GLM divergence cleaning

ID	Group	Milestone	Backends	Figure	Regression
MV-3, MV-10	Ideal MHD	M2 (#41); the MV-10 monitor M6 (#68, G1)	CPU, FNL	yes	glm-pulse

Case and scope

MV-3, the GLM pulse. A Gaussian pulse in $B_x$ along $x$ on a fluid at rest is a pure divergence error. In 1-D the pair $(B_x,\psi )$ of mixed GLM then obeys a (damped) telegraph equation decoupled from the fluid,

$${\partial }_t{B}_x+{\partial }_x\psi =0,{\partial }_t\psi +c_h^2{\partial }_x{B}_x=-\frac{c_h^2}{c_p^2}\psi ,$$

with an exact solution: the error splits into two pulses travelling at $\pm c_h$ and is damped at the rate $c_h^2/c_p^2=\alpha c_h/L$ ($\alpha$ = glm_alpha, $L$ = glm_damping_length). The oracle computes it mode by mode (d'Alembert without damping).

What MV-3 catches:

• a wrong GLM block in the face flux: $L_1$ of $B_x$ and $\psi /c_h$ against the exact solution under a bound, and the observed order;
• a wrong damping source or rate: a wrong rate is an $O(1)$ relative error (1.23 with no damping at $N=128$, against $9.2\cdot {10}^{-5}$), and the min-cell damping length is checked on its own;
• $c_h$ missing from the time step: $c_h=2$ is above the fastest wave (1.29), so it sets $\mathrm{\Delta }t$, and glm_ch_check = error must not trip;
• a wrong wall parity of $\psi$: reflecting walls ($B_x$ odd, $\psi$ even);
• a leaking none path: without cleaning $B_x$ must stay frozen, bit for bit.

MV-10, the Dedner peak. A peak in $B_x$ on a uniform magnetised flow (Dedner et al. 2002, section 5.1) has a non-zero $\mathrm{\nabla }\cdot \mathbf{B}$ from $t=0$. Without cleaning the error is only advected; GLM transports it away at $c_h$ and damps it, so $\parallel \mathrm{\nabla }\cdot \mathbf{B}{\parallel }_1$ must decay, on a uniform grid and across 2:1 AMR seams. The leg also checks the div(B) history, the derived output fields and the divb_tol monitor.

Models and numerics

Item	Choice	Code
Equations	ideal MHD with mixed GLM, $\gamma =5/3$	models; mhd_glm_flux, mhd_glm_eigenvectors, adam_flume_mhd_library.F90
Space	WENO5 flux splitting on characteristic variables; the $(B_n,\psi )$ pair is the 2×2 block of speeds $\mp c_h$	mhd_glm_face_split_fluxes, adam_flume_mhd_library.F90; compute_residuals_weno, adam_flume_cpu_object.F90
Damping	source $-(c_h^2/c_p^2)\psi$, $c_h^2/c_p^2=\alpha c_h/L$; $L$ a length or min-cell (the minimum cell size)	add_glm_damping, adam_flume_cpu_mhd_glm_kernels.F90; set_glm_damping, adam_flume_mhd_object.F90
$c_h$ check	glm_ch_check = error (MV-3, MV-10)	check_glm_ch, adam_flume_cpu_object.F90
div(B) history and monitor	max, $L_1$ and seam-local max of $\mathrm{\nabla }\cdot \mathbf{B}$ (host FD, null directions weighted zero); divb_tol on the normalised $D_{\text{flux}}$, $maxh|D_{\text{flux}}|/max|B|$	compute_divb_history, compute_dflux_norms, adam_flume_cpu_object.F90; save_divb_row, adam_flume_diagnostics_object.F90
Time	SSP Runge–Kutta, 3 stages, CFL 0.5	adam_rk_object.F90
AMR (MV-10)	2:1 seams: restriction, tricubic ghost fill, reflux	interp_seam_ghost, adam_field_object.F90; numerics
Boundaries	periodic; reflecting walls (wall-inviscid: $B_x$ odd, $\psi$ even) in the wall leg	adam_flume_bc_object.F90; boundary conditions
FNL backend	the same kernels on the device	adam_flume_fnl_object.F90, adam_flume_fnl_mhd_glm_kernels.F90

Initial and boundary conditions

MV-3 (mhd/glm-pulse):

Item	Value
Domain	$[0,1]$ along $x$; $y$ and $z$ null
Grid	the Sod grid of V1, 4 blocks along $x$, blocks of $N/4\times 2\times 2$ cells
Ranks	2
Initial condition	glm-pulse: background $(\rho ,u,v,w,p)=(1,0,0,0,1)$, $\mathbf{B}=0$; pulse in $B_x$ at $x=0.5$, width 0.1, amplitude 0.1
Cleaning	$c_h=2$

Leg	$N$	glm_alpha	glm_damping_length	$x$ boundaries	$t$
d'Alembert	64, 128, 256	0	1.0	periodic	0.3 (the pulses cross the boundary)
damped	64, 128, 256	0.5	0.1 ($c_h^2/c_p^2=10$)	periodic	0.3
min-cell	128	0.1 (rate 25.6)	min-cell	periodic	0.3
walls	128	0	1.0	wall-inviscid (both pulses reflected once)	0.4
none	128	—	—	periodic	0.3

MV-10 (mhd/divb-peak):

Item	Value
Domain	$[-0.5,1.5{]}^2$, $z$ null, periodic
Grid	the periodic grid of V3 (octree) at level 2: ${64}^2$ cells, ${16}^2$ per block, nk = 4 (the 2:1 seam fill needs 4 cells on every axis, null ones included)
AMR leg	$8^2$ cells per base block (${32}^2$), the blocks of $[-0.5,0.5{]}^2$ refined once more (a 2:1 seam the peak crosses near $t=0.375$)
Ranks	2
Initial condition	divb-peak: $\rho =1$, $u=v=1$, $w=0$, $p=6$, $\mathbf{B}=(B_x,0,1/\sqrt{4\pi })$, $B_x+=(1-(r/R{)}^2{)}^2/\sqrt{4\pi }$, $R=1/8$, at the origin
Cleaning	GLM, $c_h=5$, glm_alpha = 0.4, damping length min-cell; and none
Final time	$t=0.5$; the monitor legs 5 steps

Input files

File	Role
src/tests/flume/verification/mhd/glm-pulse/make_glm_pulse.py	writes the MV-3 input from sod/sod-x.ini
src/tests/flume/verification/mhd/glm-pulse/check.sh	runs the MV-3 legs (--np N, --divergence-control glm|eglm, --legs LIST: mv3-dalembert, mv3-damped, mv3-mincell, mv3-walls, mv3-none, the last GLM-only)
src/tests/flume/verification/mhd/glm-pulse/glm_pulse_oracle.py	the exact telegraph solution, the $L_1$ bounds, the order, $\int B_x$, the frozen none run
src/tests/flume/verification/mhd/divb-peak/make_divb_peak.py	writes the MV-10 input from conservation/amr-periodic.ini
src/tests/flume/verification/mhd/divb-peak/check.sh	runs the MV-10 legs and the monitor legs (--np N, --legs LIST: mv10-uniform, mv10-amr, mv10-monitor for the monitor runs)
src/tests/flume/verification/mhd/divb-peak/divb_peak_oracle.py	the decay, $\int \mathbf{B}$, $max|\psi |$, the derived fields, the seam column
src/tests/flume/verification/make_doc_figures.py fig_glm_pulse	draws glm-pulse.png from <runs>/glm-pulse (--only glm-pulse)

cd src/tests/flume/verification/mhd
glm-pulse/check.sh                                                     # MV-3, CPU
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl glm-pulse/check.sh    # MV-3, FNL
divb-peak/check.sh                                                     # MV-10

make_doc_runs.sh <runs> mhd writes the figure run (<runs>/glm-pulse: 128 cells, GLM, amplitude 0.1); make_doc_figures.py --runs <runs> draws it.

Results

The undamped case (glm_alpha = 0, periodic, 128 cells, $c_h=2$, $t=0.3$): $B_x$ and $\psi$ of FLUME on the exact solution, at $t=0$ and at the final time.

MV-3 (bounds: CPU baseline M2-P4a + 2%):

Check	Bound	CPU	FNL
d'Alembert, $L_1$ of $B_x$ and $\psi /c_h$, $N=64/128/256$	$1.584\cdot {10}^{-4}$ / $9.342\cdot {10}^{-6}$ / $7.487\cdot {10}^{-7}$	$1.553\cdot {10}^{-4}$ / $9.159\cdot {10}^{-6}$ / $7.340\cdot {10}^{-7}$	pass
damped, $L_1$, $N=64/128/256$	$3.393\cdot {10}^{-5}$ / $1.962\cdot {10}^{-6}$ / $1.572\cdot {10}^{-7}$	$3.327\cdot {10}^{-5}$ / $1.923\cdot {10}^{-6}$ / $1.541\cdot {10}^{-7}$	pass
observed order, both ladders	$\ge 3.0$ (SSP-33 in time)	4.1 and 3.6 (WENO5 in space still visible)	pass
min-cell, $L_1$	$1.487\cdot {10}^{-7}$	$1.458\cdot {10}^{-7}$	pass
walls, $L_1$	$1.364\cdot {10}^{-5}$	$1.337\cdot {10}^{-5}$	pass
$\int B_x$ drift (d'Alembert)	${10}^{-14}$ ($\int B_x\approx 1.8\cdot {10}^{-2}$)	$\le 5.8\cdot {10}^{-16}$	pass
none: $B_x$ last against first checkpoint	bitwise	bitwise	bitwise

MV-10 (CPU baselines M2-P4b):

Check	Bound	CPU	FNL
GLM final / initial $|\mathrm{\nabla }\cdot \mathbf{B}{|}_1$, uniform ${64}^2$ (never above the initial value)	0.1	0.057	pass
GLM / none final $|\mathrm{\nabla }\cdot \mathbf{B}{|}_1$	0.1	0.027	pass
AMR: GLM final / initial	0.2	0.091	pass
$\int B_x$, $\int B_y$, $\int B_z$ drift ($\int B_z=1.13$)	${10}^{-11}$	$\le 8.3\cdot {10}^{-13}$	pass
$max|\psi |$	$0.141$ ($c_{h}A/10$)	$3.6\cdot {10}^{-3}$ uniform, $1.3\cdot {10}^{-2}$ AMR	pass
saved pt, beta, bmag against their definitions; saved $max|\mathrm{\nabla }\cdot \mathbf{B}|$ against the history	${10}^{-12}$	exact	pass
AMR seam column: zero at step 0, never above the global maximum, positive at the end	—	pass	pass
monitor, divb_tol = 0.01, divb_error = .false. (the peak starts near 0.2)	warnings, run completes	pass	pass
monitor, divb_error = .true.	run stops with its message	pass	pass
divb_tol > 0 without [diagnostics] divb_flux	refused at initialisation	pass	pass

FNL ran every leg (work directories of both backends); the tables record the CPU values, and the backends agree to the printed digits.

Related cases: EGLM cleaning runs MV-3 under EGLM (EV-2); Orszag–Tang and field loop measure the seam div(B) of shocked and smooth flows; the seam div(B) studies of M6 are in divergence across seams.

Findings

• The observed order is that of SSP-33 in time with the WENO5 spatial error still visible (4.1 and 3.6), above the asserted 3.0.
• The $\parallel \mathrm{\nabla }\cdot \mathbf{B}{\parallel }_1$ of the Dedner peak is not monotone under GLM: a few-percent transient while the peak spreads into outgoing waves. The leg therefore asserts "never above the initial value" and the final ratio, not a monotone decay.
• On AMR the final ratio doubles (0.091 against 0.057): the 2:1 seam injects div(B) (the seam source of #29); GLM keeps it bounded.
• Since #68 G1 the divb_tol monitor acts on the normalised $D_{\text{flux}}$, which needs [diagnostics] divb_flux and GLM; a divb_tol without divb_flux is refused at initialisation instead of monitoring nothing.