Regrid round trip and input contract

ID	Group	Milestone	Backends	Figure	Regression
RG (rg0, rg1, rg2)	Runtime AMR (M5)	M5 (#74)	CPU (rg0, a host program); rg1 and rg2 CPU, FNL	no	—

Case and scope

Until M5 nothing ran amr_update after initialisation and no test ever derefined. rg0 exercises the library regrid step on its own: a block is refined, then its children derefined, on a field known in every cell; rg1 and rg2 pin the two input keys of runtime AMR.

What the case catches:

• inconsistent bookkeeping over the ranks: the redistribution moves blocks, and the leaves and the blocks summed over the ranks must agree after every step;
• an inexact transfer where an exact one is possible: a linear field must be reproduced by the refine; with the conservative prolongation the round trip must return the initial state and $\sum q\mathrm{\Delta }V$ must not change;
• a prolongation that leaves the range of the data: a steep front must stay in $(0.01,1)$, so a positive variable stays positive;
• a silent input: a negative frequency, a frequency > 0 without markers, an unknown regrid_prolongation.

The conservative prolongation

Each parent cell is split along the refined axes into children set, variable by variable, to

$$q_c=q+\varphi \sum _d\frac{{\sigma }_d{s}_d}{4},{\sigma }_d=\pm 1,$$

with $s_d$ the monotonized-central slope of the parent along axis $d$ (in parent-cell units, from its two face neighbours) and $\varphi \in [0,1]$ the largest factor that keeps every child in the range of the parent and its face neighbours (Barth and Jespersen 1989). The offsets cancel over the children, so their mean is the parent: the mean restriction undoes the prolongation exactly and $\sum q\mathrm{\Delta }V$ does not change. The MC limit bounds each axis on its own, not the sum over the axes a corner child takes; $\varphi$ closes that gap. Linear data keep $\varphi =1$ and are reproduced exactly.

Models and numerics

Item	Choice	Code
Regrid step	tree adapted with 2:1 balance, interior data prolonged or restricted, blocks redistributed, maps rebuilt	amr_update, check_blocks_number, adam_adam_object.F90; adapt, adam_field_object.F90
linear prolongation	tensor linear (weights 1/4, 3/4); the library default	refine2D, refine3D, adam_field_object.F90
conservative prolongation	limited MC slopes with the Barth–Jespersen factor $\varphi$ (#74 D-M5-2); the FLUME default	refine_conservative, adam_field_object.F90
Restriction	the mean of the children	derefine2D, derefine3D, adam_field_object.F90
[amr] regrid_prolongation	read by the library; FLUME resolves an absent key to conservative and logs it	load_from_file, prolongation_name, adam_amr_object.F90; check_runtime_amr, adam_flume_common_object.F90
[amr] frequency	0: the AMR of the initial condition only; $n>0$: a regrid every $n$ steps, which needs markers	check_runtime_amr, adam_flume_common_object.F90; input

The regrid algorithms are described in numerics.

Initial and boundary conditions

Leg	Setup
rg0	tests/amr/test_amr_regrid_roundtrip on 1, 2 and 3 ranks. Domain $[0,1{]}^3$; a realm refined uniformly to level 1: octree (ratio = 8, $8^3$ cells per block, 8 blocks) and quadtree (ratio = 4, $8\times 8\times 1$ cells, $z$ null, 4 blocks), ngc = 3. The field is set in every cell, ghosts included (a perfect ghost fill, so the transfers are tested alone). The block at the origin is refined, then its children derefined, with each prolongation
rg0 fields	linear $f=1+0.3x+0.7y+0.5z$; quadratic, $f+2x^2+1.5y^2+xy$; a steep front from 1 to 0.01, $0.505+0.495\tanh ((c-s)/0.03)$, a tanh half a coarse cell wide across $s=x+y+z=c=0.45$ on the octree, $s=x+y=c=0.3$ on the quadtree
rg1	the regression sod-x input (Sod) to $t=0.002$, 2 ranks, with [amr] frequency = 0, −1, 5 (and no markers)
rg2	the same input with [amr] regrid_prolongation absent, linear, conservative, bogus

Input files

File	Role
src/tests/amr/test_amr_regrid_roundtrip.F90	the rg0 unit test (fobis build --mode test-amr-regrid-roundtrip-gnu)
src/tests/flume/verification/regrid/check.sh --legs rg0,rg1,rg2	runs the legs (FLUME_EXE selects the backend of rg1 and rg2)
src/tests/flume/regression/sod-x/input.ini	the base input of rg1 and rg2

cd src/tests/flume/verification/regrid && ./check.sh --legs rg0,rg1,rg2                    # CPU
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl ./check.sh --legs rg1,rg2                # FNL

No figure: the legs are a transfer identity and an input contract (#85).

Results

rg0 asserts:

• the leaves and the blocks summed over the ranks agree: octree 8 → 15 → 8, quadtree 4 → 7 → 4;
• a linear field exact after the refine, with both prolongations;
• with conservative, the round trip returns the initial state and $\sum q\mathrm{\Delta }V$ is unchanged after the refine and after the round trip, on every field, to round-off (${10}^{-13}$ of the field scale); the front stays in $(0.01,1)$;
• with linear, the quadratic field drifts (the negative control).

Every leg passes on 1, 2 and 3 ranks, with the same numbers (CPU):

Field	Tree	linear: drift of $\sum q\mathrm{\Delta }V$	linear: round trip	conservative: drift	conservative: round trip
quadratic	octree	$1.3\cdot {10}^{-4}$	$3.4\cdot {10}^{-4}$	0	$4\cdot {10}^{-17}$
quadratic	quadtree	$2.7\cdot {10}^{-4}$	$3.4\cdot {10}^{-4}$	0	$2\cdot {10}^{-17}$
front	octree	$2.6\cdot {10}^{-2}$	$1.8\cdot {10}^{-1}$	$4.8\cdot {10}^{-15}$	$1\cdot {10}^{-16}$
front	quadtree	$1.8\cdot {10}^{-2}$	$1.3\cdot {10}^{-1}$	$2.6\cdot {10}^{-16}$	$1\cdot {10}^{-16}$

Drift is relative to the initial integral, after the refine; round trip is the largest $|q-f|$ over the field scale after refine then derefine.

Leg	Input	Expected	CPU	FNL
rg1	frequency = 0	runs	pass	pass
rg1	frequency = -1	fatal: "must be 0 (no runtime regridding) or positive"	pass	pass
rg1	frequency = 5, no markers	fatal: "nothing would mark a block"	pass	pass
rg2	regrid_prolongation absent	runs, resolved to conservative	pass	pass
rg2	linear, conservative	runs, taken as given	pass	pass
rg2	bogus	fatal: unknown [amr].(regrid_prolongation) "bogus"	pass	pass

The FNL column of rg1 and rg2 is read from the logs of the work-adam_flume_fnl-* directories of regrid/check.sh.

Related cases: the regrids during a FLUME run are runtime regrids; the inviscid conservation of static AMR is Euler AMR conservation.

Findings

• The linear prolongation is second order (a quadratic field's children miss by $2.6\cdot {10}^{-4}$ of the scale, the conservative one's by $1.1\cdot {10}^{-4}$) but changes the integral at every refine.
• The bound scaling is load-bearing: with $\varphi =1$ forced (a mutation run), an octree child of the front reaches $-0.094$, below zero, and the leg fails; on the quadtree the MC limit alone suffices for this front.