Jeans modes on a static AMR grid

ID	Group	Milestone	Backends	Figure	Regression
SV-10	Self-gravity	M8 (#81), P5b, P5c	CPU, FNL	yes	selfgravity-amr-stage; amr-periodic-reflux-stage (the per-stage reflux, without gravity)

Case and scope

On an AMR grid three things change in self-gravity:

• the potential comes from the multigrid solver of SV-9;
• the face gravity at the 2:1 faces is that solver's first-order mean-flux gradient;
• the energy exchange needs the seam gravity work carried by the reflux (models).

The Jeans modes of SV-3 and SV-4 check the coupling, now with a 2:1 seam the wave crosses on every side of the refined box.

What the case catches:

• a multigrid coupling that differs from the FFT one on a grid where both apply;
• a loss of order across the seams (the stable wave);
• a broken growth rate or point symmetry of the unstable mode;
• a missing or wrong seam gravity work: the semi-discrete $E+W$ is no longer exact, and the drift stops converging in $\mathrm{\Delta }t$;
• with the per-stage reflux (reflux_cadence = stage, P5c), a drift that does not recover the Runge–Kutta order;
• a solver selection that accepts what it cannot solve (isolated boundaries on a periodic grid with the multigrid solver, poisson_solver = fft on AMR).

Models and numerics

Item	Choice	Code
Equations	compressible Euler, $\gamma =5/3$, periodic self-gravity	models
Solver selection	poisson_solver = auto: the FFT solver on a uniform grid, the multigrid on an AMR grid; fft on AMR is fatal, and so is isolated on a periodic grid (the multigrid's own check; isolated boundaries on AMR are SV-13 and SV-14, #87)	set_potential, adam_flume_common_object.F90; adam_flume_gravity_object.F90
Poisson solver	multigrid (FCG + V-cycle) on the mimetic composite Laplacian, on the host; its coarsest level on the backend's FFT solver (FFTW, or cuFFT on FNL)	adam_poisson_mg_object.F90, adam_fnl_poisson_fft_object.F90
Seam gravity work	added per stage to the flux register's energy entry: the coarse side takes its own seam source back, the fine side adds the exact two-point work less its own source	add_seam_gravity_work, adam_flume_common_object.F90; accumulate_seam_fluxes, adam_flume_cpu_object.F90, adam_flume_fnl_object.F90
Reflux	reflux = .true.; reflux_cadence = step (default, end of step) or stage (P5c: the coarse seam cells corrected at every Runge–Kutta stage)	reflux_stage, adam_flume_cpu_object.F90; adam_flume_numerics_object.F90; numerics
Space	WENO5 (weno-u-5) flux splitting on characteristic variables (weno)	numerics
Time	SSP Runge–Kutta, 3 stages, CFL 0.4 (collapse: 0.4, 0.2, 0.1)	adam_rk_object.F90

Initial and boundary conditions

make_jeans.py --box refines the central box (a quarter of the period from every face; in 3-D also along $z$) one level more. The symmetry of the unstable mode survives, because the box is symmetric about the point the oracle reflects through.

Item	Value
Physics	as SV-3 / SV-4: ${\rho }_0=1$, $c=1$, $\gamma =5/3$, $\mathbf{k}=2\pi (1,1)$, every active face periodic; the stable wave $\lambda ={\lambda }_J/2$ ($G=\pi /2$), the unstable mode $\lambda =1.5{\lambda }_J$ ($G=9\pi /2$), sine-wave with wave_mode = jeans
Grid	base $4n$ (2-D, 4×4 blocks) or $2n$ (3-D, 2×2×2 blocks) for $n$ cells per block side, the box at twice the resolution; 4 ghosts
Ranks, backends	2; CPU and FNL (the multigrid runs on the host, its coarsest level on cuFFT)

Leg (jeans.sh)	Case	Base grid	Amplitude, end
sv10: multigrid against FFT	jeans2d, uniform, poisson_solver = fft and multigrid	${32}^2$	${10}^{-6}$; one period
sv10: unstable mode	jeansu2d --box, a checkpoint every 5 steps	${64}^2$	${10}^{-6}$; $t=0.5$
sv10: collapse	jeansc2d --box, CFL 0.4, 0.2, 0.1	${64}^2$	${10}^{-2}$; $t=0.4$
sv10: collapse, per-stage reflux	the same with --reflux-cadence stage	${64}^2$	${10}^{-2}$; $t=0.4$
sv10: refusals	jeans2d --solver multigrid --poisson-bc isolated; jeans2d --box --solver fft	${32}^2$	—
sv10-wave: stable wave, 2-D	jeans2d --box	${32}^2$, ${64}^2$, ${128}^2$	${10}^{-6}$; one period
sv10-wave: stable wave, 3-D	jeans3d --box	${16}^3$, ${32}^3$	${10}^{-6}$; one period
Regression selfgravity-amr-stage	jeansc2d --box --reflux-cadence stage	${32}^2$	${10}^{-2}$; 20 steps

Input files

File	Role
src/tests/flume/verification/selfgravity/make_jeans.py	the Jeans cases; --box, --solver, --poisson-bc, --reflux-cadence
src/tests/flume/verification/selfgravity/jeans.sh	legs sv10 and sv10-wave (FLUME_EXE selects the backend)
src/tests/flume/verification/selfgravity/jeans_oracle.py	compare, wave, growth, symmetry, conservation (with --tol-mass for the per-stage reflux)
src/tests/flume/regression/selfgravity-amr-stage/input.ini	the frozen per-stage collapse on the box-refined grid, ${32}^2$ base, 20 steps
src/tests/flume/regression/amr-periodic-reflux-stage/input.ini	the per-stage reflux on a triply periodic octree (no gravity)
src/tests/flume/verification/make_doc_runs.sh, group amr	sv10-2d-{8,16,32}, sv10-uniform-{8,16,32}, sv10-collapse-{0.4,0.2,0.1}, sv10-stage-{0.4,0.2,0.1}
src/tests/flume/verification/make_doc_figures.py, fig_selfgravity_amr_jeans	draws the figure (--only selfgravity-amr-jeans)

cd src/tests/flume/verification/selfgravity && ./jeans.sh --legs "sv10 sv10-wave"        # CPU
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl_fft ./jeans.sh --legs "sv10 sv10-wave"  # FNL

Results

Left: the L1 error of the stable wave falls at second order across the seams. Middle: in the collapse $E$ and $W$ trade 17 % of $|E+W|$ while $E+W$ drifts by about ${10}^{-7}$; with the end-of-step reflux (solid) the drift falls 2.8 and then 2.2 times per halving of the CFL number, with the per-stage reflux (dashed) 8 times, the SSP-RK3 order. Right: the collapse density at $t=0.4$ on the composite grid; the refined blocks are the central box.

Asserted	Bound	CPU	FNL (WSL GPU)
on a uniform ${32}^2$ grid, poisson_solver = multigrid against fft, $max|{\rho }_{mg}-{\rho }_{fft}|/({\rho }_{0}A)$ after one period	${10}^{-6}$	$3.8\cdot {10}^{-9}$	$2.7\cdot {10}^{-9}$
stable wave, order of the L1 error, finest pair, 2-D	$\ge 1.8$	1.98 (1.97, 1.98)	1.98
the same, 3-D (${16}^3\to {32}^3$)	$\ge 1.8$	2.00	2.00
unstable mode (base ${64}^2$), growth rate, relative error	${10}^{-2}$	$5.8\cdot {10}^{-4}$	$5.8\cdot {10}^{-4}$
its mirror symmetry at $t=0.5$	${10}^{-9}$	$5.7\cdot {10}^{-11}$	$4.7\cdot {10}^{-11}$
collapse (amplitude ${10}^{-2}$, base ${64}^2$, $t=0.4$), max $|$momentum$|$	${10}^{-14}$	$3.5\cdot {10}^{-18}$	$6.9\cdot {10}^{-18}$
its $E+W$ drift at CFL 0.4, 0.2, 0.1, order of the finest pair	$\ge 0.8$	$8.8\cdot {10}^{-8}$, $3.1\cdot {10}^{-8}$, $1.4\cdot {10}^{-8}$ (1.50, 1.16)	the same to 3 digits
the same with reflux_cadence = stage (P5c), order of the finest pair; mass drift	$\ge 2.5$; ${10}^{-14}$	$3.5\cdot {10}^{-8}$, $4.3\cdot {10}^{-9}$, $5.4\cdot {10}^{-10}$ (2.99, 3.00); $2.2\cdot {10}^{-16}$	the same to 3 digits; $3.3\cdot {10}^{-16}$
isolated boundaries on a periodic grid with the multigrid solver; poisson_solver = fft on AMR	refused, exit 1	refused	refused

Related cases: the uniform-grid modes are SV-3 to SV-5; the solver is SV-9; the same collapse under runtime AMR is SV-11; conservation across seams without gravity is conservation across AMR seams.

Findings

• The seam gravity work makes the energy exchange exact in space. A run with the work switched off (a temporary ablation build, not a leg) drifts by $1.03\cdot {10}^{-5}$, $9.8\cdot {10}^{-6}$, $9.6\cdot {10}^{-6}$ at the three CFL numbers, orders 0.07 and 0.02. That is a spatial leak at the seams, which no $\mathrm{\Delta }t$ removes. With the work the drift is 12 times smaller at CFL 0.4 and converges in $\mathrm{\Delta }t$.
• With the end-of-step reflux the drift is first order in $\mathrm{\Delta }t$, not third. The reflux corrects the coarse cells at the end of the step, and the stages in between advance them with their own seam fluxes. On a uniform grid (SV-4) the order is that of SSP-RK3. The per-stage reflux (reflux_cadence = stage, P5c) restores it: orders 2.99 and 3.00, and the drift at CFL 0.1 falls from $1.4\cdot {10}^{-8}$ to $5.4\cdot {10}^{-10}$.
• The multigrid's stopping rule needed a round-off floor. With a ${10}^{-6}$ mode on a unit density the source is known to ${10}^{-16}/{10}^{-6}={10}^{-10}$ of itself. The ${10}^{-10}$ relative tolerance was unreachable, and every solve ran to the 200-iteration cap. The solver now stops at $max({10}^{-10},100ϵmax|4\pi G\rho |/max|4\pi G(\rho -\overline{\rho })|)$ and warns if it hits the cap.
• The AMR wave is more accurate than the uniform one at the same base N. At base ${32}^2$ the L1 error is $1.9\cdot {10}^{-3}$ against $2.1\cdot {10}^{-3}$, because the box adds resolution. The first-order seam gravity shows in no norm the legs measure.
• Isolated boundaries on AMR came with #87: the multigrid's symmetric James step (SV-13), the polytrope on AMR (SV-14).