Conservation across 2:1 seams and forest seams

ID	Group	Milestone	Backends	Figure	Regression
VV-7	Dissipative terms (M4)	M4 (#65)	CPU, FNL	yes	seam-viscous-amr

Case and scope

A diagonal wave of large amplitude crosses 2:1 coarse-fine faces (one realm with its centre refined) and inter-realm seams (the same domain split into two realms). The conserved integrals are judged on the refined realm; the fields of the forest are judged against the single realm.

What the case catches:

• dissipative fluxes missing from the seam registers: the dissipative fluxes are added to the inviscid ones before the seam accumulation, so the Berger–Colella reflux and the inter-realm register must carry them. At a 2:1 face the coarse and the fine side compute different viscous, heat and Ohmic fluxes, and without the register the grid leaks;
• unfilled edge and corner ghosts: the dissipative stencils reach further than WENO's: the tangential derivatives read the edge and corner ghosts, which no inviscid stencil touches.

Models and numerics

Item	Choice	Code
Equations	Euler with $\mu =0.01$, $k=0.02$; ideal MHD with GLM ($c_h=3$), $B_n=1$, $\mu =0.01$, $\eta =0.01$; $\gamma =1.4$	models, GLM
Dissipative fluxes	dissipative_order = 4, added to the inviscid fluxes before the seam accumulation	add_dissipative_fluxes, adam_flume_cpu_object.F90; numerics
Inviscid fluxes	WENO5 flux splitting, characteristic variables	compute_residuals_weno, adam_flume_cpu_object.F90
Time	runge-kutta-ssp-54, CFL 0.2	adam_rk_object.F90
2:1 reflux	Berger–Colella register of the seam fluxes, [numerics] reflux on or off	accumulate_seam_fluxes, reflux_stage, adam_flume_cpu_object.F90; accumulate_coarse_flux, accumulate_fine_flux, adam_flux_register_object.F90; numerics
Coarse-fine ghosts	the library's 2:1 ghost fill	boundary conditions
Forest seams	coupling = mirror (1:1) or refined (2:1), coupling_cadence = stage_coincident; seam ghost centres wrapped by one period along a tangential axis periodic on both realms	fill_seam_from_peer_forest, adam_flume_cpu_object.F90; wrap_periodic_tangential, adam_forest_object.F90; inter-realm seam
Conservation history	volume integrals written every step	compute_conservation, adam_flume_cpu_object.F90

Initial and boundary conditions

Item	Value
Domain	$[0,1{]}^2$, $z$ null
Ranks	2
Initial condition	sine-wave at 45°, $A=0.01$ (large, so the seams carry a real flux): the acoustic wave on Euler (density, normal velocity and temperature all vary), the magnetic (Alfvén) wave on MHD-GLM; ${\rho }_0=p_0=1$, wavelength $1/\sqrt{2}$
Final time	$t=0.05$

Leg	Grid	Boundaries	Runs
refined box (conserve-*-quadtree)	quadtree, $4\times 4$ blocks of ${12}^2$ cells (48 per side at the base level), the blocks whose centroid lies in $[0.25,0.75{]}^2$ refined 2:1 (8 coarse-fine faces)	periodic in $x$ and $y$	with reflux; without; without reflux and without the dissipative coefficients (the ideal twin)
forest, mirror	the single realm split at $x=0.5$ into two realms of $4\times 4$ blocks of $6\times 12$ cells, quadtree, glued 1:1	extrapolation (outflow) in $x$, periodic in $y$	single realm; forest
forest, refined	the single run refines $x>0.5$; realm 1 the coarse half, realm 2 the fine half at one more uniform level, glued 2:1; octree ($nk=4$), because the refined coupling needs ratio 2 along every axis, null $z$ included	extrapolation in $x$, periodic in $y$	single realm; forest

An octree with null $z$ gave the same drifts as the quadtree to the printed digits at four times the cells, so the refined-box leg keeps the quadtree; the octree's 2:1 seams are exercised by the refined forest.

Input files

File	Role
src/tests/flume/verification/viscous/waves.py	make --refine ... --reflux, split (the two-realm forest), conserve (the drift oracle)
src/tests/flume/verification/multirealm/multirealm_oracle.py	the forest union against the single realm (--fields-only)
src/tests/flume/verification/viscous/check.sh --legs vv7	runs the 6 refined-box runs and the 8 forest runs (--np N; FLUME_EXE selects the backend)
src/tests/flume/verification/make_doc_figures.py --only seam-conservation	the figure, from the histories that viscous/check.sh --legs vv7 keeps (CPU)
src/tests/flume/regression/seam-viscous-amr/input.ini	the frozen Alfvén wave, $A=0.01$, MHD-GLM with $\mu$ and $\eta$, ${48}^2$ with the centre refined 2:1, $t=0.02$

cd src/tests/flume/verification/viscous && ./check.sh --legs vv7             # CPU
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl ./check.sh --legs vv7       # FNL

Results

The drift of the mass and energy integrals per step, CPU. Left and centre: the acoustic wave on Euler and the Alfvén wave on MHD-GLM across a 2:1 box; with the reflux the integrals stay at round-off, without it they drift, and so does the ideal twin without the dissipative coefficients, so the drift is the seam, not the dissipative fluxes. Right: the forests (1:1 mirror and 2:1 refined, outflow in $x$) against their single-realm runs: the summed integrals of the two realms agree to round-off at every step.

Oracle (waves.py conserve). The drift of each conserved integral is divided by $\int |q|dV$, the scale of its round-off (the momentum of a sine wave integrates to zero, so its own value is no scale; $\psi$, sourced only by the discrete $\mathrm{\nabla }\cdot \mathbf{B}$, is measured against $c_h\int |\mathbf{B}|dV$):

• with reflux, every drift $\le {10}^{-13}$;
• without reflux, the largest drift $\ge {10}^{-10}$ (the seams are exercised);
• the leak without reflux differs from the ideal twin's by $\ge {10}^{-10}$: the dissipative fluxes cross the seams unmatched, and the first item shows the register removes that too.

Case	With reflux (bound ${10}^{-13}$)	Without ($\ge {10}^{-10}$)	Dissipative minus ideal leak ($\ge {10}^{-10}$)
Euler	$\le 1.1\cdot {10}^{-17}$	$3.0\cdot {10}^{-5}$	$2.0\cdot {10}^{-6}$
MHD-GLM	$\le 1.5\cdot {10}^{-16}$	$5.9\cdot {10}^{-5}$	$5.1\cdot {10}^{-6}$

CPU and FNL (2 ranks) are identical to the printed digits except the round-off of the conserved runs, at most $1.1\cdot {10}^{-16}$ on FNL.

Forests: the union must reproduce the single realm, fields within ${10}^{-10}$ relative.

Forest	Bound	CPU	FNL
mirror and refined, Euler and MHD-GLM	${10}^{-10}$	largest difference $8.7\cdot {10}^{-12}$	$4.4\cdot {10}^{-12}$

Related cases: the accuracy across the same seams is VV-8; the inviscid conservation across AMR seams is Euler AMR conservation; the forest machinery is multi-realm runs.

Findings

• The forest comparison is not bitwise, unlike MV-14's piecewise-constant states: each realm evaluates the sine from its own block origins, one ulp apart at step 0 ($3\cdot {10}^{-15}$). The largest difference after the run sits at the outer $x$ faces, not at the seam.
• This leg found a library defect. A seam slab's edge and corner ghosts past a tangential boundary of the realm lie outside the peer domain, and were left to the boundary condition; along a periodic axis no condition fills them, and they kept their initial values. WENO never reads them, the dissipative tangential derivatives do: the mirror split went NaN at the first step (on a single realm of the same half domain the run is clean). The seam enumeration now wraps those ghost centres by one period when the axis is periodic on both realms (forest guide).