Divergence across seams

ID	Group	Milestone	Backends	Figure	Regression
AD (A2)	Seam accuracy and divergence	M6 (#68), #78	FNL (tables); CPU for the loop leg and the D_flux cross-check	yes	—

Case and scope

Every div(B) column of the MHD checks is the centred divergence of the cell values, which is not the divergence GLM and EGLM act on (#68 F4). [diagnostics] divb_flux adds D_flux, the divergence of the interface normal field read from the scheme's own $\psi$ face fluxes, the coarse 2:1 faces taking the fine fluxes (the reflux), so that it sums exactly (Gauss) over a composite grid. AD measures D_flux where the seams matter:

• cpaw: a source-free, exact nonlinear solution with $\mathrm{\nabla }\cdot \mathbf{B}=0$ that crosses two seams all the time: is the seam D_flux bounded or does it grow, and does it converge?
• loop: the field loop of MV-9 under GLM and EGLM, the case of #78 (EGLM's $⟨|B_z|⟩$ 70 times its uniform level on a static 2:1 grid): is #78 a divergence difference?

What the case catches:

• a secular growth of the seam divergence under GLM (PRISM's centred GLM lets it run away, #29);
• a seam divergence that does not converge under refinement;
• a broken D_flux instrument: its signed integral over a periodic composite domain must be round-off.

Models and numerics

Item	Choice	Code
Equations	ideal MHD, $\gamma =5/3$; mixed GLM (cpaw, loop) or EGLM (loop); glm_alpha 0.18, glm_damping_length 1.0; glm_ch 2.0 (cpaw, above the fast speed, about 1.08) or 4.0 (loop)	GLM, EGLM; add_eglm_sources, adam_flume_cpu_mhd_eglm_kernels.F90
Space	WENO5 (weno-u-5) flux splitting on characteristic variables	compute_residuals_weno, adam_flume_cpu_object.F90; mhd_glm_face_split_fluxes, mhd_eglm_face_split_fluxes, adam_flume_mhd_library.F90
Time	SSP Runge–Kutta, 5 stages (runge-kutta-ssp-54), CFL 0.4	adam_rk_object.F90
Seam operators	mean restriction, tricubic fill, reflux on (always)	numerics
D_flux	$\sum _d(F_{\psi }(i+\frac{1}{2})-F_{\psi }(i-\frac{1}{2}))/\mathrm{\Delta }{x}_d$ over $c_h^2$ (GLM) or $c_h$ (EGLM), the coarse 2:1 faces corrected to the fine fluxes; the seam column over the cells within ngc of a seam; key [diagnostics] divb_flux	compute_dflux_norms, compute_divb_history, adam_flume_cpu_object.F90 and adam_flume_fnl_object.F90; adam_flume_diagnostics_object.F90
Boundaries	periodic in $x$ and $y$; extrapolation on the null $z$ faces	boundary conditions

Initial and boundary conditions

Item	cpaw	loop
Domain	$[0,\sqrt{5}]\times [0,\sqrt{5}/2]$, null $z$	$[-1,1]\times [-0.5,0.5]$, null $z$
Initial condition	MV-6 CPAW, right polarised, inclined at $\tan \alpha =2$ (63.43°), $B_{\parallel }=1$, amplitude 0.1, wavelength 1, $\rho =1$, $p=0.1$	MV-9 field loop, radius 0.3, $A_0={10}^{-3}$, centre $(0,0)$; $(\rho ,u,v,w,p)=(1,2,1,1,1)$
Grid	quadtree (ratio = 4), $4\times 4$ base blocks of $N/4\times N/8\times 1$ cells, ngc = 3	the same
Refined region	the two middle block columns, $x\in [0.6,1.6]$: two $x$ seams the wave crosses all the time	the box $[0,1]\times [-0.5,0.5]$
Ladder	refined and uniform $N$ = 64 / 128 (--long adds 256)	refined $N=64$ against uniform $N=128$
Boundaries	periodic in $x$, $y$; extrapolation in $z$	the same
Final time	10 periods ($t=10$)	$t=1$
History cadence	every 10 steps	every step
Ranks	2	2

Input files

File	Role
src/tests/flume/verification/amr-divb/make_amr_divb.py	writes the inputs from mhd/cpaw/make_cpaw.py and mhd/field-loop/make_field_loop.py; --uniform, --periods, --cleaning, --time-max, --history
src/tests/flume/verification/mhd/amr_box.py	refine_box: the refined region, one 2:1 level at initialisation
src/tests/flume/verification/amr-divb/amr_divb_oracle.py	growth (the cpaw floors, growth and orders) and loop (a report)
src/tests/flume/verification/amr-divb/check.sh	runs the legs (--legs cpaw,loop, --long, --np N, --keep)
src/tests/flume/verification/make_doc_runs.sh <runs> divb (FNL), make_doc_figures.py --only divergence-across-seams	re-runs the field loops (GLM, EGLM; refined, without reflux, uniform) with the history every step; the CPAW panel reads the histories of check.sh

cd src/tests/flume/verification/amr-divb
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl ./check.sh          # FNL: about 12 min
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl ./check.sh --long   # adds N = 256 (30 more minutes)
./check.sh --legs loop                                               # the loop leg on the CPU

The cpaw leg costs about 2 h on the CPU ($N=128$ dominates): run it on FNL.

Oracle (amr_divb_oracle.py, reading the *-divb_history.dat columns). cpaw: per run the seam D_flux floor (the median over the second half of the run, the step-0 row excluded), its growth (the mean over the last quarter over the mean over the second quarter: about 1 when bounded) and the floor's order over the refined ladder; a uniform run has no seam, its floor is that of the global D_flux. Bounds: growth at most 1.5 on every refined run, floor order at least 1 on the finest pair. loop: the final seam and global D_flux and $⟨|B_z|⟩/A_0$, a report.

Results

FNL, 2 ranks. Left: the seam maximum of D_flux on the source-free CPAW over 10 periods, refined $N=64$ and 128 (the histories check.sh keeps), against the global maximum of the uniform $N=128$ run: the seam floor is bounded and falls 10 times from 64 to 128. Right: the A2 field loop under GLM and EGLM, the refined seam maximum against the uniform global one. No divergence map: FLUME writes no divergence field, and a centred difference of the checkpoints would read unfilled ghosts exactly at the seams.

The instrument: the signed integral of D_flux over the periodic composite domain is round-off, $\le {10}^{-17}$ every step on the refined MHD vortex, against $4\cdot {10}^{-7}$ without the fine fluxes; CPU and FNL agree to ${10}^{-11}$ relative.

cpaw (FNL, 2 ranks):

N	Seam D_flux floor	Growth	Uniform floor	Seam / uniform
64	$5.39\cdot {10}^{-4}$	0.95	$9.97\cdot {10}^{-6}$	54
128	$5.43\cdot {10}^{-5}$	0.72	$4.57\cdot {10}^{-7}$	119
256	$2.12\cdot {10}^{-5}$	0.80	$6.74\cdot {10}^{-9}$	3146

loop (FNL, 2 ranks), $t=1$:

Field loop, $t=1$	D_flux (seam, refined)	D_flux (max, uniform)	⟨|B_z|⟩/A0 refined	uniform
GLM	$2.35\cdot {10}^{-4}$	$2.03\cdot {10}^{-4}$	$1.10\cdot {10}^{-3}$	$6.86\cdot {10}^{-4}$
EGLM	$2.45\cdot {10}^{-4}$	$2.05\cdot {10}^{-4}$	$1.91\cdot {10}^{-4}$	$2.72\cdot {10}^{-6}$

Related cases: G2 varies the GLM damping length on the same cpaw leg; G3a and G3b rerun the loop leg and fix #78; the uniform-grid cleaning is GLM divergence cleaning and EGLM cleaning.

Findings

• GLM keeps the seam divergence bounded. Over 10 periods of a source-free flow the seam D_flux has no secular growth at any resolution: the upwinded $(B_n,\psi )$ pair of FLUME bounds what PRISM's centred GLM lets run away (#29; #68 F3).
• The seam floor converges at first order (orders +3.31 then +1.36: the first pair is pre-asymptotic), the $O(H)$ divergence of the $O(H^2)$ flux inconsistency of a 2:1 face (#68 F1), while the uniform floor converges at 4 to 6: the seams are the divergence error of a refined run, 3146 times the uniform level at $N=256$.
• #78 is not a divergence difference. GLM and EGLM leave the same D_flux, refined and uniform; EGLM's 250-fold smaller $B_z$ on a uniform grid comes from its source term, whose centred divergence cancels the $B_z$ the flux generates. The $B_z$ rises 70 times on the refined grid, reproduced here to the printed digits of #78.
• Attribution, corrected. A2 attributed the rise to the centred source divergence reading the second-order restricted ghosts at the seam, where the cancellation fails, and placed the fix in Phase G3 (a source divergence consistent with the face fluxes at the seams). G3a showed that the ghosts are not the cause: the reflux is. G3b fixed it by not refluxing $\mathbf{B}$ under EGLM.