Marshak wave · Octo-TIGER

The experiment

Physical setup

The cube spans −1 to +1 in normalized units, with c = 1, Planck and Rosseland extinction 100, and equal radiation/material energy capacities. Initially E = 0 and material energy is 10⁻²⁰. The material density and velocity remain fixed while its thermal energy evolves.

The left boundary supplies an incoming hemisphere flux of 1 through the FLD Robin condition cE/2 + D ∂E/∂n = 2. The right boundary is vacuum, cE/2 + D ∂E/∂n = 0; transverse faces reflect. M1 uses a moment boundary construction, so the two discrete boundary operators are not identical.

This case uses normalized units with c = 1, opacity 100, and equal material/radiation energy capacities. Hydrodynamics and gravity are disabled. The default cube spans −1 to +1, and the default final time is 1. Adaptive runs retain their actual leaf spacing in exported data.

The comparison

Comparison and interpretation

Analytic diffusion-limit reference; not an exact solution of nonlinear FLD. The Su–Olson solution supplies radiation energy, material energy, and the flux derived from its energy gradient for a semi-infinite linear diffusion problem:

∂_tE − [1/(3χ)]∂_x²E = χ(e−E), ∂_te = χ(E−e)
ξ = √3χ(x+1), τ = χt, E_ref = 4u(ξ,τ), e_ref = 4v(ξ,τ)
F_x,ref = −(4/√3)∂u/∂ξ, F_y,ref = F_z,ref = 0

Here c = 1, χ = 100, and the equal-capacity parameter is ε = 1. The left Marshak condition is u − (2/√3)∂u/∂ξ = 1. Integral representations evaluate the analytic solution numerically. Exported reference values are sampled at cell centers; numerical fields are cell averages, adding a sampling difference.

The FLD calculation instead has a nonlinear flux limiter and a finite domain with a vacuum boundary on the right. These differences matter near the early front and when radiation reaches that boundary. Slice discrepancy norms therefore combine model, boundary, quadrature, sampling, and discretization differences; they are not an exact-FLD convergence test.

What this test measures

Inspect radiation and material heating, penetration depth, nonnegative energies, the derived flux bound, and energy change after accounting for boundary transport and material exchange. Finite-domain and limiter effects matter when interpreting model differences. Short paired runs exercise the implementation and reporting pipeline; they do not establish late-time benchmark accuracy.

Measured results

All resolutions

Level	Grid	Batch status
1	16³	Ready

Analytic diffusion-limit reference comparison. The reference is exact for its stated linear diffusion equations, not for the nonlinear FLD limiter. Flux remains a derived diagnostic; no separate M1 momentum conservation test applies.

FLD backward Euler · 16³ cells

Central cell layer at z=dx/2; ray at y=z=dx/2.

FLD backward Euler with the Levermore-Pomraning limiter; stationary material and physical c = 1 in normalized units. Radiation flux is derived from the radiation-energy gradient; it is a diagnostic, not an evolved momentum variable. Analytic references describe the linear diffusion limit and need not solve nonlinear flux-limited diffusion exactly. M1 versus FLD differences compare distinct radiation models and are not analytic-reference errors.

Recorded setup and run provenance

Origin	Octo-TIGER application
Domain	Cube of side 2.000000e+00 (code length); centered at the origin
Resolution	16³ cells; finest Δx = 1.250000e-01 (code length)
Final comparison time	2.500000e-01 (code time)
Radiation model	FLD
Radiation integration	FLD backward Euler (first order in time)
Flux semantics	Derived FLD flux: F = -D grad E
Analytic reference	su-olson-linear-halfspace; linear diffusion-limit reference, not exact for the nonlinear FLD limiter
Reference norm scope	Area-weighted projected positive-side slice; no full-volume norm or convergence order inferred
Radiation backend	cpu
Physical light speed	1.000000e+00 (code speed)
Reduced-speed factor α	1.000000e+00
Radiation transport speed	1.000000e+00 (code speed)
Periodic boundaries	off
Hydrodynamics	off
Gravity	off
Implicit radiation source	on
Uniform grid	on
CFL	0.4
Timestep cap	0.025000000000000001 (code time)
Output interval	0.125 (code time)

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  "level": 1,
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  "physical_c": 1.0,
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  "radiation_equations": "stationary physical-c FLD",
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  "radiation_limiter": "levermore_pomraning",
  "radiation_model": "fld",
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    "flux": "normalized flux",
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    "time": "normalized time"
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FLD radiation energy, analytic diffusion-limit reference, and signed difference. Central cell layer at z=dx/2; ray at y=z=dx/2. · PDF

Radiation energy and derived FLD flux against analytic diffusion-limit profiles, with signed differences. The reference does not include the nonlinear flux limiter. · PDF

Radiation and material energy relative to bath energy, compared with the Su–Olson analytic linear diffusion reference. · PDF

Derived FLD flux maps

Derived FLD fx: numerical flux, analytic diffusion-reference flux, and signed difference. · PDF

Derived FLD fy: numerical flux, analytic diffusion-reference flux, and signed difference. · PDF

Derived FLD fz: numerical flux, analytic diffusion-reference flux, and signed difference. · PDF

Accepted implicit solves

Accepted steps: 10; substeps: 10; timestep retries: 0; maximum nonlinear residual: 2.669111e-11; maximum absolute coupled energy error per accepted step: 4.284886e-11.

Projected-slice differences from the analytic reference

Area-weighted L1, L2, and maximum differences on the exported positive-side slice; these are not full-volume norms or a pure discretization error. No convergence order is inferred. Energy-density and flux units are those recorded in the slice-units download.

Field	L1 difference	L2 difference	Maximum difference
er	6.587278e-03	2.563794e-02	1.025122e-01
fx	1.304488e-03	4.813163e-03	1.918009e-02
fy	2.995482e-20	1.461902e-19	1.109119e-18
fz	2.339834e-20	1.222801e-19	7.394127e-19
material_egas	6.682921e-03	2.606732e-02	1.042353e-01

Radiation energy conservation

All values below are dimensionless fractions of the fixed normalization scale at t_ref = 0.

Field	Scale	Q(t_ref)/scale	Q(t)/scale	Boundary B/scale	Source S/scale	Residual R/scale	Maximum \|R\|/scale
er	E_bath V	0.000000e+00	2.979603e-03	-5.844643e-03	-2.865040e-03	-9.432713e-12	9.432713e-12

Radiation energy after signed boundary transport and material exchange. Flux moments are not conserved variables in FLD. · PDF

Saved-state movie

Marshak · L1 · FLD backward Euler

3 stored states; 3 shown. Zero-order hold; no solution interpolation. Playback holds each saved numerical state over its time interval and includes endpoint holds.

Snapshot times: 0, 0.125, 0.25 normalized time.

Arrows show derived FLD diagnostic flux -D grad(E); radiation momentum is not evolved. Arrows have equal length and show direction only.

Final-time profiles compare FLD with a linear diffusion-limit analytic reference, not an exact solution of nonlinear flux-limited diffusion. The movie shows numerical states only.

Central z=0+ radiation energy slice with faint cell mesh and equal-length flux direction arrows. Arrows show derived FLD diagnostic flux -D grad(E); radiation momentum is not evolved. Fixed color limits; actual stored Silo times held without solution interpolation. Final-time profiles compare FLD with a linear diffusion-limit analytic reference, not an exact solution of nonlinear flux-limited diffusion. The movie shows numerical states only. The maps show the full simulation domain without cropping.

Download MP4 · Rendering and timing metadata · Full run report

All campaign movies

Data and reproducibility

Effective configuration Run metadata Solver log Numerical/reference/difference slice CSV Slice units FLD diagnostics Accepted solve history CSV Slice reference-difference norms CSV Radiation energy history CSV Energy budget summary CSVFinal numerical Silo (local file; excluded from staged site)

Reading FLD diagnostics

The profiles include an analytic linear diffusion-limit reference for radiation energy and flux, plus material energy in Marshak. The reference omits the nonlinear limiter; the Marshak reference also assumes a semi-infinite domain. Reported slice discrepancies combine these model differences with discretization and reference-evaluation errors. Numerical flux is derived as F = −D∇E and satisfies |F| ≤ cE; it is not an evolved M1 momentum variable.

The radiation energy ledger checks R = E(t) − E(t_ref) + B − S, with outward boundary energy B and material-to-radiation exchange S. Its fixed normalization is initial radiation energy, or bath energy times domain volume for cold Marshak heating. Coupled gas-plus-radiation errors per accepted implicit step are recorded separately in the solver diagnostics.

Nonlinear residuals, substeps, and retries are measured solver diagnostics. The short pilot checks the implementation and reporting pipeline; comparison of models is distinct from an accuracy benchmark or speedup at matched error.

Sources & provenance

Where this problem comes from

Su & Olson (1996), “Benchmark results for the non-equilibrium Marshak diffusion problem,” JQSRT 56, 337–351, gives the nonequilibrium diffusion benchmark. The local reference is implemented in src/test_problems/marshak/marshak.cpp; the run records source hashes for reproducibility.

Levermore & Pomraning (1981), “A flux-limited diffusion theory,” ApJ 248, 321–334, derives flux-limited diffusion and its classical diffusion limit. The saved run's reference metadata distinguishes the analytic linear diffusion equations from the nonlinear limiter evolved by this solver.

Radiation method context

M1 RK1 evolves radiation energy and flux moments with the retained first-order radiation update. FLD evolves radiation energy and stationary material thermal energy with a coupled backward-Euler solve; its flux F = −D∇E is reconstructed from energy with the Levermore–Pomraning limiter. The scalar implicit transport solve uses composite-grid multigrid and an iterative linear solver. Both time integrators are first order.

The FLD runner currently supports the stationary Gaussian scattering pulse and Marshak heating test on the CPU at physical light speed. Streaming wave/front, equilibrium sphere, and moving radiation hydrodynamics are unsupported for FLD. Flux diagnostics in FLD are not independently evolved M1 moments and have no separate M1 momentum conservation test. A difference between paired M1 and FLD solutions is a model comparison, not an analytic error. ADER2 has been removed from the current solver; historical result files retain their original provenance.

Skinner & Ostriker (2013) and Hanawa & Audit (2014) provide context for the M1 equations. The implementation and saved executable/source hashes identify the actual numerical methods used in this batch.

Reproducing this batch

Descriptions document the reviewed source implementation. Older or edited solver versions can use different profile constants or averaging rules; these are not all encoded in run.ini. Use the run metadata, executable hash, and saved batch source hashes to identify the implementation behind a result. A source path or descriptive formula alone is not proof of the executable's contents.

Batch settings and source hashes Batch status