Physical setup
The periodic cube starts with E = 1 in the left half and E = 10−10 in the right half, using exact cell averages at the edges. The field is uniform in the transverse directions. The domain, initial energy, recorded scattering extinction, and absence of emission match the M1 streaming-front problem. Gas dynamics and gravity are disabled.
M1 initializes a directed streaming flux. FLD instead derives its flux from the energy gradient. At zero extinction and nonzero gradient, the Levermore–Pomraning vacuum limit is F = −cE ∇E/|∇E|. The new thin runs use explicit positive extinction with optical depth χL = 10−4, giving finite D at flat positive-energy regions. Legacy zero-opacity runs retain their saved settings. A spatially varying periodic field therefore has oppositely directed fluxes on opposite slopes; it cannot preserve M1's single beam direction.
The standard runner uses a uniform cubic grid with hydrodynamics and gravity disabled. Its CGS box spans −3 × 10¹⁰ to +3 × 10¹⁰ cm on each axis, with c = 2.99792458 × 10¹⁰ cm/s and default final time 4 s. Opt-in adaptive runs are identified below; N = INX × 2level then describes their configured finest equivalent resolution. Each completed run below lists its recorded parameters.
Comparison and interpretation
No analytic spatial solution of this nonlinear thin-opacity FLD problem is supplied. The translated M1 sinusoid or top-hat solves zero-opacity directed M1 transport; it is only a limit reference at positive opacity and does not solve the FLD closure. A constant-diffusion heat solution is not substituted. Accordingly, FLD exports numerical profiles without analytic-reference or error columns.
An exact integral reference follows from the energy equation and periodic boundaries:
The exact initial and conserved mean is ⟨E⟩ = (1 + 10−10)/2, so the total radiation energy is L³(1 + 10−10)/2.
What this test measures
The report compares the complete recorded energy history with that exact integral and checks E ≥ 0 and |F| ≤ cE on the exported slice. These necessary checks do not validate the spatial profile. Resolution and timestep studies measure numerical sensitivity; paired M1/FLD curves measure closure differences, not analytic errors. FLD does not evolve or conserve independent flux moments, so M1 momentum-ledger tests do not apply.
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.
Momentum plots are omitted for this fixed-material problem. The prescribed medium acts as an external momentum reservoir, so radiation momentum exchange does not imply that a compensating gas momentum was evolved.
Reading FLD diagnostics
Gaussian, sphere, and Marshak profiles include analytic linear diffusion-limit references; Marshak also includes material energy. Those references omit the nonlinear limiter, and Marshak assumes a semi-infinite domain. Slice discrepancies combine these model differences with discretization and reference-evaluation errors. Thin positive-opacity wave/front cases instead have numerical-only profiles and exact periodic energy-integral checks. 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 radiation sources S, including material exchange and any prescribed emitter. 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
The shared initial energy comes from streamingWave or streamingFront in octotiger/test_problems/radiation/profiles.hpp. The vacuum flux follows directly from the large-R limit λ(R)R → 1 of the implemented limiter. The integrated-energy reference follows from the divergence theorem and periodic boundaries.
Levermore & Pomraning (1981), “A flux-limited diffusion theory,” ApJ 248, 321–334, derives flux-limited diffusion and its classical diffusion limit. Saved reference metadata records whether an analytic profile is supplied and distinguishes its 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 uses backward-Euler radiation diffusion/material exchange with the Levermore–Pomraning limiter. Moving cases add first-order conservative gas transport and radiation force/work; their reconstructed lab-frame flux includes (E I+P) v as well as −D∇E. The scalar implicit transport solve uses composite-grid multigrid and an iterative linear solver. Both time integrators are first order.
The suite includes Gaussian scattering, Marshak heating, a Gaussian emitter, explicit thin positive-opacity wave/front comparisons, and moving-gas shock/blast problems on CPU at physical light speed. A configured case does not establish that a solve converged; only validated completed runs appear as results. Flux diagnostics in FLD 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.