthese velocity space gradients, and significant spatial averaging to reduce the noisein post-processing may mix energization processes occurring in different regions ofconfiguration space, thus making it more challenging to determine the specifics ofthe energy exchange between the plasma and the electromagnetic fields. We nowturn to another application which reveals a different utility of the continuum kineticdiscretization: the phase space dynamics themselves being sensitive to phase spaceresolution. We consider here an extension of the benchmark studied in Section 4.2.6, thephase space dynamics of filamentation-type instabilities. Recall in Figure 4.18 forthe parameters chosen for the benchmark that the oblique, 45 ◦ , mode had a growthrate within 20-30 percent of the two-stream. This may not be similar enough toaffect the dynamics under more general perturbations of all modes in the system forthis parameter regime, v th e /u y = 1 / v th e = 0 . c . But the evolution the competitionof all the modes present, as would occur in the astrophysical systems where thesemodes are present, is likely to have an effect on the dynamics. For example, we canask whether the full spectrum of modes vying for dominance under more generalconditions affects the efficiency of magnetic field growth from the unstable beams ofplasma, a question of vital importance for the origins of the cosmological magneticfield [Schlickeiser and Shukla, 2003, Lazar et al., 2009].272f we survey the parameter space more extensively, we find that these obliquemodes can have comparable growth rates to the two-stream instability as the ratio ofthe thermal velocity to the drift speed is reduced and the beams are made colder—see Figure 5.9. Although some parameters, e.g., v th e /u d = 0 .
5, clearly show that thetwo-stream instability is the fastest growing mode and there is not much competitionfor the fastest growing mode in the system, we can expect that the competition couldbe quite significant as the beams become colder and multiple modes spanning a widerange of angles saturate at similar times.To study the competition between all of these modes, two-stream, oblique, andfilamentation, we set-up a similar phase space domain to Section 4.2.6, two configura-tion space and two velocity space dimensions (2X2V) with a drifting electron-protonplasma. The protons are taken to be a stationary, charge-neutralizing backgroundas before, and the electrons are initialized as two drifting Maxwellians, Eq. (4.40).We repeat this initial electron distribution here for clarity, f e ( x, y, v x , v y ) = m e n πT e exp (cid:18) − m e ( v x ) + ( v y − u d ) T e (cid:19) + m e n πT e exp (cid:18) − m e ( v x ) + ( v y + u d ) T e (cid:19) . The electromagnetic fields are initialized as a bath of fluctuations in the electric andmagnetic fields in the two configuration space dimensions, i.e., B z ( t = 0) = , (cid:88) n x ,n y =0 ˜ B n x ,n y sin (cid:18) πn x xL x + 2 πn y yL y + ˜ φ n x ,n y (cid:19) , (5.17)where ˜ B n x ,n y and ˜ φ n x ,n y are random amplitudes and phases respectively. The electricfields, E x ( t = 0) and E y ( t = 0), are initialized similarly to Eq. (5.17), and all three273igure 5.9: Contour plot of the angle of the fastest-growing mode in the parameterspace of v th e /u d and u d /c (top panel). θ = 90 ◦ corresponds to a pure two-streammode, and θ = 0 ◦ corresponds to a pure filamentation mode. Red crosses correspondto the four simulations presented. Growth rates versus wavenumber (bottom panels)of different modes for the hot (right panel) and cold (left panel) cases for u d = 0 . c .We can see in the hot case, v th e /u d = 0 .
5, that the two-stream instability is thefastest growing mode, while when we make the beams colder, v th e /u d = 0 .
1, theoblique modes for a variety of angles have comparable growth rates to the puretwo-stream instability. 274ynamically important electromagnetic fields in this two dimensional geometry aregiven equal average energy densities, (cid:104) (cid:15) E x / (cid:105) = (cid:104) (cid:15) E y / (cid:105) = (cid:104) B z / µ (cid:105) ≈ − E K ,where E K is the initial total electron energy.We focus on four particular simulations, whose parameters are indicated byred crosses in Figure 5.9. The drift velocity is fixed at u d = 0 . c , but we vary thetemperature of the beams by choosing v th e /u d ∈ { . , . , . , . } . The boxsizes, respectively, are L x /d e ∈ { . , . , . , . } and L y /d e ∈ { . , . , . , . } ,where d e is the electron inertial length, d e = c/ω pe . Box sizes L x = 2 π/k max ◦ and L y = 2 πm/k max ◦ are chosen to be roughly equal, L x ≈ L y , while fitting a singlefastest-growing wavelength of the filamentation instability and an integer number, m ≈ k max ◦ /k max ◦ , of two-stream modes. The configuration space boundary conditionsare periodic, and the velocity space boundary conditions are zero-flux. The velocityspace extents are varied for each simulation to contain the phase space evolutionof the instabilities in the nonlinear regime, [ − u d , u d ] to [ − u d , u d ] . Likewise,we vary the resolution in configuration and velocity space to obtain convergence,from 32 × to 64 × . All simulations use piecewise quadratic Serendipitypolynomials.We plot in Figure 5.10 the evolution of the magnetic field energy, (cid:15) B , andelectric field energy, (cid:15) E , normalized to the initial total energy of the electrons. Wecompare in Figure 5.10 the results of the four simulations in 2X2V (solid lines), wheretwo-stream, oblique, and filamentation modes are allowed to grow and competewith each other, with the results of similar 1X2V simulations (dashed lines) varying v th e /u d , but which only support the filamentation instability. We see that, while275igure 5.10: Growth and saturation of magnetic field (top panel) and electric field(bottom panel) energies normalized by the initial total electron energy for beamswith drift velocity u d = 0 . c at different temperatures. Solid lines correspond to2X2V simulations with initial random modes which drive two-stream, oblique andfilamentation modes, while dashed lines correspond to 1X2V simulations which onlysupport pure filamentation modes. We can see clearly the effect of the higher di-mensionality and competition between the different modes, since for all 1X2V sim-ulations, regardless of the ratio of v th e /u d , a magnetic field grows and saturates,whereas the growth of a magnetic field is sensitive to this ratio of v th e /u d whenthe two-stream, oblique, and filamentation modes are allowed to compete with eachother in two configuration space dimensions.276he 1X2V simulations robustly grow a magnetic field from the free energy of theunstable beams of plasma and the formation of current filaments from this freeenergy, irrespective of this ratio of v th e /u d and the temperature of the beams, thesituation is quite different in two configuration space dimensions, wherein the variousmodes are permitted to compete with each other.In 2X2V, the initial growth phase is quite different from the corresponding1X2V simulations. In 2X2V, we see the growth of both magnetic and electric fluc-tuations due to the combination of unstable oblique and two-stream modes. Theoblique modes in particular are what lead to the growth of both electric and mag-netic field fluctuations, as the two-stream instability would only grow an electricfield, and the filamentation instability is much more slowly growing than the otherinstabilities. Following saturation, potential wells formed by the saturation of two-stream and oblique modes, the tilted current filaments of oblique modes, and thevertical, i.e., uniform in y , current filaments associated with the potentially still-growing filamentation instability all nonlinearly interact and vie for dominance.To understand this interplay between the formation of current filaments andpotential wells by the various instabilities, we examine the electromagnetic fields andparticle distribution functions of the two limiting cases, v th e /u d = 0 .
5, the hot case,and v th e /u d = 0 .
1, the cold case. We plot in Figures 5.11 and 5.12 the evolution ofthe hot case in the early and late nonlinear stages of the plasma. Likewise, he coldcase is presented in Figures 5.13 and 5.14.In the hot case, in the early nonlinear stage, we see the formation of the two-stream modes with their quasi-one dimensional structure in E y , uniform in x and277igure 5.11: t = 60 ω − pe and t = 100 ω − pe snapshots of the evolution of the hotcase. We see the initial development of the two-stream instability and roll-up of thedistribution function, before the electron tubes formed by the two-stream instabilityare destroyed by the more slowly growing filamentation instability.278igure 5.12: t = 150 ω − pe and t = 300 ω − pe snapshots of the evolution of the hot case.In the deep nonlinear phase we observe the development of a temperature anisotropyin the distribution function, which provides a secondary free energy source for thesecular Weibel instability. The growth of the secular Weibel instability from thetemperature anisotropy ultimately supports a saturated magnetic field.279igure 5.13: t = 30 ω − pe and t = 50 ω − pe snapshots of the evolution of the cold case.We observe significantly more structure in the electromagnetic fields compared tothe hot case in Figure 5.11, as a variety of oblique modes all growth in tandem withthe two-stream instability. These additional modes also lead to additional phasespace structure, in contrast to the simple plateaus in v y which formed in the hotcase. 280igure 5.14: t = 100 ω − pe and t = 175 ω − pe snapshots of the evolution of the cold case.The saturated oblique modes have now given their energy back to the electrons ina much more isotropic fashion than a pure two-stream mode, leading to almost zerotemperature anisotropy. Without a temperature anisotropy to provide free energyto the Weibel instability, the magnetic field collapses, and we observe no saturatedmagnetic field structure. 281ultiple wavelengths of the fastest growing mode in y . While there is some initialmagnetic field present due to the growing oblique modes, the dynamics are domi-nated at this stage by the electrostatic two-stream instability. As the two-streammodes saturate, we see the roll-up in phase space in the y − v y reduced distributionfunctions shown. Importantly, in the early nonlinear stage, the more slowly grow-ing filamentation instability arises and fractures the saturated two-stream modes.We thus have the beginnings of magnetic field growth due to the presence of thefilamentation instability.However, the sustained growth of the magnetic field arises due to the presenceof a secondary instability in the hot case. The fast saturation of the two-streaminstability, along with the disruption and release of the stored electrostatic energyfrom the saturated two-stream modes by the filamentation instability, heats theelectrons primarily in one direction in velocity space, v y , because the electrostatictwo-stream instability is fundamentally one-dimensional. But this leads to a tem-perature anisotropy in the electron distribution, as can be seen forming in the latenonlinear evolution of the hot case in Figure 5.12. This temperature anisotropy pro-vides a source of free energy for the secular Weibel instability [Weibel, 1959], and asaturated magnetic field. We can clearly see this temperature anisotropy by inspec-tion of the electron distribution function in v x − v y in the late nonlinear time, as thedistribution function is visibly broadened in v y . Note that the magnetic energy sat-urates at (cid:15) B ∼ − , near the Alfv´en-limited regime, ρ e ∼ m e u d / ( eB z ) ∼ d e ∼ L x ,and enters a steady-state oscillation at the magnetic bounce frequency, agreeingclosely with previous particle-in-cell studies [Fonseca et al., 2003, Silva et al., 2003,282ishikawa et al., 2003, 2005, Kato and Takabe, 2008, Kumar et al., 2015, Takamotoet al., 2018] and 1X2V simulations [Califano et al., 1998, Cagas et al., 2017b].The cold case is strikingly different, as we see that the two-stream mode isnow competing with a spectrum of oblique modes in the early nonlinear stage inFigure 5.13. The electric and magnetic fields are much more structured, and whilea single oblique mode is relatively dominant, we see that the distribution functionstructure from the initial saturation of the instabilities is not as simple as the roll-upand formation of electron tubes observed in the hot case. Critically, the saturationof a spectrum of oblique modes at similar times leads to a heating of the electronsin a roughly isotropic fashion, as can be seen in the v x − v y cuts in Figure 5.14. Thisisotropic energization means that there is no temperature anisotropy to providefree energy for the Weibel instability to sustain the growing magnetic field. Themagnetic field that has grown ultimately collapses as the oblique modes damp onthe electrons, giving their energy back to the electrons.We can explicitly quantify this difference in the anisotropy after these insta-bilities have gone nonlinear. In Figure 5.15, we find that the spatially averagedtemperature anisotropy, ¯ A , drops from a large initial value in both the hot, ¯ A = 5,and cold, ¯ A = 101, cases to some residual value as the instabilities present growoff this effective temperature anisotropy, where the spatially averaged temperatureanisotropy is defined as¯ A = (cid:90) L y (cid:90) L x (cid:82) ( v y − u y ) f ( x, y, v ) d v (cid:82) ( v x − u x ) f ( x, y, v ) d v d x , (5.18)where u x,y are the flows in the x and y dimensions respectively. We note the evolution283igure 5.15: Effective temperature anisotropy of the hot case (red) and cold case(blue) over time. The effective temperature anisotropy starts at a finite value be-cause of the initial beams in v y and then decreases as the beam-driven instabilitiesare excited. For the hot case, the temperature anisotropy reduces to a finite value,off which the secular Weibel instability can ultimately feed. In the cold case, theeffective temperature anisotropy decreases to a value close to one, i.e., close toisotropy, and thus there is no free energy source for the secular Weibel instabilityto grow and support a saturated magnetic field.of the temperature anisotropy in the hot case, where we observe a decrease in theanisotropy from ¯ A = 5 to a finite value, ¯ A ≈
2, that explains the source of free energyfor the secular Weibel instability that ultimate supports the saturated magneticfield. The cold case on the other hand, has functionally no temperature anisotropyafter nonlinear saturation, having collapsed from the large effective temperatureanisotropy of two cold beams, ¯ A = 101, to ¯ A ≈ . p3d [Zeiler et al., 2002]. We initialize the simulations in exactly the same way as thecontinuum VM-FP simulations using Gkeyll , we specify two drifting Maxwelliansfor the electrons, and a bath of fluctuations in the electromagnetic fields givenby Eq. (5.17). The particle-in-cell simulations are performed using linear particleinterpolants (triangle shaped particles), and the number of particles per cell is variedto determine the effect that particle noise has on the solution.We can see that indeed, particle noise does appear to lead to a saturatedmagnetic field state. Further, the convergence to the continuum, grid-based methodis slow, as it requires a considerable number of particles to recreate the behavior285igure 5.16: Comparison of the integrated magnetic field energy between a numberof particle-in-cell simulations, varying the particles-per-cell, and the
Gkeyll
VM-FPsimulation of the cold case. In the limit of large particle-per-cell counts, the particle-in-cell simulations agree with the continuum kinetic result, but as the number ofparticles-per-cell is decreased, a saturated magnetic field appears.of the collapsing magnetic field. The saturated magnetic field in the low particlecount simulations is a result of “quasi-thermal” noise in the sampling of the currentto produce the magnetic field. Essentially, in the same way that particle noise canmanifest as fluctuations in the electric field due to errors in the sampling of thedensity of the particle distribution function [Langdon, 1979], so too can these errorsmanifest in the current, giving rise to and supporting a magnetic field.Given the fact that the low particle count simulations saturate at what appearsto be the noise floor of the simulations, we can potentially improve the comparisonby filtering the particle-in-cell data using a simple low pass filter at the largestwavenumber fluctuations in the box. We plot the same comparison between our286ontinuum VM-FP simulation of the cold case, and the two extreme particle counts,with and without filtering, in Figure 5.17. The improvement from a low-pass filterFigure 5.17: Comparison of the integrated magnetic field energy between the largestand smallest particle-per-cell counts, with and without a low pass filter, and the
Gkeyll
VM-FP simulation of the cold case. We can see that the filter does allowfor the recovery of the collapsing magnetic field in the low particle-per-cell count,adding credibility to the interpretation that the saturated magnetic field is due tonoise.adds further credibility to the interpretation that the saturated magnetic field inthe low particle count particle-in-cell calculations arises due to counting noise.Importantly, while these isolated simulations can be improved with filtering,it does not eliminate the possibility that particle noise is at least partially respon-sible for the lack of agreement between the
Gkeyll results presented here and otherparticle-in-cell studies [Kato and Takabe, 2008]. While filtering as a post-processingstep works robustly for this problem set-up, where the plasma is perturbed and al-287owed to evolve, a driven simulation in which the plasma instabilities are constantlybeing excited may be polluted by this same noise that we can see in the non-filteredcase. It is much more difficult to filter the noise in-situ, and thus the dynamics maybe affected by the magnetic field attempting to collapse due to the electron instabil-ities, but being unable to, under the stress of a constant injection of noise-polluted,unstable fluctuations. Given the sensitivity of the overall dynamics and magneticfield growth to parameter regimes of relevance in astrophysical plasmas, it is vitalthat care be taken when resolving the phase space evolution of these instabilities.Further details of this comparison can be found in Juno et al. [2020].We conclude this section having presented a series of simulations of unstableplasmas, in which novel behavior in the competition between beam-driven instabili-ties was found in the limit of the beam temperature and the ratio v th e /u d decreasing.While the continuum DG VM-FP solver described in this thesis recovers the resultsof previous kinetic studies when the electron beams are hot, we find that the secularWeibel instability can feed off the residual temperature anisotropy remaining fromsaturated two-stream modes, the picture changes dramatically as the beams growcolder. The oblique modes that exist between the filamentation instability and two-stream instability become as fast growing as, or faster than, the two-stream insta-bility, significantly complicating the initial nonlinear phase. Without the dominanttwo-stream mode in the early nonlinear saturation, the electrons are ultimately en-ergized quasi-isotropically, leading to a collapse of the temperature anisotropy andlack of a free energy source to support a saturated magnetic field.We attempted to replicate this result in analogous particle-in-cell simulations288nd found that the result is sensitive to the particle noise arising from the numberof particles per cell employed in the simulation. Simulations with very few particlesper cell attain saturated magnetic field states arising from the presence of quasi-thermal noise in the magnetic field, i.e., sampling error in the computation of thecurrent from the particles discretizing the distribution function. While these errorscan be mitigated with filtering in this isolated system, we emphasize that recoveringthe behavior of these instabilities in a driven context, such as a collisionless shock,may be more challenging. We thus argue for the utility of the continuum approachpresented in this thesis as a means of obtaining an accurate solution for plasmadynamics that are sensitive to phase space resolution.289 hapter 6: Summary and Future Work We have presented in this thesis the derivation, implementation, and appli-cation of a discretization of the Vlasov–Maxwell–Fokker–Planck (VM-FP) systemof equations which uses the discontinuous Galerkin (DG) finite element method tonumerically integrate the VM-FP equation system on a phase space grid. In con-trast to traditional particle-based approaches to the numerical integration of thekinetic equation, this approach provides a high fidelity representation of the particledistribution function, free of the counting noise inherent to Monte Carlo methods.This unpolluted discrete representation of the particle dynamics in the full phasespace affords new opportunities for analysis of the plasma processes present directlyin phase space, and makes new problems accessible by increasing the signal-to-noiseratio. We identified and solved a number of analytic and numerical challenges through-out this thesis. We showed what is required in the mathematical formulation of theDG algorithm for the discrete VM-FP system of equations to retain important prop-erties of the continuous system, such as conservation of mass and energy. In theimplementation stage, we noted that the direct discretization of the VM-FP systemof equations was rife with difficulties owing to the high dimensional nature of the290quation system. Importantly, we noted that standard means of lowering the costof DG schemes would be catastrophic for the discretization of the VM-FP systemof equations, as numerical integration errors that could reduce the computationalcomplexity of the algorithm would inevitably destroy the implicit properties of theVM-FP system of equations, such as conservation of energy. We designed a schemefree of aliasing errors in the integration, and further formulated a basis set forour DG scheme using orthonormal, modal polynomials that sparsified the resultingtensor-tensor convolutions.We benchmarked the implementation of the DG VM-FP solver against a largesuite of tests, and numerically demonstrated the analytically proved properties ofthe scheme. The DG VM-FP solver was then deployed to study the energization ofplasmas in fundamental plasma processes such as collisionless shocks as well as thedetails of the nonlinear saturation of beam-driven instabilities. Using the increasedphase space resolution afforded to us by a continuum discretization of the VM-FPsystem of equations, we were able to directly diagnose the energization processessuch as shock-drift acceleration in phase space. Likewise, we were able to identifya new parameter regime as the unstable beams became colder for the saturationof filamentation-type instabilities. In this cold parameter regime, we observed nosaturated magnetic field as a result of the competition between additional unstablemodes that could grow more quickly in the cold beam parameter regime. We drewspecial attention to this result, as analogous particle-in-cell simulations of this sys-tem found saturated magnetic fields when using low numbers of particles per celldue to particle noise. 291here are a number of avenues of future research to build off the algorithmicand physics work presented in this thesis. The methods in this thesis can be ex-tended to other kinetic systems, for example the relativistic Vlasov-Maxwell systemof equations. In addition, it is worth exploring whether the recovery procedure de-scribed in Chapters 2 and 3 for the diffusion operator in the Fokker–Planck equationcan also be applied to other components of the update, e.g., the discretization ofMaxwell’s equations. Given some of the challenges in discretizing Maxwell’s equa-tions, especially in the choice of numerical flux function, an alternative approachthat reconstructs continuous functions at the interface could be particularly power-ful. On the physics side, we have demonstrated that the field-particle correlation,combined with our continuum discretization of the VM-FP system of equations,provides a particularly useful way to characterize the energization processes presentin phase space, but we have only scratched the surface of what can be done. Evenamongst the benchmarks presented, for example the lower hybrid drift instabilityand magnetic pumping, identifying the phase space energization signatures of theseprocesses would further build a Rosetta stone for assistance in interpreting future nu-merical and observational solutions. We can also extend the study of filamentation-type instabilities to include the proton dynamics as well as inhomogeneities in thebeams, e.g., if the two beams have different densities.But we conclude noting the power and utility of our continuum VM-FP solverin the
Gkeyll simulation framework, and emphasize that there is an enormousarray of problems that can be tackled with this solver, especially if one requires292igh phase space resolution. 293 ppendix A: Proofs of the Properties of the ContinuousVlasov–Maxwell–Fokker–Planck System of Equations
Proof of Proposition 1 (The collisionless Vlasov–Maxwell system ofequations conserves mass.)
Proof.
If we multiply the conservation equation form of the collisionless Vlasovequation, Eq. (1.68), by the mass of the particle, integrate over the phase spacedomain K , and apply the divergence theorem, we obtain, ddt (cid:18) m s (cid:90) K f s d z (cid:19) = − m s (cid:73) ∂K α s f s dS = 0 , (A.1)by our assumed boundary conditions. Note that this proposition holds individuallyfor each species s in the plasma as we are not including the effects of source termssuch as ionization or recombination in our system. Proof of Proposition 2 (The collisionless Vlasov–Maxwell system ofequations conserves the L norm of the particle distribution function.) Proof.
We first multiply the conservation equation form of the collisionless Vlasovequation, Eq. (1.68), by the distribution function f s and integrate over the full phase294pace to obtain ddt (cid:18) (cid:90) K f s d z (cid:19) = − (cid:73) ∂K α s f s dS + (cid:90) K ∇ z f s · α s f s d z , (A.2)where we have used the chain rule, f s ddt f s = 12 ddt (cid:0) f s (cid:1) , (A.3)to simplify the left hand side and integration by parts to rewrite the right handside. We can again use our assumed boundary conditions to eliminate the surfaceintegral, and the product rule to rewrite the volume integral, ∇ z f s · α s f s = ∇ z · (cid:18) α s f s (cid:19) −
12 ( ∇ z · α s ) f s = ∇ z · (cid:18) α s f s (cid:19) , (A.4)since phase space is incompressible, ∇ z · α s = (cid:18) ∇ x · v , q s m s ∇ v · [ E + v × B ] (cid:19) = 0 . (A.5)But, since we can rewrite the volume term as a total derivative, we can again applythe divergence theorem and use boundary conditions to eliminate the remainder ofthe right hand side, ddt (cid:18) (cid:90) K f s d z (cid:19) = 0 . This completes the proof. As with the conservation of particles, the conservation ofthe L norm by the collisionless Vlasov–Maxwell system holds individually for eachspecies s in the system. Proof of Proposition 3 (The collisionless Vlasov–Maxwell system of quations conserves the entropy density S = − f ln( f ) of the system.) Proof.
Again using the conservation equation form of the collisionless Vlasov equa-tion, Eq. (1.68), multiplying by − ln f s , and integrating over phase space we obtain ddt (cid:20)(cid:90) K − f s ln( f s ) d z (cid:21) = (cid:73) ∂K ln( f s ) ( α s f s ) dS − (cid:90) K ∂f s ∂t + ∇ z ln( f s ) · α s f s , (A.6)where we have again used the chain rule to rewrite the time derivative, − ln( f s ) ∂∂t f s = ∂f s ∂t − ∂ ln( f s ) f s ∂t , (A.7)since ∂ ln( f s ) ∂t = ∂f s ∂t f s . (A.8)We have also again used integration by parts on the right hand side of Eq. (1.68)and can eliminate the surface integral with our boundary conditions in phase space.Using the chain rule and the incompressibility of phase space, Eq. (A.5), we find ∇ z ln( f s ) · α s f s = α s · ∇ z f s = ∇ z · ( α s f s ) , (A.9)but this expression means the the right hand side is simply the collisionless Vlasovequation, Eq. (1.68), which is equal to zero, completing the proof, ddt (cid:18)(cid:90) K − f s ln( f s ) d z (cid:19) = 0 . As before with mass conservation and conservation of the L norm, conservationof entropy holds independently for each species in the collisionless Vlasov–Maxwellsystem. 296 roof of Proposition 4 (The collisionless Vlasov–Maxwell system ofequations conserves the total, particles plus fields, momentum.) Proof.
We begin by multiplying Eq. (1.65) by m s v , summing over species, and in-tegrating over phase space to obtain (cid:90) K (cid:88) s m s v ∂f s ∂t d z (cid:124) (cid:123)(cid:122) (cid:125) (cid:82) Ω (cid:80) s ∂ M s∂t d x = − (cid:90) K (cid:88) s m s v ∇ x · ( v f s ) d z − (cid:90) K (cid:88) s m s v ∇ v · (cid:20) q s m s ( E + v × B ) f s (cid:21) d z . (A.10)Since the velocity coordinate does not depend on configuration space, we can bring m s v inside the divergence in the first term on the right side, apply the divergencetheorem, and eliminate this term by our configuration space boundary conditions.For the second term on the right hand side, we can use integration by parts to movethe velocity divergence onto m s v , eliminating the surface term using our boundarycondition in velocity space, − (cid:90) K (cid:88) s m s v ∇ v · (cid:20) q s m s ( E + v × B ) f s (cid:21) d z = (cid:90) K (cid:88) s q s ∇ v v · ( E + v × B ) f s d z , = (cid:90) Ω ρ c E + J × B d x , (A.11)where we have used the fact that ∇ v v = ←→ I and the definitions of the charge densityand current density, Eqns. 1.63–1.64, to perform the integral over velocity space. Tomake further progress, we consider Maxwell’s equations. Taking the cross-productof Eq. (1.59) with (cid:15) E , the cross-product of Eq. (1.60) with B /µ , and subtracting297he resulting equations we obtain (cid:15) ∂∂t ( E × B ) + (cid:15) E × ( ∇ x × E ) (cid:124) (cid:123)(cid:122) (cid:125) ( ∇ x E ) · E − ( E ·∇ x ) E + 1 µ B × ( ∇ x × B ) (cid:124) (cid:123)(cid:122) (cid:125) ( ∇ x B ) · B − ( B ·∇ x ) B = − J × B , (A.12)for the evolution of the electromagnetic momentum density, (cid:15) E × B . Now, for anyvector field A we have ( ∇ A ) · A = ∇| A | / , (A.15)( A · ∇ ) A = ∇ · ( AA ) − A ∇ · A . (A.16)Using these vector identities and the divergence Eqns. (1.61) and (1.62) to replace ∇ x · E = ρ c /(cid:15) and ∇ x · B = 0 gives (cid:15) ∂∂t ( E × B ) + ∇ x (cid:18) (cid:15) | E | + 12 µ | B | (cid:19) − ∇ x · (cid:18) (cid:15) EE + 1 µ BB (cid:19) + (cid:37) c E = − J × B . (A.17)We recognize the spatial gradients and divergences in Eq. (A.17) to be acting on theMaxwell stress tensor, ←→ σ = (cid:15) (cid:18) EE − | E | ←→ I (cid:19) + 1 µ (cid:18) BB − | B | ←→ I (cid:19) . (A.18)So, inserting Eq. (A.17) into Eq. (A.11) and using configuration space boundaryconditions to eliminate the total derivatives of the Maxwell stress tensor gives our The electromagnetic momentum density is also commonly written as p EM = S c , (A.13)where S is the Poynting flux, S = 1 µ E × B , (A.14)and c is the speed of light, c = 1 / √ (cid:15) µ . ddt (cid:32)(cid:90) Ω (cid:88) s M s + (cid:15) E × B d x (cid:33) = 0 . We emphasize that the linear momentum is a conserved vector quantity. In otherwords, only the corresponding components of the particle and electromagnetic mo-mentum can be exchanged, e.g., the x particle momentum can be exchanged withthe x component of the electromagnetic momentum. Of course the stress tensor forthe particles, ←→ S s = (cid:90) ∇ x · ( vv f s ) d v , (A.19)can move momentum between the various components of the particle momentumdensity, and likewise the Maxwell stress tensor can move momentum between thevarious components of the electromagnetic momentum density. But when the par-ticles and electromagnetic fields exchange momentum, they do so component bycomponent. Proof of Proposition 5 (The collisionless Vlasov–Maxwell system ofequations conserves the total, particles plus fields, energy.)
Proof.
We proceed in a similar fashion to our proof of momentum conservation, butwe now multiply Eq. (1.65) by 1 / m s | v | , sum over species, and integrate over phase299pace to obtain (cid:90) K (cid:88) s m s | v | ∂f s ∂t d z (cid:124) (cid:123)(cid:122) (cid:125) (cid:82) Ω (cid:80) s ∂ E s∂t d x = − (cid:90) K (cid:88) s m s | v | ∇ x · ( v f s ) d z − (cid:90) K (cid:88) s m s | v | ∇ v · (cid:20) q s m s ( E + v × B ) f s (cid:21) d z . (A.20)Since the velocity coordinate does not depend on configuration space, we can move1 / m s | v | inside the configuration space divergence, forming a total derivative andallowing us to use the divergence theorem and boundary conditions to eliminatethis term. As before with momentum conservation, we use integration by parts andvelocity space boundary conditions on the second term on the right hand side, − (cid:90) K (cid:88) s m s | v | ∇ v · (cid:20) q s m s ( E + v × B ) f s (cid:21) d z = (cid:90) K q s ∇ v | v | · ( E + v × B ) f s d z = (cid:90) Ω J · E d z , (A.21)where we have used the fact that v · ( v × B ) = 0 by properties of the cross productto eliminate the magnetic field term. To make further progress, we again examineMaxwell’s equations. Taking the dot product of Eq. (1.60) with E /µ , the dotproduct of Eq. (1.59) with B /µ , and adding the resulting equations gives us ∂∂t (cid:18) (cid:15) | E | + 12 µ | B | (cid:19) + 1 µ [ B · ( ∇ x × E ) − E · ( ∇ x × B )] (cid:124) (cid:123)(cid:122) (cid:125) = ∇ x · ( E × B ) = − J · E . (A.22)Using this result in Eq. (A.21), along with configuration space boundary conditionsto eliminate the divergence of the Poynting flux, gives the total energy conservation300aw, ddt (cid:32)(cid:90) Ω (cid:88) s E s + (cid:15) | E | + 12 µ | B | d x (cid:33) = 0 . Proof of Proposition 6 (The Fokker–Planck equation conserves mass.)
Proof.
If we multiply Eq. (1.66) by the mass of the particle and integrate over phasespace, just as with Proposition 1, we can use the boundary conditions in velocityspace to obtain ddt (cid:18) m s (cid:90) K f cs d z (cid:19) = (cid:73) ∂K ν s (cid:20) ( v − u s ) f s + T s m s ∇ v f s (cid:21) dS = 0 . (A.23)Because we are not including particle sources such as ionization and recombination,this conservation relation holds for each plasma species. Importantly, because theFokker-Planck operator only involves derivatives in velocity space, this conservationis local , (cid:90) K \ Ω m s ∂f cs ∂t d v = ∂ρ s ∂t = 0 , (A.24)i.e., the Fokker-Planck collision operator does not change the local mass (or number)density in configuration space. Proof of Proposition 7 (The Fokker–Planck equation conserves theparticle momentum.)
Proof.
If we first multiply Eq. (1.66) by m s v and integrate over phase space, we can301ntegrate the collision operator by parts once to obtain ddt (cid:18)(cid:90) K m s v f cs d z (cid:19) = (cid:73) ∂K m s v ν s (cid:20) ( v − u s ) f + T s m s ∇ v f s (cid:21) dS − (cid:90) K m s ν s ∇ v v · (cid:20) ( v − u s ) f s + T s m s ∇ v f s (cid:21) d z . (A.25)We can eliminate the surface integral with our boundary conditions in velocity space.Recall that ∇ v v = ←→ I , so the volume integral simplifies to (cid:90) K m s ∇ v v · (cid:20) ( v − u s ) f s + T s m s ∇ v f s (cid:21) d z = (cid:90) K m s ( v − u ) f s d z , = (cid:90) Ω ( M s − m s n s u s ) d x = 0 , (A.26)where we have dropped the velocity independent collision frequency for notationalconvenience. In our simplification to Eq. (A.26) we have used the fact that thediffusion coefficient, T s /m s , does not depend on velocity space to write what remainsof the diffusion term as a total derivative, which upon integrating the total derivativeand using the boundary conditions in velocity space, eliminates the diffusion term.Eq. (A.26) completes the proof. As with conservation of mass in Proposition 6, sincethe Fokker–Planck collision operator only includes derivatives in velocity space, wecan construct a local conservation law, (cid:90) K \ Ω m s v ∂f cs ∂t d v = ∂ M s ∂t = 0 , (A.27)i.e., the Fokker–Planck collision operator does not change the local momentum den-sity in configuration space. Proof of Proposition 8 (The Fokker–Planck equation conserves the article energy.)
Proof.
In analogy with Proposition 7, we multiply Eq. (1.66) by 1 / m s | v | , integrateover phase space, and use integration by parts to obtain ddt (cid:18)(cid:90) K m s | v | f cs d z (cid:19) = (cid:73) ∂K m s | v | ν s (cid:20) ( v − u s ) f + T s m s ∇ v f s (cid:21) dS − (cid:90) K m s ν s ∇ v | v | · (cid:20) ( v − u s ) f s + T s m s ∇ v f s (cid:21) d z . (A.28)As before, we can eliminate the surface integral with our boundary conditions invelocity space. Using the fact that ∇ v | v | = 2 v , the volume integral can be rewrittenas (cid:90) K m s v · (cid:20) ( v − u s ) f s + T s m s ∇ v f s (cid:21) d z = (cid:90) K (cid:2) m s (cid:0) | v | − v · u s (cid:1) f s + T s v · ∇ v f s (cid:3) d z , = (cid:90) Ω E s − m s n s | u s | − n s T s d x = 0 , (A.29)where we have dropped the velocity independent collision frequency for notationalconvenience and used integration by parts and the velocity space boundary condi-tions to simplify (cid:90) K T s v · ∇ v f s d z = (cid:90) K T s f s ( ∇ v · v ) d z = (cid:90) Ω n s T s d x . (A.30)Eq. (A.29) completes the proof. We note that as with conservation of mass in Propo-sition 6 and conservation of momentum in Proposition 7, since the Fokker–Planckcollision operator only includes derivatives in velocity space, we can construct a local (cid:90) K \ Ω m s | v | ∂f cs ∂t d v = ∂ E s ∂t = 0 , (A.31)i.e., the Fokker–Planck collision operator does not change the local energy densityin configuration space. Proof of Proposition 9 (The Fokker–Planck equation satisfies theSecond Law of Thermodynamics and leads to a non-decreasing entropydensity S = − f ln( f ) .) Proof.
Defining the total entropy as S s = − (cid:90) K f s ln f s d z , (A.32)and taking the time derivative of the total entropy, we have ∂ S s ∂t = − (cid:90) K ∂f s ∂t [ln( f s ) + 1] d z . (A.33)We can rewrite the Fokker–Planck operator as a flux in velocity space, ∂f cs ∂t = ∇ v · F , (A.34)where F = ( v − u s ) f s + T s m s ∇ v f s , (A.35)and we have dropped the velocity independent collision frequency ν s for notationalconvenience without loss of generality. Because we have already proved the collision-less component of the VM-FP system of equations does not change the entropy of the304ystem, Proposition 3, we need only consider the contribution of the Fokker–Planckequation to the evolution of the total entropy, ∂ S s ∂t = − (cid:90) K ∇ v · F [ln( f s ) + 1] d z . (A.36)We can integrate the flux by parts and use our boundary conditions in velocity spaceto obtain ∂ S s ∂t = (cid:90) K f s ∇ v f s · F d z . (A.37)We now substitute ∇ v f s = m s T s [ F − ( v − u s ) f s ] , (A.38)into Eq. (A.37) to obtain, ∂ S s ∂t = (cid:90) K m s T s (cid:20) | F | f s − ( v − u s ) · F (cid:21) d z . (A.39)Using the definition of F , the second term in this equation becomes (cid:90) K ( v − u s ) · F d z = (cid:90) K ( | v | − u s · v + | u s | ) f s + T s m s ( v − u s ) · ∇ v f s d z = (cid:90) Ω m s E s − n s | u s | + n s | u s | − n s T s m s d x = 0 , (A.40)where we have used integration by parts on the ∇ v f s term. Hence, ∂ S s ∂t = (cid:90) K m s T s f s | F | d z ≥ , (A.41)as long as f s ≥ . Given the preceding discussion, we can also define a velocity And ν s > s s ( x , t ) = − (cid:90) K \ Ω f s ( x , v , t ) ln( f s ( x , v , t )) d v , (A.42)which is a monotonically increasing function, ∂s s ( x , t ) ∂t = (cid:90) K \ Ω m s T s ( x , t ) 1 f s ( x , v , t ) | F ( x , v , t ) | d v ≥ , (A.43)since the Fokker–Planck operator only involves derivatives in velocity space. Inother words, the collision operator leads to non-decreasing entropy at each point inconfiguration space, and further mixing in configuration space is required to attaina global maximum entropy state. We might be unsurprised by this statement, asthe entropy increase in velocity space corresponds to the second of Bogoliubov’stimescales, while the entropy increase in all of phase space corresponds to the thirdof Bogoliubov’s timescales. Proof of Corollary 1 (The maximum entropy solution to the Fokker–Planck collision operator is the Maxwellian velocity distribution.)
Proof.
By Proposition 9, we know that the entropy is a monotonically increasingfunction. But, if the entropy is a monotonically increasing function in time, and theentropy is a well-defined quantity, i.e., Eq. (A.32) is not a divergent integral, thenthe extremum of the entropy must necessarily maximize the entropy. Thus, we needonly find when ∂ S s ∂t = 0 . (A.44)306he time evolution of the entropy vanishes when F = 0 , (A.45)i.e., ∇ v f s = − m s T s ( v − u s ) f s . (A.46)Solving for the distribution function f s , we find f s = A exp (cid:18) − m s | v − u s | T s (cid:19) , (A.47)where A is some constant of integration. To find the constant of integration, weexploit the requirement that the integral over velocity space of the distributionfunction must by definition give the density, n s = (cid:90) K \ Ω A exp (cid:18) − m s | v − u s | T s (cid:19) d v , (A.48)which means A = n s (cid:18) m s πT s (cid:19) , (A.49)where the integral over each velocity direction naturally gives a factor of (cid:112) πT s /m s . Further discussions of the Maxwellian velocity distribution and itsconnection to thermodynamic equilibrium.
Eq. (A.44) is often referred to as the principle of detailed balance. To giveourselves physical intuition for what it means for the time evolution of the entropy307o vanish, we must consider what we mean by the plasma being in thermodynamicequilibrium. A useful way to define equilibrium is that every process ongoing in theplasma is exactly compensated by its reverse, e.g., every Coulomb collision a particlein the plasma experiences is exactly balanced by an equal and opposite Coulombcollision. The contribution of Coulomb collisions to the plasma’s dynamics wouldthen vanish. But the contribution of Coulomb collisions vanishing was exactly therequirement for entropy production to disappear. Inevitably, the velocity distribu-tion function for which Coulomb collisions are “in balance” defines our equilibriumstate and the state of maximum entropy.There are additional subtleties worth mentioning; for example, we have usedthe total entropy vanishing to derive the Maxwellian as the maximum entropy dis-tribution, but the plasma is free to be a different Maxwellian at each point in con-figuration space since the density, flow, and temperature may vary in space. In thiscase, the entropy density can be maximized at a given configuration space location,but the total entropy may not yet be maximized. For example, a spatially varyingMaxwellian may itself be unstable and drive the system to a still higher entropystate.We wish to make one additional note about the interconnection between theMaxwellian velocity distribution, the Fokker–Planck equation, and the entropy. TheMaxwellian velocity distribution is actually the naturally arising weight functionwhen considering additional properties of the Fokker–Planck operator in Eq. (1.66).For example, we can show that the Fokker–Planck operator is self-adjoint, i.e., for308rbitrary functions g ( x , v , t ), f ( x , v , t ), (cid:18) g, ∂f c ∂t (cid:19) f M = (cid:18) f, ∂g c ∂t (cid:19) f M , (A.50)with the inner product defined as( f, g ) f M = (cid:90) K \ Ω f M f g d v . (A.51)Note that ( · , · ) f M is a bilinear operator taking two arguments, defined by the integralequation in Eq. (A.51). Here, we consider only the integrals over velocity space forsimplicity and f M is the Maxwellian for which the collision operator vanishes. Notethat we have dropped the species subscript. Integrating Eq. (A.50) by parts we get (cid:18) g, ∂f c ∂t (cid:19) f M = − (cid:90) K \ Ω ∇ v (cid:18) gf M (cid:19) · (cid:20) ( v − u ) f + Tm ∇ v f (cid:21) d v . (A.52)We have the identity Tm f M ∇ v (cid:18) ff M (cid:19) = ( v − u ) f + Tm ∇ v f. (A.53)Using this identity leads to (cid:18) g, ∂f c ∂t (cid:19) f M = − Tm (cid:90) K \ Ω f M ∇ v (cid:18) gf M (cid:19) · ∇ v (cid:18) ff M (cid:19) d v . (A.54)This equation is symmetric in f and g from which the self-adjoint property follows.As an aside, the self-adjoint property indicates that the eigenvalues of theoperator are all real and hence all solutions are damped. In other words, the Fokker–Planck operator in the VM-FP system of equations does not support any oscillatorymodes. One can show that the eigenfunctions of the operator Eq. (1.66) are simply309he multi-dimensional tensor Hermite functions [Grant and Feix, 1967, Hammettet al., 1993, Harris, 2004, Anderson and O’Neil, 2007a, Patarroyo, 2019] and eachmode is damped proportional to the mode number.We can use the self-adjoint property to discuss the behavior of the distributionfunction squared, f , at least in this norm with the Maxwellian weight. If we set g = f in Eq. (A.54) we get (cid:90) K \ Ω ff M ∂f c ∂t d v = ddt (cid:90) K \ Ω
12 ( f c ) f M d v = − Tm (cid:90) K \ Ω f M ∇ v (cid:18) ff M (cid:19) · ∇ v (cid:18) ff M (cid:19) d v ≤ , (A.55)which shows that the Fokker–Planck operator will decay f /f M integrated overvelocity space. But what about f , the L norm, without the Maxwellian weight?We previously discussed the L norm of the collisionless component of theVM-FP system of equations in Proposition 2, showing it is a conserved quantity inthe evolution of the distribution function from the collisionless part of the VM-FPsystem of equations. We proceed in a similar fashion to Proposition 2, but now withthe Fokker–Planck equation, ddt (cid:90) K \ Ω f d v = − (cid:90) K \ Ω ∇ v f · (cid:20) ( v − u ) f + Tm ∇ v f (cid:21) d v , (A.56)where we have already integrated by parts once and used our velocity space boundaryconditions to eliminate the surface term. We now write the first term as ∇ v f · ( v − u ) f = ∇ v (cid:18) f (cid:19) · ( v − u ) = v · ∇ v (cid:18) f (cid:19) − ∇ v · (cid:18) u f (cid:19) . (A.57)The second term is a total derivative and will vanish on upon the use of the diver-310ence theorem and our velocity space boundary conditions. This procedure leaves ddt (cid:90) K \ Ω f d v = − (cid:90) K \ Ω v · ∇ v (cid:18) f (cid:19) + Tm |∇ v f | d v . (A.58)Performing integration by parts on the first term we obtain ddt (cid:90) K \ Ω f d v = (cid:90) K \ Ω f − Tm |∇ v f | d v . (A.59)For a Maxwellian, the right-hand side vanishes, ddt (cid:90) K \ Ω f M d v = (cid:90) K \ Ω f M − Tm (cid:18) − m ( v − u ) T f M (cid:19) d v , = (cid:90) K \ Ω f M − mT | v − u | f M d v = 0 , (A.60)but one can construct perturbations on the Maxwellian that may change the sign.To see this, perform a perturbation around a Maxwellian f = f M + δf to get thevariation, δ ddt (cid:90) K \ Ω f d v = (cid:90) K \ Ω (cid:18) f M − Tm ∇ v f M (cid:19) δf d v , = (cid:90) K \ Ω (cid:18) − m | v − u | T (cid:19) f M δf d v . (A.61)Clearly, δf can be of any sign. This result shows that the L norm is not monotonicand the Maxwellian is not the extremum of the L norm. Physically, as the dragvelocity v − u is compressible, the contribution from the drag term cannot be turnedinto a total derivative. The compressibility of the drag term is in contrast to thecollisionless case, in which the phase-space velocity is incompressible and hence thephase-space integrated f is constant.We have focused on these additional properties of the Fokker–Planck collision311perator—the operator is self-adjoint and decays f /f M , but not f —to make theconnection between the Maxwellian velocity distribution and the entropy productionof the operator even more explicit. Proposition 9, that the VM-FP system of equa-tions obeys the Second Law of Thermodynamics, and Corollary 1, that the VM-FPsystem of equations obeys Boltzmann’s H-theorem, are inseparable, and Corollary 1naturally follows from Proposition 9. The fact that the Maxwellian velocity distri-bution is then a natural weight function for discussing additional properties of thecollision operator in the VM-FP system of equations should thus be unsurprising,and we cannot avoid including this weight function when discussing the behavior ofquantities such as the distribution function squared, f .We will conclude this discussion with one final way to think about the con-nection between the Maxwellian velocity distribution, entropy production, and the1 /f M weighting of the inner product. 1 /f M naturally arises when measuring howmuch a distribution function deviates away from a Maxwellian in terms of entropy.In other words, writing f = f M + δf , then the entropy S [ f ] = − (cid:82) K f ln( f ) d v as afunctional of f can be written as, S [ f M + δf ] = S [ f M ] − (1 / (cid:90) ( δf ) /f M d v + . . . , (A.62)through second order. This expansion is consistent with the result that any smalldeviation, δf (cid:28) f M , away from a Maxwellian is a state of lower entropy. Note that,to derive this, we have made use of (cid:90) v p δf d v = 0 for p = 0 , , , (A.63)312ecause the Maxwellian f M has the same zeroth through second moments as f byconstruction. In other words, any finite zeroth through second moments in δf couldjust be absorbed into the Maxwellian f M , and f M redefined. This norm for δf isequivalent to a norm on the total f , plus a constant, since (cid:90) f /f M d v = (cid:90) ( f M + δf ) /f M d v = n + (cid:90) ( δf ) /f M d v , (A.64)where the density n = (cid:82) f d v is conserved by the collision operator. This result that S [ f M + δf ] = constant − (1 / (cid:90) f /f M d v + . . . , (A.65)shows a relationship between the collision operator causing the entropy to be neverdecreasing and the 1 /f M -weighted norm to be never increasing.313 ibliography What is ITER?, February, 20th, 2020. URL .T. Abel, O. Hahn, and R. Kaehler. Tracing the dark matter sheet in phase space.
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