MHD with a zero field

ID	Group	Milestone	Backends	Figure	Regression
MV-2	Ideal MHD	M2 (#41, P3)	CPU, FNL	no	uniform-amr-mhd

Case and scope

The MHD face kernels (the $7\times 7$ Roe–Balsara core at the face average, with the decoupled $B_n$ or the GLM pair) replace the Euler ones whenever physical_model = mhd-ideal. With $\mathbf{B}=0$ the MHD system is the Euler one plus inert fields, which gives exact expectations before any MHD-specific test (MV-4). Both variants, divergence_control = none and glm, run two legs:

1. Sod along $x$, $y$, $z$ through the MHD path: $L_1(\rho )$ against the exact Riemann solution within the M1 bound of V1; the $y$ and $z$ runs equal the $x$ run bit for bit after the permutation of the axes; $\mathbf{B}$ (and $\psi$) stay exactly zero;
2. a uniform MHD state ($\rho$, $\mathbf{u}$, $p$ and $\mathbf{B}$ all non-zero) on the periodic box of V3:
   • single level: the last checkpoint equals the step-0 state bit for bit (the MHD fluxes of equal states cancel);
   • with the 2:1 refined octant: equal to round-off, ${10}^{-14}$ relative.

What the case catches: an MHD kernel that does not reduce to Euler at $\mathbf{B}=0$, a direction-dependent defect of the MHD eigensystem, a spurious source of $\mathbf{B}$ or $\psi$, and MHD fluxes of equal states that do not cancel, on a uniform grid and across 2:1 faces.

Models and numerics

Item	Choice	Code
Equations	ideal MHD, $\gamma =c_p/c_v=1.4$ (the $c_p,c_v$ of the Euler base inputs, kept by make_input.py)	models
Space	WENO5 (weno-u-5) flux splitting on characteristic variables; the 7×7 Roe–Balsara core at the arithmetic face average	compute_residuals_weno, adam_flume_cpu_object.F90; mhd_face_average, mhd_eigenvectors, mhd_face_split_fluxes, mhd_glm_face_split_fluxes, adam_flume_mhd_library.F90; numerics
Cleaning	none, or GLM with $c_h=3$, glm_alpha = 0.18, glm_damping_length = 1, glm_ch_check = warning	models; add_glm_damping, adam_flume_cpu_mhd_glm_kernels.F90
Time	SSP Runge–Kutta, 3 stages, CFL 0.5	adam_rk_object.F90
AMR (uniform, AMR leg)	geometric box marker on the octant; 2:1 ghost restriction and interpolation; reflux	mark_by_geometry, adam_flume_common_object.F90; interp_seam_ghost, adam_field_object.F90; boundary conditions
Boundaries	Sod: extrapolation; uniform: periodic	boundary conditions, periodic
FNL backend	the same kernels on the device	adam_flume_fnl_object.F90

Initial and boundary conditions

Every input is derived from a committed Euler input by make_input.py: physical_model = mhd-ideal, an [mhd] section, $n_v=8$ (none) or 9 (glm), and bx, by, bz added to every region and inflow state.

Leg	Base input	Grid	State	Boundaries	Steps
Sod $x$, $y$, $z$	sod/sod-{x,y,z}.ini	as V1: 64 blocks of $50\times 4\times 4$ cells along the active axis, one level	riemann-problem: $(\rho ,u,p)=(1,0,1)$ / $(0.125,0,0.1)$, $\mathbf{B}=0$	extrapolation	$t=0.2$ (174 steps none, 241 steps glm: the $c_h$ bound sets $\mathrm{\Delta }t$)
uniform, single	conservation/amr-periodic.ini, markers_number = 0	$[0,1{]}^3$, octree level 1: 8 blocks of $8^3$ cells	uniform, $s=0$: $(\rho ,u,v,w,p)=(1,0.3,-0.2,0.1,1)$, $\mathbf{B}=(0.8,0.5,-0.3)$	periodic	10
uniform, AMR	conservation/amr-periodic.ini	the octant $[0,0.5{]}^3$ refined once more: 15 blocks, 2:1 faces on every side of the octant	the same	periodic	10

Ranks: 2 in every leg.

Input files

File	Role
src/tests/flume/verification/mhd/zero-field/make_input.py	derives the MHD input from an Euler one (--divergence-control, --b, --set section.key=value)
src/tests/flume/verification/sod/sod-{x,y,z}.ini	the Sod base inputs
src/tests/flume/verification/conservation/amr-periodic.ini	the V3 base input of the uniform legs
src/tests/flume/verification/mhd/zero-field/check.sh	runs every leg (--np N, --legs LIST: mv2-sod, mv2-uniform; FLUME_EXE selects the backend)
src/tests/flume/verification/sod/sod_oracle.py	$L_1(\rho )$ against the exact solution (--l1-max 3.31e-03), the $y$/$z$ against $x$ bitwise check
src/tests/flume/verification/mhd/plumbing/mhd_plumbing_oracle.py	--zero ($\mathbf{B}$, $\psi$ exactly zero), --steady (last checkpoint against step 0)

cd src/tests/flume/verification/mhd/zero-field && ./check.sh         # CPU
FLUME_EXE=$PWD/../../../../../../exe/adam_flume_fnl ./check.sh       # FNL

Results

A contract leg: no figure. Work directories of both backends are kept (zero-field/work-adam_flume_{cpu,fnl}-np2-*) with logs and histories; the checkpoints are deleted after the oracles and the oracle lines are not kept, so the cell-level verdicts below rest on the check.sh run. The values quoted are read from the kept histories:

Check	Bound	CPU	FNL
Sod, $L_1(\rho )$, both variants	$3.31\cdot {10}^{-3}$ (V1)	not kept	not kept
Sod, $y$ and $z$ against $x$	bitwise	not kept	not kept
Sod, $\mathbf{B}$ (and $\psi$) in the last checkpoint	exactly zero	not kept; $\int B_x,\int B_y,\int B_z$ (and $\int \psi$) are 0 in every history row, and the glm div(B) history is 0 at every step	the same
uniform, single level: last against step-0 checkpoint	bitwise	not kept; every volume integral constant to the printed digit, $\int \psi =0$	the same
uniform, 2:1 octant: last against step-0 checkpoint	${10}^{-14}$ relative	not kept; integrals constant to the printed digit, $max|\int \psi |=1.2\cdot {10}^{-18}$	integrals constant, $max|\int \psi |=2.0\cdot {10}^{-19}$

The regression case uniform-amr-mhd freezes the uniform AMR leg (GLM, 10 steps; regression/run.sh).

Related cases: MHD plumbing checks the time step, outputs and restart of the same model; Sod shock tube is the Euler reference; Ryu–Jones 2a (MV-4) is the first test with a non-zero field; conservation across AMR seams (V3) is the base of the uniform legs and MHD conservation (MV-11) its MHD counterpart.

Findings

• The Sod legs are not bitwise against the Euler path: at $\mathbf{B}=0$ the MHD eigenspaces are degenerate and WENO acts in a different characteristic basis, so the expectation is the V1 accuracy bound, not equality.
• The uniform state across the 2:1 patch is exact only to round-off. The cause is the interface, not the MHD kernels: the coarse–fine ghost fill reproduces a constant only to round-off for general values, and the Euler path does the same (a uniform Euler state with $u,v,w=0.8,0.5,-0.3$ drifts by $1.3\cdot {10}^{-15}$ on this grid, while the committed all-ones state happens to be exact; check.sh header).
• The work directories *-none-uniform and *-glm-uniform (CPU, dated before the others) predate the single/AMR split; check.sh no longer writes them.