Crash: A Block-Adaptive-Mesh Code for Radiative Shock Hydrodynamics - Implementation and Verification

1. INTRODUCTION

• Radiation hydrodynamics describes radiation interacting with matter through emission, absorption, and scattering, and heating or accelerating plasma at high radiation energy density.Radiation intensity resolves energy by propagation direction and frequency.Source: §1 · p. 1
• CRASH combines an Eulerian grid with block-based adaptive mesh refinement for radiation-hydrodynamic simulations.The paper describes radiation transport approaches ranging from moment and variable Eddington tensor methods to diffusion models.Source: §1 · p. 2
• CRASH extends BATS-R-US with radiation, equation-of-state, and opacity libraries; both codes are publicly available through SWMF or as stand-alone codes.The implementation discussion focuses on radiation in plasmas without magnetic fields, though much of it can extend to MHD plasmas.Source: §1 · p. 2
• Verification covers radiation and electron heat conduction on non-uniform 1D, 2D, and 3D slab meshes and in axially symmetric geometry.The paper also presents an adaptively refined, multi-material radiation-hydrodynamic simulation and reports scaling up to 1000 processors.Source: §1 · p. 3

2. EQUATIONS OF RADIATION HYDRODYNAMICS IN DENSE PLASMAS

• The model targets non-relativistic, multi-material high-energy-density plasmas in LTE, with separate electron and radiation temperatures and gray or multigroup flux-limited diffusion.The radiation model is non-equilibrium diffusion; electrons conduct heat and emit and absorb radiation.Source: §2 · p. 3

2.1. Radiation Transport

• Radiation transport evolves group energy densities, with compression-driven frequency shifts represented as conservative advection along the frequency axis.The diffusion model assumes an isotropic radiation field, while flux limiting is used for free-streaming conditions.Source: §2.1 · p. 4

2.2. Hydrodynamics

• Separate ion and electron internal-energy equations represent their possible temperature disequilibrium, collisional energy transfer, electron heat conduction, and electron-radiation exchange.A total-energy equation is also used to recover correct jump conditions at hydrodynamic shocks.Source: §2.2 · p. 6

2.3. Level Sets and Material Identification

• Material interfaces are tracked with signed level-set functions, and when numerical errors make multiple functions positive, the code selects the largest value.The authors describe this selection as a simple approximation and identify more sophisticated approaches as future work.Source: §2.3 · p. 7

2.4. Equation of State and Opacities

• Material-specific equation-of-state and opacity lookup tables supply thermodynamic quantities and group-averaged opacities used during simulations.The EOS tables use log electron temperature and ion density; opacity tables use log density and electron temperature.Source: §2.4 · pp. 7–8

2.5. Flux Limited Diffusion

• Flux-limited diffusion bounds radiation transport in the optically thin limit, while a flux-limited electron heat-flux model substitutes for computationally expensive Fokker-Planck solutions.The electron heat-flux model limits transport when the temperature scale length approaches the electron mean free path.Source: §2.5 · p. 8

3. THE NUMERICAL METHOD

• The discretized system includes density, momentum, total and electron internal energies, radiation group energy, and material level-set equations.The equations are time integrated using operator splitting.Source: §3 · pp. 8–9
• Operator splitting advances the system through hydrodynamic advection and pressure, radiation advection in frequency space, then implicit diffusion and energy exchange.Frequency advection is split from the hydro update because CRASH is built around BATS-R-US across 1D, 2D, and 3D.Source: §3 · p. 9

3.1. Hydro Solve

• The hydro step has conservative total-energy and non-conservative pressure variants that can be combined in a hybrid simulation.The conservative variant advances with shock-capturing schemes and applies EOS corrections after each Runge-Kutta stage.Source: §3.1 · p. 9, §3.1.1 · p. 11

3.1.1. Conservative

• CRASH implements HLLE, Rusanov, and exact-Riemann Godunov shock-capturing schemes, and generalizes HLLE for radiation, level sets, and an EOS.The method uses finite-volume cell averages and a two-stage Runge-Kutta hydro update.Source: §3.1.1 · pp. 9, 11
• The conservative scheme splits electron energy into ideal and extra parts to extend constant-γ shock-capturing solvers to spatially varying electron γ.The extra energy is advected conservatively, and the EOS recovers the true electron pressure while conserving total energy.Source: §3.1.1 · pp. 9–10

3.1.2. Non-Conservative Pressure Equations

• CRASH can switch formulations using criteria such as steep pressure-gradient shock detection, with the user selecting the switching threshold.This supports combining shock-capturing behavior with pressure balance at material interfaces.Source: §3.1.2 · p. 12

3.2. Frequency Advection

• For sufficiently many frequency groups, each group energy is approximated by photon frequency times spectral energy density times logarithmic group spacing.The grid is uniformly spaced in log frequency.Source: §3.2 · p. 12
• The frequency-advection update uses interpolated group energies at group boundaries and a one-stage second-order upwind discretization.The limited slope uses the superbee limiter; when the CFL number exceeds one, the update is sub-cycled.Source: §3.2 · pp. 12–13
• Frequency advection is conservative linear advection in log-frequency, with physical flux Fg−1/2 = uνEg−1/2.The advection speed is uν = −(γr −1)∇·u.Source: §3.2 · p. 12
• Zero radiation flux at the frequency-domain edges prevents radiation leakage from those boundaries.The numerical flux is based on the log-frequency advection speed and interpolated group energy.Source: §3.2 · pp. 12–13

3.3.1. Coupled Implicit Scheme

• The coupled formulation can solve all G + 2 variables together or reduce the solve to G + 1 equations when ion heat conduction is absent.The reduced system is described as more efficient, especially for few groups such as gray diffusion; adding ion heat conduction requires the full coupled system.Source: §3.3.1 · p. 13
• The coupled scheme solves a linearized system for radiation groups and electron energy-like variables, using Krylov solvers and block preconditioning.The matrix is generally nonsymmetric with multigroup Planck weights; for gray diffusion it is symmetric positive definite and can use preconditioned conjugate gradients.Source: §3.3.1 · pp. 14–15
• The coupled update conserves total energy to round-off error, but excessively large time steps can produce negative ion or electron internal energy.When the radiation-like variable drops far below its starting value, the advance may be retried with a smaller step.Source: §3.3.1 · p. 15
• A temperature-based reformulation has an SPD system, but its conservative group-energy update can fail for overly large time steps.The authors also implement Crank-Nicolson for verification tests with time-constant coefficients.Source: §3.3.1 · pp. 15–16

3.3.2. Decoupled Implicit Scheme

• The decoupled scheme solves each radiation group and electron temperature independently to reduce the coupled system’s preconditioning and memory burden.It assumes radiation-electron exchange does not make the electron temperature change much in a step, as when electron energy density greatly exceeds radiation energy density.Source: §3.3.2 · p. 16
• The decoupled conservative update preserves total energy to round-off error and always uses less memory, but is not guaranteed to run faster.It reduces preconditioning and Krylov-solver time while generally requiring more parallel message passing.Source: §3.3.2 · p. 17

3.4. Boundary Conditions

• The main code provides fixed, zero-gradient extrapolation, periodic, and reflective boundary conditions, while also permitting user-specified types.Source: §3.4 · p. 17
• Radiation boundaries use zero or fixed incoming flux instead of zero-gradient extrapolation.The condition assumes no external radiation sources and that outflowing radiation does not return; simple zero-gradient extrapolation can make the diffusion problem ill-posed.Source: §3.4 · p. 17

4. CODE VERIFICATION

• Verification testing covers hydrodynamics, multimaterial advection, gray and multigroup radiation diffusion, heat conduction, and full-system laboratory configurations.Tests run daily against analytic or semi-analytic solutions where possible, with grid or time convergence assessed as needed.Source: §4 · p. 18

4.1. Error Assessment

• The relative L1 and maximum errors quantify numerical solutions against reference solutions, which are coarsened to match the numerical grid when needed.Source: §4.1 · pp. 18–19

4.2.1. Su-Olson Test

• The Su-Olson test’s final radiation and material temperatures agree well with the semi-analytical solution.The test uses fixed time steps and Crank-Nicolson integration with time-independent matrix coefficients; its grid-convergence comparison uses a high-resolution CRASH numerical reference.Source: §4.2.1 · p. 19

4.2.2. Lowrie’s Non-equilibrium Radhydro Solutions

• The Mach 1.05 radiative-shock test achieves second-order grid convergence for both conservative and non-conservative hydrodynamic schemes.Radiation smooths the hydro shock, permitting the non-conservative pressure formulation; the test advances a rotated shock across an AMR resolution change.Source: §4.2.2 · p. 20

4.2.3. Double Light Front

• The double light-front test shows a 2/3 convergence rate, consistent with numerical diffusion of the discontinuities.The test checks group diffusion and flux limiters by comparing two fronts against the analytic light-speed position.Source: §4.2.3 · p. 21

4.2.4. Relaxation of Radiation Energy Test

• With 80 logarithmically spaced groups, the radiation-energy relaxation test demonstrates second-order temporal convergence under the large-specific-heat assumption.That assumption keeps material temperature and the Planckian spectrum constant.Source: §4.2.4 · p. 22

4.3.1. Uniform Heat Conduction in rz-geometry

• Second-order convergence is demonstrated for uniform heat conduction in rz-geometry using a non-uniform grid and an analytical reference solution.The temperature profile diffuses toward a more uniform state over time.Source: §4.3.1 · p. 22

4.3.2. Reinicke Meyer-ter Vehn Test

• The Reinicke–Meyer-ter-Vehn solution is closely reproduced, with errors largest at the shock and heat-front discontinuities.The rz-grid preserves the spherical symmetry of the solution without significant distortion; convergence is first order because of the shock and heat front.Source: §4.3.2 · p. 23

4.3.3. Heat Conduction Version of Lowrie’s Test

• First-order relative L1 convergence is observed for the Mach 5 heat-conduction test, with no spurious oscillations near the shock or precursor.The test recasts Lowrie and Edwards’ non-equilibrium gray-diffusion shock tube to include electron heat conduction and ion-electron energy exchange.Source: §4.3.3 · pp. 24–25
• The Lowrie test reaches at most first-order convergence because both the shock and heat precursor are discontinuous.The ion-temperature spike is spatially narrow and requires many grid cells to resolve fully.Source: §4.3.3 · p. 25

4.4. Full System Tests

• The 3D full-system simulation captures radiative effects including a wall shock, plastic ablation, and a radiative precursor ahead of the primary shock.The setup uses flux-limited diffusion with 30 groups and dynamic mesh refinement; HYADES supplies the initial condition after the 1.1 ns laser pulse.Source: §4.4 · pp. 25–26

4.5. Parallel Performance

• Good strong scaling is obtained up to 1024 cores, while performance saturates at higher core counts.The fixed-size test uses five materials, 30 radiation groups, separate electron and ion temperatures, and approximately 2.6 million cells.Source: §4.5 · p. 26

• CRASH extends BATS-R-US with gray or multigroup radiation diffusion using flux-limited diffusion and the Block Adaptive Tree Library for highly resolved solutions.The combination includes new radiation-transfer and heat-conduction libraries, plus equation-of-state and multigroup opacity solvers.Source: §5 · pp. 26–27
• Separate electron pressure and flux-limited electron heat conduction address electron-ion temperature disequilibrium at hydrodynamic shocks.The electron heat flux is limited by the free-streaming heat flux.Source: §5 · p. 27
• An operator split advances hydrodynamics, frequency-space radiation advection, then implicit radiation diffusion, heat conduction, and energy exchange.Krylov solvers with a BILU preconditioner scale to 500 or 1000 processors; a multilevel preconditioner is future work for scaling beyond 1000.Source: §5 · p. 27
• Verification tests cover gray and multigroup radiation diffusion and heat conduction in 1D, 2D, and 3D slab and 2D rz geometry.The paper also presents a 3D multi-material radiative shock propagating through an elliptic nozzle for future validation studies.Source: §5 · p. 27
• The radiation-transfer implementation is readily available for MHD simulations, supporting prospective validation with laboratory-astrophysics experiments or astrophysical plasma simulations.Source: §5 · p. 27

A. DISCRETIZATION OF THE DIFFUSION OPERATOR AT RESOLUTION CHANGES

• On uniform meshes, the diffusion operator uses a finite-volume method, central differences for the flux gradient, and arithmetic averaging of cell-centered diffusion coefficients at faces.Generalizing this discretization to resolution changes is less straightforward.Source: A. DISCRETIZATION OF THE DIFFUSION… · p. 27
• At resolution changes, CRASH replaces Edwards (1996)'s weighted harmonic average of diffusion coefficients with an arithmetic average for the interface flux densities.The cited Edwards approach requires strong flux continuity and a one-sided difference for the diffusion-flux gradient.Source: A. DISCRETIZATION OF THE DIFFUSION… · pp. 27–28
• Verification tests show that this change produces properly propagating radiative precursor and shock fronts; generalizations to 1D and 3D are straightforward.Source: A. DISCRETIZATION OF THE DIFFUSION… · p. 28

B. RZ-GEOMETRY

• In rz geometry, radial cell-face area and cell volume are made proportional to the distance r from the symmetry axis.Source: B. RZ-GEOMETRY · p. 28
• The rz formulation also modifies the r component of the momentum equation.Source: B. RZ-GEOMETRY · p. 28
• The radial momentum correction reflects that the pressure term is a gradient, not a divergence.Source: B. RZ-GEOMETRY · p. 28