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.
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
This problem is available for M1 RK1 only in the current runner. No FLD result is claimed.
No exact analytic solution is supplied. Ensman (1994), Figure 13 at t = 7.5 × 103 s, provides a numerical comparison target. Figure 10 also shows evolution at 860, 4000, 7500, and 13000 s. Differences of geometry, equation of state, opacity, or boundaries change those profiles.
Approximate analytic shock scales
For γ = 5/3, μ = 0.5, and imposed wall-frame speed U, a strong, gas-pressure-dominated adiabatic baseline gives ρ2/ρ1 = 4, shock-relative upstream speed u1 = 4U/3, wall-frame shock speed s = U/3, and T2 ≈ U²/(3R), where R = kB/(μmH) = 1.65022 × 108 erg/(g K).
For U = 20 km/s the baseline is s ≈ 6.67 km/s and T2 ≈ 8080 K. 8080 K is not the predicted plateau of the cooled radiative shock. Radiative losses can increase compression above four and change both speed and plateau temperature; Ensman's radiating calculation has a plateau near 4000 K in Figure 15, versus about 8000 K for its adiabatic comparison.
In the approximate steady supercritical, gas-pressure-dominated diffusion picture, T− ≈ T2 and T+ ≈ (3 − γ)T2 = 4T2/3. T− and T+ are the gas temperatures immediately ahead of and at the spike; these relations use the cooled downstream plateau, not the adiabatic estimate. They are qualitative checks, not exact M1 reference curves or pass/fail tolerances.
What this test measures
Look for precursor heating to the downstream equilibrium temperature, the narrow gas spike, and the asymmetric radiation flux. Ensman's developed solution includes negative radiation flux behind the shock. Refine space and time to distinguish a resolved precursor and spike from numerical smearing; a short smoke run cannot demonstrate these late-time features.
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.
Unsupported by the stationary FLD backend; no FLD run is planned for this problem.
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, radiation realizability |F| ≤ cE, 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.
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.