Equilibrium Sphere: 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: diffusion-limit analytic reference. Steady diffusion solution for a Gaussian emitter. Finite-flux nonlinear M1 and quadrature errors can produce a reference-error floor.

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	1.011927e-05	2.440502e-05	1.440835e-04
Radiation flux x	erg/(cm^2 s)	7.754557e+04	1.627591e+05	1.245258e+06
Radiation flux y	erg/(cm^2 s)	7.754557e+04	1.627591e+05	1.245258e+06
Radiation flux z	erg/(cm^2 s)	2.811047e+04	5.144538e+04	2.981943e+05

Euler (first order) versus diffusion-limit analytic reference

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

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	4.883502e-05	1.102998e-04	6.424667e-04
Radiation flux x	erg/(cm^2 s)	1.051779e+06	2.570489e+06	1.630999e+07
Radiation flux y	erg/(cm^2 s)	1.051779e+06	2.570489e+06	1.630999e+07
Radiation flux z	erg/(cm^2 s)	1.240354e+06	2.653502e+06	1.630999e+07

Solver-reported full-volume reference norms

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	2.019307e-05	5.151584e-05	6.424667e-04
Radiation flux x	erg/(cm^2 s)	2.596979e+05	9.097040e+05	1.630999e+07
Radiation flux y	erg/(cm^2 s)	2.596979e+05	9.097040e+05	1.630999e+07
Radiation flux z	erg/(cm^2 s)	2.596979e+05	9.097040e+05	1.630999e+07

Source slice CSV · Run metadata

Coupled ADER2 versus diffusion-limit analytic reference

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

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	3.904556e-05	8.779870e-05	5.082125e-04
Radiation flux x	erg/(cm^2 s)	9.792466e+05	2.460068e+06	1.601179e+07
Radiation flux y	erg/(cm^2 s)	9.792466e+05	2.460068e+06	1.601179e+07
Radiation flux z	erg/(cm^2 s)	1.212244e+06	2.603727e+06	1.601179e+07

Solver-reported full-volume reference norms

Field	Units	L₁	L₂	L∞
Radiation energy density	erg/cm^3	1.593405e-05	4.090040e-05	5.082125e-04
Radiation flux x	erg/(cm^2 s)	2.350431e+05	8.708244e+05	1.601179e+07
Radiation flux y	erg/(cm^2 s)	2.350431e+05	8.708244e+05	1.601179e+07
Radiation flux z	erg/(cm^2 s)	2.350431e+05	8.708244e+05	1.601179e+07

Source slice CSV · Run metadata

Quantitative comparison and input hashes (JSON)

Level 3

Euler (first order): pending.

Coupled ADER2: pending.

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.