All problem comparisons · Campaign
Marshak: M1 RK1 and FLD backward Euler
Saved-state movies
Movies use stored numerical states, held without solution interpolation.
M1 reference: Su–Olson diffusion reference. The supplied Su–Olson radiation/material reference solves the linear thermal-capacity diffusion problem in normalized units. The saved parameter values define its match to the source model; diffusion and finite-speed M1 transport can differ. Reference values are evaluated at cell centers, without an additional cell-volume average.
FLD reference: Su–Olson linear diffusion analytic reference. The semi-infinite Su–Olson solution describes constant diffusion and equal linear radiation/material heat capacities. It is a diffusion-limit reference for FLD. Flux limiting, the finite-domain far boundary, cell-center sampling, and discretization can all contribute to its measured discrepancy.
M1 and FLD solve different transport closures. Both updates are first order in time. A small model discrepancy does not establish accuracy against an exact solution. M1 evolves radiation flux; FLD reconstructs diagnostic flux from its energy gradient. Flux differences are not errors in a shared radiation-momentum conservation law. FLD here supports stationary ideal gas with physical light speed on the CPU.
Level 0
M1 RK1: complete results, parameters and reference discussion
FLD backward Euler: complete results, parameters and reference discussion
matched. No other refinement level or physical time is substituted for a missing partner.
Matched profile coordinates and values (CSV)
FLD minus M1: model discrepancy on the saved slice
| Field | Units | L₁ | L₂ | L∞ |
|---|---|---|---|---|
| Radiation energy density | normalized energy density | 6.340422e-02 | 1.736498e-01 | 4.908954e-01 |
| Radiation flux x | normalized flux | 4.481618e-03 | 1.217133e-02 | 3.439687e-02 |
| Radiation flux y | normalized flux | 9.413098e-24 | 3.326411e-23 | 1.602721e-22 |
| Radiation flux z | normalized flux | 5.795993e-21 | 3.058975e-20 | 2.312365e-19 |
| Material energy density | normalized energy density | 6.121183e-02 | 1.678301e-01 | 4.744607e-01 |
M1 RK1: model-specific reference diagnostics
Su–Olson diffusion reference. The supplied Su–Olson radiation/material reference solves the linear thermal-capacity diffusion problem in normalized units. The saved parameter values define its match to the source model; diffusion and finite-speed M1 transport can differ. Reference values are evaluated at cell centers, without an additional cell-volume average.
Area-weighted reference errors on this model’s original leaf slice.
| Field | Units | L₁ | L₂ | L∞ |
|---|---|---|---|---|
| Radiation energy density | normalized energy density | 6.979463e-02 | 1.916485e-01 | 5.418220e-01 |
| Radiation flux x | normalized flux | 4.579875e-03 | 1.239782e-02 | 3.503160e-02 |
| Radiation flux y | normalized flux | 0.000000e+00 | 0.000000e+00 | 0.000000e+00 |
| Radiation flux z | normalized flux | 0.000000e+00 | 0.000000e+00 | 0.000000e+00 |
| Material energy density | normalized energy density | 6.735545e-02 | 1.851361e-01 | 5.234262e-01 |
Solver-reported full-volume reference norms
| Field | Units | L₁ | L₂ | L∞ |
|---|---|---|---|---|
| Radiation energy density | normalized energy density | 6.979463e-02 | 1.916485e-01 | 5.418220e-01 |
| Radiation flux x | normalized flux | 4.579875e-03 | 1.239782e-02 | 3.503160e-02 |
| Radiation flux y | normalized flux | 0.000000e+00 | 0.000000e+00 | 0.000000e+00 |
| Radiation flux z | normalized flux | 0.000000e+00 | 0.000000e+00 | 0.000000e+00 |
| Material energy density | normalized energy density | 6.735545e-02 | 1.851361e-01 | 5.234262e-01 |
Source slice CSV · Run metadata
FLD backward Euler: model-specific reference diagnostics
Su–Olson linear diffusion analytic reference. The semi-infinite Su–Olson solution describes constant diffusion and equal linear radiation/material heat capacities. It is a diffusion-limit reference for FLD. Flux limiting, the finite-domain far boundary, cell-center sampling, and discretization can all contribute to its measured discrepancy.
Area-weighted reference errors on this model’s original leaf slice.
| Field | Units | L₁ | L₂ | L∞ |
|---|---|---|---|---|
| Radiation energy density | normalized energy density | 6.390453e-03 | 1.800540e-02 | 5.092660e-02 |
| Radiation flux x | normalized flux | 9.825686e-05 | 2.306628e-04 | 6.347323e-04 |
| Radiation flux y | normalized flux | 9.413098e-24 | 3.326411e-23 | 1.602721e-22 |
| Radiation flux z | normalized flux | 5.795993e-21 | 3.058975e-20 | 2.312365e-19 |
| Material energy density | normalized energy density | 6.143659e-03 | 1.731205e-02 | 4.896552e-02 |
Level 1
M1 RK1: complete results, parameters and reference discussion
FLD backward Euler: complete results, parameters and reference discussion
matched. No other refinement level or physical time is substituted for a missing partner.
Matched profile coordinates and values (CSV)
FLD minus M1: model discrepancy on the saved slice
| Field | Units | L₁ | L₂ | L∞ |
|---|---|---|---|---|
| Radiation energy density | normalized energy density | 5.115297e-02 | 1.903576e-01 | 7.593460e-01 |
| Radiation flux x | normalized flux | 4.393539e-03 | 1.640291e-02 | 6.545852e-02 |
| Radiation flux y | normalized flux | 2.995482e-20 | 1.461902e-19 | 1.109119e-18 |
| Radiation flux z | normalized flux | 2.339834e-20 | 1.222801e-19 | 7.394127e-19 |
| Material energy density | normalized energy density | 4.960891e-02 | 1.850360e-01 | 7.382484e-01 |
M1 RK1: model-specific reference diagnostics
Su–Olson diffusion reference. The supplied Su–Olson radiation/material reference solves the linear thermal-capacity diffusion problem in normalized units. The saved parameter values define its match to the source model; diffusion and finite-speed M1 transport can differ. Reference values are evaluated at cell centers, without an additional cell-volume average.
Area-weighted reference errors on this model’s original leaf slice.
| Field | Units | L₁ | L₂ | L∞ |
|---|---|---|---|---|
| Radiation energy density | normalized energy density | 5.774023e-02 | 2.159714e-01 | 8.618582e-01 |
| Radiation flux x | normalized flux | 3.300516e-03 | 1.167124e-02 | 4.627843e-02 |
| Radiation flux y | normalized flux | 0.000000e+00 | 0.000000e+00 | 0.000000e+00 |
| Radiation flux z | normalized flux | 0.000000e+00 | 0.000000e+00 | 0.000000e+00 |
| Material energy density | normalized energy density | 5.629181e-02 | 2.110790e-01 | 8.424837e-01 |
Solver-reported full-volume reference norms
| Field | Units | L₁ | L₂ | L∞ |
|---|---|---|---|---|
| Radiation energy density | normalized energy density | 5.774023e-02 | 2.159714e-01 | 8.618582e-01 |
| Radiation flux x | normalized flux | 3.300516e-03 | 1.167124e-02 | 4.627843e-02 |
| Radiation flux y | normalized flux | 0.000000e+00 | 0.000000e+00 | 0.000000e+00 |
| Radiation flux z | normalized flux | 0.000000e+00 | 0.000000e+00 | 0.000000e+00 |
| Material energy density | normalized energy density | 5.629181e-02 | 2.110790e-01 | 8.424837e-01 |
Source slice CSV · Run metadata
FLD backward Euler: model-specific reference diagnostics
Su–Olson linear diffusion analytic reference. The semi-infinite Su–Olson solution describes constant diffusion and equal linear radiation/material heat capacities. It is a diffusion-limit reference for FLD. Flux limiting, the finite-domain far boundary, cell-center sampling, and discretization can all contribute to its measured discrepancy.
Area-weighted reference errors on this model’s original leaf slice.
| Field | Units | L₁ | L₂ | L∞ |
|---|---|---|---|---|
| Radiation energy density | normalized energy density | 6.587278e-03 | 2.563794e-02 | 1.025122e-01 |
| Radiation flux x | normalized flux | 1.304488e-03 | 4.813163e-03 | 1.918009e-02 |
| Radiation flux y | normalized flux | 2.995482e-20 | 1.461902e-19 | 1.109119e-18 |
| Radiation flux z | normalized flux | 2.339834e-20 | 1.222801e-19 | 7.394127e-19 |
| Material energy density | normalized energy density | 6.682921e-03 | 2.606732e-02 | 1.042353e-01 |
How these comparisons are measured
All field values retain their saved units. Central profiles share x coordinates and the positive row adjacent to y = z = 0. Adaptive leaf-cell averages are prolonged as constants onto the finest common Cartesian x-y partition, without extrapolation or time interpolation. Different leaf sizes may average different z extents adjacent to the plane. These are saved finite-volume slices.
Slice L₁ is mean absolute discrepancy, L₂ is root-mean-square discrepancy, and L∞ is maximum absolute discrepancy, with area weighting. These are two-dimensional diagnostics; separately labeled solver reference norms cover the full volume. Each model is measured against its own saved reference. A diffusion-limit error includes departures of the nonlinear closure from that limit.
The saved Su–Olson reference uses cell-center point values without a cell-volume average. Reference differences for both models include sampling, closure, finite-domain, and discretization effects; Su–Olson is not an exact nonlinear M1 or flux-limited FLD solution.