Physical setup
The source is q(r) = ℒ exp(−r²/w²)/(π3/2w³), with constant positive Rosseland extinction χ and zero Planck absorption. The material stays fixed and receives no recoil or heating. The source deposits its exact cell-integrated radiation energy during each accepted timestep.
The initial radiation energy is a cell average of the steady linear-diffusion equilibrium. On the finite cube, each boundary uses Dirichlet energy sampled from the same equilibrium, averaged over the boundary face by four-point Gauss–Legendre quadrature in each tangential direction. FLD derives its own initial and evolved flux from its nonlinear limiter; no independent flux is prescribed. The recorded parameters give the source luminosity, width, background, and opacity.
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. For D₀ = c/(3χ), the stationary equations ∇·Fref = q and Fref = −D₀∇Eref give
Fref(x) = [ℒ/(4πr³)] [erf(s) − 2s exp(−s²)/√π] x, s = r/w
The expressions have finite limits at the origin, with zero flux there. Initial and comparison cell averages use four-point Gauss–Legendre quadrature on each axis. The reference is time independent and solves the linear diffusion problem with the prescribed source and boundary energy.
Nonlinear FLD uses λ(R) rather than 1/3, so finite-gradient limiter effects can move the numerical state away from this linear equilibrium. The reference flux is −D₀∇Eref, not the nonlinear limiter applied to Eref. Reported slice differences include that model mismatch, spatial and temporal error, boundary discretization, and quadrature error.
What this test measures
Inspect equilibrium drift, radial symmetry, positivity, the derived flux bound, and the energy budget. The finite cube intercepts only a fraction of the Gaussian's all-space luminosity ℒ: the integrated source is ℒ[erf(L/(2w))]³. The change in stored energy plus outward boundary transport must match this in-domain injection. The imposed equilibrium boundary can exchange energy with the domain; it is not a vacuum boundary. Slice differences do not imply an exact nonlinear-FLD convergence order.
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 analytic Gaussian-emitter solution, its volume quadrature, and its exact source cell integral are implemented by equilibriumSphere, sphereAverage, and bulbSourceAverage in octotiger/test_problems/radiation/profiles.hpp. The construction follows by integrating the enclosed spherical source luminosity and then the linear diffusion gradient.
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.