Gaussian Pulse: Euler and ADER2

Runtime revision and continuation

Revision 6 with Newton rejection classification and physical initialization for damped Radau recovery. A collectively rejected AP trial is restored and retried at the same timestep using conservative HLL fluxes for all radiation moments; existing timestep halving remains if both trials fail. Physical guards, source equations, tolerances and iteration limits are unchanged. HLL fallback can change diffusion-limit accuracy on affected steps; no universal second-order or AP accuracy is claimed. Same physical model with a qualified realizability recovery change. An AP trial rejection can trigger a conservative whole-step HLL fallback. This may change diffusion-limit accuracy on fallback steps; no universal order or AP claim is made. Earlier completed results retain their original runtime and source hashes. Approved comparisons spanning revisions are explicitly qualified; their differences include runtime-revision effects. Checkpoint continuations retain states from both runtime segments and are excluded from clean timing claims. Conservation diagnostics reset at the checkpoint. Displayed new-ledger budgets cover only the resumed segment, with its own starting totals. The original raw ledger is retained separately; its abandoned post-checkpoint tail is not used.

Per-run runtime identities and qualifications

Reference: linearized analytic reference. Fourier telegraph-equation reference for isotropic, linearized radiation moments. Nonlinear M1 can retain a closure-model error at finite amplitude.

Coupled ADER2 accuracy remains under investigation. A completed run or a small difference does not establish uniform second-order accuracy.

Revision 6 validation limits. Streaming-wave temporal-order thresholds failed at limiter transitions even with resolved references. Optically thick radiation temporal probes remain unresolved under the unchanged 1e-12 measurement guard. Strict stiff coupled temporal tests have not been rerun for revision 6 and remain an open limitation.

Level 2

Euler (first order): complete results, parameters and reference discussion

Coupled ADER2: 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)

ADER2 minus Euler: slice discrepancy

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	1.810764e-05	4.407825e-05	2.589600e-04
Radiation flux x	erg/(cm^2 s)	1.312539e+05	2.901424e+05	1.340358e+06
Radiation flux y	erg/(cm^2 s)	1.312539e+05	2.901424e+05	1.340358e+06
Radiation flux z	erg/(cm^2 s)	3.649823e+04	1.291577e+05	7.901392e+05

Euler (first order) versus linearized analytic reference

Area-weighted errors on this method’s original leaf slice.

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	1.986134e-05	4.524319e-05	1.813365e-04
Radiation flux x	erg/(cm^2 s)	1.954307e+05	4.705469e+05	2.474439e+06
Radiation flux y	erg/(cm^2 s)	1.954307e+05	4.705469e+05	2.474439e+06
Radiation flux z	erg/(cm^2 s)	1.047773e+05	2.469561e+05	1.324834e+06

Solver-reported full-volume reference norms

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	7.385814e-06	2.283424e-05	1.813365e-04
Radiation flux x	erg/(cm^2 s)	6.159883e+04	2.171852e+05	2.474439e+06
Radiation flux y	erg/(cm^2 s)	6.159883e+04	2.171852e+05	2.474439e+06
Radiation flux z	erg/(cm^2 s)	6.159883e+04	2.171852e+05	2.474439e+06

Source slice CSV · Run metadata

Coupled ADER2 versus linearized analytic reference

Area-weighted errors on this method’s original leaf slice.

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	3.401671e-05	7.788240e-05	4.402965e-04
Radiation flux x	erg/(cm^2 s)	2.516497e+05	5.396765e+05	2.114973e+06
Radiation flux y	erg/(cm^2 s)	2.516497e+05	5.396765e+05	2.114973e+06
Radiation flux z	erg/(cm^2 s)	1.328580e+05	3.425122e+05	2.114973e+06

Solver-reported full-volume reference norms

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	1.182842e-05	3.635906e-05	4.402965e-04
Radiation flux x	erg/(cm^2 s)	7.760839e+04	2.453516e+05	2.114973e+06
Radiation flux y	erg/(cm^2 s)	7.760839e+04	2.453516e+05	2.114973e+06
Radiation flux z	erg/(cm^2 s)	7.760839e+04	2.453516e+05	2.114973e+06

Source slice CSV · Run metadata

Quantitative comparison and input hashes (JSON)

Level 3

Euler (first order): complete results, parameters and reference discussion

Coupled ADER2: 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)

ADER2 minus Euler: slice discrepancy

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	6.197743e-06	1.459393e-05	8.875497e-05
Radiation flux x	erg/(cm^2 s)	4.753814e+04	1.140957e+05	7.046102e+05
Radiation flux y	erg/(cm^2 s)	4.753814e+04	1.140957e+05	7.046102e+05
Radiation flux z	erg/(cm^2 s)	4.199641e+03	1.442272e+04	1.234253e+05

Euler (first order) versus linearized analytic reference

Area-weighted errors on this method’s original leaf slice.

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	3.100099e-06	5.949939e-06	2.410265e-05
Radiation flux x	erg/(cm^2 s)	2.757679e+04	6.376372e+04	5.374988e+05
Radiation flux y	erg/(cm^2 s)	2.757679e+04	6.376372e+04	5.374988e+05
Radiation flux z	erg/(cm^2 s)	1.215476e+04	2.834376e+04	1.534115e+05

Solver-reported full-volume reference norms

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	1.065226e-06	3.016476e-06	2.410265e-05
Radiation flux x	erg/(cm^2 s)	7.716388e+03	2.585419e+04	5.374988e+05
Radiation flux y	erg/(cm^2 s)	7.716388e+03	2.585419e+04	5.374988e+05
Radiation flux z	erg/(cm^2 s)	7.716388e+03	2.585419e+04	5.374988e+05

Source slice CSV · Run metadata

Coupled ADER2 versus linearized analytic reference

Area-weighted errors on this method’s original leaf slice.

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	4.452250e-06	1.148332e-05	7.413785e-05
Radiation flux x	erg/(cm^2 s)	4.160655e+04	1.094324e+05	7.344223e+05
Radiation flux y	erg/(cm^2 s)	4.160655e+04	1.094324e+05	7.344223e+05
Radiation flux z	erg/(cm^2 s)	1.544277e+04	4.130455e+04	2.768368e+05

Solver-reported full-volume reference norms

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	1.565707e-06	5.503553e-06	7.745741e-05
Radiation flux x	erg/(cm^2 s)	1.078147e+04	4.286486e+04	7.344223e+05
Radiation flux y	erg/(cm^2 s)	1.078147e+04	4.286486e+04	7.344223e+05
Radiation flux z	erg/(cm^2 s)	1.078147e+04	4.286486e+04	7.344223e+05

Source slice CSV · Run metadata

Quantitative comparison and input hashes (JSON)

Level 4

Euler (first order): pending.

Coupled ADER2: pending.

Quantitative comparison and input hashes (JSON)

How these comparisons are measured

All field values use the saved units (CGS for the seven original problems; normalized units for Marshak). The 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, with no extrapolation or time interpolation. Different leaf sizes may average different z extents adjacent to the plane. The plotted curves therefore represent the saved finite-volume slices.

Slice L₁ is mean absolute difference, L₂ is root-mean-square difference, and L∞ is maximum absolute difference, with area weighting. These are two-dimensional slice diagnostics. The separately labeled solver norms cover the full volume.

Each method uses its own saved cell-average reference where supplied; finer meshes can change the sampled reference.