← Octo-TIGER / All test problemsMixed-frame radiation hydrodynamics

Radiating blast wave

Gas density, momentum and thermal energy evolve with radiation transport, radiation force and paired gas/radiation work.

FLD method scope. Grey radiation at physical light speed on CPU, backward-Euler diffusion/material exchange, and the Levermore–Pomraning limiter. Moving cases include first-order gas transport, radiation force/work, and a derived lab-frame flux with radiation enthalpy advection. Where supplied, analytic diffusion-limit references omit nonlinear-limiter and boundary-model effects. Thin streaming cases have no supplied analytic spatial FLD profile and use exact energy integrals instead. Paired M1 RK1 and FLD timings use different stability limits and do not establish performance at matched error.

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/(4πcRout²), 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.

FLD uses a first-order conservative gas/radiation update and implicit nonlinear diffusion/material exchange at physical light speed. The limiter supplies radiation pressure. The stored flux is the derived lab-frame flux Flab = −D∇E + (EI+P)·v; it is not an independently evolved radiation momentum field.

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.

The comparison

Comparison and interpretation

No independent analytic or digitized reference profile is supplied. Published benchmark figures identify the physical setup, but their curves are not embedded and these Cartesian adaptations are not identical to the original geometries. M1/FLD field differences compare transport closures and discretizations, not error against an exact shared solution.

What this test measures

Check gas/radiation total energy including boundary flux, nonnegative density/thermal/radiation energies, shock or blast development, and resolution/timestep sensitivity. The comoving diffusive flux is bounded by cE; the lab-frame flux additionally contains enthalpy advection and can exceed cE. These tests do not by themselves establish dynamic-diffusion asymptotic accuracy.

Measured results

All resolutions

LevelGridBatch status
232³Ready

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.

32³ cells · actual final time 2.200000e+04 s

Central cell layer at z=dx/2; ray at y=z=dx/2.

First-order mixed-frame gas/radiation evolution with physical light speed. Derived lab-frame flux includes radiation enthalpy advection. No independent analytic or digitized reference profile is supplied; M1 differences are model discrepancies.

Mixed-frame FLD with first-order gas transport and backward-Euler diffusion/material exchange; physical c = 29979245800 cm/s. Radiation flux is the derived lab-frame diffusion plus enthalpy-advection flux; it is not an independently evolved momentum variable. No independent analytic or digitized FLD reference profile is supplied. M1 versus FLD differences compare distinct radiation models and are not analytic-reference errors.

Recorded setup and run provenance
OriginOcto-TIGER application
DomainCube of side 2.000000e+13 cm; centered at the origin
Resolution32³ cells; finest Δx = 6.250000e+11 cm
Final comparison time2.200000e+04 s
Radiation modelFLD
Radiation integrationFLD backward Euler (first order in time)
Flux semanticsDerived lab-frame FLD flux: F = -D grad E + (E I + P) v
Analytic referenceNo independent analytic or digitized benchmark profile supplied
Reference norm scopeNo analytic-profile norms or convergence orders
Radiation backendcpu
Physical light speed2.997925e+10 cm/s
Reduced-speed factor α1.000000e+00
Radiation transport speed2.997925e+10 cm/s
Periodic boundariesoff
Hydrodynamicson
Gravityoff
Implicit radiation sourceon
Uniform gridon
CFL0.4
Timestep cap110 s
Output interval183.33333333333334 s
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    ],
    "substeps": 381
  },
  "inx": 8,
  "length": 20000000000000.0,
  "level": 2,
  "movie_capture": {
    "hard_dt": 110.0,
    "odt": 183.33333333333334,
    "requested_snapshots": 121,
    "steps_per_output_check": 1
  },
  "origin": "Octo-TIGER application",
  "physical_c": 29979245800.0,
  "radiation_algorithm_version": 2,
  "radiation_backend": "cpu",
  "radiation_equations": "mixed-frame physical-c FLD radiation hydrodynamics",
  "radiation_flux_semantics": "derived_fld_lab_flux",
  "radiation_frame": "mixed",
  "radiation_hydro_scheme": "first_order_conservative",
  "radiation_integrator": "backward_euler",
  "radiation_limiter": "levermore_pomraning",
  "radiation_model": "fld",
  "radiation_reduced_c_factor": 1.0,
  "reference_exact_for_solver": false,
  "reference_kind": "unavailable",
  "reference_model": "none",
  "reference_sampling": "none",
  "status": "complete",
  "time": 22000.0,
  "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": "First-order mixed-frame gas/radiation evolution with physical light speed. Derived lab-frame flux includes radiation enthalpy advection. No independent analytic or digitized reference profile is supplied; M1 differences are model discrepancies."
}
Numerical gas temperature, radiation temperature, and density. Central cell layer at z=dx/2; ray at y=z=dx/2.
Numerical gas temperature, radiation temperature, and density. Central cell layer at z=dx/2; ray at y=z=dx/2. · PDF
Numerical profiles; no reference curves are supplied. Central cell layer at z=dx/2; ray at y=z=dx/2.
Numerical profiles; no reference curves are supplied. Central cell layer at z=dx/2; ray at y=z=dx/2. · PDF

Saved-state movie

Radiating Blast · L2 · FLD backward Euler

121 stored states; 121 shown. Physical time: 0–22000 s. Playback: 60 s.

Zero-order hold; no solution interpolation. Playback holds each saved numerical state over its time interval and includes endpoint holds.

All 121 snapshot times

Snapshot times: 0, 220, 440, 660, 770, 990, 1210, 1320, 1540, 1650, 1870, 2090, 2310, 2420, 2640, 2860, 2970, 3190, 3300, 3520, 3740, 3850, 4070, 4290, 4510, 4620, 4840, 5060, 5170, 5390, 5610, 5720, 5940, 6050, 6270, 6490, 6600, 6820, 7040, 7150, 7370, 7590, 7700, 7920, 8140, 8360, 8470, 8690, 8910, 9020, 9240, 9460, 9570, 9790, 10010, 10120, 10340, 10560, 10670, 10890, 11110, 11220, 11440, 11660, 11770, 11990, 12100, 12320, 12540, 12650, 12870, 13090, 13200, 13420, 13640, 13750, 13970, 14190, 14300, 14520, 14740, 14850, 15070, 15290, 15400, 15620, 15840, 15950, 16170, 16390, 16610, 16720, 16940, 17160, 17270, 17490, 17710, 17820, 18040, 18260, 18370, 18590, 18810, 18920, 19140, 19360, 19470, 19690, 19910, 20020, 20240, 20460, 20570, 20790, 21010, 21120, 21340, 21560, 21670, 21890, 22000 s.

Arrows show derived lab-frame FLD flux -D grad(E) + (E I + P)·v, including radiation enthalpy advection; radiation momentum is not an independently evolved field. Arrows have equal length and show direction only.

Movies show numerical mixed-frame radiation hydrodynamics. No independent analytic or digitized benchmark profile is supplied.

Central z=0+ radiation energy slice with faint cell mesh and equal-length flux direction arrows. Arrows show derived lab-frame FLD flux -D grad(E) + (E I + P)·v, including radiation enthalpy advection; radiation momentum is not an independently evolved field. Gray density contours and a matching density/velocity panel. Fixed color limits; actual stored Silo times held without solution interpolation. Movies show numerical mixed-frame radiation hydrodynamics. No independent analytic or digitized benchmark profile is supplied. The maps show the full simulation domain without cropping.

Download MP4 · Rendering and timing metadata · Full run report

All campaign movies

Implicit solve and energy balances

Radiation-relative imbalance and the raw accepted-step coupled error are reported separately. A gas-dominated shock can have a much larger radiation-relative imbalance than its residual normalized by gas plus radiation energy.

{
  "accepted_steps": 200,
  "coupled_energy_normalization": {
    "basis": "initial full-volume gas plus radiation energy",
    "maximum_step_conservation_error_normalized": 3.5455864298313923e-13,
    "note": "Radiation-relative residuals remain in radiation_energy_budget. Scaling a radiation-only residual by total energy is not a separate proof of combined gas-plus-radiation conservation.",
    "radiation_equation_maximum_residual_normalized": 2.483568208075503e-11,
    "value": 1.0004870049690307e+47
  },
  "history_end_time": 22000.0,
  "history_start_time": 0.0,
  "maximum_nonlinear_residual": 9.978487510918042e-11,
  "maximum_step_coupled_conservation_error": 3.547313148040848e+34,
  "radiation_energy_budget": {
    "boundary": 5.6808611736687314e+45,
    "boundary_fraction": 106322.38520541495,
    "field": "er",
    "final": 6.902258391223411e+46,
    "final_fraction": 1291819.238358591,
    "history_end_time": 22000.0,
    "history_scope": "full_run",
    "history_start_time": 0.0,
    "initial": 5.343052794285326e+40,
    "initial_fraction": 1.0,
    "integral_units": "erg",
    "max_abs_residual": 2.4847777181337625e+36,
    "max_normalized_error": 4.650483185925773e-05,
    "normalization_basis": "abs(initial_energy)",
    "normalization_scale": 5.343052794285326e+40,
    "normalization_time": 0.0,
    "normalized_error": 4.117880647825457e-05,
    "normalized_residual": -4.117880647825457e-05,
    "raw_change": 6.902253048170617e+46,
    "residual": -2.2002053701897277e+36,
    "source": 7.470339165757511e+46,
    "source_fraction": 1398140.6236051847,
    "zero_initial_flux_relative_tolerance": 1.4210854715202004e-14
  },
  "reference_exact_for_solver": false,
  "reference_kind": "unavailable",
  "reference_model": "none",
  "reference_norm_scope": "unavailable; numerical radiation hydrodynamics",
  "retries": 181,
  "slice_physical_bounds": {
    "flux_bound_note": "Lab-frame flux includes enthalpy advection; its norm can exceed cE. Hydro slice validation uses the conservative (c+2|v|)E envelope.",
    "flux_semantics": "derived_fld_lab_flux",
    "maximum_reduced_flux": 0.8053177915925555,
    "minimum_radiation_energy": 4999.546026019604,
    "scope": "projected-positive-side-slice"
  },
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}

Data and reproducibility

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, the recorded-frame radiation flux envelope, 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.

Levermore & Pomraning (1981), “A flux-limited diffusion theory,” ApJ 248, 321–334, derives flux-limited diffusion and its classical diffusion limit. Saved reference metadata records whether an analytic profile is supplied and distinguishes its equations from the nonlinear limiter evolved by this solver.

Radiation method context

M1 RK1 evolves radiation energy and flux moments with the retained first-order radiation update. FLD uses backward-Euler radiation diffusion/material exchange with the Levermore–Pomraning limiter. Moving cases add first-order conservative gas transport and radiation force/work; their reconstructed lab-frame flux includes (E I+P) v as well as −D∇E. The scalar implicit transport solve uses composite-grid multigrid and an iterative linear solver. Both time integrators are first order.

The suite includes Gaussian scattering, Marshak heating, a Gaussian emitter, explicit thin positive-opacity wave/front comparisons, and moving-gas shock/blast problems on CPU at physical light speed. A configured case does not establish that a solve converged; only validated completed runs appear as results. Flux diagnostics in FLD 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.