Physical setup
The normalized cube spans −1 ≤ x,y,z ≤ 1. The material density and momentum are fixed; radiation exchanges energy with the material through a linear equilibrium law Eeq = e. The propagation speed is c = 1, absorption and extinction are both χ = 100, and the default final time is t = 1. Equal material and radiation energy capacities correspond to ε = 1. Gravity and hydrodynamic transport are disabled.
This is a heating test in normalized units, rather than an ideal-gas temperature benchmark. Plots show E/4 and e/4. The saved metadata and source record the actual boundary treatment and reference quadrature.
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
The Su–Olson reference is the solution of a linear, gray, nonequilibrium diffusion problem in a semi-infinite stationary medium:
∂te = χ(E − e)
With ξ = √3χ(x + 1), τ = χt, u = E/4, and v = e/4, the reference has u − (2/√3)∂ξu = 1 at the illuminated boundary and zero initial u and v. The existing integral quadrature supplies radiation energy, material energy, and diffusion flux. Reference values are evaluated at cell centers; the exported comparison does not apply an additional cell-volume average.
What this test measures
Inspect the penetration depth and delay between radiation and material heating. The full-volume L1, L2, and maximum norms include material energy as well as all four radiation moments. Nonlinear M1 has finite propagation speed and a closure different from diffusion near the illuminated boundary and early front. Consequently these norms include model and boundary differences; they are not an exact-M1 order test. A zero transverse-flux error has no logarithmic convergence order. The numerical boundary combines a prescribed incoming half-isotropic bath with outgoing moments from the interior M1 intensity. Its finite-grid ghost-cell/HLL realization and early causal transient can differ from the diffusion boundary solution.
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.
Errors, budgets, and movies
Every norm uses the full three-dimensional domain, with cell-volume weighting. Here e = numerical − reference and V is the domain volume:
p = ln(errorcoarse / errorfine) / ln(Δxcoarse / Δxfine)
Orders require two completed runs with matching physical time, configuration, units, executable, and origin. An exact zero error has no logarithmic order. The central slice and line profiles are visual diagnostics: the report slice is the cell layer at z = Δx/2, and its line profile is at y = z = Δx/2.
Conservation is assessed using the measured domain integrals and the signed cumulative boundary and source budgets:
A labeled checkpoint continuation uses its checkpoint as t_ref, with counters and fixed normalization starting there. Its reported residuals cover that resumed segment only; original histories remain separate.
B is outward transport and S is the source change. Conservation plots and this page's budget tables show fractions of the initial totals: Q(t)/D, [Q(0) − B(t) + S(t)]/D, and the signed residual R(t)/D. The denominator D = |Q(0)| is fixed for the entire history. It never grows with the current total or accumulated transport/source budgets. A negative initial flux therefore starts at −1, and the plotted fraction retains its sign.
A component whose initial net flux is zero or negligible cannot be divided by its own initial total. It instead uses the fixed scale D = c |E₀|, where E₀ is the initial integrated radiation energy. This fallback is labeled on its panels and in the table. “Negligible” means |Q(0)|/(c |E₀|) ≤ 64ε ≈ 1.421085 × 10−14, with ε the double-precision machine epsilon. This prevents roundoff left by cancellation in a symmetric field from becoming the denominator. For the initially cold Marshak case, the prescribed bath sets D = Ebath V for energy and D = c Ebath V for flux, with Ebath = 4/c. These scales remain fixed even though radiation enters the initially empty domain.
Raw histories and dimensional summary columns remain in physical units: energy integrals in erg, physical-flux integrals in erg cm/s for CGS runs. Dividing a flux integral by c² gives radiation momentum. Fraction-history CSV files are dimensionless except for their time column; summary files record each denominator and its basis. Maximum fractional residuals use the same fixed denominator as the final residual.
For runs labeled RSLA, these budgets check the modified radiation equations. They do not establish conservation of physical gas-plus-radiation energy. With reduced-speed factor α, the closed-system coupled invariants are Egas + Er/α and gas momentum + F/(αc²), with physical c in the closure. Compare multiple α values before interpreting an RSLA result as an approximation to physical-c evolution.
Movies show the numerical Silo states with a fixed color range for each movie. Recorded states are held for their actual simulation-time gaps; intermediate solutions are not interpolated. The 3D option shows the domain's exterior surface. Movies and plots retain their recorded physical units.
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.
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.