A small Gaussian perturbation evolves in a fixed scattering medium. A Fourier solution of the telegraph equation supplies the reference.
Includes short-duration runs. These saved results use 32³ cells and end at 0.5 s. Only one spatial resolution is shown, so these results do not measure a convergence order. Runs shorter than the benchmark duration do not establish developed shock or blast structure. Full-duration benchmark validation remains separate.
gaussian pulse | EULER | level 2 | AMR
Central z=0+ radiation energy slice with faint cell mesh and equal-length flux direction arrows. Gaussian energy excess above the recorded background uses a labeled logarithmic floor. Fixed color limits; actual stored Silo times held without solution interpolation. The maps show the full simulation domain without cropping. Initial periodic pulse expansion through 0.5 s; the pulse remains in a periodic domain.
The initial state is a smooth Gaussian energy excess of amplitude A and width w over a uniform background Ebg, with zero flux. The Gaussian is periodized on the three-dimensional cube. A prescribed, stationary scattering medium damps flux at rate cχ, while there is no energy source. Gas motion, recoil, and heating are absent from this radiation-only setup.
The standard CGS templates give Ebg = 1 erg/cm³, A = 0.01 erg/cm³, w = 6 × 10⁹ cm, and χ = 1.6666666667 × 10−10 cm−1. The saved settings below take precedence for a particular run.
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 × 2level then describes their configured finest equivalent resolution. Each completed run below lists its recorded parameters.
The reference solves the telegraph equation. Linearizing the moment equations about an isotropic radiation background gives P = EI/3 and
For each periodic Fourier wavevector k = 2πj/L, the generator solves the damped oscillator analytically. Define α = cχ/2, λ² = α² − c²|k|²/3, and the mode propagators
Negative λ² uses the equivalent real sine/cosine expressions; λ = 0 uses B = te−αt. The zero mode keeps the mean energy constant. Stable exponential expressions in telegraphMode avoid cancellation and overflow. Flux relaxation and propagation speed c/√3 are retained.
The C++ generator first forms Gaussian cell averages using error-function integrals, with image shifts −4 through +4 on each axis. It transforms these values with FFTW, removes every Nyquist plane on the even grid so odd spectral derivatives remain real, and evolves each retained mode. Both initial and final arrays use this same filter. The inverse transform is divided by N³ once, and the uniform background is added to the energy. The resulting OTRAD001 binary stores all four physical fields at the two times.
The numerical solver evolves nonlinear M1; the telegraph solution is its small-perturbation, isotropic-pressure reference. At finite amplitude, measured error includes a closure-model difference as well as spatial and temporal discretization. Refinement can therefore reach a model-error floor. Initial spectral truncation also depends on resolution. The prescribed source damps flux at rate cχ; its numerical integration depends on the recorded method, as described below.
In the periodic domain total energy should remain constant. Flux damping belongs in the signed source budget when assessing flux-integral conservation.
| Level | Grid | Batch status |
|---|---|---|
| 2 | 32³ | Ready |
Convergence requires at least two completed, comparable resolutions.
| Grid | Field | Norm | Error | Order | Error units |
|---|---|---|---|---|---|
| AMR levels 1–2; 4096 active cells; finest equivalent 32³ | er | L1 | 7.385814e-06 | — | erg/cm³ |
| AMR levels 1–2; 4096 active cells; finest equivalent 32³ | er | L2 | 2.283424e-05 | — | erg/cm³ |
| AMR levels 1–2; 4096 active cells; finest equivalent 32³ | er | Linf | 1.813365e-04 | — | erg/cm³ |
| AMR levels 1–2; 4096 active cells; finest equivalent 32³ | fx | L1 | 6.159883e+04 | — | erg/(cm² s) |
| AMR levels 1–2; 4096 active cells; finest equivalent 32³ | fx | L2 | 2.171852e+05 | — | erg/(cm² s) |
| AMR levels 1–2; 4096 active cells; finest equivalent 32³ | fx | Linf | 2.474439e+06 | — | erg/(cm² s) |
| AMR levels 1–2; 4096 active cells; finest equivalent 32³ | fy | L1 | 6.159883e+04 | — | erg/(cm² s) |
| AMR levels 1–2; 4096 active cells; finest equivalent 32³ | fy | L2 | 2.171852e+05 | — | erg/(cm² s) |
| AMR levels 1–2; 4096 active cells; finest equivalent 32³ | fy | Linf | 2.474439e+06 | — | erg/(cm² s) |
| AMR levels 1–2; 4096 active cells; finest equivalent 32³ | fz | L1 | 6.159883e+04 | — | erg/(cm² s) |
| AMR levels 1–2; 4096 active cells; finest equivalent 32³ | fz | L2 | 2.171852e+05 | — | erg/(cm² s) |
| AMR levels 1–2; 4096 active cells; finest equivalent 32³ | fz | Linf | 2.474439e+06 | — | erg/(cm² s) |
| Origin | Octo-TIGER application |
|---|---|
| Domain | Cube of side 6.000000e+10 cm; centered at the origin |
| Resolution | AMR levels 1–2; 4096 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 | 5.000000e-01 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 | on |
| Hydrodynamics | off |
| Gravity | off |
| Implicit radiation source | on |
| Uniform grid | off |
| CFL | 0.4 |
| Timestep cap | 0.0050000000000000001 s |
| Output interval | 0.025000000000000001 s |
| Background energy density | 1 erg/cm³ |
| Scattering coefficient χ | 1.6666666666666666e-10 cm⁻¹ |
| Gaussian width w | 6000000000 cm |
| Pulse amplitude A | 0.01 erg/cm³ |
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"mesh": {
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"levels": {
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}
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},
"geometry": "Positive-side leaf cells touching geometric z=0; cell centers have z=leaf_dx/2",
"native_slice_cells": 256,
"projection": "Piecewise-constant leaf cell averages onto finest display pixels; raw CSV retains native cells",
"slice_cells_by_level": {
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}
},
"mesh_description": "Adaptive Cartesian leaves; cells/dx denote configured finest equivalent resolution",
"min_level": 1,
"movie_capture": {
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"norms": {
"er": {
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"L2": 2.283424e-05,
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},
"fx": {
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},
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"origin": "Octo-TIGER application",
"physical_c": 29979245800.0,
"radiation_ader_version": 0,
"radiation_backend": "cpu",
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"status": "complete",
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"units": {
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"energy_density": "erg/cm^3",
"flux": "erg/(cm^2 s)",
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"luminosity": "erg/s",
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}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.
Gaussian energy is shown as E − Ebg on a logarithmic scale. Values at or below 8.571540e-10 use the display floor, including zero and negative excesses. Raw values, signed errors, and error norms are unchanged.
No completed movie is available for this resolution.
All values below are dimensionless fractions of the fixed normalization 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 | 1.000000e+00 | 0.000000e+00 | 0.000000e+00 | -1.667907e-16 | 1.667907e-16 |
| fx | c |E₀| (zero/negligible initial flux) | 0.000000e+00 | 2.026908e-22 | 0.000000e+00 | 1.438314e-22 | 5.885946e-23 | 1.766284e-22 |
| fy | c |E₀| (zero/negligible initial flux) | 0.000000e+00 | 8.091611e-23 | 0.000000e+00 | -2.473160e-22 | 3.282321e-22 | 3.282321e-22 |
| fz | c |E₀| (zero/negligible initial flux) | 0.000000e+00 | 4.563882e-22 | 0.000000e+00 | 3.739950e-23 | 4.189887e-22 | 4.189887e-22 |
Every norm uses the full three-dimensional domain, with cell-volume weighting. Here e = numerical − reference and V is the domain volume:
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:
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 = Ebath V for energy and D = c Ebath V for flux, with Ebath = 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.
The suite derives its reference from the linearized two-moment equations above. The implementation is telegraphMode in profiles.hpp, the ReferenceData reader/writer in reference.hpp, and radiation_results/cpp/gen_radiation_reference.cpp. The generator was recovered from the September 13 source patch and adapted to the September 17 interfaces while retaining its Fourier construction.
Frigo & Johnson (2005), “The Design and Implementation of FFTW3”, describes the transform library. The Gaussian and its telegraph reference are the suite's construction; FFTW supplies the discrete transforms.
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.
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.