James A. Rossmanith

A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equations

The Vlasov-Poisson equations describe the evolution of a collisionless plasma, represented through a probability density function (PDF) that self-interacts via an electrostatic force. One of the main difficulties in numerically solving this system is the severe time-step restriction that arises from parts of the PDF associated with moderate-to-large velocities. The dominant approach in the plasma physics community for removing these time-step restrictions is the so-called particle-in-cell (PIC) method, which discretizes the distribution function into a set of macro-particles, while the electric field is represented on a mesh. Several alternatives to this approach exist, including fully Lagrangian, fully Eulerian, and so-called semi-Lagrangian methods. The focus of this work is the semi-Lagrangian approach, which begins with a grid-based Eulerian representation of both the PDF and the electric field, then evolves the PDF via Lagrangian dynamics, and finally projects this evolved field back onto the original Eulerian mesh. In particular, we develop in this work a method that discretizes the 1+1 Vlasov-Poisson system via a high-order discontinuous Galerkin (DG) method in phase space, and an operator split, semi-Lagrangian method in time. Second-order accuracy in time is relatively easy to achieve via Strang operator splitting. With additional work, using higher-order splitting and a higher-order method of characteristics, we also demonstrate how to push this scheme to fourth-order accuracy in time. We show how to resolve all of the Lagrangian dynamics in such a way that mass is exactly conserved, positivity is maintained, and high-order accuracy is achieved. The Poisson equation is solved to high-order via the smallest stencil local discontinuous Galerkin (LDG) approach. We test the proposed scheme on several standard test cases.

An Unstaggered, High-Resolution Constrained Transport Method for Magnetohydrodynamic Flows

The ideal magnetohydrodynamic (MHD) equations are important in modeling phenomena in a wide range of applications, including space weather, solar physics, laboratory plasmas, and astrophysical fluid flows. Numerical methods for the MHD equations must confront the challenge of producing approximate solutions that remain accurate near shock waves and that satisfy a divergence-free constraint on the magnetic field. Failure to accomplish this often leads to unphysical solutions. In this paper, a high-resolution wave propagation method is developed that utilizes a novel constrained transport technique to keep the magnetic field divergence-free. This approach is based on directly solving the magnetic potential equation in conjunction with a new limiting strategy to obtain a nonoscillatory magnetic field. It is demonstrated in this work that an unstaggered definition of the divergence is the correct one to use in the case of wave propagation methods. Therefore, we solve the magnetic potential equation on the same grid as the MHD equations; hence the usual grid staggering that is found in constrained transport methods is eliminated. We demonstrate through truncation error analysis and direct numerical simulation that the resulting method is second order accurate in space and time for smooth solutions and nonoscillatory near shocks and other discontinuities. The resulting numerical method has been implemented as an extension to the clawpack software package and can be freely downloaded from the Web.

An unstaggered constrained transport method for the 3D ideal magnetohydrodynamic equations

Numerical methods for solving the ideal magnetohydrodynamic (MHD) equations in more than one space dimension must either confront the challenge of controlling errors in the discrete divergence of the magnetic field, or else be faced with nonlinear numerical instabilities. One approach for controlling the discrete divergence is through a so-called constrained transport method, which is based on first predicting a magnetic field through a standard finite volume solver, and then correcting this field through the appropriate use of a magnetic vector potential. In this work we develop a constrained transport method for the 3D ideal MHD equations that is based on a high-resolution wave propagation scheme. Our proposed scheme is the 3D extension of the 2D scheme developed by Rossmanith [J.A. Rossmanith, An unstaggered, high-resolution constrained transport method for magnetohydrodynamic flows, SIAM J. Sci. Comput. 28 (2006) 1766], and is based on the high-resolution wave propagation method of Langseth and LeVeque [J.O. Langseth, R.J. LeVeque, A wave propagation method for threedimensional hyperbolic conservation laws, J. Comput. Phys. 165 (2000) 126]. In particular, in our extension we take great care to maintain the three most important properties of the 2D scheme: (1) all quantities, including all components of the magnetic field and magnetic potential, are treated as cell-centered; (2) we develop a high-resolution wave propagation scheme for evolving the magnetic potential; and (3) we develop a wave limiting approach that is applied during the vector potential evolution, which controls unphysical oscillations in the magnetic field. One of the key numerical difficulties that is novel to 3D is that the transport equation that must be solved for the magnetic vector potential is only weakly hyperbolic. In presenting our numerical algorithm we describe how to numerically handle this problem of weak hyperbolicity, as well as how to choose an appropriate gauge condition. The resulting scheme is applied to several numerical test cases.

A superconductive undulator with a period length of 3.8 mm

During recent years several attempts have been undertaken to decrease the period length of undulators to the millimetre range. In this paper a novel type of in-vacuum undulator is described which is built using superconductive wires. The period length of this special device is 3.8 mm. In principle, it is possible to decrease this period length even further. A 100-period-long undulator has been built and will be tested with a beam in the near future.

Finite difference weighted essentially non-oscillatory schemes with constrained transport for 2D ideal Magnetohydrodynamics

Summary form only given. A novel algorithm based on a high order finite difference adaptive WENO scheme [2] will be presented to solve the 2D ideal Magnetohydrodynamics (MHD) equations. The algorithm will use the unstaggered Constrained Transport technique from the magnetic potential advection constrained transport method [2] to satisfy the divergence-free constraint of the magnetic field. However the treatment of our algorithm is significantly different from [2]. The new feature will be (1) the method is finite difference type, (2) high order in both time (4th-order) and space (3rd-order), (3) all the conservative quantities are essentially non-oscillatory, (4) Adaptive Mesh Refinement will be used as the base framework to increase resolution. Convergence study will be done on the smooth problem. 2D/2.5D benchmark problems such as rotated Brio-Wu shock tube, Orszag-Tang and Cloud-shock interaction will be presented. We expect our algorithm is robust, essentially non-oscillatory and can capture shock waves and discontinuities well.

A high-resolution constrained transport method with adaptive mesh refinement for ideal MHD

Abstract We consider a high-resolution finite volume method that utilizes a constrained transport technique to keep the magnetic field divergence-free. This scheme is based on the wave propagation method of LeVeque [J. Comp. Phys. 131 (1997) 327] and makes use of the constrained transport framework of Evans and Hawley [Astrophys. J. 332 (1988) 659]. This method differs from other constrained transport methods in that it is based on directly solving the magnetic potential equation in conjunction with a special limiting strategy to obtain a non-oscillatory magnetic field. It is formulated without the use of a staggered grid and implemented so that no spatial averaging is needed in the update. We present in this work an adaptive mesh refinement implementation of this method using the amrclaw software package developed by Berger and LeVeque [SIAM J. Numer. Anal. 35 (1998) 2298]. The proposed method is tested on a two-dimensional cloud-shock interaction problem.

A High-Order Unstaggered Constrained-Transport Method for the Three-Dimensional Ideal Magnetohydrodynamic Equations Based on the Method of Lines

Numerical methods for solving the ideal magnetohydrodynamic (MHD) equations in more than one space dimension must confront the challenge of controlling errors in the discrete divergence of the magnetic field. One approach that has been shown successful in stabilizing MHD calculations are constrained-transport (CT) schemes. CT schemes can be viewed as predictor-corrector methods for updating the magnetic field, where a magnetic field value is first predicted by a method that does not exactly preserve the divergence-free condition on the magnetic field, followed by a correction step that aims to control these divergence errors. In Helzel, Rossmanith, and Taetz [J. Comput. Phys., 230 (2011), pp. 3803--3829] the authors presented an unstaggered CT method for the MHD equations on three-dimensional Cartesian grids. In this approach an evolution equation for the magnetic potential is solved during each time step and a divergence-free update of the magnetic field is computed by taking the curl of the magnetic pot...

Positivity-Preserving Discontinuous Galerkin Methods with Lax---Wendroff Time Discretizations

This work introduces a single-stage, single-step method for the compressible Euler equations that is provably positivity-preserving and can be applied on both Cartesian and unstructured meshes. This method is the first case of a single-stage, single-step method that is simultaneously high-order, positivity-preserving, and operates on unstructured meshes. Time-stepping is accomplished via the Lax–Wendroff approach, which is also sometimes called the Cauchy–Kovalevskaya procedure, where temporal derivatives in a Taylor series in time are exchanged for spatial derivatives. The Lax–Wendroff discontinuous Galerkin (LxW-DG) method developed in this work is formulated so that it looks like a forward Euler update but with a high-order time-extrapolated flux. In particular, the numerical flux used in this work is a convex combination of a low-order positivity-preserving contribution and a high-order component that can be damped to enforce positivity of the cell averages for the density and pressure for each time step. In addition to this flux limiter, a moment limiter is applied that forces positivity of the solution at finitely many quadrature points within each cell. The combination of the flux limiter and the moment limiter guarantees positivity of the cell averages from one time-step to the next. Finally, a simple shock capturing limiter that uses the same basic technology as the moment limiter is introduced in order to obtain non-oscillatory results. The resulting scheme can be extended to arbitrary order without increasing the size of the effective stencil. We present numerical results in one and two space dimensions that demonstrate the robustness of the proposed scheme.

A class of residual distribution schemes and their relation to relaxation systems

Residual distributions (RD) schemes are a class of high-resolution finite volume methods for unstructured grids. A key feature of these schemes is that they make use of genuinely multidimensional (approximate) Riemann solvers as opposed to the piecemeal 1D Riemann solvers usually employed by finite volume methods. In 1D, LeVeque and Pelanti [R.J. LeVeque, M. Pelanti, A class of approximate Riemann solvers and their relation to relaxation schemes, J. Comput. Phys. 172 (2001) 572] showed that many of the standard approximate Riemann solver methods (e.g., the Roe solver, HLL, Lax-Friedrichs) can be obtained from applying an exact Riemann solver to relaxation systems of the type introduced by Jin and Xin [S. Jin, Z.P. Xin, Relaxation schemes for systems of conservation-laws in arbitrary space dimensions, Commun. Pure Appl. Math. 48 (1995) 235]. In this work we extend LeVeque and Pelantis results and obtain a multidimensional relaxation system from which multidimensional approximate Riemann solvers can be obtained. In particular, we show that with one choice of parameters the relaxation system yields the standard N-scheme. With another choice, the relaxation system yields a new Riemann solver, which can be viewed as a genuinely multidimensional extension of the local Lax-Friedrichs scheme. This new Riemann solver does not require the use Roe-Struijs-Deconinck averages, nor does it require the inversion of an mxm matrix in each computational grid cell, where m is the number of conserved variables. Once this new scheme is established, we apply it on a few standard cases for the 2D compressible Euler equations of gas dynamics. We show that through the use of linear-preserving limiters, the new approach produces numerical solutions that are comparable in accuracy to the N-scheme, despite being computationally less expensive.

A Wave Propagation Algorithm for the Solution of PDEs on the Surface of a Sphere

Large-scale geophysical flows are governed by partial differential equations on the surface of a sphere. In this paper we present a high-resolution finite volume method using gnomonic grid mappings to solve equations relevant to geophysical fluid dynamics. The method is a generalization of the wave propagation algorithm of CLAWPACK for domains which lie on curved manifolds. We show that in this finite volume context it becomes possible to regularize the singularities arising from the gnomonic mapping; and thus, it becomes possible to compute the solution to various hyperbolic conservation laws on the surface of a sphere in a globally conservative and accurate way. With a slight modification, this approach can also be used to solve equations on a circular domain