MHD conservation across AMR seams

ID	Group	Milestone	Backends	Figure	Regression
MV-11	Ideal MHD	M2 (#41, P7a)	CPU, FNL	yes	—

Case and scope

The V3 case of conservation across AMR seams (a triply periodic box, one refined octant, 6 coarse–fine faces, a uniform flow with a 1% seeded density/pressure perturbation) run through the MHD path with the uniform field $\mathbf{B}=(0.8,0.5,-0.3)$. The perturbation drives the induction, so the flux register carries the $\mathbf{B}$ and $\psi$ rows at every seam. Nothing crosses the periodic boundary, so every volume integral of the conservative state must stay constant to round-off when the Berger–Colella reflux is on.

Four legs:

1. GLM without damping (glm_alpha = 0), reflux on: all 9 integrals ($\rho$, $\rho \mathbf{u}$, $E$, each component of $\mathbf{B}$, $\psi$) within ${10}^{-13}$; $\psi$ starts at zero, so its drift is relative to the largest initial integral;
2. the same, reflux off: the negative control must drift by at least ${10}^{-10}$, so the seams are exercised;
3. GLM with damping (glm_alpha = 0.18), reflux on: the 8 physical integrals within ${10}^{-13}$; $\int \psi$ is reported, not bounded;
4. no divergence control (none, $n_v=8$), reflux on: the 8 integrals within ${10}^{-13}$.

Every run must register the 6 coarse–fine faces (registered intra-realm AMR seam faces: +6 in the log).

What the case catches: a reflux register without the $\mathbf{B}$ or $\psi$ rows, a wrong seam flux of the MHD or GLM kernels, a vacuous test (the negative control), and a seam layout that registers fewer faces than the six of the octant.

Models and numerics

Item	Choice	Code
Equations	ideal MHD, $\gamma =c_p/c_v=1.4$ (the V3 input's); none ($n_v=8$) or mixed GLM ($n_v=9$)	models
Space	WENO5 (weno-u-5) flux splitting on characteristic variables	compute_residuals_weno, adam_flume_cpu_object.F90; mhd_face_split_fluxes, mhd_glm_face_split_fluxes, adam_flume_mhd_library.F90
Cleaning	GLM, $c_h=3$, glm_damping_length = 1, glm_alpha = 0 or $0.18$, glm_ch_check = warning; or none	models; add_glm_damping, adam_flume_cpu_mhd_glm_kernels.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; interp_seam_ghost, adam_field_object.F90; boundary conditions
Reflux	Berger–Colella, stage fluxes weighted by the SSP coefficients, every conservative row	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
Integrals	volume integrals of the conservative variables, every step	compute_conservation, adam_flume_cpu_object.F90
Boundaries	periodic on all six faces	boundary conditions
FNL backend	the same kernels on the device	adam_flume_fnl_object.F90

Initial and boundary conditions

Item	Value
Base input	conservation/amr-periodic.ini (V3), turned into MHD by mhd/zero-field/make_input.py
Domain	$[0,1{]}^3$
Grid	octree, one uniform level (8 blocks of $8^3$ cells), the octant $[0,0.5{]}^3$ refined once more: 15 blocks, 6 coarse–fine faces, ngc = 3
Ranks	2
Initial condition	uniform: $(\rho ,u,v,w,p)=(1,1,1,1,1)$, $\mathbf{B}=(0.8,0.5,-0.3)$, $\psi =0$, seeded density/pressure perturbation $s=0.01$
Boundaries	periodic on all six faces
Final time	20 steps ($t=0.0347$)

Leg	divergence_control	glm_alpha	reflux	Bound
glm-reflux	glm	0	on	all 9 integrals $\le {10}^{-13}$
glm-noreflux	glm	0	off	max drift $\ge {10}^{-10}$ (negative control)
glm-damped-reflux	glm	0.18	on	the 8 physical integrals $\le {10}^{-13}$; $\psi$ reported
none-reflux	none	—	on	the 8 integrals $\le {10}^{-13}$

Input files

File	Role
src/tests/flume/verification/conservation/amr-periodic.ini	the V3 base input
src/tests/flume/verification/mhd/zero-field/make_input.py	derives the MHD input (--divergence-control, --b 0.8 0.5 -0.3, --set mhd.glm_alpha=…, --set numerics.reflux=…)
src/tests/flume/verification/mhd/conservation/check.sh	runs the four legs and checks the six registered faces (--np N, --legs LIST: mv11-glm for legs 1 and 2, mv11-damped, mv11-none; FLUME_EXE selects the backend)
src/tests/flume/verification/conservation/conservation_oracle.py	the drifts (--conserved, --max-drift, --leaky, --min-drift, --exclude psi)
src/tests/flume/verification/make_doc_figures.py (fig_mhd_conservation)	the figure, from the kept CPU conservation histories of mhd/conservation/ (--only mhd-conservation)

cd src/tests/flume/verification/mhd/conservation && ./check.sh        # CPU
FLUME_EXE=$PWD/../../../../../../exe/adam_flume_fnl ./check.sh        # FNL
cd ../.. && python3 make_doc_figures.py --out ../../../../docs/public/flume --only mhd-conservation

Results

One panel per CPU leg (GLM with reflux, damped GLM with reflux, GLM without reflux, no cleaning with reflux): $|I(t)-I(0)|/|I(0)|$ of every volume integral against the step ($\psi$ absolute), floored at ${10}^{-17}$.

Maximum drift over the 20 steps, computed from the kept conservation histories (mhd/conservation/work-adam_flume_{cpu,fnl}-np2-<leg>/amr-periodic-conservation_history.dat) with the oracle's definition (relative to $|I(0)|$; $\psi$ relative to the largest initial integral, $\int E=4.490$):

Leg	Bound	CPU	FNL
glm-reflux, 8 physical integrals	${10}^{-13}$	0 (constant to the printed digit)	$\le 3.7\cdot {10}^{-16}$ ($B_z$)
glm-reflux, $\psi$	${10}^{-13}$	$3.8\cdot {10}^{-20}$	$9.5\cdot {10}^{-20}$
glm-noreflux, max over the integrals	$\ge {10}^{-10}$	$1.69\cdot {10}^{-5}$ ($\rho u$); $1.5\cdot {10}^{-6}$ to $1.7\cdot {10}^{-5}$ over the 9	the same to 3 digits
glm-damped-reflux, 8 physical integrals	${10}^{-13}$	$\le 2.2\cdot {10}^{-16}$	$\le 3.7\cdot {10}^{-16}$
glm-damped-reflux, $\psi$ (reported)	—	$6.0\cdot {10}^{-11}$ ($|\int \psi |=2.70\cdot {10}^{-10}$)	the same
none-reflux, 8 integrals	${10}^{-13}$	0	$\le 3.7\cdot {10}^{-16}$
registered coarse–fine faces, every leg, both ranks	6	6	6

Related cases: conservation across AMR seams is the Euler V3 base; zero field (MV-2) runs the same box with a uniform MHD state; GLM divergence cleaning checks the damping; the M6 integral-drift study near seams is integral drift.

Findings

• With reflux the MHD integrals are constant to round-off on both backends, $\psi$ included: the register carries the $\mathbf{B}$ and $\psi$ rows.
• With damping, $\int \psi$ is not conserved: the source $-k\psi$ couples the Runge–Kutta stages. The seam-flux mismatch of stage $s$ reaches ${\psi }^{n+1}$ through the later stages' damping, so it enters with weight $b_s(1-O(k\mathrm{\Delta }t))$ while the register corrects $b_s$; the remainder is $O(k\mathrm{\Delta }t)$ of the uncorrected leak. Measured at M2-P7a (check.sh header): $|\int \psi |=2.7\cdot {10}^{-11}$, $2.7\cdot {10}^{-10}$, $1.3\cdot {10}^{-9}$ for glm_alpha 0.018, 0.18, 0.9, linear in the rate; without the seams it stays at ${10}^{-21}$. The kept damped histories give $2.70\cdot {10}^{-10}$ at 0.18, as recorded.
• The check.sh header records the M2-P7a CPU drift with reflux as $\le 4.7\cdot {10}^{-14}$ ($B_z$ the largest) and $7\cdot {10}^{-6}$ to $1.7\cdot {10}^{-5}$ without. The kept CPU histories show exact constancy with reflux and $1.5\cdot {10}^{-6}$ ($B_x$) to $1.7\cdot {10}^{-5}$ without; the $7\cdot {10}^{-6}$ lower end matches $E$ ($7.07\cdot {10}^{-6}$), not the $\mathbf{B}$ integrals.
• The bounds hold on FNL as well. The FNL drifts with reflux are one to two ulps, not zero, and the FNL step-0 integrals already differ from the CPU ones in the last digit ($\int B_z=-0.29999999999999993$ against $-0.29999999999999999$): the two backends do not compute the integrals with the same arithmetic.