#78 is the reflux, not the restricted ghosts

ID	Group	Milestone	Backends	Figure	Regression
G3a	Seam studies	M6 (#68), #78	FNL only	yes	—

Case and scope

A2 traced EGLM's 70-fold $⟨|B_z|⟩$ on a static 2:1 grid (#78) to the centred divergence of its Derigs source reading the second-order restricted ghosts. Phases C to E gave the seams sixth-order ghosts (C1, D1) and a reflux policy (E2), so G3a asks first whether #78 survives them, and so which of the two, the ghosts or the reflux, causes it.

Models and numerics

Item	Choice	Code
Equations	ideal MHD, $\gamma =5/3$, GLM or EGLM, glm_ch 4.0, glm_alpha 0.18, glm_damping_length 1.0	GLM, EGLM; add_eglm_sources, adam_flume_cpu_mhd_eglm_kernels.F90; add_eglm_sources_dev, adam_flume_fnl_mhd_eglm_kernels.F90
Space, time	WENO5 flux splitting on characteristic variables; SSP Runge–Kutta 5 stages, CFL 0.4	as in AD
Seam operators	default (mean + tricubic) or lagrange6 + quintic	C1, D1
Reflux	on (always), troubled (flags no face of this smooth loop, so it equals the reflux off bit for bit), or off	apply_reflux_to_stage_forest, flag_troubled_seam_cells, adam_flume_fnl_object.F90

Initial and boundary conditions

Item	Value
Case	the A2 field loop of AD: $[-1,1]\times [-0.5,0.5]$, quadtree, the box $[0,1]\times [-0.5,0.5]$ refined one 2:1 level
Initial condition	field loop, radius 0.3, $A_0={10}^{-3}$; $(\rho ,u,v,w,p)=(1,2,1,1,1)$
Grid	refined $N=64$ against uniform $N=128$
Boundaries	periodic in $x$, $y$; extrapolation in $z$
Final time	$t=1$
Ranks	2, FNL

Input files

File	Role
src/tests/flume/verification/amr-divb/make_amr_divb.py loop	the field loop inputs (--cleaning glm or eglm, --seam-ghost-fill, --seam-restriction, --reflux-policy, --no-reflux)
src/tests/flume/verification/amr-divb/amr_divb_oracle.py loop	the $⟨|B_z|⟩/A_0$ report
src/tests/flume/verification/make_doc_runs.sh <runs> divb (FNL), make_doc_figures.py --only field-loop-seams	re-runs the six field loops with a checkpoint every 20 steps

Results

FNL, today's code (after G3b): the A2 field loop under GLM and EGLM, refined $N=64$ with and without reflux, and uniform $N=128$, a checkpoint every 20 steps. Left: the magnetic energy; right: $⟨|B_z|⟩/A_0$. The re-runs end at exactly the values of the tables (EGLM $9.11\cdot {10}^{-6}$ with the reflux, $9.31\cdot {10}^{-6}$ without, $2.72\cdot {10}^{-6}$ uniform; GLM $1.10\cdot {10}^{-3}$, $1.18\cdot {10}^{-3}$, $6.86\cdot {10}^{-4}$). The pre-G3b reflux of B (70 times the uniform level) is a code state that no longer exists, so it is not drawn.

Seam operators, reflux	EGLM $⟨|B_z|⟩/A_0$ (× uniform)	GLM (× uniform)
uniform N = 128	$2.72\cdot {10}^{-6}$ (1)	$6.86\cdot {10}^{-4}$ (1)
default, always	$1.91\cdot {10}^{-4}$ (70)	$1.10\cdot {10}^{-3}$ (1.6)
default, off (= troubled)	$9.31\cdot {10}^{-6}$ (3.4)	$1.18\cdot {10}^{-3}$ (1.7)
lagrange6 + quintic, always	$2.89\cdot {10}^{-4}$ (106)	$1.12\cdot {10}^{-3}$ (1.6)
lagrange6 + quintic, off (= troubled)	$3.14\cdot {10}^{-5}$ (11.5)	$1.56\cdot {10}^{-3}$ (2.3, run with troubled)

Related cases: AD is the A2 attribution this corrects; G3b is the fix; the uniform-grid EGLM is EGLM cleaning; the loop itself is field loop.

Findings

• The reflux causes #78. With the default operators and no reflux, EGLM's $B_z$ is 3.4 times its uniform level instead of 70; higher-order ghosts do not help (106 with the reflux, 11.5 without). GLM does not care (1.6 to 2.3 times in every row). The A2 attribution to the restricted ghosts is wrong.
• Mechanism (inferred, not measured directly): EGLM's Derigs source $-\mathrm{\nabla }\cdot \mathbf{B}(0,\mathbf{B},\mathbf{u}\cdot \mathbf{B},\mathbf{u},\dots )$ cancels, on a uniform grid, the $B_z$ that the flux divergence generates. The reflux corrects a coarse skin cell's state by the flux mismatch $F_c-\sum F_f$ after the step, but the source that step used saw the coarse fluxes' divergence: the correction changes the discrete divergence the fluxes imply without the matching source term, so the cancellation fails exactly at the refluxed cells.
• Workaround before G3b: EGLM with reflux_policy = troubled keeps #78 at 3.4 times on smooth fields (the reflux is applied only where a shock or the limiter flags a face), at the conservation price measured in E2.
• The fix: G3b: the EGLM reflux leaves $\mathbf{B}$ alone (3.3 times).