Streaming Wave: 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: exact analytic solution. Exact nonlinear M1 periodic-advection cell averages.

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	3.974018e-02	4.133068e-02	4.785348e-02
Radiation flux x	erg/(cm^2 s)	6.368198e+08	6.623069e+08	7.668320e+08
Radiation flux y	erg/(cm^2 s)	9.552297e+08	9.934603e+08	1.150248e+09
Radiation flux z	erg/(cm^2 s)	3.184099e+08	3.311534e+08	3.834160e+08

Euler (first order) versus exact analytic solution

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

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	5.075526e-02	6.418457e-02	1.191588e-01
Radiation flux x	erg/(cm^2 s)	8.133318e+08	1.028531e+09	1.909470e+09
Radiation flux y	erg/(cm^2 s)	1.219998e+09	1.542796e+09	2.864204e+09
Radiation flux z	erg/(cm^2 s)	4.066659e+08	5.142654e+08	9.547348e+08

Solver-reported full-volume reference norms

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	5.075526e-02	6.418457e-02	1.191588e-01
Radiation flux x	erg/(cm^2 s)	8.133318e+08	1.028531e+09	1.909470e+09
Radiation flux y	erg/(cm^2 s)	1.219998e+09	1.542796e+09	2.864204e+09
Radiation flux z	erg/(cm^2 s)	4.066659e+08	5.142654e+08	9.547348e+08

Source slice CSV · Run metadata

Coupled ADER2 versus exact analytic solution

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

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	7.740758e-02	9.514244e-02	1.633612e-01
Radiation flux x	erg/(cm^2 s)	1.240424e+09	1.524618e+09	2.617795e+09
Radiation flux y	erg/(cm^2 s)	1.860636e+09	2.286927e+09	3.926693e+09
Radiation flux z	erg/(cm^2 s)	6.202120e+08	7.623089e+08	1.308898e+09

Solver-reported full-volume reference norms

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	7.740758e-02	9.514244e-02	1.633612e-01
Radiation flux x	erg/(cm^2 s)	1.240424e+09	1.524618e+09	2.617795e+09
Radiation flux y	erg/(cm^2 s)	1.860636e+09	2.286927e+09	3.926693e+09
Radiation flux z	erg/(cm^2 s)	6.202120e+08	7.623089e+08	1.308898e+09

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	3.523321e-02	3.847428e-02	7.191846e-02
Radiation flux x	erg/(cm^2 s)	5.645974e+08	6.165342e+08	1.152463e+09
Radiation flux y	erg/(cm^2 s)	8.468961e+08	9.248013e+08	1.728695e+09
Radiation flux z	erg/(cm^2 s)	2.822987e+08	3.082671e+08	5.762316e+08

Euler (first order) versus exact analytic solution

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

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	4.639050e-02	5.446053e-02	1.012121e-01
Radiation flux x	erg/(cm^2 s)	7.433884e+08	8.727072e+08	1.621881e+09
Radiation flux y	erg/(cm^2 s)	1.115083e+09	1.309061e+09	2.432821e+09
Radiation flux z	erg/(cm^2 s)	3.716942e+08	4.363536e+08	8.109404e+08

Solver-reported full-volume reference norms

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	4.639050e-02	5.446053e-02	1.012121e-01
Radiation flux x	erg/(cm^2 s)	7.433884e+08	8.727072e+08	1.621881e+09
Radiation flux y	erg/(cm^2 s)	1.115083e+09	1.309061e+09	2.432821e+09
Radiation flux z	erg/(cm^2 s)	3.716942e+08	4.363536e+08	8.109404e+08

Source slice CSV · Run metadata

Coupled ADER2 versus exact analytic solution

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

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	2.988865e-02	3.420566e-02	6.205922e-02
Radiation flux x	erg/(cm^2 s)	4.789530e+08	5.481313e+08	9.944730e+08
Radiation flux y	erg/(cm^2 s)	7.184295e+08	8.221969e+08	1.491709e+09
Radiation flux z	erg/(cm^2 s)	2.394765e+08	2.740656e+08	4.972365e+08

Solver-reported full-volume reference norms

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	2.988865e-02	3.420566e-02	6.205922e-02
Radiation flux x	erg/(cm^2 s)	4.789530e+08	5.481313e+08	9.944730e+08
Radiation flux y	erg/(cm^2 s)	7.184295e+08	8.221969e+08	1.491709e+09
Radiation flux z	erg/(cm^2 s)	2.394765e+08	2.740656e+08	4.972365e+08

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.