Limited coarse-to-fine fills

ID	Group	Milestone	Backends	Figure	Regression
D2	Seam studies	M6 (#68)	FNL (crossings); CPU, FNL (AO legs)	yes	—

Case and scope

The unlimited quintic fill of D1 reaches sixth order on smooth flows but worsens a shock entering a finer region. D2 asks whether a limited 6-point fill keeps the order and brings the shock crossings of A3 back to tricubic's level (the Phase D exit). Neither did, so D2 then ran every fill on the c2f crossings to find out what the shock-crossing defect is not.

Models and numerics

Item	Choice	Code
seam_ghost_fill = weno6	per direction the WENO-Z combination of the cubics of the 4-point sub-stencils that contain the ghost's interval (the C3 restriction's kernel, now shared); linear weights reproducing the Lagrange interpolant of their union (all positive); sub-stencils holding a jump dropped; the clamped linear interpolation of the interval when none is left. Order 6 at the centred anchor positions, 5 one cell from the donor block's edge, 4 at it	seam_interpolate_weno6, seam_fill_weno6_1d, seam_weno6_core, adam_seam_interpolation_library.F90
seam_ghost_fill = quintic-mp	the D1 quintic clamped per direction to a Suresh–Huynh monotonicity-preserving bound: the interval's values and the curvature-corrected linear value $L+\theta (\theta -1)/2d^{M4}$, widened by the curvature term; at the block edge, where the target lies a quarter cell outside the footprint, the extrapolation is allowed only when the edge-side intervals hold no jump. Order 6 at every target	seam_interpolate_quintic_mp, seam_fill_quintic_mp_1d, seam_minmod4, adam_seam_interpolation_library.F90
The other fills compared	injection (order 1), restriction-compatible (order 2, exact $R\circ P=I$ with the mean restriction), tricubic (4, default), quintic (6)	seam_interpolate_compatible, seam_interpolate_tricubic, seam_interpolate_quintic, adam_seam_interpolation_library.F90; interp_seam_ghost, adam_field_object.F90
Key	[amr] seam_ghost_fill, read in initialize; both limited fills opt-in, the default stays tricubic	adam_adam_object.F90
Restriction	lagrange6 on the crossings; weno6 (euler-ww) or lagrange6 (euler-mp) on AO	C1, C3c
Scheme, time, reflux	as in AO (reflux off) and AS (reflux on)	numerics

Initial and boundary conditions

Run	Setup
Crossings	the AS c2f crossings, Sod $N=200$ and RJ2a $N=256$, lagrange6 restriction, reflux on, every fill; FNL, 2 ranks
AO euler-ww	the AO Euler vortex, weno6 restriction and fill, reflux off, $N$ = 64 / 128 / 256
AO euler-mp	the AO Euler vortex, lagrange6 + quintic-mp, reflux off, $N$ = 64 / 128 / 256
PV-5	the limited blast across 2:1 seams of the positivity limiter, under both limited fills
GP	the ghost probe on a cubic field, under both limited fills

Input files

File	Role
src/tests/flume/verification/amr-order/check.sh --legs euler-ww,euler-mp	the AO legs: orders $\ge 4.8$ / 4.5 (euler-ww) and $\ge 5.5$ (euler-mp), the errors at most the baseline plus 2 %
src/tests/flume/verification/amr-shock/make_amr_shock.py	the crossings (--seam-ghost-fill, --seam-restriction)
src/tests/flume/verification/ghosts/check.sh --seam-ghost-fill F	GP
src/tests/flume/verification/make_doc_runs.sh <runs> seams, make_doc_figures.py --only limited-fills	re-runs the crossings d-{quintic,weno6,quintic-mp}-{c2f,f2c} (CPU)
src/tests/flume/verification/multirealm/check.sh	leg 6 (--legs leg6) runs weno6 and quintic-mp across a refined inter-realm seam

cd src/tests/flume/verification/amr-order && ./check.sh --legs euler-ww,euler-mp

Results

Sod at $N=200$, lagrange6, reflux on, CPU: the plateau behind the shock with the quintic, weno6 and quintic-mp fills (and the tricubic, c2f), coarse to fine and fine to coarse. The fills differ by at most about ${10}^{-3}$ in density, against a defect of $2.5\cdot {10}^{-2}$ of the jump: the defect is not the fill.

Every fill on the c2f crossings (FNL, 2 ranks, lagrange6 restriction; plateau and band defects as in AS):

Fill (order)	Sod plateau $\rho$	Sod band $\rho$	Sod shock (cells)	RJ2a plateau $\rho$	RJ2a band $\rho$
injection (1)	$2.98\cdot {10}^{-2}$	$7.91\cdot {10}^{-3}$	0.173	$2.49\cdot {10}^{-2}$	$2.16\cdot {10}^{-4}$
restriction-compatible (2)	$2.32\cdot {10}^{-2}$	$5.31\cdot {10}^{-3}$	0.164	$1.98\cdot {10}^{-2}$	$1.28\cdot {10}^{-4}$
tricubic (4)	$2.46\cdot {10}^{-2}$	$6.33\cdot {10}^{-3}$	0.158	$1.41\cdot {10}^{-2}$	$1.03\cdot {10}^{-4}$
quintic (6)	$2.63\cdot {10}^{-2}$	$7.95\cdot {10}^{-3}$	0.159	$2.30\cdot {10}^{-2}$	$0.68\cdot {10}^{-4}$
weno6 (6, limited)	$3.01\cdot {10}^{-2}$	$7.75\cdot {10}^{-3}$	0.178	$1.48\cdot {10}^{-2}$	$1.52\cdot {10}^{-4}$
quintic-mp (6, limited)	$3.14\cdot {10}^{-2}$	$8.82\cdot {10}^{-3}$	0.173	$2.47\cdot {10}^{-2}$	$6.25\cdot {10}^{-4}$

Smooth flows (AO, $N$ = 128 → 256):

Leg	L1 order	L∞ order	L1 at N = 256	L∞ at N = 256
euler-ww (weno6 + weno6)	5.32	5.50	$8.60\cdot {10}^{-8}$	$6.61\cdot {10}^{-6}$
euler-mp (lagrange6 + quintic-mp)	6.10	6.33	$6.04\cdot {10}^{-8}$	$4.77\cdot {10}^{-6}$
euler-q6 (lagrange6 + quintic, D1)	6.11	6.35	$6.03\cdot {10}^{-8}$	$4.78\cdot {10}^{-6}$

PV-5 passes with no inadmissible stage under both limited fills; GP holds a cubic field to round-off with either fill.

Related cases: D1 is the unlimited fill; E0 isolates the cause of the crossing defect; the baselines are AO and AS.

Findings

• The defect has no trend with the fill's order, and limiting makes it worse. The best fill differs by case and by metric (restriction-compatible on Sod, tricubic on the RJ2a plateau, quintic on its band); every fill but the limited ones sits within roughly $\pm 30\mathrm{\%}$ of tricubic, and the two limited fills are the worst on Sod, their shock furthest off. The plateau defect behind a crossing shock is not an overshoot of the fill, so no fill can be held to "no worse than tricubic".
• The best Sod result belongs to the one fill with exact $R\circ P=I$: the defect points to the consistency of the fill with the restriction at the crossing, and to the reflux (A3: reflux on/off moves it by up to 30 %), which is Phase E's ground, not the fill's order. E0 tested both and ruled both out.
• On smooth flows both limited fills keep their order. euler-ww has the weno6 restriction's accuracy (the fill's 6-point WENO does not lose to the tricubic); euler-mp equals euler-q6: the MP bound never clips the vortex.