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. Opt-in adaptive runs are identified below; N = INX × 2^level then describes their configured finest equivalent resolution. Each completed run below lists its recorded parameters.

The comparison

Comparison and interpretation

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 prescribed source adds Δt q̄ to energy and damps flux at rate cχ. Its numerical integration depends on the recorded method, as described below. 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	—³	Ready

Convergence requires at least two completed, comparable resolutions.

Full-volume norms and orders

Grid	Field	Norm	Error	Order	Error units
AMR levels 1–2; 32768 active cells; finest equivalent 32³	er	L1	2.019307e-05	—	erg/cm³
AMR levels 1–2; 32768 active cells; finest equivalent 32³	er	L2	5.151584e-05	—	erg/cm³
AMR levels 1–2; 32768 active cells; finest equivalent 32³	er	Linf	6.424667e-04	—	erg/cm³
AMR levels 1–2; 32768 active cells; finest equivalent 32³	fx	L1	2.596979e+05	—	erg/(cm² s)
AMR levels 1–2; 32768 active cells; finest equivalent 32³	fx	L2	9.097040e+05	—	erg/(cm² s)
AMR levels 1–2; 32768 active cells; finest equivalent 32³	fx	Linf	1.630999e+07	—	erg/(cm² s)
AMR levels 1–2; 32768 active cells; finest equivalent 32³	fy	L1	2.596979e+05	—	erg/(cm² s)
AMR levels 1–2; 32768 active cells; finest equivalent 32³	fy	L2	9.097040e+05	—	erg/(cm² s)
AMR levels 1–2; 32768 active cells; finest equivalent 32³	fy	Linf	1.630999e+07	—	erg/(cm² s)
AMR levels 1–2; 32768 active cells; finest equivalent 32³	fz	L1	2.596979e+05	—	erg/(cm² s)
AMR levels 1–2; 32768 active cells; finest equivalent 32³	fz	L2	9.097040e+05	—	erg/(cm² s)
AMR levels 1–2; 32768 active cells; finest equivalent 32³	fz	Linf	1.630999e+07	—	erg/(cm² s)

This problem: norms/orders CSV JSON

AMR levels 1–2; 32768 active cells; finest equivalent 32³ / level 2

Recorded setup and run provenance

Origin	Octo-TIGER application
Domain	Cube of side 6.000000e+10 cm; centered at the origin
Resolution	AMR levels 1–2; 32768 active cells; finest equivalent 32³; finest Δx = 1.875000e+09 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+00 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	off
Gravity	off
Implicit radiation source	on
Uniform grid	off
CFL	0.4
Timestep cap	0.040000000000000001 s
Output interval	0.20000000000000001 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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Final fields and profiles

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.

Numerical, reference, and signed-error energy slices. 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/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

No completed movie is available for this resolution.

Conservation

All values below are dimensionless fractions of the fixed normalization scale at t_ref = 0 s.

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

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:

A labeled checkpoint continuation uses its checkpoint as t_ref, with counters and fixed normalization starting there. Its reported residuals cover that resumed segment only; original histories remain separate.

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. For the initially cold Marshak case, the prescribed bath sets D = E_bath V for energy and D = c E_bath V for flux, with E_bath = 4/c. These scales remain fixed even though radiation enters the initially empty domain.

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.

For runs labeled RSLA, these budgets check the modified radiation equations. They do not establish conservation of physical gas-plus-radiation energy. With reduced-speed factor α, the closed-system coupled invariants are Egas + Er/α and gas momentum + F/(αc²), with physical c in the closure. Compare multiple α values before interpreting an RSLA result as an approximation to physical-c evolution.

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. 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