Subcritical radiative shock

Physical setup

This test evolves gas and radiation together. The intended subcritical regime has a radiative precursor with T_gas,pre < T_downstream; 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 v_x = −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 × 2^level.

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.

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 = a_rT_r⁴. 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

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.