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.
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:
ξ = √3χ(x+1), τ = χt, Eref = 4u(ξ,τ), eref = 4v(ξ,τ)
Fx,ref = −(4/√3)∂u/∂ξ, Fy,ref = Fz,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.
All resolutions
No completed run is available for this problem in this batch. Its description is available above; results will appear here as runs finish.
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.
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.