James Paul Holloway

One-dimensional Riemann solvers and the maximum entropy closure

Abstract The maximum entropy closure for the two-moment approximation of the neutron transport equation is presented. We use a robust Roe-type Riemann solver to solve the resulting moment equations. We also present three boundary conditions to use with this method. The ghost cell method effectively implements the Mark boundary condition by placing phantom cells just outside the physical system. This method is extremely simple to implement and gives reasonable results. The boundary Eddington factor method implements the Marshak boundary condition. While it yields good results at boundaries with incoming neutrons, it does not do so well at vacuum boundaries. The partial numerical flux method is an extension of the Marshak boundary condition, allowing us to specify extra angular information about the incoming neutron distribution. The neutron flux calculations with this method are generally the best out of the three boundary conditions presented here. Several simple steady-state and time-dependent problems illustrate the qualities, both good and bad, of the maximum entropy closure.

CRASH: A BLOCK-ADAPTIVE-MESH CODE FOR RADIATIVE SHOCK HYDRODYNAMICS-IMPLEMENTATION AND VERIFICATION

We describe the Center for Radiative Shock Hydrodynamics (CRASH) code, a block-adaptive-mesh code for multi-material radiation hydrodynamics. The implementation solves the radiation diffusion model with a gray or multi-group method and uses a flux-limited diffusion approximation to recover the free-streaming limit. Electrons and ions are allowed to have different temperatures and we include flux-limited electron heat conduction. The radiation hydrodynamic equations are solved in the Eulerian frame by means of a conservative finite-volume discretization in either one-, two-, or three-dimensional slab geometry or in two-dimensional cylindrical symmetry. An operator-split method is used to solve these equations in three substeps: (1) an explicit step of a shock-capturing hydrodynamic solver; (2) a linear advection of the radiation in frequency-logarithm space; and (3) an implicit solution of the stiff radiation diffusion, heat conduction, and energy exchange. We present a suite of verification test problems to demonstrate the accuracy and performance of the algorithms. The applications are for astrophysics and laboratory astrophysics. The CRASH code is an extension of the Block-Adaptive Tree Solarwind Roe Upwind Scheme (BATS-R-US) code with a new radiation transfer and heat conduction library and equation-of-state and multi-group opacity solvers. Both CRASH and BATS-R-US are part of the publicly available Space Weather Modeling Framework.

The theory and simulation of relativistic electron beam transport in the ion-focused regime.

Several recent experiments involving relativistic electron beam (REB) transport in plasma channels show two density regimes for efficient transport; a low‐density regime known as the ion‐focused regime (IFR) and a high‐pressure regime. The results obtained in this paper use three separate models to explain the dependency of REB transport efficiency on the plasma density in the IFR. Conditions for efficient beam transport are determined by examining equilibrium solutions of the Vlasov–Maxwell equations under conditions relevant to IFR transport. The dynamic force balance required for efficient IFR transport is studied using the particle‐in‐cell (PIC) method. These simulations provide new insight into the transient beam front physics as well as the dynamic approach to IFR equilibrium. Nonlinear solutions to the beam envelope are constructed to explain oscillations in the beam envelope observed in the PIC simulations but not contained in the Vlasov equilibrium analysis. A test particle analysis is also devel...

Abel's inversion applied to experimental spectroscopic data with off axis peaks

Abstract Many mathematical techniques for solving Abels integral equation have been proposed over the years in the literature. Often these methods handle test functions with known solutions and Gaussian type profiles quite accurately. For experimental data and non-Gaussian shapes, however, these methods are inadequate. An experimental study of the uniformity of the plasma emission on the Gaseous Electronics Conference Reference Cell has as its goal to provide immediate feedback regarding the plasma uniformity. Therefore a rapid, accurate, and sturdy computational method of data analysis is required. The first round of experiments indicate off axis or donut shaped plasmas within the reference cell. The analysis and conclusions are presented herein.

Application of preconditioned GMRES to the numerical solution of the neutron transport equation

The generalized minimal residual (GMRES) method with right preconditioning is examined as an alternative to both standard and accelerated transport sweeps for the iterative solution of the diamond differenced discrete ordinates neutron transport equation. Incomplete factorization (ILU) type preconditioners are used to determine their effectiveness in accelerating GMRES for this application. ILU(τ), which requires the specification of a dropping criteria τ, proves to be a good choice for the types of problems examined in this paper. The combination of ILU(τ) and GMRES is compared with both DSA and unaccelerated transport sweeps for several model problems. It is found that the computational workload of the ILU(τ)-GMRES combination scales nonlinearly with the number of energy groups and quadrature order, making this technique most effective for problems with a small number of groups and discrete ordinates. However, the cost of preconditioner construction can be amortized over several calculations with different source and/or boundary values. Preconditioners built upon standard transport sweep algorithms are also evaluated as to their effectiveness in accelerating the convergence of GMRES. These preconditioners show better scaling with such problem parameters as the scattering ratio, the number of discrete ordinates, and the number of spatial meshes. These sweeps based preconditioners can also be cast in a matrix free form that greatly reduces storage requirements.

Prediction and Computer Model Calibration Using Outputs From Multifidelity Simulators

Computer simulators are widely used to describe and explore physical processes. In some cases, several simulators are available, each with a different degree of fidelity, for this task. In this work, we combine field observations and model runs from deterministic multifidelity computer simulators to build a predictive model for the real process. The resulting model can be used to perform sensitivity analysis for the system, solve inverse problems, and make predictions. Our approach is Bayesian and is illustrated through a simple example, as well as a real application in predictive science at the Center for Radiative Shock Hydrodynamics at the University of Michigan. The Matlab code that is used for the analyses is available from the online supplementary materials.

A reconstruction algorithm for a spatially resolved plasma optical emission spectroscopy sensor

Abstract The reconstruction algorithm used for a new spatially resolved plasma optical emission spectroscopy sensor is described. The sensor has a wedge shaped field of view which is rotated horizontally across the plasma. The resulting signal as a function of angle defines an ill-posed linear problem that must be solved to determine the emissivity as a function of radius in the plasma. This problem is solved by modified Tikhonov regularization using a finite difference regularizer which discourages rapid variation in the reconstructed emission. The optimal regularization parameter is determined by minimizing the product of the norm of the residual and the norm of this regularizer. The robustness of the algorithm against noise introduced into idealized data is demonstrated, both for a standard Abel inversion problem and for test data based on the sensor model. The algorithm is used to reconstruct emitted power density profiles and to perform actinometry in a Lam TCP 9400 plasma processing tool; these reconstructions show qualitatively correct behaviors.

Spectral velocity discretizations for the Vlasov-Maxwell equations

Abstract Two Hermite based spectral methods are examined. A method based on an asymmetric expansion in Hermite polynomials offers theoretical advantages over other spectral basis sets for the treatment of the velocity variables in the Vlasov-Maxwell equations of plasma kinetic theory. It provides for exact conservation of energy, momentum, and particles (in a fully discrete system, not just as the discretization is refined), and this conservation is not effected by the use of velocity space filters during time-stepping. The asymmetric Hermite method exactly solves the spatially uniform problem (plasma oscillations), and both Hermite based methods exactly recover the temporal behavior of retained moments for free streaming in a periodic system. Yet the cost of these Hermite methods scales only linearly with the number of degrees for freedom, just like a finite difference method, and in fact the number of runtime operations required per degree of freedom in the asymmetric method can be essentially identical...

Theory of radiative shocks in the mixed, optically thick-thin case

A theory of radiating shocks that are optically thick in the downstream (postshock) state and optically thin in the upstream (preshock) state, which are called thick-thin shocks, is presented. Relations for the final temperature and compression, as well as the postshock temperature and compression as a function of the shock strength and initial pressure, are derived. The model assumes that there is no radiation returning to the shock from the upstream state. Also, it is found that the maximum compression in the shock scales as the shock strength to the 1/4 power. Shock profiles for the material downstream of the shock are computed by solving the fluid and radiation equations exactly in the limit of no radiation returning to the shock. These profiles confirm the validity and usefulness of the model in that limit.

A quasilinear implicit Riemann solver for the time-dependent Pn equations

Abstract An implicit Riemann solver for the one- and two-dimensional time-dependent spherical harmonics approximation (Pn) to the linear transport equation is presented. This spatial discretization scheme is based on cell-averaged quantities and uses a monotonicity-preserving high resolution method to achieve second-order accuracy (away from extreme points in the solution). Such a spatial scheme requires a nonlinear method of reconstructing the slope within a spatial cell. We have devised a means of creating an implicit (in time) method without the necessity of a nonlinear solver. This is done by computing a time step using a first-order scheme and then, based on that solution, reconstructing the slope in each cell, an implementation that we justify by analyzing the model equation for the method. This quasilinear approach produces smaller errors in less time than both a first-order scheme and a method that solves the full nonlinear system using a Newton-Krylov method.