Radiating blast wave · Octo-TIGER

The experiment

Physical setup

The blast problem evolves an initially localized energy deposition with coupled gas and radiation. Its useful diagnostics include shock position, gas and radiation temperatures, density, and outward radiation flux.

Ensman's original problem is a spherical shell from 10¹² to 10¹³ cm with ρ(r) = 10⁻⁶(r/10¹² cm)⁻⁷ g/cm³. The shell is initially in radiative equilibrium with luminosity 10³⁴ erg/s. An energy of 10⁴⁷ erg is deposited in the innermost zone, followed by zero inner luminosity and velocity. The opacity uses the electron-scattering coefficient as pure absorption. This stratified spherical setup is distinct from a uniform-medium Cartesian thermal bomb.

Spherical blast on a Cartesian mesh. The initial density, gas and radiation temperatures, radiation flux, and Gaussian energy deposition depend only on radius. The computational box spans ±10¹³ cm and contains a filled core of radius R₀ = 10¹² cm. Density is 10⁻⁶[max(r,R₀)/R₀]⁻⁷ g/cm³. The 10⁴⁷ erg gas-energy deposition is a Gaussian of width R₀. Exact error-function cell integrals and finite-box normalization preserve its total at every uniform-grid resolution. The initial radiation field uses an LTE diffusion envelope based on luminosity 10³⁴ erg/s and outer E = L/(4πcR_out²), with a regularized core. A streaming floor E ≥ (1 + 10⁻¹²)|F|/c keeps the approximate atmosphere realizable while preserving its radial flux. The central luminosity is removed when evolution starts. The default duration is 22000 s.

The filled core, Gaussian deposition width, Cartesian grid, M1 closure, streaming floor, and approximate LTE diffusion initialization are deliberate adaptations. They differ from the original hollow shell, innermost Lagrangian-zone deposition, and radiation-transfer relaxation.

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.

The comparison

Comparison and interpretation

No analytic radiative blast profile is supplied. Ensman (1994), Figures 17–22, follows the original numerical solution from energy deposition through radiative breakout and later shell formation.

What the Sedov–Taylor limit can tell us

For an energy-conserving point explosion into a cold, uniform medium with negligible radiative losses, the spherical similarity solution gives

R_shock(t) = ξ(γ) [E₀t²/ρ₀]^1/5, v_shock = 2R_shock/(5t)

This is a useful separate adiabatic benchmark. It is not an analytic solution of the present radiating blast: the finite Gaussian deposition, filled core, steep r⁻⁷ envelope, initial radiation field, and escaping luminosity introduce different scales and energy transport. Substituting the envelope exponent into a simple power-law Sedov formula does not provide a valid reference for this truncated atmosphere. Quantitative validation here requires conservation and resolution studies plus an independent radiation-hydrodynamics solution with the same setup.

What this test measures

Track the shock and radiation precursor separately, together with energy exchange and boundary losses. In Ensman's original calculation the precursor breaks out near 5800 s and the luminosity peaks near 9600 s. These times depend on that paper's shell and boundary conditions; they are not pass/fail thresholds for an adapted setup.

Measured results

All resolutions

Level	Grid	Batch status
2	—³	Ready
3	—³	Ready

These are numerical solutions without a supplied analytic reference. No error norms or convergence orders are inferred from the profiles. Short runs used for implementation checks do not establish the published late-time shock or blast structure.

AMR levels 1–2; 32768 active cells; finest equivalent 32³ · actual final time 4.000000e+03 s

AMR: positive-side cells touching z=0, projected as constant leaf averages. Centers have z=leaf_dx/2; the ray similarly touches y=0. This is not a common cell-center plane. Raw CSV preserves native cells; display CSV repeats coarse values on finest pixels, retaining source dx/z. Full-volume norms use actual leaf volumes; uniform-grid convergence orders are not reported.

Early-time implementation check. This run stops before the configured benchmark time and does not validate the developed shock or blast.

Physical-light-speed radiation equations (alpha = 1); c = 29979245800 cm/s.

Recorded setup and run provenance

Origin	Octo-TIGER application
Domain	Cube of side 2.000000e+13 cm; centered at the origin
Resolution	AMR levels 1–2; 32768 active cells; finest equivalent 32³; finest Δx = 6.250000e+11 cm
Slice geometry	AMR: positive-side cells touching z=0, projected as constant leaf averages. Centers have z=leaf_dx/2; the ray similarly touches y=0. This is not a common cell-center plane. Raw CSV preserves native cells; display CSV repeats coarse values on finest pixels, retaining source dx/z. Full-volume norms use actual leaf volumes; uniform-grid convergence orders are not reported.
Final comparison time	4.000000e+03 s
Radiation integration	Legacy first-order radiation update
Radiation backend	cpu
Physical light speed	2.997925e+10 cm/s
Reduced-speed factor α	1.000000e+00
Radiation transport speed	2.997925e+10 cm/s
Periodic boundaries	off
Hydrodynamics	on
Gravity	off
Implicit radiation source	on
Uniform grid	off
CFL	0.4
Timestep cap	20 s
Output interval	100 s

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Numerical gas temperature, radiation temperature, and density. AMR: positive-side cells touching z=0, projected as constant leaf averages. Centers have z=leaf_dx/2; the ray similarly touches y=0. This is not a common cell-center plane. Raw CSV preserves native cells; display CSV repeats coarse values on finest pixels, retaining source dx/z. Full-volume norms use actual leaf volumes; uniform-grid convergence orders are not reported. · PDF

Numerical profiles; no reference curves are supplied. AMR: positive-side cells touching z=0, projected as constant leaf averages. Centers have z=leaf_dx/2; the ray similarly touches y=0. This is not a common cell-center plane. Raw CSV preserves native cells; display CSV repeats coarse values on finest pixels, retaining source dx/z. Full-volume norms use actual leaf volumes; uniform-grid convergence orders are not reported. · PDF

Movies

No completed movie is available for this resolution.

Data and reproducibility

Effective configuration Run metadata Raw conservation segment Solver log Numerical slice CSV Data units Final numerical Silo

AMR levels 2–3; 262144 active cells; finest equivalent 64³ · actual final time 4.000000e+03 s

Early-time implementation check. This run stops before the configured benchmark time and does not validate the developed shock or blast.

Physical-light-speed radiation equations (alpha = 1); c = 29979245800 cm/s.

Recorded setup and run provenance

Origin	Octo-TIGER application
Domain	Cube of side 2.000000e+13 cm; centered at the origin
Resolution	AMR levels 2–3; 262144 active cells; finest equivalent 64³; finest Δx = 3.125000e+11 cm
Slice geometry	AMR: positive-side cells touching z=0, projected as constant leaf averages. Centers have z=leaf_dx/2; the ray similarly touches y=0. This is not a common cell-center plane. Raw CSV preserves native cells; display CSV repeats coarse values on finest pixels, retaining source dx/z. Full-volume norms use actual leaf volumes; uniform-grid convergence orders are not reported.
Final comparison time	4.000000e+03 s
Radiation integration	Legacy first-order radiation update
Radiation backend	cpu
Physical light speed	2.997925e+10 cm/s
Reduced-speed factor α	1.000000e+00
Radiation transport speed	2.997925e+10 cm/s
Periodic boundaries	off
Hydrodynamics	on
Gravity	off
Implicit radiation source	on
Uniform grid	off
CFL	0.4
Timestep cap	20 s
Output interval	100 s

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      "field": "fz",
      "final": 8.410690143329055e+38,
      "final_fraction": 5.251335710043138e-13,
      "initial": 5.818318184641287e+28,
      "initial_fraction": 3.632750884258176e-23,
      "integral_units": "erg cm/s",
      "max_abs_residual": 1.12098235250986e+40,
      "max_normalized_error": 6.999015012736139e-12,
      "normalization_basis": "c * abs(initial_energy)",
      "normalization_scale": 1.601628729856992e+51,
      "normalized_error": 6.032693252293504e-12,
      "normalized_residual": 6.032693252293504e-12,
      "raw_change": 8.410690142747223e+38,
      "residual": 9.66213483128769e+39,
      "source": -5.720997293858278e+39,
      "source_fraction": -3.5719871823035423e-12,
      "zero_initial_flux_relative_tolerance": 1.4210854715202004e-14
    }
  ],
  "dx": 312500000000.0,
  "executable_sha256": "fc0e982bc397d82e25c7105cff6a97a514f09c57439e579587f4ddf6f6a8fa14",
  "inx": 8,
  "length": 20000000000000.0,
  "level": 3,
  "max_level": 3,
  "mesh": {
    "full_mesh": {
      "last_regrid_step": 200.0,
      "last_regrid_time": 4000.0,
      "leaf_blocks": 512,
      "leaf_cells": 262144,
      "levels": {
        "3": {
          "leaf_blocks": 512,
          "leaf_cells": 262144
        }
      }
    },
    "geometry": "Positive-side leaf cells touching geometric z=0; cell centers have z=leaf_dx/2",
    "native_slice_cells": 4096,
    "projection": "Piecewise-constant leaf cell averages onto finest display pixels; raw CSV retains native cells",
    "slice_cells_by_level": {
      "3": 4096
    }
  },
  "mesh_description": "Adaptive Cartesian leaves; cells/dx denote configured finest equivalent resolution",
  "min_level": 2,
  "movie_capture": {
    "hard_dt": 20.0,
    "odt": 100.0,
    "requested_snapshots": 41,
    "steps_per_output_check": 5
  },
  "origin": "Octo-TIGER application",
  "physical_c": 29979245800.0,
  "radiation_ader_version": 0,
  "radiation_backend": "cpu",
  "radiation_equations": "physical-c",
  "radiation_integrator": "euler",
  "radiation_reduced_c_factor": 1.0,
  "reference_kind": "numerical-only",
  "status": "complete",
  "study_config_sha256": "a7ffc569ea95d062e7f4d4b96786783a2c678d14b32981f7e48f1a25899973e6",
  "time": 4000.0,
  "transport_c": 29979245800.0,
  "units": {
    "chi": "1/cm",
    "energy_density": "erg/cm^3",
    "flux": "erg/(cm^2 s)",
    "length": "cm",
    "luminosity": "erg/s",
    "mass": "g",
    "source": "erg/(cm^3 s)",
    "system": "CGS",
    "time": "s"
  },
  "validation_scope": "Early-time implementation check. This run stops before the configured benchmark time and does not validate the developed shock or blast."
}

Movies

No completed movie is available for this resolution.

Data and reproducibility

Effective configuration Run metadata Raw conservation segment Solver log Numerical slice CSV Data units Final numerical Silo

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.

Sources & provenance

Where this problem comes from

Ensman (1994), ApJ 424, 275, §5.3, pp. 286–290, Figures 17–22; original paper PDF. Published profiles are numerical evidence, not an analytic reference. The local runs retain their actual numerical parameters and elapsed physical time.

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.

Batch settings and source hashes Batch status