Conservation across AMR seams

ID	Group	Milestone	Backends	Figure	Regression
V3, RV-4	Euler	M1 (#35), M3 (#47)	CPU, FNL	yes	amr-periodic-reflux, amr-periodic-reflux-stage

Case and scope

A finite-volume update conserves only if every interior face flux leaves one cell and enters its neighbour. At a 2:1 coarse–fine face the two sides compute different fluxes (different stencils, different ghost fills), so the grid leaks mass, momentum and energy unless the Berger–Colella reflux replaces the coarse flux with the restricted fine one. The case is a triply periodic box with one refined octant: nothing crosses the boundary, so the five volume integrals must stay constant to round-off.

What the case catches:

• a missing or wrong reflux at any of the 6 coarse–fine faces, periodic ones included;
• a reflux register that does not hold the flux the step used: FLUME accumulates the seam fluxes of every Runge–Kutta stage weighted by its SSP coefficient;
• a vacuous test: the run without reflux is the negative control and must drift well above round-off, otherwise the case does not exercise the seams;
• block layouts the 2:1 seams cannot handle, refused at initialization: an odd cell count along a refined axis (#39; a 2:1 child boundary in the middle of a parent cell gave NaN silently), and a count below $2n_{gc}$ (#66; the outer coarse ghost layers would be restricted from the fine block's own, stale ghosts).

Models and numerics

Item	Choice	Code
Equations	compressible Euler, ideal gas, $\gamma =1.4$	models
Space, weno	WENO5 (weno-u-5) flux splitting on characteristic variables	compute_residuals_weno, adam_flume_cpu_object.F90
Space, weno-riemann (RV-4)	WENO5 interpolation (characteristic), HLLC, the 6th-order flux correction with the WENO-weight sensor	compute_residuals_riemann, adam_flume_cpu_object.F90
Time	SSP Runge–Kutta, 3 stages (runge-kutta-ssp-33), CFL 0.5	adam_rk_object.F90
AMR	geometric box marker, one more level on the octant; 2:1 ghost restriction and interpolation	mark_by_geometry, adam_flume_common_object.F90; boundary conditions
Reflux	Berger–Colella, stage fluxes weighted by the SSP coefficients	accumulate_seam_fluxes, adam_flume_cpu_object.F90; accumulate_coarse_flux, accumulate_fine_flux, adam_flux_register_object.F90; apply_reflux_corrections, adam_forest_object.F90; numerics
Block-cell checks	odd counts and counts below $2n_{gc}$ along a refined axis refused	check_amr_block_cells, adam_realm_object.F90
Boundaries	periodic on all six faces	boundary conditions
Integrals	volume integrals of the conservative variables, every step	compute_conservation, adam_flume_cpu_object.F90

Initial and boundary conditions

Item	Value
Domain	$[0,1{]}^3$
Grid	octree, one uniform level (8 blocks of $8^3$ cells), then the octant $[0,0.5{]}^3$ refined once more: 15 blocks, 6 coarse–fine faces (the log must report registered intra-realm AMR seam faces: +6), ngc = 3
Ranks	2
State	uniform flow $(\rho ,u,v,w,p)=(1,1,1,1,1)$ with a 1% seeded density and pressure perturbation (s = 0.01), so every seam carries a flux
Initial condition	uniform
Boundaries	periodic on all six faces
Final time	20 steps

Leg	Change	Expected
reflux on	—	the five integrals constant within ${10}^{-13}$
reflux off	reflux = .false.	drift of at least ${10}^{-10}$ (negative control)
odd cells	ni = 7	refused at initialization ([grid].(ni)=+7 is odd: 2:1 refinement)
thin blocks	ni = 4 ($<2n_{gc}$)	refused at initialization ([grid].(ni)=+4 is less than 2*ngc)

Input files

File	Role
src/tests/flume/verification/conservation/amr-periodic.ini	the case
src/tests/flume/verification/conservation/check.sh	V3: the four legs (--np N, --legs LIST: v3-reflux for legs 1 and 2, v3-odd, v3-thin; FLUME_EXE selects the backend)
src/tests/flume/verification/conservation/conservation_oracle.py	the drift of the integrals (--conserved, --leaky); --compare for two runs cell by cell
src/tests/flume/verification/riemann-flux/check.sh --legs conservation	RV-4: the reflux on and off legs on weno-riemann
src/tests/flume/verification/make_doc_figures.py (fig_conservation)	the figure, from the conservation histories of conservation/

cd src/tests/flume/verification/conservation && ./check.sh               # CPU
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl ./check.sh              # FNL
cd ../riemann-flux && ./check.sh --legs conservation                     # RV-4, HLLC

Results

Relative drift of the five volume integrals against time, reflux on (solid) and off (dashed).

Run	Bound	CPU	FNL
reflux on, weno	${10}^{-13}$	$\le 6\cdot {10}^{-15}$	$2.2\cdot {10}^{-16}$
reflux on, weno-riemann HLLC	${10}^{-13}$	0	$2.2\cdot {10}^{-16}$
reflux off	$\ge {10}^{-10}$	$1$–$2\cdot {10}^{-5}$	$1$–$2\cdot {10}^{-5}$
ni = 7	refused	refused	refused
ni = 4	refused	refused	not run

Related cases: Shock over a cylinder and Multi-realm runs rely on the same reflux; the dissipative fluxes across 2:1 seams are checked in VV-7.

Findings

• With the stage fluxes weighted by the SSP coefficients the register holds the flux the committed step used, and the integrals stay at round-off on 1 and 2 ranks (measured at M1-P5).
• Without the reflux the integrals drift by $1$–$2\cdot {10}^{-5}$ in 20 steps: the seams carry flux, and the reflux leg is not vacuous.
• The MHD version (MV-11, mhd/conservation) adds the three components of $\mathbf{B}$ and $\psi$: drift $\le 4.7\cdot {10}^{-14}$.
• The two refused layouts were silent before their checks: NaN for the odd count, rank-dependent results for the thin block (measured 0.2 in density between 1 and 2 ranks on the step forest, see Forward-facing step).
• The ni = 4 leg has no FNL run on record (no FNL work directory).