Equilibrium sphere · Octo-TIGER

The experiment

Physical setup

A Gaussian light source is centered in a stationary medium of prescribed opacity χ. It emits total luminosity ℒ over all space, with a smooth width w:

q(r) = ℒ exp(−r²/w²) / (π^3/2 w³), r = |x|

The domain starts from the steady diffusion solution on top of background E_bg. Boundaries are nonperiodic: ghost cells are fixed to cell averages of that same analytic equilibrium. The incoming background is part of this boundary state. The gas remains fixed; the source adds radiation energy and the medium damps flux.

The standard CGS templates use w = 6 × 10⁹ cm, χ = 1.6666666667 × 10⁻¹⁰ cm⁻¹, E_bg = 1 erg/cm³, and ℒ = 2.698132122 × 10²⁹ erg/s. This is a smooth Gaussian emitter. The saved settings below give the actual values used.

The standard runner uses a uniform cubic grid with hydrodynamics and gravity disabled. Its CGS box spans −3 × 10¹⁰ to +3 × 10¹⁰ cm on each axis, with c = 2.99792458 × 10¹⁰ cm/s and default final time 4 s. A level has N = INX × 2^level cells per side. Each completed run below lists its recorded parameters; template defaults are not substituted for missing metadata.

The comparison

How the reference is computed

In steady isotropic diffusion, D = c/(3χ), ∇ · F = q, and F = −D∇E. Integrating the enclosed Gaussian luminosity gives

E(r) = E_bg + [3χℒ/(4πc)] erf(r/w)/r
F(x) = [ℒ/(4πr³)] [erf(s) − 2s e^−s²/√π] x, s = r/w

The apparent singularities are removable: E(0) = E_bg + 3χℒ/(2π^3/2cw) and F(0) = 0. The implementation uses series expansions near the origin.

Initial, boundary, and comparison states are cell averages evaluated with four-point Gauss–Legendre quadrature on each axis (64 samples per cell). The smooth-field averages therefore have quadrature error. The Gaussian source has an exact cell integral, evaluated as a product of three error-function differences:

q̄ = (ℒ/Δx³) ∏_d=x,y,z ½ {erf[(x_d + Δx/2)/w] − erf[(x_d − Δx/2)/w]}

The source step adds Δt q̄ to energy and divides flux by 1 + cχΔt. The steady reference itself is independent of time.

What this test measures

This is a diffusion-limit equilibrium test. Nonlinear M1 has an anisotropic pressure at finite flux, so the diffusion solution need not remain exactly stationary in that model. Errors can reflect closure differences, transport/source splitting, discretization, and quadrature. The plots measure equilibrium drift and Cartesian directional effects as resolution changes.

Energy injected inside the finite cube should balance its outward transport and the change in stored energy. ℒ is the all-space luminosity; use the measured in-domain source budget rather than assuming the finite cube receives exactly ℒ. Symmetry can make net flux integrals very small, which is why their conservation normalization includes an energy-based scale.

Measured results

All resolutions

Level	Grid	Batch status
2	32³	Ready
3	64³	Ready
4	128³	Waiting

Full-volume convergence for energy and all three flux components · PDF

Full-volume norms and orders

Grid	Field	Norm	Error	Order	Error units
32³	er	L1	2.019307e-05	—	erg/cm³
32³	er	L2	5.151584e-05	—	erg/cm³
32³	er	Linf	6.424667e-04	—	erg/cm³
32³	fx	L1	2.596979e+05	—	erg/(cm² s)
32³	fx	L2	9.097040e+05	—	erg/(cm² s)
32³	fx	Linf	1.630999e+07	—	erg/(cm² s)
32³	fy	L1	2.596979e+05	—	erg/(cm² s)
32³	fy	L2	9.097040e+05	—	erg/(cm² s)
32³	fy	Linf	1.630999e+07	—	erg/(cm² s)
32³	fz	L1	2.596979e+05	—	erg/(cm² s)
32³	fz	L2	9.097040e+05	—	erg/(cm² s)
32³	fz	Linf	1.630999e+07	—	erg/(cm² s)
64³	er	L1	4.417055e-06	2.192704e+00	erg/cm³
64³	er	L2	1.156422e-05	2.155348e+00	erg/cm³
64³	er	Linf	3.464097e-04	8.911424e-01	erg/cm³
64³	fx	L1	5.365054e+04	2.275170e+00	erg/(cm² s)
64³	fx	L2	2.134352e+05	2.091599e+00	erg/(cm² s)
64³	fx	Linf	6.129317e+06	1.411958e+00	erg/(cm² s)
64³	fy	L1	5.365054e+04	2.275170e+00	erg/(cm² s)
64³	fy	L2	2.134352e+05	2.091599e+00	erg/(cm² s)
64³	fy	Linf	6.129317e+06	1.411958e+00	erg/(cm² s)
64³	fz	L1	5.365054e+04	2.275170e+00	erg/(cm² s)
64³	fz	L2	2.134352e+05	2.091599e+00	erg/(cm² s)
64³	fz	Linf	6.129317e+06	1.411958e+00	erg/(cm² s)

This problem: norms/orders CSV JSON

32³ cells / level 2

Recorded setup and run provenance

Origin	Octo-TIGER application
Domain	Cube of side 6.000000e+10 cm; centered at the origin
Resolution	32³ cells; level 2; Δx = 1.875000e+09 cm
Final comparison time	4.000000e+00 s
Light speed	2.997925e+10 cm/s
Periodic boundaries	off
Hydrodynamics	off
Gravity	off
Implicit radiation source	on
Uniform grid	on
CFL	0.4
Timestep cap	0.013333333333333332 s
Output interval	0.066666666666666666 s
Background energy density	1 erg/cm³
Scattering coefficient χ	1.6666666666666666e-10 cm⁻¹
Gaussian width w	6000000000 cm
All-space luminosity ℒ	2.6981321220000001e+29 erg/s

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    "monopole_device_kernel_type": "OFF",
    "multipole_device_kernel_type": "OFF",
    "n_species": "1",
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    "periodic": "off",
    "problem": "RADIATION_EQUILIBRIUM_SPHERE",
    "rad_implicit": "on",
    "rad_test_amplitude": "0.01",
    "rad_test_background": "1",
    "rad_test_chi": "1.6666666666666666e-10",
    "rad_test_luminosity": "2.6981321220000001e+29",
    "rad_test_width": "6000000000",
    "radiation": "on",
    "stop_step": "100000",
    "stop_time": "4",
    "unigrid": "on",
    "xscale": "30000000000"
  },
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    "max_level": "2",
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  "inx": 8,
  "length": 60000000000.0,
  "level": 2,
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  "norms": {
    "er": {
      "L1": 2.019307e-05,
      "L2": 5.151584e-05,
      "Linf": 0.0006424667
    },
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      "L2": 909704.0,
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    },
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      "L1": 259697.9,
      "L2": 909704.0,
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    },
    "fz": {
      "L1": 259697.9,
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      "Linf": 16309990.0
    }
  },
  "origin": "Octo-TIGER application",
  "status": "complete",
  "time": 4.0,
  "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"
  }
}

Final fields and profiles

Numerical, reference, and signed-error energy slices at z = Δx/2 · PDF

Numerical/reference line profiles and signed errors for all four radiation fields · PDF

All three flux maps

fx: numerical, reference, and signed-error slice · PDF

fy: numerical, reference, and signed-error slice · PDF

fz: numerical, reference, and signed-error slice · PDF

Movies

er · slice normal to z · 61 recorded snapshots · 2.000000e+01 s playback. MP4 · Movie metadata

Conservation

All values below are dimensionless fractions of the fixed initial scale.

Field	Scale	Q(0)/scale	Q(t)/scale	Boundary B/scale	Source S/scale	Residual R/scale	Maximum \|R\|/scale
er	\|Q(0)\|	1.000000e+00	9.999980e-01	4.929480e-03	4.927470e-03	-6.682576e-17	1.143749e-16
fx	c \|E₀\| (zero/negligible initial flux)	4.213966e-22	-1.294250e-19	7.633235e-19	6.364062e-19	-2.929058e-21	3.024223e-21
fy	c \|E₀\| (zero/negligible initial flux)	7.866070e-22	2.500680e-19	-6.217881e-18	-6.032618e-18	6.401787e-20	6.714955e-20
fz	c \|E₀\| (zero/negligible initial flux)	-6.011925e-22	3.922922e-20	-1.293542e-18	-1.293144e-18	3.943254e-20	4.048357e-20

Integrated radiation budgets and their transport/source-corrected residuals · PDF

Data and reproducibility

Effective configuration Run metadata Solver log Paired slice values CSV Slice units Raw conservation history Budget summary CSV Conservation fractions CSV Fraction units Conservation units L1 norms L2 norms L∞ norms Final numerical Silo Reference Silo

64³ cells / level 3

Recorded setup and run provenance

Origin	Octo-TIGER application
Domain	Cube of side 6.000000e+10 cm; centered at the origin
Resolution	64³ cells; level 3; Δx = 9.375000e+08 cm
Final comparison time	4.000000e+00 s
Light speed	2.997925e+10 cm/s
Periodic boundaries	off
Hydrodynamics	off
Gravity	off
Implicit radiation source	on
Uniform grid	on
CFL	0.4
Timestep cap	0.013333333333333332 s
Output interval	0.066666666666666666 s
Background energy density	1 erg/cm³
Scattering coefficient χ	1.6666666666666666e-10 cm⁻¹
Gaussian width w	6000000000 cm
All-space luminosity ℒ	2.6981321220000001e+29 erg/s

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      "integral_units": "erg cm/s",
      "max_abs_residual": 1.158478586259052e+23,
      "max_normalized_error": 1.7642823482705478e-20,
      "normalization_basis": "c * abs(initial_energy)",
      "normalization_scale": 6.566287915279887e+42,
      "normalized_error": 7.130733338106675e-21,
      "normalized_residual": 7.130733338106675e-21,
      "raw_change": -5.9166086942016016e+23,
      "residual": 4.682244814509327e+22,
      "source": 5.43447997940511e+24,
      "source_fraction": 8.276335198094136e-19,
      "zero_initial_flux_relative_tolerance": 1.4210854715202004e-14
    }
  ],
  "dx": 937500000.0,
  "executable_sha256": "698ae293307532d5aedaf86a49381319eb1a911b3ef40a33cc91e74c2e089de8",
  "inx": 8,
  "length": 60000000000.0,
  "level": 3,
  "movie_capture": {
    "hard_dt": 0.013333333333333332,
    "odt": 0.06666666666666667,
    "requested_snapshots": 61,
    "steps_per_output_check": 5
  },
  "norms": {
    "er": {
      "L1": 4.417055e-06,
      "L2": 1.156422e-05,
      "Linf": 0.0003464097
    },
    "fx": {
      "L1": 53650.54,
      "L2": 213435.2,
      "Linf": 6129317.0
    },
    "fy": {
      "L1": 53650.54,
      "L2": 213435.2,
      "Linf": 6129317.0
    },
    "fz": {
      "L1": 53650.54,
      "L2": 213435.2,
      "Linf": 6129317.0
    }
  },
  "origin": "Octo-TIGER application",
  "status": "complete",
  "time": 4.0,
  "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"
  }
}

Final fields and profiles

All three flux maps

Movies

er · slice normal to z · 61 recorded snapshots · 2.000000e+01 s playback. MP4 · Movie metadata

Conservation

All values below are dimensionless fractions of the fixed initial scale.

Field	Scale	Q(0)/scale	Q(t)/scale	Boundary B/scale	Source S/scale	Residual R/scale	Maximum \|R\|/scale
er	\|Q(0)\|	1.000000e+00	9.999996e-01	4.927891e-03	4.927470e-03	1.542133e-17	1.111621e-16
fx	c \|E₀\| (zero/negligible initial flux)	-1.247334e-21	4.206999e-19	-6.010757e-18	-5.589253e-18	4.431688e-22	1.908576e-21
fy	c \|E₀\| (zero/negligible initial flux)	3.989221e-22	1.753010e-20	8.378489e-20	9.896043e-20	1.955632e-21	2.278035e-20
fz	c \|E₀\| (zero/negligible initial flux)	-7.304208e-23	-9.017888e-20	9.248701e-19	8.276335e-19	7.130733e-21	1.764282e-20

Data and reproducibility

This problem: all conservation summaries CSV JSON

Reading the results

Errors, budgets, and movies

Every norm uses the full three-dimensional domain, with cell-volume weighting. Here e = numerical − reference and V is the domain volume:

L1 = Σ |e| ΔV / V, L2 = √(Σ e² ΔV / V), L∞ = max |e|
p = ln(error_coarse / error_fine) / ln(Δx_coarse / Δx_fine)

Orders require two completed runs with matching physical time, configuration, units, executable, and origin. An exact zero error has no logarithmic order. The central slice and line profiles are visual diagnostics: the report slice is the cell layer at z = Δx/2, and its line profile is at y = z = Δx/2.

Conservation is assessed using the measured domain integrals and the signed cumulative boundary and source budgets:

R(t) = Q(t) − Q(0) + B(t) − S(t)

B is outward transport and S is the source change. Conservation plots and this page's budget tables show fractions of the initial totals: Q(t)/D, [Q(0) − B(t) + S(t)]/D, and the signed residual R(t)/D. The denominator D = |Q(0)| is fixed for the entire history. It never grows with the current total or accumulated transport/source budgets. A negative initial flux therefore starts at −1, and the plotted fraction retains its sign.

A component whose initial net flux is zero or negligible cannot be divided by its own initial total. It instead uses the fixed scale D = c |E₀|, where E₀ is the initial integrated radiation energy. This fallback is labeled on its panels and in the table. “Negligible” means |Q(0)|/(c |E₀|) ≤ 64ε ≈ 1.421085 × 10⁻¹⁴, with ε the double-precision machine epsilon. This prevents roundoff left by cancellation in a symmetric field from becoming the denominator. A nonzero initial radiation energy is required.

Raw histories and dimensional summary columns remain in physical units: energy integrals in erg, physical-flux integrals in erg cm/s for CGS runs. Dividing a flux integral by c² gives radiation momentum. Fraction-history CSV files are dimensionless except for their time column; summary files record each denominator and its basis. Maximum fractional residuals use the same fixed denominator as the final residual.

Movies show the numerical Silo states with a fixed color range for each movie. Recorded states are held for their actual simulation-time gaps; intermediate solutions are not interpolated. The 3D option shows the domain's exterior surface. Movies and plots retain their recorded physical units.

Sources & provenance

Where this problem comes from

This smooth Gaussian-emitter test is defined by equilibriumSphere, sphereAverage, and bulbSourceAverage in octotiger/test_problems/radiation/profiles.hpp. Its reference follows by solving the steady diffusion equation for that source. The available implementation does not identify it as a verbatim published sphere benchmark, so the provenance here is the suite's analytic construction.

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. The suite's fixed-medium damping 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.

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