Shock-crossing defect is a capturing error

ID	Group	Milestone	Backends	Figure	Regression
E0	Seam studies	M6 (#68)	FNL; CPU re-runs agree to the printed digits	yes	—

Case and scope

A3 found an $O(1)$, non-converging defect behind a shock that crosses a 2:1 seam; D2 showed that no coarse-to-fine fill reduces it and that limiting the fill makes it worse. E0 asks whether the defect comes from the consistency of the fill with the restriction (only restriction-compatible with the mean restriction has $R\circ P=I$ exactly), from the reflux, or from neither. The answer decides whether a seam operator can fix it.

Models and numerics

Item	Choice	Code
Equations	Sod: compressible Euler, $\gamma =1.4$; RJ2a: ideal MHD, $\gamma =5/3$, no divergence control	models
Space, time	WENO5 flux splitting on characteristic variables; SSP Runge–Kutta 3 stages, CFL 0.5	as in AS
Fill	restriction-compatible (order 2) or tricubic (order 4)	seam_interpolate_compatible, seam_interpolate_tricubic, adam_seam_interpolation_library.F90
Restriction	mean or lagrange6	restrict_seam_high_order, adam_field_object.F90
Reflux	on or off	apply_reflux_to_stage_forest, adam_flume_cpu_object.F90 and adam_flume_fnl_object.F90

Initial and boundary conditions

Item	Value
Cases	the AS c2f crossings: Sod $N=200$ (and 400), RJ2a $N=256$; octree, $x>0.75$ refined, $t=0.2$, extrapolation on all faces
Arms	every combination of fill (restriction-compatible, tricubic) × restriction (mean, lagrange6) × reflux (on, off)
Ranks	2, FNL

Input files

File	Role
src/tests/flume/verification/amr-shock/make_amr_shock.py	the crossings (--seam-ghost-fill, --seam-restriction, --no-reflux)
src/tests/flume/verification/amr-shock/amr_shock_oracle.py	the plateau and band defects, the mass and the shock position

No check.sh leg runs E0.

Results

*The table below as bars (make_doc_figures.py --only capturing-error): no combination of coarse-to-fine fill, restriction and reflux removes the defect; the largest change is the reflux on Sod with the tricubic fill. *

The CPU re-runs (make_doc_runs.sh <runs> seams): left, the plateau after the c2f crossing with the tricubic and the compatible fills at $N$ = 200 and 400, the bump behind the face the same at both resolutions; right, the defect of every arm, which the CPU reproduces to the printed digits of the FNL table below (2.32, 1.82, 2.46, 3.10, 3.17 $\cdot {10}^{-2}$; at $N=400$: 2.35 and 2.14 $\cdot {10}^{-2}$, band $1.71\cdot {10}^{-3}$).

Plateau density defect behind the shock:

Fill + restriction	Sod, reflux on	Sod, reflux off	RJ2a, reflux on	RJ2a, reflux off
compatible + mean ($R\circ P=I$)	$2.32\cdot {10}^{-2}$	$1.82\cdot {10}^{-2}$	$1.97\cdot {10}^{-2}$	$1.83\cdot {10}^{-2}$
compatible + lagrange6	$2.32\cdot {10}^{-2}$	$1.85\cdot {10}^{-2}$	$1.98\cdot {10}^{-2}$	$1.84\cdot {10}^{-2}$
tricubic + mean	$2.46\cdot {10}^{-2}$	$3.10\cdot {10}^{-2}$	$1.41\cdot {10}^{-2}$	$1.68\cdot {10}^{-2}$
tricubic + lagrange6	$2.46\cdot {10}^{-2}$	$3.17\cdot {10}^{-2}$	$1.41\cdot {10}^{-2}$	$1.68\cdot {10}^{-2}$

Sod with the reflux on at $N=400$: $2.35\cdot {10}^{-2}$ (tricubic) and $2.14\cdot {10}^{-2}$ (compatible), while the band defect next to the seam converges ($6.35\cdot {10}^{-3}\to 1.71\cdot {10}^{-3}$, order 1.9). With the reflux off the mass drifts (${10}^{-5}$ to ${10}^{-4}$) and the shock lands 0.06 to 0.22 cells off.

Related cases: AS found the defect; D2 ruled out the fill's order; E2 proceeds with the reflux policy on its own merit.

Findings

• The restriction is irrelevant (mean and lagrange6 within 0.5 % in every case).
• Consistency does not control it: the $R\circ P=I$ pair is best on Sod and worse than tricubic on RJ2a.
• The reflux does not cause it: off, it moves the defect by up to 30 % either way and costs conservation.
• It does not converge: $N$ = 200 to 400 lowers it by 5 to 8 %.
• So it is not a seam-operator defect. The evidence points to the start-up error of a captured shock whose numerical structure, a few cells wide, is squeezed into finer cells (an interpretation, not measured directly): the transition emits a spurious wave (the plateau entropy defect is about 1.8 %) whose amplitude scales with the shock's strength, not with the spacing.
• Remedy: keep shocks off the seams by refining ahead of them (a gradient or Löhner marker with a buffer and a short regrid frequency, M5), which numerics now recommends. Phase E (the reflux policy) proceeds on its own merit.