Gaussian pulse

Physical setup

The initial state is a smooth Gaussian energy excess of amplitude A and width w over a uniform background E_bg, 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 E_bg = 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. 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.

How the reference is computed

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.

What this test measures

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 fixed-medium source step damps flux by Fⁿ⁺¹ = Fⁿ/(1 + cχΔt), so source integration contributes temporal error.

In the periodic domain total energy should remain constant. Flux damping belongs in the signed source budget when assessing flux-integral conservation.

All resolutions

Convergence requires at least two completed, comparable resolutions.

Full-volume norms and orders

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	on
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
Pulse amplitude A	0.01 erg/cm³

{
  "background": 1.0,
  "c": 29979245800.0,
  "case": "gaussian_pulse",
  "cells": 32,
  "command": [
    "/home/dmarce1/workspace/octotiger/release/octotiger",
    "--config_file=/home/dmarce1/workspace/octotiger/radiation_results/results/live-20260917-084723-426280/gaussian_pulse/l2/run.ini",
    "--hpx:threads=12"
  ],
  "comparison_signature": {
    "atomic_mass": "1",
    "atomic_number": "1",
    "cfl": "0.4",
    "code_to_cm": "1",
    "code_to_g": "1",
    "code_to_s": "1",
    "disable_analytic": "off",
    "disable_diagnostics": "on",
    "gravity": "off",
    "hard_dt": "0.013333333333333332",
    "hydro": "off",
    "hydro_device_kernel_type": "OFF",
    "hydro_host_kernel_type": "LEGACY",
    "monopole_device_kernel_type": "OFF",
    "multipole_device_kernel_type": "OFF",
    "n_species": "1",
    "omega": "0",
    "periodic": "on",
    "problem": "RADIATION_GAUSSIAN_PULSE",
    "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"
  },
  "config": {
    "atomic_mass": "1",
    "atomic_number": "1",
    "cfl": "0.4",
    "code_to_cm": "1",
    "code_to_g": "1",
    "code_to_s": "1",
    "datadir": "/home/dmarce1/workspace/octotiger/radiation_results/results/live-20260917-084723-426280/gaussian_pulse/l2/",
    "disable_analytic": "off",
    "disable_diagnostics": "on",
    "disable_output": "off",
    "gravity": "off",
    "hard_dt": "0.013333333333333332",
    "hydro": "off",
    "hydro_device_kernel_type": "OFF",
    "hydro_host_kernel_type": "LEGACY",
    "max_level": "2",
    "min_level": "2",
    "monopole_device_kernel_type": "OFF",
    "multipole_device_kernel_type": "OFF",
    "n_species": "1",
    "odt": "0.066666666666666666",
    "omega": "0",
    "periodic": "on",
    "problem": "RADIATION_GAUSSIAN_PULSE",
    "rad_implicit": "on",
    "rad_reference": "/home/dmarce1/workspace/octotiger/radiation_results/results/live-20260917-084723-426280/gaussian_pulse/l2/reference.bin",
    "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"
  },
  "conservation": [
    {
      "boundary": 0.0,
      "field": "er",
      "final": 2.1601202758847313e+32,
      "initial": 2.1601202758847317e+32,
      "integral_units": "erg",
      "max_abs_residual": 3.602879701896397e+16,
      "max_normalized_error": 1.6679069874573287e-16,
      "normalization_scale": 2.1601202758847317e+32,
      "normalized_error": 1.6679069874573287e-16,
      "raw_change": -3.602879701896397e+16,
      "residual": -3.602879701896397e+16,
      "source": 0.0
    },
    {
      "boundary": 0.0,
      "field": "fx",
      "final": 4.467570830351532e+18,
      "initial": 0.0,
      "integral_units": "erg cm/s",
      "max_abs_residual": 7.210893507367996e+20,
      "max_normalized_error": 1.1135005745780917e-22,
      "normalization_scale": 6.475877670831218e+42,
      "normalized_error": 8.97579631752976e-23,
      "raw_change": 4.467570830351532e+18,
      "residual": 5.8126158950620044e+20,
      "source": -5.767940186758489e+20
    },
    {
      "boundary": 0.0,
      "field": "fy",
      "final": -7.205759403792794e+16,
      "initial": 0.0,
      "integral_units": "erg cm/s",
      "max_abs_residual": 7.479488189144372e+20,
      "max_normalized_error": 1.154976756715717e-22,
      "normalization_scale": 6.475877670831218e+42,
      "normalized_error": 3.3607947316949714e-23,
      "raw_change": -7.205759403792794e+16,
      "residual": -2.176409555923066e+20,
      "source": 2.1756889799826866e+20
    },
    {
      "boundary": 0.0,
      "field": "fz",
      "final": 2.3851063626554147e+19,
      "initial": 0.0,
      "integral_units": "erg cm/s",
      "max_abs_residual": 1.2784188118231532e+21,
      "max_normalized_error": 1.9741243995102527e-22,
      "normalization_scale": 6.475877670831218e+42,
      "normalized_error": 1.4171585250907128e-22,
      "raw_change": 2.3851063626554147e+19,
      "residual": -9.177345248663049e+20,
      "source": 9.415855884928591e+20
    }
  ],
  "dx": 1875000000.0,
  "executable_sha256": "3b9a570a5bbeebf503ec10ea24db4a3f8116a46d7906760c82be91bb89e10f7a",
  "inx": 8,
  "length": 60000000000.0,
  "level": 2,
  "movie_capture": {
    "hard_dt": 0.013333333333333332,
    "odt": 0.06666666666666667,
    "requested_snapshots": 61,
    "steps_per_output_check": 5
  },
  "norms": {
    "er": {
      "L1": 2.983871e-07,
      "L2": 3.652412e-07,
      "Linf": 9.418889e-07
    },
    "fx": {
      "L1": 1244.727,
      "L2": 1315.622,
      "Linf": 2282.908
    },
    "fy": {
      "L1": 1244.727,
      "L2": 1315.622,
      "Linf": 2282.908
    },
    "fz": {
      "L1": 1244.727,
      "L2": 1315.622,
      "Linf": 2282.908
    }
  },
  "origin": "Octo-TIGER application",
  "reference_sha256": "12e2eb0a3533d2357d8d21b3945c84cc2289dcf1e4680185aacba02a7d517f33",
  "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

Gaussian energy is shown as E − Ebg on a logarithmic scale. Values at or below 7.218479e-11 use the display floor, including zero and negative excesses. Raw values, signed errors, and error norms are unchanged.

All three flux maps

Movies

Conservation

Field	Q(0)	Q(t)	Boundary B	Source S	Residual R	Final |R|/scale	Maximum |R|/scale
er	2.160120e+32	2.160120e+32	0.000000e+00	0.000000e+00	-3.602880e+16	1.667907e-16	1.667907e-16
fx	0.000000e+00	4.467571e+18	0.000000e+00	-5.767940e+20	5.812616e+20	8.975796e-23	1.113501e-22
fy	0.000000e+00	-7.205759e+16	0.000000e+00	2.175689e+20	-2.176410e+20	3.360795e-23	1.154977e-22
fz	0.000000e+00	2.385106e+19	0.000000e+00	9.415856e+20	-9.177345e+20	1.417159e-22	1.974124e-22

Data and reproducibility

Effective configurationRun metadataSolver logPaired slice values CSVSlice unitsRaw conservation historyBudget summary CSVConservation unitsL1 normsL2 normsL∞ normsGaussian reference binaryFinal numerical SiloReference Silo

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:

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. Energy integrals have units erg in CGS; physical-flux integrals have units erg cm/s. Dividing a flux integral by c² gives radiation momentum. Energy residuals are normalized by max(|E₀|, |E|, |B_E|, |S_E|). A flux component uses the maximum of its corresponding terms and c times that energy scale, so a zero net flux does not cause division by zero. Summaries show the final residual and the largest normalized residual over the saved history.

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.

Where this problem comes from

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.

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.