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
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.
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
Skinner & Ostriker (2013), “A Two-moment Radiation Hydrodynamics Module in Athena Using a Time-explicit Godunov Method,” ApJS 206, 21, provides context for the two-moment M1 system and source integration. For the radiation-only Gaussian and sphere cases, Euler uses backward-Euler damping: Fn+1 = F*/(1 + cχΔt), where F* is the flux after transport. This uses the algebra of their equations (42b)–(43) with θ = 1 and prescribed χ. Their paper neglects scattering; the suite reuses that damping formula for its prescribed scattering medium. ADER2 revision 6 uses the same two-stage Radau IIA local integration for all eight problems. Transport is evaluated on the old state, a predicted time average and a corrected time average, with reconstruction recomputed each time. Prescribed-medium constraints are applied within the implicit stages: the Gaussian and sphere tests hold the material fixed; Marshak holds density and velocity fixed while thermal energy evolves; streaming disables interaction. The coupled problems evolve gas and radiation together on one shared timestep. This implementation does not yet include gravity, and uniform stiff coupled second-order accuracy remains unresolved. Historical revision 5 used exponential affine-forcing solutions for prescribed scattering, SDIRK2 for stationary Marshak heating, and Radau stages for coupled hydro–radiation. Those saved results retain their original method revision and do not validate revision 6. Here the formulas use physical light speed; reduced-speed runs retain their recorded transport speed.
The solver's M1 implementation also cites Hanawa & Audit (2014), “Reformulation of the M1 model of radiative transfer,” JQSRT 145, 9–16. These references describe the numerical/physical framework; the exact test definitions are the source functions identified above.
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.