Order across static 2:1 seams

ID	Group	Milestone	Backends	Figure	Regression
AO (A1, C1)	Seam accuracy and divergence	M6 (#68)	CPU, FNL	yes	—

Case and scope

A smooth vortex starts centred on the corner of a refined quadrant, so its core straddles an $x$ seam, a $y$ seam and their corner, and the free stream $(1,1)$ carries it into the fine quadrant. The exact solution is the convected vortex, so the error is measured cell by cell, split into the seam band and the interior.

Every smooth order test (V2, MV-5 to MV-7 in order of accuracy) runs on a uniform grid; VV-8 measures a seam with dissipative waves. No test measured the inviscid order of a smooth feature crossing a seam, where the #68 audit (F1) predicts that the mean restriction of the coarse ghosts and the Berger–Colella reflux of point-value fluxes each cap the order at 2.

What the case catches:

• a loss of order at the seams, in the band and in the regions the flow reaches after crossing a seam;
• a degradation of the seam operators: the finest refined errors are bounded by the CPU baseline plus 2 %, so an improvement of the transfers or of the reflux passes and a degradation fails;
• a broken instrument: the uniform control (no refined quadrant) must reach design order.

Phases C (high-order restriction), D (quintic prolongation) and E (reflux policy) of #68 are measured against these baselines; the legs euler-l6, euler-w6, euler-q6, euler-ww, euler-mp and euler-tq are their entries in the same ladder.

Models and numerics

Item	Choice	Code
Equations, euler	compressible Euler, ideal gas, $\gamma =c_p/c_v=1.4$	models
Equations, mhd	ideal MHD with mixed GLM cleaning, $\gamma =5/3$, glm_ch 3.0, glm_alpha 0.18, glm_damping_length 1.0	models; mhd_glm_face_split_fluxes, adam_flume_mhd_library.F90
Space	WENO5 (weno-u-5) flux splitting on characteristic variables	compute_residuals_weno, adam_flume_cpu_object.F90; compute_face_split_fluxes, adam_flume_euler_library.F90
Time	SSP Runge–Kutta, 5 stages (runge-kutta-ssp-54), CFL 0.4	adam_rk_object.F90
Fine-to-coarse ghosts	[amr] seam_restriction: mean (default, the 2:1 mean), lagrange6 (C1), weno6 (C3)	update_ghost_local, restrict_seam_high_order, adam_field_object.F90; seam_restriction_code, SEAM_W_LAGRANGE6, adam_seam_interpolation_library.F90; key read in initialize, adam_adam_object.F90
Coarse-to-fine ghosts	[amr] seam_ghost_fill: tricubic (default), quintic (D1), weno6, quintic-mp (D2)	interp_seam_ghost, adam_field_object.F90; seam_interpolate_tricubic, seam_interpolate_quintic, adam_seam_interpolation_library.F90
Reflux	Berger–Colella, [numerics] reflux = .true.; reflux_policy = always (default) or troubled (euler-tq)	accumulate_seam_fluxes, apply_reflux_to_stage_forest, flag_troubled_seam_cells, adam_flume_cpu_object.F90; reduce_fine_sums, adam_flux_register_object.F90
Boundaries	periodic in $x$ and $y$; extrapolation on the null $z$ faces	boundary conditions
FNL backend	the same kernels on the device; the restriction rows in the local ghost kernel (C2)	adam_flume_fnl_object.F90; update_ghost_local_gpu_dev, adam_fnl_field_kernels.F90

The seam operators are described in numerics.

C1, the 6-point restriction ([amr] seam_restriction = lagrange6, #68 C1, C2). The coarse seam ghosts take the tensor 6-point Lagrange interpolation of the fine cells to the coarse centre instead of their mean: exact 1-D weights $k/256$, the footprint shifted inside the donor block at its edge, as B3 recommends. A row whose fine pair does not lie inside its donor block falls back to the mean. On FNL the restriction runs in the device twin of C2.

Initial and boundary conditions

Item	euler	mhd
Domain	$[0,1{]}^2$, null $z$	$[-7,7{]}^2$, null $z$
Initial condition	V2 isentropic vortex: radius 0.07, strength 5, centre $(0.5,0.5)$; free stream $(\rho ,u,v,p)=(1,1,1,1)$	MV-7 magnetised vortex: radius 1, $\kappa =1$, $\mu =1$, centre $(0,0)$; background $(\rho ,u,v,p)=(1,1,1,1)$, $\mathbf{B}=0$
Base grid	quadtree (ratio = 4), $4\times 4$ base blocks of $N/4\times N/4\times 1$ cells, ngc = 3; $N$ = 64 / 128 / 256	the same
Refined region	quadrant $[0.5,1{]}^2$: four base blocks refined one 2:1 level at initialisation (primitive-box marker, max_level = 3, no regridding)	quadrant $[0,7{]}^2$, the same
Uniform control	the same case without the box (max_level = 2), the coarse spacing	the same
Boundaries	periodic in $x$, $y$; extrapolation in $z$	the same
Final time	$t=0.1$ (CFL 0.4; V2's 0.1 gives the same errors to five digits: uniform $N=64$ L1 $8.2184\cdot {10}^{-5}$ both)	$t=0.5$
Ranks	2	2
euler-octree (on request)	the euler leg on an octree (ratio = 8, nk = 4), $N$ = 64 / 128; it refines along the null $z$ too, about 16 times the cost of the quadtree	—

Legs of the euler ladder (each: refined and uniform runs, $N$ = 64 / 128 / 256):

Leg	seam_restriction	seam_ghost_fill	Reflux	Phase
euler	mean	tricubic	on, always	A1
euler-l6	lagrange6	tricubic	off	C1
euler-w6	weno6	tricubic	off	C3, C3c
euler-q6	lagrange6	quintic	off	D1
euler-ww	weno6	weno6	off	D2
euler-mp	lagrange6	quintic-mp	off	D2
euler-tq	lagrange6	quintic	on, troubled	E2

Input files

File	Role
src/tests/flume/verification/amr-order/make_amr_order.py	writes the inputs from the V2 input vortex/vortex-n{064,128,256}.ini (euler) or mhd/vortex/make_mhd_vortex.py (mhd); --ratio, --uniform, --no-reflux, --seam-restriction, --seam-ghost-fill, --reflux-policy, --time-max, --it-save
src/tests/flume/verification/mhd/amr_box.py	refine_box: the refined quadrant, one 2:1 level at initialisation
src/tests/flume/verification/amr-order/amr_order_oracle.py	the per-region errors against the exact vortex, the orders, the bounds
src/tests/flume/verification/make_doc_figures.py --only order-across-seams	the figure, from the kept work directories of amr-order/check.sh --keep (CPU)
src/tests/flume/verification/amr-order/check.sh	runs the legs (--legs euler,mhd,euler-octree,euler-l6,euler-w6,euler-q6,euler-ww,euler-mp,euler-tq, default all but euler-octree; --np N, --keep; FLUME_EXE selects the backend)

cd src/tests/flume/verification/amr-order && ./check.sh             # CPU, default legs (all but euler-octree)
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl ./check.sh         # FNL
./check.sh --legs euler-octree                                      # the octree leg, on request

Oracle (amr_order_oracle.py). Each cell of the last checkpoint is compared with the exact convected vortex: the density for Euler; for MHD the eight conservative variables, combined as MV-7 does ($\sqrt{\sum _v{\overline{e}}_v^2}$ of the per-variable means). The error is split into the seam band, the cells within ngc of their own spacing from the boundary of the refined quadrant (periodic images included), and the interior, the rest; a uniform run is all interior. For each region: L1 (the volume-weighted mean of $|e|$), L∞ and the orders of successive runs.

Bound	Leg	Value
finest refined L1 (all, band) and L∞ (all)	every leg but euler-octree	at most the CPU baseline plus 2 %
uniform control, interior L1 order	every leg	$\ge 4.7$
refined L1 / L∞ order, finest pair	euler-l6, euler-w6, euler-ww	$\ge 4.8$ / $\ge 4.5$
refined L1 / L∞ order, finest pair	euler-q6, euler-mp, euler-tq	$\ge 5.5$ / $\ge 5.5$

Cost of euler-l6: about 10 min on the CPU, 2 on FNL.

Results

CPU, from the work directories amr-order/check.sh --legs euler,euler-l6,euler-q6 --keep leaves: $L_1(\rho )$ over all cells and $L_{\mathrm{\infty }}(\rho )$ in the seam band against $N$ (the 128 → 256 orders in the legend), today's operators (mean + tricubic, reflux on) capped at 2.01 and 1.46, lagrange6 without reflux at the uniform control, lagrange6 + quintic below it; and the density error maps at $N=128$ (the refined quadrant in white): today's error concentrates on the seams, lagrange6 + quintic leaves the vortex's own.

Baselines (A1). CPU and FNL identical to every printed digit, 2 ranks; orders of the finest pair, $N$ = 128 → 256:

Leg	Refined L1 at N = 256	Uniform L1 at N = 256	L1 order: all / band / interior	L∞ order	Uniform L1 order
euler	$4.05\cdot {10}^{-6}$	$8.38\cdot {10}^{-8}$	2.01 / 1.71 / 1.91	1.46	5.79
mhd	$1.83\cdot {10}^{-6}$	$2.62\cdot {10}^{-8}$	2.07 / 1.85 / 1.92	1.89	5.34

C1, the 6-point restriction (euler-l6). CPU and FNL (the device twin of C2) agree to every printed digit, and on the same run to $3\cdot {10}^{-14}$:

N = 256	L1 all	L∞ all	Orders 128 → 256, L1 / L∞
euler (mean, reflux on, today)	$4.05\cdot {10}^{-6}$	–	2.01 / 1.46
euler-l6 (lagrange6, reflux off)	$8.55\cdot {10}^{-8}$	$6.36\cdot {10}^{-6}$	5.33 / 5.53
uniform at the coarse spacing	$8.38\cdot {10}^{-8}$	$5.34\cdot {10}^{-6}$	5.79 / 7.38

The other legs of the ladder report on their study pages: euler-w6 in C3c, euler-q6 in D1, euler-ww and euler-mp in D2, euler-tq in E2.

Related cases: AS is the shock baseline of the same seam operators; AL logs the drift of this vortex without the reflux; B1 reproduces the caps in a 2-D prototype; the uniform-grid orders are V2 and MV-5 to MV-7.

Findings

• The seams cap the whole solution at second order, not only the band. The error made at a seam is carried into the interior by the flow, so every region a feature reaches after crossing a seam is second order. The plan's expectation of "design order in the interior" holds only for regions the flow does not reach from a seam.
• On this smooth flow the refinement makes the solution worse. At $N=256$ the refined composite error is 48 times (Euler) and 70 times (MHD) the error of the uniform run at the coarse spacing, which reaches design order.
• Euler's L∞ order is 1.46, below the 2 of the 1-D prototype in #68 (Appendix A). The vortex crosses the seam corner, where the two seam directions meet; the 2-D prototype of Phase B was to check whether the corner explains it (B1 and B2 found it does not).
• C1: the restriction was the seam error. With the 6-point restriction and without the reflux the refined composite error equals the uniform coarse run's, 47 times below today's: at these resolutions the restriction was the seam error, and the tricubic prolongation's $O(h^4)$ term has not shown yet.
• C1: the reflux caps the order by itself. With the reflux on, the restriction changes nothing (band L∞ order 1.43 against 1.46 at $N$ = 64 → 128), as B1 predicts. Reaching the high order with the reflux kept at the shocks is Phase E's job (E2).
• No restriction row fell back to the mean: every fine pair lies inside its donor block.
• C1 cost on FNL: about 16 % of the wall time of this run ($N=256$, reflux off: 56 s against 48 s with the mean; WSL, indicative only): the restriction rows read 36 cells instead of 4 on this quadtree, in the same kernels as the copies.