Jim E. Morel

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.

Advances in Discrete-Ordinates Methodology

In 1968, Bengt Carlson and Kaye Lathrop published a comprehensive review on the state of the art in discrete-ordinates (SN) calculations [10]. At that time, SN methodology existed primarily for reactor physics simulations. By today’s standards, those capabilities were limited, due to the less-developed theoretical state of SN methods and the slower and smaller computers that were then available. In this chapter, we review some of the major advances in SN methodology that have occurred since 1968. These advances, combined with the faster speeds and larger memories of today’s computers, enable today’s SN codes to simulate problems of much greater complexity, realism, and physical variety. Since 1968, several books and reviews on general numerical methods for SN simulations have been published [32, 46, 71], but none of these covers the advanced work done during the past 20 years.

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.

Properties of the implicitly time-differenced equations of thermal radiation transport

Numerical simulations of thermal radiation transport (TRT) use varying levels of implicitness to discretize the time variable t. The degree of implicitness generally depends on the strength of the coupling between the radiation and matter. In this paper we use a contraction mapping method to show that if all terms in the TRT equations except possibly the opacity are discretized implicitly, then for any @Dt>0, the time-discretized TRT equations (i) have a unique solution, which (ii) satisfies the maximum principle and (iii) preserves the equilibrium (thick) diffusion limit. No other available time-discretization of the TRT equations has all these desirable properties. Numerical results are included to illustrate the theoretical predictions.

Linear multifrequency-grey acceleration recast for preconditioned Krylov iterations

The linear multifrequency-grey acceleration (LMFGA) technique is used to accelerate the iterative convergence of multigroup thermal radiation diffusion calculations in high energy density simulations. Although it is effective and efficient in one-dimensional calculations, the LMFGA method has recently been observed to significantly degrade under certain conditions in multidimensional calculations with large discontinuities in material properties. To address this deficiency, we recast the LMFGA method in terms of a preconditioned system that is solved with a Krylov method (LMFGK). Results are presented demonstrating that the new LMFGK method always requires fewer iterations than the original LMFGA method. The reduction in iteration count increases with both the size of the time step and the inhomogeneity of the problem. However, for reasons later explained, the LMFGK method can cost more per iteration than the LMFGA method, resulting in lower but comparable efficiency in problems with small time steps and weak inhomogeneities. In problems with large time steps and strong inhomogeneities, the LMFGK method is significantly more efficient than the LMFGA method.

Numerical analysis of time integration errors for nonequilibrium radiation diffusion

Numerical analysis of time integration errors for nonequilibrium radiation diffusion is considered. Two first-order implicit time integration methods are studied. Asymptotic analysis and modified equation analysis are applied to both time integration methods. Numerical experiments are used to highlight the results of the analysis. Asymptotic analysis is used to highlight the source of temperature spiking when a hot radiation wave propagates into a cold material. Modified equation analysis is used to provide insight into the thermal wave speed coming from the two different first-order methods.

Angular Multigrid Preconditioner for Krylov-Based Solution Techniques Applied to the Sn Equations with Highly Forward-Peaked Scattering

It is well known that the diffusion synthetic acceleration (DSA) methods for the Sn equations become ineffective in the Fokker-Planck forward-peaked scattering limit. In response to this deficiency, Morel and Manteuffel (1991) developed an angular multigrid method for the 1-D Sn equations. This method is very effective, costing roughly twice as much as DSA per source iteration, and yielding a maximum spectral radius of approximately 0.6 in the Fokker-Planck limit. Pautz, Adams, and Morel (PAM) (1999) later generalized the angular multigrid to 2-D, but it was found that the method was unstable with sufficiently forward-peaked mappings between the angular grids. The method was stabilized via a filtering technique based on diffusion operators, but this filtering also degraded the effectiveness of the overall scheme. The spectral radius was not bounded away from unity in the Fokker-Planck limit, although the method remained more effective than DSA. The purpose of this article is to recast the multidimensional PAM angular multigrid method without the filtering as an Sn preconditioner and use it in conjunction with the Generalized Minimal RESidual (GMRES) Krylov method. The approach ensures stability and our computational results demonstrate that it is also significantly more efficient than an analogous DSA-preconditioned Krylov method.

A Least-Squares Transport Equation Compatible with Voids

Standard second-order self-adjoint forms of the transport equation, such as the even-parity, odd-parity, and self-adjoint angular flux equation, cannot be used in voids. Perhaps more important, they experience numerical convergence difficulties in near-voids. Here we present a new form of a second-order self-adjoint transport equation that has an advantage relative to standard forms in that it can be used in voids or near-voids. Our equation is closely related to the standard least-squares form of the transport equation with both equations being applicable in a void and having a nonconservative analytic form. However, unlike the standard least-squares form of the transport equation, our least-squares equation is compatible with source iteration. It has been found that the standard least-squares form of the transport equation with a linear-continuous finite-element spatial discretization has difficulty in the thick diffusion limit. Here we extensively test the 1D slab-geometry version of our scheme with respect to void solutions, spatial convergence rate, and the intermediate and thick diffusion limits. We also define an effective diffusion synthetic acceleration scheme for our discretization. Our conclusion is that our least-squares Sn formulation represents an excellent alternative to existing second-order Sn transport formulations.

Nonlinear variants of the TR/BDF2 method for thermal radiative diffusion

We apply the Trapezoidal/BDF2 (TR/BDF2) temporal discretization scheme to nonlinear grey radiative diffusion. This is a scheme that is not well-known within the radiation transport community, but we show that it offers many desirable characteristics relative to other second-order schemes. Several nonlinear variants of the TR/BDF2 scheme are defined and computationally compared with the Crank-Nicholson scheme. It is found for our test problems that the most accurate TR/BDF2 schemes are those that are fully iterated to nonlinear convergence, but the most efficient TR/BDF2 scheme is one based upon a single Newton iteration. It is also shown that neglecting the contributions to the Jacobian matrix from the cross-sections, which is often done due to a lack of smooth interpolations for tabular cross-section data, has a significant impact upon efficiency.

Manufactured Solutions in the Thick Diffusion Limit

Abstract Spatially analytic SN solutions currently exist only under very limited circumstances. For cases in which analytical solutions may not be available, one can turn to manufactured solutions to test the properties of spatial transport discretization schemes. In particular, we show it is possible to use a manufactured solution to conduct such tests in the thick diffusion limit, even though the computed solution is independent of the problem characteristics. We show that a diffusion limit scaling with a manufactured solution source term results in an expression that is valid in the diffusion limit, though it is not of the standard form used in asymptotic diffusion limit analysis. We then derive a necessary, but not sufficient, condition that must be satisfied in order for a spatial discretization of the transport equation to preserve the thick diffusion limit. This condition is stated in terms of the difference between a numerically computed scalar flux solution compared against a known scalar flux. For a sufficiently diffusive problem and optically thick mesh cells, the necessary condition states that if a spatial discretization of the SN equations has the thick diffusion limit, the norm of the difference in the two solutions must converge to zero with decreasing mesh cell spacing. Based on the first observation that the diffusion limit holds for a manufactured solution source term, the known solution can conveniently be taken to be a manufactured solution in a mesh refinement numerical experiment to check whether a spatial discretization satisfies this condition. We present computational examples that verify our analysis and illustrate the expediency of this approach.