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The RAGE radiation-hydrodynamic code

Michael Gittings, Robert Weaver, Michael Clover, Thomas Betlach, Nelson Byrne, Robert Coker, Edward Dendy, Robert Hueckstaedt, Kim New, W Rob Oakes, Dale Ranta, Ryan Stefan

arXiv:0804.1394v1physics.comp-phphysics.plasm-ph

TL;DR

RAGE addresses the need for scalable radiation-hydrodynamics simulations with robust treatment of hydrodynamics, adaptive meshes, and radiation–material coupling. It combines a parallel Eulerian framework with Godunov hydrodynamics, adaptive refinement, and iterative nonlinear coupling, enabling considerably larger timesteps and highly resolved problems. The paper also documents verification and validation efforts, while identifying scope limits in the MMEOS and self-gravity components.

  • Problem

    RAGE targets high-deformation radiation-hydrodynamics problems requiring multidimensional, massively parallel simulations and robust coupled hydrodynamic and radiation algorithms.

  • Method

    The paper presents a parallel adaptive Eulerian code combining Godunov hydrodynamics, adaptive mesh refinement, gray or multigroup diffusion, and iterative nonlinear radiation–material coupling.

  • Results

    Considerably larger timesteps enable RAGE to run more highly resolved problems within time and resource constraints.

  • Takeaways & Limitations

    The nonlinear implicit treatments extend the practical resolution achievable by the adaptive radiation-hydrodynamics code under fixed time and resource constraints.

  • Takeaways & Limitations

    MMEOS is not easily adapted to separate ion and electron temperatures, while self-gravity currently supports only three-dimensional non-periodic problems.

Abstract

from arXiv · show

We describe RAGE, the ``Radiation Adaptive Grid Eulerian'' radiation-hydrodynamics code, including its data structures, its parallelization strategy and performance, its hydrodynamic algorithm(s), its (gray) radiation diffusion algorithm, and some of the considerable amount of verification and validation efforts. The hydrodynamics is a basic Godunov solver, to which we have made significant improvements to increase the advection algorithm's robustness and to converge stiffnesses in the equation of state. Similarly, the radiation transport is a basic gray diffusion, but our treatment of the radiation-material coupling, wherein we converge nonlinearities in a novel manner to allow larger timesteps and more robust behavior, can be applied to any multi-group transport algorithm.

Introduction

RAGE extends the SAGE Eulerian hydrodynamics code with radiation transport for high-deformation, radiative-flow problems. Its development emphasizes massively parallel execution, adaptive meshing, robust hydrodynamics, nonlinear radiation–material coupling, and extensive verification and validation.

  • Code scope: RAGE is a multidimensional, multimaterial, massively parallel Eulerian code that couples the Euler equations with gray or multigroup radiation diffusion.It uses locally adaptive computational cells with unit aspect ratio.
  • Numerical methods: RAGE introduces nonlinear radiation–material energy coupling with exponential heating differencing and iterative treatment of nonlinear terms.The treatment is intended to support larger timesteps and reduce computer time.
  • Verification and validation: Verification and validation efforts assess algorithmic correctness and suitability across a broad range of problems of interest to Los Alamos.
  • Code scope: RAGE builds on SAGE by adding gray and multigroup radiation packages for problems where radiative energy transport matters.
  • Development and parallelism: The code was redesigned from a CRAY vector Fortran-77 implementation into parallel, modular software using Fortran-90/95, C interfaces, MPI, and domain decomposition.
  • Development and parallelism: 600 million computation cells are currently simulated in three dimensions using 2000 processors, while the stated target is 10,000 processors for a 1-billion-cell problem.

1 SETUP and the RAGE Computational Grid

RAGE constructs adaptive octree grids from geometric descriptions using point-containment queries, material-property sampling, and constrained local refinement. Geometry libraries support CAD-derived and intrinsic models while preserving controlled mesh transitions.

  • Computational grid: The RAGE grid is an octree in which adaptation bisects each zone in every dimension and uses separate resolutions for material gradients and boundaries.
  • Geometry setup: Grid generation requires a point-containment test, with region priorities determining which geometric region contains each queried point.
  • Mesh constraints: Adjacent cells may differ by at most one refinement level, enforcing a 2:1 edge-length ratio across the mesh.
  • Geometry setup: RAGE estimates finest-cell material properties through multiple queries analogous to Monte Carlo integration over density and temperature or alternate state variables.
  • Geometry setup: Spica enables grids from OSO regions and STL-based CAD geometry, while intrinsic support includes spheres, cylinders, cones, prisms, tori, and perturbed boundaries.
  • Geometry setup: A Pro/ENGINEER engine model is processed through OSO and Spica into a RAGE grid whose illustrated level-1-to-level-5 transition reflects the 2:1 constraint.Typical well-resolved grids may descend to level 8.

2 Data Structures and Parallel Implementation

RAGE combines distributed cell and face data structures with hidden communication layers, load balancing, and parallel I/O. Its weak-scaling tests show good performance across large processor counts, with communication causing the main degradation.

  • Parallel implementation: RAGE uses a MIMD parallel model in which MPI communication is hidden behind application-level token-library routines.
  • Data structures: Cells are globally indexed across a Hilbert or column-major ordering, while active leaf cells drive most physics calculations and face arrays support finite-volume operations.
  • Communication: Clone structures provide local ghost-cell copies of adjacent off-processor cells for distributed face-based calculations.
  • Communication: Face-based communication follows gather, compute, and scatter stages using clone_get and clone_put operations.
  • Parallel I/O: Parallel I/O supports MPI-IO and asynchronous bulkio libraries, with low-level data management hidden from physics developers.
  • Parallel scaling: Timing tests report approximately 20-fold performance growth across systems spanning roughly a decade of hardware generations.
  • Parallel scaling: Performance degrades by about a factor of 2 at thousands of processors, primarily because neighboring-process communication and collective operations increase.

3 Adaptive Mesh Refinement Overview

RAGE uses continuous, cell-based adaptive mesh refinement to resolve shocks, interfaces, and gradients while preserving global conservation and manageable mesh transitions. Refinement is evaluated every cycle, with conservative rules governing refinement, coarsening, and load balance.

  • AMR motivation: The AMR method was developed for time-dependent shock hydrodynamics requiring fine resolution at shock fronts far from other resolved features.
  • AMR design: Global conservation motivated coarse boundaries that coincide exactly with fine-mesh boundaries rather than arbitrarily oriented patches.
  • AMR design: Cell-based refinement bisects each dimension and limits adjacent cells to a one-level difference, unlike the patch-based approach of Berger et al.
  • Adaptive Mesh Refinement: RAGE’s Continuous Adaptive Mesh Refinement evaluates every cell on every cycle and refines or coarsens locally in space and time.Its overhead is empirically about 20% of total runtime.
  • AMR algorithm: The two-phase CAMR algorithm separates initial setup refinement from dynamic refinement during each timestep.
  • AMR algorithm: Refinement is triggered by activity tests ahead of shocks, small velocity and pressure changes, and first-order estimates of local spatial variation.
  • AMR implementation: Load leveling is applied only when a processor’s cell count exceeds the average by a preset tolerance, avoiding redistribution on every cycle.

4 Hydro

RAGE’s hydrodynamics combines an approximate Riemann solver with Lagrangian-inspired advection, iterative stabilization, and tensor artificial viscosity. The scheme is second-order on smooth flows and first-order on discontinuities, with documented accuracy behavior and limitations in shocked multidimensional problems.

  • 4.1 Hydrodynamic algorithm: RAGE uses a two-shock or single-intermediate-state approximate Riemann solver with ADX to compute face impulse and work.The Riemann solution also determines a donor-zone Lagrangian particle trajectory for advection.
  • 4.1 Hydrodynamic algorithm: Iterating volume changes, sound speeds, and the Riemann solver enables semi-implicit handling of collapsing bubbles and strongly shocked zones.The coupling through common neighbors is intended to prevent unphysical results from independent updates.
  • 4.5 Accuracy and Limitations: The method is 2nd-order accurate in space and time for smooth problems and 1st-order accurate for discontinuities or shocks.Smooth wave tests produce convergence ratios near 2, while shock problems are closer to 0.75; a Mach 10 shock shows nearly factor-of-2 error reduction under refinement.
  • 4.3 Riemann Solver: Velocity and pressure gradients are included in reconstructing face mass flux, extending standard Riemann-solver logic beyond pointwise states.RAGE treats the Riemann solution as a half-time point value and varies the velocity jump along the particle path.
  • 4.3.2 Artificial Viscosity: Tensor artificial viscosity adds axis-aligned error that vanishes in smooth flows, symmetrizing multidimensional numerical errors and reducing carbuncle growth.A double Mach reflection calculation demonstrates its effect on vortical flow near the Mach stem.
  • 4.5.2 Grid Imprinting and Grid Seeding: In spherical adiabatic compression, density L1 error remains near machine accuracy until first-order error enters from the outer pseudo-inflow boundary.For shocked converging shells, grid-seeded perturbations can drive significant instability growth after shock reflection near the origin.

5 Multi-Material Equations of State and Mixed Cells

RAGE treats all computational cells as potentially mixed-material zones and uses MMEOS to enforce pressure and material-temperature equilibrium. Preprocessed inverted tables make mixed-cell EOS evaluation scale linearly with the number of materials, although the approach is difficult to extend to separate ion and electron temperatures.

  • Mixed-cell assumptions: RAGE assumes that mixed cells contain multiple materials in pressure and material-temperature equilibrium.This provides a unique mixed-cell temperature and pressure under the code’s modeling assumptions.
  • MMEOS method: RAGE preprocesses pure-material equations of state into inverted vm(P, T) and em(P, T) tables for repeated mixed-cell lookups.The preprocessor supports SESAME and several analytic EOS forms.
  • MMEOS method: MMEOS constructs total volume and energy by mass-weighting each material’s specific-volume and specific-energy tables.The inputs include material masses, total mass, zone volume, and total energy.
  • MMEOS method: The mixed-cell inversion iterates across bounding isotherms and pressures until the equilibrium state matching the cell’s total energy is found.MMEOS interpolates along isotherms and updates pressure using weighted thermodynamic derivatives.
  • Performance: Mixed-EOS lookup time is linear in the number of materials, unlike pressure-balancing methods that can scale quadratically.Despite this favorable scaling, MMEOS can consume half the runtime of small pure-hydrodynamics problems with five materials.
  • Limitations: MMEOS is not easily adapted to the 2-T regime because separate ion and electron temperatures require higher-dimensional tables and additional constraints.The extension would require arrays such as vm(Ti, Te, Ptot) and separate energy-related tables.

6 Gravity

RAGE supports constant, analytic, and self-gravity models, incorporating gravitational effects into its Godunov hydrodynamics. Its self-gravity solver uses a hierarchical multipole approximation with an error criterion, but currently applies only to 3D non-periodic problems.

  • Gravity models: RAGE incorporates constant acceleration, spatially varying analytic potentials, and self-gravity, usable in combination.Analytic options include single- and binary-star potentials and a power-law star-cluster potential.
  • Self-Gravity: Self-gravity determines potential from the global mass distribution by solving Poisson’s equation.Unlike local hyperbolic hydrodynamics, Poisson’s equation requires a global implicit calculation.
  • Self-Gravity: RAGE uses the Salmon–Warren–Winckelmans hierarchical multipole method to reduce direct-sum gravity cost from O(N^2) toward approximately O(N log N).Distant cells are grouped and represented through multipole expansions.
  • Self-Gravity: The treewalk applies the multipole approximation when its MAC is satisfied and recursively opens source cells otherwise.The process continues until the criterion is met or leaf cells are reached.
  • Self-Gravity: RAGE uses a quadrupole approximation with p = 2 and typically finds mac tol = .01 sufficient as a relative error criterion.Dropping the B4 term simplifies the critical-radius calculation but makes that radius larger than strictly necessary.
  • Limitations: The self-gravity solver currently supports only three-dimensional problems with non-periodic boundary conditions.Two-dimensional solutions and periodic boundaries are under development, and parameter selection requires user vigilance.

7 Radiation-Matter Energy Exchange in RAGE

RAGE treats radiation–matter energy exchange with an asymmetric time discretization: backward differencing for radiation diffusion and exact ODE integration for material energy. This exponential differencing produces more accurate relaxation and supports larger timesteps through nonlinear iteration.

  • Coupling strategy: RAGE’s radiation–matter coupling produces exponential relaxation, accommodates varying CV and Planck opacity, and solves the resulting nonlinear equations with a novel technique.The approach is designed for high-temperature energy exchange and is compared with alternative treatments.
  • Physical model: RAGE models radiation with gray diffusion, while opacities, mean free paths, material properties, and radiation temperature derive from the equation of state and opacity data.The gray treatment evolves radiation energy density E; spectral information affects µ and λ but is not directly calculated here.
  • Time integration: Each timestep separates hydro, material heat conduction, and radiation phases, with advection and radiation-pressure terms treated explicitly before implicit radiation diffusion.The radiation phase uses the hydro-updated radiation energy as its initial value and solves the diffusion term implicitly.
  • Time differencing: The radiation diffusion equation uses a backward difference that is only first-order accurate in time but stable and robust, while matter energy is integrated as an ODE.Treating the matter equation as an ODE permits the actual time dependence of Φ, θ, and dependent variables to be evaluated during the timestep.
  • Nonlinear solve: The coupled radiation and matter equations remain implicit and nonlinear because material properties and the coupling coefficient depend on evolving temperature and opacity.Integration of the matter equation supplies e+ so the radiation and matter equations can be solved simultaneously.

7.4 Inverting ∆t(Φ+)

RAGE’s exponential treatment of radiation–matter exchange is extended from constant-opacity idealizations to temperature-dependent material properties using selective opacity subcycling. This permits larger global timesteps while handling sharply varying and nonmonotonic opacity behavior.

  • 7.4.1 An Ideal Case: For constant τ, exponential differencing relaxes as exp(−∆t/τ), whereas backward differencing relaxes as 1/(1 + ∆t/τ), transferring more energy at intermediate timesteps.Both schemes agree to first order in ∆t/τ, but exponential differencing approaches equilibrium more closely as ∆t/τ grows.
  • 7.4.1 An Ideal Case: Aluminum data motivate the common µ ∼ θ−3 approximation, but in the 10–30 eV range both µ and CV increase with θ and τ varies approximately as θ−5/2.Thus constant-opacity or constant-specific-heat assumptions are not universally valid.
  • 7.4.2 Φ in the General Case: RAGE detects substantial changes in τ and subdivides only affected cells’ timesteps so material properties remain reasonably constant within each subcycle.The subcycle factors combine multiplicatively, preserving the Φ+ relation even when α is no longer a simple exponential.
  • 7.4.2 Φ in the General Case: Selective subcycling avoids constraining the entire mesh to stiff zones and can permit an order-of-magnitude temperature change where backward differencing would require 5–10% control.The procedure is usually rare and avoids a diffusion-operator inversion, making its cost small relative to diffusion solves in 2D or 3D.
  • 7.4.3 A Practical Illustration of the Iterated Φ: Oxygen opacity rises by almost 10^3 between 1 and 10 eV before falling, a regime where subcycling can traverse an opacity peak correctly.Without subcycling, keeping τ within 10% would require temperature changes no larger than 1.66% because τ ∼ θ−6.
  • 7.4.3 A Practical Illustration of the Iterated Φ: At the radiative Courant timestep, opacity subcycling enables the oxygen Marshak-wave calculation, whereas 2% temperature controls would require 186 timesteps per zone.The comparison uses a 2000-timestep converged result; the leading-edge width still reflects the diffusion scheme’s O(∆t) accuracy.

7.5 The Equations for E+ and e+.

RAGE closes the radiation and material-energy update through a nonlinear system for E+ and e+, while using face-temperature diffusion coefficients to improve Marshak-wave behavior. Its AMR interface treatment favors local coupling and currently accepts formal spatial-accuracy loss at scale changes.

  • 7.5 The Equations for E+ and e+.: The equation of state maps Φ+ to θ+ and e+, while forward-time diffusion coefficients complete the nonlinear coupled system solved for E+ and the remaining variables.The system advances radiation and material energy from t− to t+ at every spatial point and timestep.
  • Spatial discretization: RAGE’s rectilinear unit-aspect-ratio cells simplify spatial differencing, with an O(∆x) face gradient producing O(∆x2) zone-centered divergences.Only the gradient component normal to a face contributes to its divergence.
  • Spatial discretization: For diffusion interfaces, RAGE enforces equal left, right, and net centroid-to-centroid fluxes to determine face quantities.Optical depths and centroid-to-interface distances enter the resulting face-flux calculation.
  • Face treatment: Using zone-center diffusion coefficients can suppress Marshak-wave radiation flow, so RAGE estimates a common face temperature before evaluating each side’s coefficients and optical depths.The coefficients can still differ because adjacent cells may have different densities or constituents.
  • AMR interfaces: Across AMR scale changes, RAGE splats central values across larger cells, formally reducing spatial accuracy but keeping the diffusion matrix coupled only across shared interfaces.The error is mitigated because AMR places scale changes where gradients are relatively small.

7.7 Solution Techniques for Non-Linear Systems

RAGE advances radiation–matter diffusion through an iterative nonlinear solve that combines estimated coefficients, a linearized coupled-energy solve, and corrective cell-wise rebalancing. Tests report close agreement with a reference calculation, accurate endpoint behavior across large timesteps when opacity subcycling is allowed, and generally one or two iterations, while very large timesteps remain non-guaranteed.

  • Iterative solution: RAGE’s timestep iteration has three steps: estimate the advanced state, solve a coefficient-linearized radiation diffusion system, then correct each cell’s material-energy balance.The corrective step preserves each cell’s total energy from the linear solve while enforcing the material energy equation.
  • Iterative solution: The linearized solution conserves energy exactly, subject to equation-of-state inversion and linear-solver accuracy.
  • Linear solver: A LAMG algebraic multigrid preconditioner typically speeds the linear solver by a factor of five or more versus point-Jacobi preconditioning on moderate-to-large 2D and 3D meshes.
  • Iterative solution: Normally, one or two iterations suffice, with termination currently based on sufficiently small changes in total cell energy.RAGE is also investigating termination based on the nonlinear residual.
  • Verification: RAGE and Ψtc produce temperatures in very close agreement for the same zoning and timesteps in a comparison problem.The comparison uses RAGE histograms and a Ψtc curve at 20 ns.
  • Verification: Two 10^-8 s timesteps reach the correct electron temperature at a given time when opacity subcycling is allowed, although intermediate-time information is lost.
  • Verification: RAGE’s exponential differencing keeps radiation and material temperatures in equilibrium in the self-similar test, aside from slight early-time disequilibrium near the origin.The comparison is affected because the radiation package cannot include the extra 4aθ^3 contribution to total specific heat.
  • Limitations: Very large-timestep results are described as non-divergent rather than accurate, because backward diffusion differencing is first-order and convergence lacks a theorem.The approach permits radiation matrix solves without forcing other physics packages onto the smaller effective timestep.

8 Verification and Validation for SAGE and RAGE

SAGE and RAGE are assessed through analytic and semi-analytic verification problems, regression tests, and quantitative comparisons with experiments. The reported results include adaptive-mesh accuracy and timing benefits, gas-curtain agreement, and gas-cylinder velocity agreement within 10–15 percent.

  • Verification: The verification suite includes Sedov, Guderley, hydrodynamic, radiation-hydrodynamic, LTE, non-LTE, and Marshak-wave problems.
  • Verification: Adaptive meshing achieves the same accuracy in less time and with fewer zones than fixed meshes in the Sedov comparison.Fixed-mesh L1 errors scale consistently with Δx^0.6–0.75 for shock problems, while shockless problems behave like Δx^2.
  • Verification: Many verification problems are included in LANL regression tests as part of software quality assurance.
  • Validation: The gas-curtain validation initializes the code with a measured, background-subtracted and smoothed density profile to compare calculated and experimental shock-tube results at approximately 450 ms.
  • Validation: Validation efforts span shock-tube, laser-driven, pulsed-power, laser-induced-shock, and asteroid-impact or tsunami applications.
  • Validation: Gas-cylinder density images agree when diffuse measured profiles are used, and the corresponding velocity data agree to within 10–15 percent.Fog tracers provide velocity-field measurements from two closely spaced images.

9 Summary

RAGE combines Godunov-based adaptive hydrodynamics with implicit gray radiation diffusion and extensive verification and validation. Its radiation coupling supports large timesteps by controlling radiation-energy changes rather than material-temperature changes, though some timestep and splitting controls remain under development.

  • Verification and validation: The code and its packages are subjected to substantial analytic verification and experimental validation efforts.
  • Hydrodynamics: RAGE’s Godunov hydrodynamics uses a second-order alternating-direction explicit method that determines the maximum timestep.Implicit anti-cavitation logic handles equation-of-state stiffnesses without timestep control based on relative density or mass-fraction changes.
  • Radiation diffusion: The gray radiation package uses first-order backward Euler in time and second-order spatial differencing, with exact material-energy integration for opacities proportional to T^-3.Other opacity behaviors are handled to specified accuracy through subcycling, followed by linearization and Newton-Raphson rebalancing.
  • Radiation diffusion: RAGE’s radiation treatment is effectively maximally implicit: accuracy depends on relative radiation-energy changes, independent of material-temperature changes.
  • Timestep control: The default timestep control targets relative total-energy change, with an example tolerance of sie pct = 0.2.Material-temperature control remains possible, but the splitting logic is intended to move toward material- and radiation-energy controls.
  • Practical consequence: Implicit coupling and Newton-Raphson corrections allow timesteps considerably larger than traditional operator-split temperature-change limits, enabling more highly resolved problems within resource constraints.

A A Less Ideal Opacity

For power-law opacity, exponential differencing can be treated analytically, reducing the update to a relation that determines the advanced radiation variable from known previous-state quantities. A less ideal opacity case yields seemingly more reasonable results across multiple initial parameter values.

  • Analytic treatment: Exponential differencing remains analytically tractable when the optical-depth parameter has a power-law dependence on temperature and radiation energy.
  • Analytic treatment: The hypergeometric-function relation uses known Φ−, λ, Φ∗, τ∗, and Δt to determine Φ+ as a function of total energy, potentially through numerical root finding.
  • Analytic treatment: When λ is a multiple of 1/4, the hypergeometric function can be expressed using standard elementary functions.
  • Energy coupling: The parameter α governs energy transfer between matter and radiation during a timestep, and its behavior is examined for the less ideal opacity case.
  • Results: Figure 33 compares three values of φ−; the φ− = 0 curve matches the exponential curve from Figure 18, while the more sophisticated treatment gives seemingly more reasonable results.

B Alternative Face Temperature Calculations

RAGE evaluates several alternatives for calculating face temperatures in gray diffusion, including optical-depth and spatial interpolation assumptions. The authors report that the main approaches model Marshak-wave problems well, while a newer averaging method may be preferable for two-temperature opacities.

  • Alternative interpolation assumptions: RAGE considers optical-depth linearity and spatial linearity as alternative assumptions for interpolating face temperatures.The optical-depth approach applies to θ^4, while spatial approaches interpolate material temperature or related quantities.
  • Opacity dependence: For fully ionized material, free-free opacity scales with temperature as T^-1/2 before frequency averaging produces a two-temperature opacity.The resulting mean free path scales approximately as T^1/2 when the weak electron-temperature dependence is ignored.
  • Flux construction: The resulting gray-diffusion flux is approximately proportional to ∇E^2 after ignoring quarter-power factors.This motivates a finite-difference flux based on differences in E^2 across adjacent zones.
  • Observed behavior: All three established face-temperature approaches tested by the authors model Marshak-wave problems well.The authors had not examined the final averaging method in detail, but expected it to resemble the linear-in-space average of θ^4 in most cases.

C Improved Difference Scheme for T-Cells

RAGE improves its difference scheme at T-cell interfaces by enforcing piecewise continuity and flux continuity across nonconforming faces. Corrections substantially reduce errors in Cartesian and cylindrical steady conduction tests, although partial corrections can be severely inaccurate.

  • Improved difference scheme for T-cells: The scheme combines optical-depth-weighted energy values and face optical depths to construct a common flux across adjacent zones.In three dimensions, the face optical depth sums contributions from the central zone and four smaller neighboring zones.
  • Improved difference scheme for T-cells: At T-cell interfaces, RAGE enforces piecewise continuity of radiation energy using face-area-weighted values across the adjoining zones.The relation EA = a1EB + a3EC uses normalized interface areas, with geometry-dependent area fractions.
  • Verification results: 10^-13 L∞ temperature error is achieved in Cartesian geometry after including face effects, versus approximately 0.5% with the uncorrected scheme.The machine-accuracy result requires correcting the face optical depth and flux divergence, without modifying the matrix.
  • Verification results: In cylindrical geometry, full correction reduces the L∞ error to approximately 0.1% and restores O(∆x^2) behavior.Correcting face optical depth and flux divergence without correcting the matrix instead produces errors comparable to the temperature, exceeding 100%.
  • Implementation boundary: The authors defer partial fixes until the implementation is verified to run equally well on multiple processors and on one processor.They explicitly judge doing nothing preferable to applying an incompletely validated correction.

D Spillman’s variant of the Marshak/Milne Boundary Condition

Spillman’s boundary-condition variant introduces a user-specified vacuum length to bridge optically thin and thick regimes. It avoids vanishing flux in optically thick boundary zones, unlike the standard Marshak/Milne or Dirichlet treatment, but large adjacent-zone optical depth can suppress intended inflow under the standard condition.

  • Boundary-condition construction: Spillman’s variant uses a specified vacuum boundary length λV together with the zone mean free path λZ to define the boundary treatment.The zone optical depth is computed as ∆τZ = ∆sZf/λZ.
  • Boundary-condition construction: The Spillman boundary flux depends on an exponential optical-depth term and a factor involving the ratio ξ.Its limiting behavior changes according to the relative sizes of λZ, λV, and the zone width.
  • Limiting behavior: In optically thin zones, Spillman retains Milne-like behavior even when the user chooses λV poorly.The authors describe this as a robust feature of the variant’s thin-zone limit.
  • Boundary limitation: A standard Milne inflow condition can produce flux approaching zero when the boundary-adjacent zone has enormous optical depth.The authors identify this as contrary to the inflow expected when the adjacent zone is thinner.
  • Limiting behavior: In optically thick zones, Spillman’s numerical prefactor remains within roughly a factor of two of the optically thin Milne result instead of vanishing.The exponential term vanishes in this limit, while the denominator remains between unity and 4/3.
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