Physical setup
This test evolves gas and radiation together. The intended subcritical regime has a radiative precursor with Tgas,pre < Tdownstream; radiation and gas temperatures need not agree in the precursor or at the shock spike.
Ensman's original calculation used a moving spherical piston at 6 × 105 cm/s in a shell spanning 8 × 1011 to 8.7 × 1011 cm. Initially ρ = 7.78 × 10−10 g/cm³, the absorption coefficient is 3.115 × 10−10 cm−1, and the 300 equal-mass zones begin at 10 K at the outer edge, rising by 0.25 K per zone inward. The gas uses γ = 5/3 and fully ionized hydrogen (mean molecular weight μ = 0.5).
Local Cartesian adaptation. The box spans −3.5 × 1010 to +3.5 × 1010 cm. Gas moves toward a reflecting left wall at vx = −6 × 105 cm/s, with fixed inflow at the right face and zero-gradient transverse boundaries. Temperature varies linearly from 85 K at the wall to 10 K at the inflow. Absorption follows χ = 3.115 × 10−10(ρ / 7.78 × 10−10 g/cm³) cm−1. The default duration is 38300 s. A stationary planar wall replaces the spherical moving piston; the M1 moment closure replaces the original 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 profile is supplied for the coupled problem. Ensman (1994), Figure 9 at t = 3.83 × 104 s, is an original numerical comparison target. Its curves are not embedded here, digitized, or used to manufacture error norms.
Approximate analytic shock scales
Let U be the imposed wall-frame inflow speed and R = kB/(μmH) = 1.65022 × 108 erg/(g K). A strong, gas-pressure-dominated adiabatic shock with negligible upstream pressure has
T2 ≈ 2(γ − 1)u1²/[R(γ + 1)²] = U²/(3R)
Here u1 is the upstream speed relative to the shock and s is the outward shock speed in the wall frame. For U = 6 km/s, this gives s ≈ 2 km/s and T2 ≈ 727 K. These are approximate conservation-law scales, not an analytic transient profile or an error threshold. Upstream heating, finite Mach number, and radiative energy losses modify the jump.
What this test measures
Inspect the radiation precursor, the difference between gas and radiation temperatures, the gas temperature overshoot, and the shock speed. A subcritical shock has gas immediately upstream cooler than the downstream equilibrium gas. The physical regime must be assessed after sufficient evolution; the case name alone does not demonstrate that the regime has been reached.
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), “Test Problems for Radiation and Radiation-Hydrodynamics Codes,” ApJ 424, 275, §§5.1–5.2, pp. 283–285, describes the original radiative shocks. Original paper PDF. The original spherical geometry, piston boundary, and time must be matched before claiming quantitative agreement with its figures.
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.