Physical setup
The supercritical case couples hydrodynamics, absorption, emission, and radiation momentum exchange. In a developed supercritical shock the gas immediately upstream is heated approximately to the downstream equilibrium temperature, while a narrow gas temperature spike remains at the front.
Ensman's original shell, density, opacity, initial temperature gradient, and ideal-gas parameters match the subcritical calculation. The piston speed is increased to 2 × 106 cm/s. The local Cartesian adaptation uses the same box, temperature gradient, density-dependent absorption, reflecting left wall, right inflow, and transverse boundaries as the local subcritical case, with vx = −2 × 106 cm/s and default duration 7500 s. A planar wall and M1 moment closure replace the original moving spherical piston and transfer treatment.
FLD uses a first-order conservative gas/radiation update and implicit nonlinear diffusion/material exchange at physical light speed. The limiter supplies radiation pressure. The stored flux is the derived lab-frame flux Flab = −D∇E + (EI+P)·v; it is not an independently evolved radiation momentum field.
These cases enable gas dynamics and radiation coupling on a Cartesian grid; gravity is disabled. Uniform runs are the default; adaptive runs are identified in their recorded setup. Each completed run records its actual configuration, time, units, and executable hash below. The configured finest equivalent cells per side are N = INX × 2level.
Comparison and interpretation
No independent analytic or digitized reference profile is supplied. Published benchmark figures identify the physical setup, but their curves are not embedded and these Cartesian adaptations are not identical to the original geometries. M1/FLD field differences compare transport closures and discretizations, not error against an exact shared solution.
What this test measures
Check gas/radiation total energy including boundary flux, nonnegative density/thermal/radiation energies, shock or blast development, and resolution/timestep sensitivity. The comoving diffusive flux is bounded by cE; the lab-frame flux additionally contains enthalpy advection and can exceed cE. These tests do not by themselves establish dynamic-diffusion asymptotic 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 the numerical profiles
Gas and radiation temperatures are distinct diagnostics; the radiation temperature is defined by E = arTr4. The central ray lies at y = z = Δx/2 and is offset from the exact origin by half a cell in each transverse direction. The two-dimensional plane is z = Δx/2.
Check positivity, the recorded-frame radiation flux envelope, conservation with the appropriate boundary budget, thermal equilibration, shock speed and precursor structure, and sensitivity to resolution and timestep. A finite, positive short run is an implementation check, not evidence that a subcritical or supercritical state has developed. Comparisons to published figures also require matching geometry, opacities, boundaries, and physical time.
Where this problem comes from
Ensman (1994), ApJ 424, 275, §5.2, Figures 10–15; original paper PDF. The subcritical and supercritical labels describe physical regimes, rather than universal velocity thresholds independent of the surrounding medium.
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.