Physical setup
The Gaussian energy and fixed material state use the same initial energy cell averages as the M1 case. The prescribed scattering opacity is constant, the boundaries are periodic, and there is no radiation energy source, gas motion, or material heating. M1 starts with zero evolved flux; FLD instead derives its initial flux from the initial energy gradient, so the models do not share an independent flux initial condition.
The saved run settings specify the Gaussian background, amplitude, width, opacity, domain, and actual final time. The Fourier generator stores the initial energy and the analytic linear diffusion evolution, with reference flux derived from that reference energy.
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
Analytic diffusion-limit reference; not an exact solution of nonlinear FLD. The FLD reference solves the periodic constant-coefficient heat equation, rather than the M1 telegraph equation:
δÊ(k,t) = δÊ(k,0) exp(−D₀|k|²t)
F̂ref = −iD₀kδÊ(k,t)
The generator transforms the same periodized Gaussian cell averages used to initialize the simulation and removes the same Nyquist planes. Each retained mode evolves analytically; the constant background is unchanged. Reference flux is the analytic gradient of those modes, including at the initial time. Thus the reference is exact for linear diffusion of the represented, filtered initial energy, up to floating-point evaluation.
The actual solver uses D = cλ(R)/χ, where R = |∇E|/(χE) and λ(R) = (2+R)/(6+3R+R²). Only as R → 0 does λ → 1/3. Reference differences at finite R include this model difference as well as spatial and temporal errors. The reported L1, L2, and maximum differences cover the exported projected slice, not the full three-dimensional volume.
What this test measures
Inspect energy spreading, positivity, the bound |F| ≤ cE, and periodic energy conservation. Compare resolutions and timesteps separately. A paired M1/FLD curve difference includes a difference between physical closures and transient flux relaxation, not only discretization error. Initial spectral filtering also depends on resolution.
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 periodic heat-equation Fourier solution above is implemented by radiation_results/cpp/gen_radiation_reference.cpp --model fld. The generator shares initial energy construction with M1, but evolves each mode by diffusion instead of the M1 telegraph propagator. The ReferenceData reader uses the resulting cell-average arrays.
Frigo & Johnson (2005), “The Design and Implementation of FFTW3”, describes the transform library used for the Fourier construction.
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.