FLD backward Euler · 32³ cells
Central cell layer at z=dx/2; ray at y=z=dx/2.
FLD backward Euler with the Levermore-Pomraning limiter; stationary material and physical c = 29979245800 cm/s. Radiation flux is derived from the radiation-energy gradient; it is a diagnostic, not an evolved momentum variable. Analytic references describe the linear diffusion limit and need not solve nonlinear flux-limited diffusion exactly. M1 versus FLD differences compare distinct radiation models and are not analytic-reference errors.
Recorded setup and run provenance
| Origin | Octo-TIGER application |
|---|---|
| Domain | Cube of side 6.000000e+10 cm; centered at the origin |
| Resolution | 32³ cells; finest Δx = 1.875000e+09 cm |
| Final comparison time | 2.000000e-01 s |
| Radiation model | FLD |
| Radiation integration | FLD backward Euler (first order in time) |
| Flux semantics | Derived FLD flux: F = -D grad E |
| Analytic reference | steady-gaussian-emitter; linear diffusion-limit reference, not exact for the nonlinear FLD limiter |
| Reference norm scope | Area-weighted projected positive-side slice; no full-volume norm or convergence order inferred |
| 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 | on |
| CFL | 0.4 |
| Timestep cap | 0.001 s |
| Output interval | 0.0016666666666666668 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 |
{
"background": 1.0,
"c": 29979245800.0,
"case": "equilibrium_sphere",
"cells": 32,
"command": [
"/ddnB/work/dmarce1/octotiger-m1-fld-20261014/campaign/control/launch-solver",
"/ddnB/work/dmarce1/octotiger-m1-fld-20261014/build/octotiger",
"--config_file=/ddnB/work/dmarce1/octotiger-m1-fld-20261014/campaign/runs/l2-equilibrium_sphere-fld/equilibrium_sphere/l2/run.ini",
"--hpx:threads=64"
],
"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": "on",
"disable_diagnostics": "on",
"eos": "IDEAL",
"gravity": "off",
"hard_dt": "0.001",
"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": "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",
"radiation_algorithm_version": 1,
"radiation_integrator": "backward_euler",
"radiation_limiter": "levermore_pomraning",
"radiation_model": "fld",
"radiation_reduced_c_factor": "1",
"stop_step": "100000",
"stop_time": "0.20000000000000001",
"unigrid": "on",
"xscale": "30000000000"
},
"config": {
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"atomic_number": "1",
"cfl": "0.4",
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"code_to_s": "1",
"datadir": "/ddnB/work/dmarce1/octotiger-m1-fld-20261014/campaign/runs/l2-equilibrium_sphere-fld/equilibrium_sphere/l2/",
"disable_analytic": "on",
"disable_diagnostics": "on",
"disable_output": "off",
"eos": "IDEAL",
"gravity": "off",
"hard_dt": "0.001",
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"hydro_device_kernel_type": "OFF",
"hydro_host_kernel_type": "LEGACY",
"max_level": "2",
"min_level": "2",
"monopole_device_kernel_type": "OFF",
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"n_species": "1",
"odt": "0.0016666666666666668",
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"dx": 1875000000.0,
"executable_sha256": "ca38f12e338eed0b7074dbb15c70c089a8d31d1c12aa87f5b044c42824721f0b",
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"L2": 5.0452557376075305e-05,
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"linear_residual": 8.921859554371925e-12,
"nonlinear_iterations": 2,
"nonlinear_residual": 8.33300371781581e-16,
"retries": 0,
"substeps": 1,
"t": 0.19300000000000014,
"vcycles": 6
},
{
"boundary_energy": -2.7089359498352698e+26,
"conservation_error": -3.784718828767108e+15,
"dt": 0.001,
"linear_iterations": 6,
"linear_residual": 8.921794477807483e-12,
"nonlinear_iterations": 2,
"nonlinear_residual": 8.332997419739063e-16,
"retries": 0,
"substeps": 1,
"t": 0.19400000000000014,
"vcycles": 6
},
{
"boundary_energy": -2.7089097795685487e+26,
"conservation_error": -181722628189533.37,
"dt": 0.001,
"linear_iterations": 6,
"linear_residual": 8.921936760476588e-12,
"nonlinear_iterations": 2,
"nonlinear_residual": 8.332991165826692e-16,
"retries": 0,
"substeps": 1,
"t": 0.19500000000000015,
"vcycles": 6
},
{
"boundary_energy": -2.708883814413418e+26,
"conservation_error": -1.6573411648885417e+15,
"dt": 0.001,
"linear_iterations": 6,
"linear_residual": 8.921846141139414e-12,
"nonlinear_iterations": 2,
"nonlinear_residual": 8.332984955650726e-16,
"retries": 0,
"substeps": 1,
"t": 0.19600000000000015,
"vcycles": 6
},
{
"boundary_energy": -2.708858051476376e+26,
"conservation_error": -393623693401156.8,
"dt": 0.001,
"linear_iterations": 6,
"linear_residual": 8.921930882476033e-12,
"nonlinear_iterations": 2,
"nonlinear_residual": 8.332978788788466e-16,
"retries": 0,
"substeps": 1,
"t": 0.19700000000000015,
"vcycles": 6
},
{
"boundary_energy": -2.708832487916598e+26,
"conservation_error": 1.6864256162639175e+15,
"dt": 0.001,
"linear_iterations": 6,
"linear_residual": 8.921896207650911e-12,
"nonlinear_iterations": 2,
"nonlinear_residual": 8.332972664822386e-16,
"retries": 0,
"substeps": 1,
"t": 0.19800000000000015,
"vcycles": 6
},
{
"boundary_energy": -2.7088071209445273e+26,
"conservation_error": 348949312949752.1,
"dt": 0.0009999999999998621,
"linear_iterations": 6,
"linear_residual": 8.921983683980922e-12,
"nonlinear_iterations": 2,
"nonlinear_residual": 8.33296658334007e-16,
"retries": 0,
"substeps": 1,
"t": 0.19900000000000015,
"vcycles": 6
}
],
"substeps": 200
},
"inx": 8,
"length": 60000000000.0,
"level": 2,
"movie_capture": {
"hard_dt": 0.001,
"odt": 0.0016666666666666668,
"requested_snapshots": 121,
"steps_per_output_check": 1
},
"origin": "Octo-TIGER application",
"physical_c": 29979245800.0,
"radiation_algorithm_version": 1,
"radiation_backend": "cpu",
"radiation_equations": "stationary physical-c FLD",
"radiation_flux_semantics": "derived_fld_flux",
"radiation_integrator": "backward_euler",
"radiation_limiter": "levermore_pomraning",
"radiation_model": "fld",
"radiation_reduced_c_factor": 1.0,
"reference_exact_for_solver": false,
"reference_kind": "linear-diffusion-limit",
"reference_model": "steady-gaussian-emitter",
"reference_sampling": "cell averages",
"status": "complete",
"time": 0.2,
"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": "Analytic steady constant-D Gaussian-emitter reference. Nonlinear FLD limiter, boundary sampling and discretization can cause equilibrium drift; these are slice discrepancies, not exact nonlinear-FLD error norms."
}
Derived FLD flux maps
Accepted implicit solves
Accepted steps: 200; substeps: 200; timestep retries: 0; maximum nonlinear residual: 1.250134e-15; maximum absolute coupled energy error per accepted step: 5.263206e+15.
Projected-slice differences from the analytic reference
Area-weighted L1, L2, and maximum differences on the exported positive-side slice; these are not full-volume norms or a pure discretization error. No convergence order is inferred. Energy-density and flux units are those recorded in the slice-units download.
| Field | L1 difference | L2 difference | Maximum difference |
|---|---|---|---|
| er | 1.541166e-05 | 5.045256e-05 | 4.264541e-04 |
| fx | 4.691296e+05 | 9.879944e+05 | 5.123159e+06 |
| fy | 4.691296e+05 | 9.879944e+05 | 5.123159e+06 |
| fz | 1.325364e+05 | 3.701645e+05 | 2.113611e+06 |
Radiation energy 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 | 9.999982e-01 | 2.481570e-04 | 2.463735e-04 | -7.429547e-16 | 7.554444e-16 |
Saved-state movie
Equilibrium Sphere · L2 · FLD backward Euler
121 stored states; 121 shown. Physical time: 0–0.2 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, 0.002, 0.004, 0.005, 0.007, 0.009, 0.01, 0.012, 0.014, 0.015, 0.017, 0.019, 0.02, 0.022, 0.024, 0.025, 0.027, 0.029, 0.03, 0.032, 0.034, 0.035, 0.037, 0.039, 0.04, 0.042, 0.044, 0.045, 0.047, 0.049, 0.05, 0.052, 0.054, 0.055, 0.057, 0.059, 0.06, 0.062, 0.064, 0.065, 0.067, 0.069, 0.07, 0.072, 0.074, 0.075, 0.077, 0.079, 0.08, 0.082, 0.084, 0.085, 0.087, 0.089, 0.09, 0.092, 0.094, 0.095, 0.097, 0.099, 0.1, 0.102, 0.104, 0.105, 0.107, 0.109, 0.11, 0.112, 0.114, 0.115, 0.117, 0.119, 0.12, 0.122, 0.124, 0.125, 0.127, 0.129, 0.13, 0.132, 0.134, 0.135, 0.137, 0.139, 0.14, 0.142, 0.144, 0.145, 0.147, 0.149, 0.15, 0.152, 0.154, 0.155, 0.157, 0.159, 0.16, 0.162, 0.164, 0.165, 0.167, 0.169, 0.17, 0.172, 0.174, 0.175, 0.177, 0.179, 0.18, 0.182, 0.184, 0.185, 0.187, 0.189, 0.19, 0.192, 0.194, 0.195, 0.197, 0.199, 0.2 s.
Arrows show derived FLD diagnostic flux -D grad(E); radiation momentum is not evolved. Arrows have equal length and show direction only.
Final-time profiles compare FLD with a linear diffusion-limit analytic reference, not an exact solution of nonlinear flux-limited diffusion. The movie shows numerical states only.
Central z=0+ radiation energy slice with faint cell mesh and equal-length flux direction arrows. Arrows show derived FLD diagnostic flux -D grad(E); radiation momentum is not evolved. Fixed color limits; actual stored Silo times held without solution interpolation. Final-time profiles compare FLD with a linear diffusion-limit analytic reference, not an exact solution of nonlinear flux-limited diffusion. The movie shows numerical states only. The maps show the full simulation domain without cropping.
Download MP4 · Rendering and timing metadata · Full run report
Data and reproducibility
Effective configurationRun metadataSolver logNumerical/reference/difference slice CSVSlice unitsFLD diagnosticsAccepted solve history CSVSlice reference-difference norms CSVRadiation energy history CSVEnergy budget summary CSVFinal numerical Silo (local file; excluded from staged site)