Quintic coarse-to-fine fill

ID	Group	Milestone	Backends	Figure	Regression
D1	Seam studies	M6 (#68)	CPU, FNL (table: FNL)	yes	—

Case and scope

With the 6-point restriction (C1) and the reflux off, the smooth AO vortex reaches the uniform coarse error, but its order (5.33) is that of the tricubic coarse-to-fine fill's $O(h^4)$ term waiting to show. D1 asks whether a 6-point (quintic) fill brings the composite to the scheme's order, and what it does to a shock crossing a 2:1 face (the Phase D exit: AO L1 order $\ge 4.7$ and L∞ $\ge 4.5$, and A3 no worse than tricubic).

Models and numerics

Item	Choice	Code
Method	[amr] seam_ghost_fill = quintic: the tensor 6-point Lagrange interpolation at the fine ghost's centre, 1-D weights $k/8192$ (exact in binary); the footprint shifted inside the coarse donor block at its edge as the tricubic one is (blocks of at least 6 cells along the refined axes); along an unrefined axis only the anchor is read	seam_interpolate_quintic, SEAM_W_QUINTIC, seam_shift_anchor_pos, adam_seam_interpolation_library.F90; interp_seam_ghost, adam_field_object.F90
Anchor shifts	carried in the fill meta of every map row, bits 18–26	seam_meta_pack, adam_seam_interpolation_library.F90; make_local_map_ghost_cell, adam_maps_object.F90
FNL	summed straight from the device field, no per-thread footprint	update_ghost_local_gpu_dev, adam_fnl_field_kernels.F90
Inter-realm	the same fill on the inter-realm rows	adam_forest_object.F90
Restriction	lagrange6 in every run below	C1
Key	read in initialize; the default stays tricubic	adam_adam_object.F90
Scheme, time, reflux	as in AO and AS: WENO5 flux splitting; reflux off on AO, on for the crossings	numerics

Initial and boundary conditions

Run	Setup
AO euler-q6	the AO Euler vortex, quadtree, $N$ = 64 / 128 / 256, lagrange6 + quintic, reflux off
Shock crossings	the AS Sod ($N=200$) and RJ2a ($N=256$) c2f and f2c crossings, lagrange6 restriction, reflux on, the null axes widened to 8 cells
PV-5	the limited Balsara–Spicer blast across 2:1 seams of the positivity limiter, split and HLLD
GP	the ghost probe on a degree-5 field
Multirealm leg 6	a refined inter-realm seam, bitwise against the single-realm run (multi-realm runs)

All runs on 2 ranks unless stated.

Input files

File	Role
src/tests/flume/verification/amr-order/check.sh --legs euler-q6	the AO ladder: orders $\ge 5.5$ and the errors at most the baseline plus 2 %
src/tests/flume/verification/amr-shock/make_amr_shock.py	the crossings (--seam-restriction, --seam-ghost-fill)
src/tests/flume/verification/ghosts/check.sh --seam-restriction lagrange6 --seam-ghost-fill quintic --degree 5	GP on a degree-5 field
src/tests/flume/verification/make_doc_figures.py --only quintic-fill	the figure, from the probe runs of ghosts/check.sh --degree 5 --keep (default operators, then lagrange6 + quintic) and the kept AO work directories (CPU)
src/tests/flume/verification/multirealm/check.sh	leg 6 (--legs leg6), every seam operator across a refined inter-realm seam
src/tests/flume/verification/mhd/positivity/check.sh	PV-5 (--legs pv5, not a default leg)

cd src/tests/flume/verification/amr-order && ./check.sh --legs euler-q6
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl ./check.sh --legs euler-q6

Results

CPU. Left and centre: the ghost probe on a degree-5 field, box3d (octree) and channel2d (quadtree), 2 ranks, the largest error per ghost group: mean + tricubic misses the field by up to $5\cdot {10}^{-5}$ in every group a seam operator fills (the expected failure: exact to degree 1 and 3), lagrange6 + quintic matches it to ${10}^{-14}$. Right: the AO ladder without reflux, lagrange6 with the tricubic and the quintic fill.

FNL, 2 ranks (plateau values in units of ${10}^{-2}$; band values absolute):

Check	lagrange6 + tricubic	lagrange6 + quintic
AO euler, reflux off, N = 256: L1 all / band, Linf	$8.55\cdot {10}^{-8}$ / $2.61\cdot {10}^{-7}$, $6.36\cdot {10}^{-6}$	$6.03\cdot {10}^{-8}$ / $1.74\cdot {10}^{-7}$, $4.78\cdot {10}^{-6}$
AO orders L1 / Linf (128 → 256)	5.33 / 5.53	6.11 / 6.35
Sod c2f plateau $\rho$ / $p$, band $\rho$	$2.457$ / $0.691$, $0.633\cdot {10}^{-2}$	$2.631$ / $0.858$, $0.795\cdot {10}^{-2}$
Sod f2c plateau $\rho$ / $p$, band $\rho$	$1.575$ / $1.015$, $0.343\cdot {10}^{-2}$	$1.573$ / $0.985$, $0.338\cdot {10}^{-2}$
RJ2a c2f plateau $\rho$ / $p$, band $\rho$	$1.410$ / $0.622$, $1.03\cdot {10}^{-4}$	$2.297$ / $0.986$, $0.68\cdot {10}^{-4}$
PV-5 split / HLLD: most faces limited, inadmissible stages	46336 / 17920, 0	45644 / 17920, 0

Backend agreement: np 1 / 2 / 3 bitwise on the CPU, np 1 / 2 on FNL, CPU against FNL $2.7\cdot {10}^{-13}$; the AO euler-q6 ladder agrees on CPU and FNL to every printed digit.

Related cases: D2 limits this fill; E2 keeps its order with the reflux on; the baselines are AO and AS.

Findings

• The smooth order reaches the scheme's. With the reflux off, the 6-point restriction and the quintic fill, AO euler is sixth order (6.11 / 6.35) and its error at $N=256$ is below the uniform coarse run's ($8.38\cdot {10}^{-8}$): the Phase D exit (L1 $\ge 4.7$, L∞ $\ge 4.5$) holds, and the 2:1 seams no longer cost accuracy on this flow.
• Unlimited, it worsens a shock entering a finer region. On the c2f crossings the plateau defect behind the shock grows (Sod +7 % in $\rho$, +24 % in $p$; RJ2a +63 % / +59 %): the shock's coarse values reach the fine ghosts through a 6-point stencil with larger overshoots than the 4-point one. f2c is unchanged, shock positions are the same, mass is exact, and PV-5 passes with no inadmissible stage.
• The Phase D exit requires A3 no worse than tricubic, which is the job of D2's limited variants; until then quintic is an opt-in for smooth flows.
• Coverage: AO leg euler-q6 asserts the order; GP checks the fill on a degree-5 field; multirealm leg 6 runs it across a refined inter-realm seam, bitwise.