Self-gravity under runtime AMR

ID	Group	Milestone	Backends	Figure	Regression
SV-11	Self-gravity	M8 (#81), P5c	CPU, FNL	yes	—

Case and scope

Under runtime AMR the grid, and with it the solver, change during the run. Every regrid must:

• set up the right solver: the FFT on a uniform grid, the multigrid on an AMR one;
• solve $\mathrm{\Phi }$ from the regridded state;
• rebuild the flux register the seam gravity work rides on.

What changes and what does not. The regrid itself conserves $\rho$, $\rho \mathbf{u}$ and $E$ (conservative prolongation and restriction). $W=\frac{1}{2}\sum \rho \mathrm{\Phi }V$, though, is the energy of the grid's own discrete Laplacian, so a regrid that changes the grid changes $W$, and $E+W$ jumps by a truncation-sized amount. Between regrids the scheme stays exact (SV-10). The leg separates the two.

What the case catches:

• a regrid that keeps the old solver or the old potential: the step changes of $E+W$ between regrids leave the time-integration level;
• a jump of $E+W$ off a regrid step (a leak that is not the change of the discrete $W$);
• a momentum or mass leak through the regrid.

Isolated boundaries under runtime AMR, refused here until #87, are verified with the polytrope (SV-14), with the refusal of a regrid that refines a boundary block.

Models and numerics

Item	Choice	Code
Equations	compressible Euler, $\gamma =5/3$, periodic self-gravity	models
Regrid	every 5 steps ([amr] frequency = 5), one level above the base, a Löhner estimator on the density (mode 4): refine above 0.3, derefine below 0.1	regrid_forest, adam_flume_cpu_object.F90, adam_flume_fnl_object.F90; numerics
Solver after a regrid	set_potential sets the solver up on the new grid (FFT if the leaves are at one level, multigrid otherwise) and solves $\mathrm{\Phi }$ from the regridded state; the FNL backend copies it to the device	set_potential, adam_flume_common_object.F90; refresh_potential, adam_flume_fnl_object.F90
Seam gravity work	in the per-stage reflux	add_seam_gravity_work, adam_flume_common_object.F90; reflux_stage, adam_flume_cpu_object.F90
Space, time	WENO5 flux splitting (weno), SSP-RK3, CFL 0.4	numerics

Initial and boundary conditions

Item	Value
Case	jeansc2d: the collapse of SV-4, $G=9\pi /2$, amplitude ${10}^{-2}$, the box $[1/8,9/8{]}^2$ (sine-wave, wave_mode = jeans)
Grid	base ${32}^2$ (4×4 blocks of $8^2$, two uniform levels), at most one level more by the regrid; 4 ghosts
Boundaries	every active face periodic
Reflux	per stage (reflux_cadence = stage)
Ranks	2
End	$t=0.4$ (the figure's run: a checkpoint every 10 steps)

Input files

File	Role
src/tests/flume/verification/selfgravity/make_jeans.py	--regrid N writes the Löhner marker and [amr] frequency
src/tests/flume/verification/selfgravity/jeans.sh	leg sv11 (FLUME_EXE selects the backend)
src/tests/flume/verification/selfgravity/jeans_oracle.py	regrid: the grid must change; every jump of $E+W$ (a step change above 100 times the median) on a regrid step; the other step changes, the momentum and the mass bounded
src/tests/flume/verification/make_doc_runs.sh, group amr	sv11-regrid, a checkpoint every 10 steps
src/tests/flume/verification/make_doc_figures.py, fig_selfgravity_amr_regrid	draws the figure (--only selfgravity-amr-regrid)

cd src/tests/flume/verification/selfgravity && ./jeans.sh --legs sv11                       # CPU
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl_fft ./jeans.sh --legs sv11                   # FNL

Results

Left: $E+W$ drifts at the time integration's level until the regrid at $t=0.35$ refines the ridge (16 → 52 blocks), then jumps once and stays flat. Right: the final density, the refined blocks along the ridge.

Asserted	Bound	CPU	FNL (WSL GPU)
the grid changes (blocks of the first and the last checkpoint)	must differ	16 → 52	16 → 52
every jump of $E+W$ on a regrid step	yes	one jump, at step 60: $-2.3\cdot {10}^{-4}$	the same, $-2.3\cdot {10}^{-4}$
the largest other step change of $E+W$, relative	${10}^{-6}$	$7.6\cdot {10}^{-8}$	$7.6\cdot {10}^{-8}$
max $|$momentum$|$; mass drift	${10}^{-14}$; ${10}^{-14}$	$2.9\cdot {10}^{-18}$; 0	$4.7\cdot {10}^{-18}$; $1.1\cdot {10}^{-16}$

Related cases: the same collapse on a static AMR grid is SV-10, on a uniform grid SV-4; runtime regridding without gravity is runtime regrids.

Findings

• The regrid jump is the change of the discrete $W$. At the regrid $E$ is unchanged and $W$ moves by $1.4\cdot {10}^{-3}$ of itself, the size of the $O(h^2)$ truncation at ${32}^2$. It does not converge in a resolution ladder, because the refinement happens at different stages of the collapse (at base ${64}^2$ the ridge crosses the threshold later, when it is sharper). The jumps are reported, not bounded.
• Correcting $E$ at each regrid would make the regrid conserve $E+W$; it is not done.
• The solver follows the grid. With refine and derefine thresholds of 0.1 and 0.03 the same collapse refines everywhere by $t=0.27$, so the grid becomes uniform again and the FFT solver takes over. That run passes the same oracle, with jumps at steps 40 and 50.
• No regression case freezes self-gravity under runtime AMR; selfgravity-amr-stage covers the per-stage reflux on a static grid.