MHD plumbing

ID	Group	Milestone	Backends	Figure	Regression
— (mhd/plumbing, M2-P1)	Ideal MHD	M2 (#41, P1)	CPU, FNL	no	—

Case and scope

M2-P1 wired the ideal-MHD model through FLUME end to end: the input predicate (state width and names), the MHD conversions and fast magnetosonic speed, the per-model kernel instances, the initial- and boundary-condition keys and the wall rule, the outputs and the restart. The leg checks that plumbing, for both variants divergence_control = none ($n_v=8$) and glm ($n_v=9$, with $\psi$), on a seeded uniform state:

1. time step: one step of the unperturbed uniform state ($s=0$). The step must equal $\mathrm{CFL}/\sum _d(|u_d|+c_{f,d})/\mathrm{\Delta }{x}_d$, with the fast speed computed by the oracle independently of the Fortran; under GLM it is bounded by $\mathrm{CFL}/(c_h\sum _{d}1/\mathrm{\Delta }{x}_d)$, the active bound here (M2-P4);
2. auxiliary fields: the saved MHD auxiliaries equal the values recomputed from the saved conservative fields;
3. restart: 5 steps, a restart and 5 more steps of the seeded ($s=0.05$) state equal the continuous 10-step run bit for bit, the residual and conservation histories byte for byte, and every model variable (bx, by, bz [, psi]) is saved;
4. refusals (variant none): an unknown divergence_control and isentropic-vortex with MHD stop with their error messages.

What the case catches: a wrong fast-speed formula, a wrong $\mathbf{B}$ index, a wrong auxiliary layout on the device (the dt check exercises the device auxiliary and fast-speed kernels on FNL), a kernel instance of the wrong model, $c_h$ missing from the time step, a missing $\mathbf{B}$ or $\psi$ field in the output, a restart that loses $\psi$, and a silently accepted invalid input.

The retired zero-residual legs

At M2-P1 the MHD face fluxes did not exist yet, so the residual was exactly zero and the leg also required constant integrals and 10 steps equal to 1 step. Since M2-P3 the face fluxes are live and those legs are retired; the exact steady state is checked by MV-2 (zero field, uniform state, bitwise). The oracle's module docstring still describes the P1 state ("no face fluxes yet"); its --static mode is no longer called by check.sh.

Models and numerics

Item	Choice	Code
Equations	ideal MHD, $\gamma =c_p/c_v=2.5/1.5=5/3$; none ($n_v=8$) or mixed GLM ($n_v=9$)	models; initialize, adam_flume_physics_object.F90
GLM	$c_h=3$, glm_alpha = 0.18, glm_damping_length = 1, glm_ch_check = warning	models; adam_flume_mhd_object.F90
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
Time step	$\mathrm{CFL} / \max \sum_d (	u_d
Time	SSP Runge–Kutta, 3 stages, CFL 0.8	adam_rk_object.F90
Auxiliary fields	$\rho ,u,v,w,p,T,H,a,B_x,B_y,B_z$ from the conservative state	compute_q_aux, adam_flume_cpu_object.F90; mhd_conservative_to_auxiliary, adam_flume_mhd_library.F90; models
Output and restart	HDF5 checkpoints; restart files	save_xh5f, save_restart_files, load_restart_files, adam_flume_common_object.F90
Refusals	unknown divergence_control; isentropic-vortex requires euler	load_from_file, adam_flume_mhd_object.F90, adam_flume_ic_object.F90
Boundaries	extrapolation, wall-inviscid, inflow (with $\mathbf{B}$)	adam_flume_bc_object.F90; boundary conditions
FNL backend	the same kernels on the device	adam_flume_fnl_object.F90

Initial and boundary conditions

Item	Value
Domain	$[0,1{]}^3$
Grid	octree, two uniform levels (iu_ref_levels = 2): 64 blocks of $8^3$ cells, ${32}^3$ cells, $\mathrm{\Delta }x=1/32$, ngc = 3 (both ranks own blocks, so the restart check is meaningful)
Ranks	2
Initial condition	uniform: $(\rho ,u,v,w,p)=(1,0.3,-0.2,0.1,1)$, $\mathbf{B}=(0.8,0.5,-0.3)$, seeded density/pressure perturbation $s=0.05$ ($s=0$ in the dt check)
Boundaries	$x$: extrapolation; $y$: wall-inviscid; $z_{min}$: inflow with the region state; $z_{max}$: extrapolation
Steps	dt check 1; auxiliary check 10; restart check 5 + 5 against 10

Check	Variant (leg)	Change	Expected
dt	none, glm	$s=0$, it_max = 1	step 1 equals the oracle's dt to ${10}^{-13}$ relative
aux	none, glm	—	auxiliaries equal the recomputed ones to ${10}^{-12}$ relative
restart	none, glm	restart_save = 5, then restart = .true.	bitwise fields, identical histories, model variables saved
refused	none	divergence_control = bogus	unknown [mhd].(divergence_control) "bogus"
refused	none	type = isentropic-vortex	isentropic-vortex requires [physics].(physical_model) = euler

Input files

File	Role
src/tests/flume/verification/mhd/plumbing/mhd-uniform-none.ini	the case, divergence_control = none
src/tests/flume/verification/mhd/plumbing/mhd-uniform-glm.ini	the case, divergence_control = glm
src/tests/flume/verification/mhd/plumbing/check.sh	runs every leg (--np N, --build, --legs LIST: none, glm, refused; FLUME_EXE selects the backend); a leg is a variant, none and glm each run the dt, aux and restart checks
src/tests/flume/verification/mhd/plumbing/mhd_plumbing_oracle.py	--dt, --aux, --compare; also --zero and --steady used by MV-2

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

Results

A contract leg: no figure. The work directories of both backends are kept (plumbing/work-adam_flume_{cpu,fnl}-np2-*), with their logs and histories; the checkpoints are deleted, and the oracle's printed lines are not kept. The table quotes what those files still show:

Check	Bound	CPU	FNL
dt of step 1, none (history *-dt)	oracle $4.7620653358110355\cdot {10}^{-3}$, ${10}^{-13}$ relative	$4.7620653358110363\cdot {10}^{-3}$ ($1.7\cdot {10}^{-16}$)	the same digits
dt of step 1, glm: the $c_h$ bound $0.8/(3\cdot 3\cdot 32)$ is active ($c_h\cdot 3=9$ against $\sum _d(|u_d|+c_{f,d})=5.25$)	oracle $2.7777777777777778\cdot {10}^{-3}$	$2.7777777777777779\cdot {10}^{-3}$	the same digits
restart: residual and conservation histories against the continuous run (*-A vs *-B, cmp)	byte-identical	identical, both variants	identical, both variants
auxiliary fields against the recomputed ones	${10}^{-12}$ relative	not kept	not kept
restart: interior fields, every variable, model variables present	bitwise	not kept	not kept
unknown divergence_control	refused	unknown [mhd].(divergence_control) "bogus"; expected one of glm, none	the same
isentropic-vortex with MHD	refused	[initial_conditions].(type) = isentropic-vortex requires [physics].(physical_model) = euler	the same

The dt oracle values are recomputed from the oracle's formula and the input; the measured values are the step-1 rows of the kept conservation histories.

Related cases: zero field (MV-2) took over the exact-steady-state legs; GLM divergence cleaning verifies the $c_h$ time-step bound and the damping physically.

Findings

• The fast-speed time step agrees with the independent textbook formula to one ulp on both backends, and the GLM step is set by $c_h$, so a $c_h$ missing from the time step would fail the leg.
• The restart reproduces the continuous run's histories byte for byte on both backends and both variants.
• The work directories *-none-C and *-glm-C (dated before the others) belong to the retired "10 steps = 1 step" leg; check.sh no longer writes them.
• The auxiliary and bitwise-field verdicts rest on the check.sh run; the checkpoints they read are deleted after the oracle, so this page cannot re-derive them.