Field loop advection

ID	Group	Milestone	Backends	Figure	Regression
MV-9	Ideal MHD	M2 (#41); --agree #79	CPU, FNL	yes	field-loop

Case and scope

A weak magnetic loop, $A_z=A_0(R-r)$ with $A_0={10}^{-3}$, $R=0.3$ (plasma $\beta \approx {10}^6$), is advected obliquely across the periodic box (Gardiner & Stone 2005; Mignone & Tzeferacos 2010, section 4.4.1). The field is divergence-free pointwise but jumps at $r=R$, and a collocated scheme cannot keep the discrete div(B) zero there. With the out-of-plane velocity $w=1$ the equations give

$${\partial }_t{B}_z=w\mathrm{\nabla }\cdot \mathbf{B},$$

so the generated $⟨|B_z|⟩$ measures the divergence error that GLM leaves, and the decay of the magnetic energy measures the numerical dissipation of the loop.

What the case catches:

• a divergence error above the GLM level: $⟨|B_z|⟩/A_0$ under a bound and decreasing under refinement;
• excess dissipation: $E_B(T)/E_B(0)$ above a bound;
• a seam div(B) source that does not decay (--amr): the seam div(B) must fall from its peak;
• backend disagreement on $\mathbf{B}$ (--agree): the CPU and FNL runs compared field by field.

Models and numerics

Item	Choice	Code
Equations	ideal MHD, $\gamma =5/3$	models
Space	WENO5 flux splitting on characteristic variables; weno-riemann under --numerics	mhd_glm_face_split_fluxes, mhd_eigenvectors_core, adam_flume_mhd_library.F90; numerics
Divergence control	GLM, $c_h=4$ (glm_ch_check = error), glm_alpha = 0.18, glm_damping_length = 1.0	models
Time	SSP Runge–Kutta, 5 stages (runge-kutta-ssp-54), CFL 0.4	adam_rk_object.F90
AMR (--amr)	primitive-box marker at initialisation; 2:1 restriction, tricubic ghost fill, reflux; seam div(B) column of the history	interp_seam_ghost, adam_field_object.F90; compute_divb_history, adam_flume_cpu_object.F90
Boundaries	periodic	boundary conditions
FNL backend	the same kernels on the device	adam_flume_fnl_object.F90

Initial and boundary conditions

Item	Value
Domain	$[-1,1]\times [-0.5,0.5]$, $z$ null
Grid	the vortex grid of V2 (quadtree, $4\times 4$ blocks) stretched to the box, $N\times N/2$ square cells, ladder $N=64,128$
Grid, --amr	ladder $N=32,64$, $[0,1]\times [-0.5,0.5]$ refined once (2:1 seams at $x=0$ and $x=1$, crossed twice by the loop); octree with null $z$, nk = 4
Ranks	2
Initial condition	field-loop: $\rho =1$, $p=1$, $\mathbf{v}=(2,1,1)$, loop at the origin, $R=0.3$, $A_0={10}^{-3}$
Boundaries	periodic on $x$ and $y$
Final time	$t=2$ (two crossings along each axis); --amr $t=1$ (one crossing)

Input files

File	Role
src/tests/flume/verification/mhd/field-loop/make_field_loop.py	writes the input from vortex/vortex-n064.ini (--cells, --time-max, --glm-ch, --vz, --refine-box)
src/tests/flume/verification/mhd/field-loop/check.sh	runs the ladder (--amr, --agree, --numerics SPEC, --legs mv9)
src/tests/flume/verification/mhd/field-loop/field_loop_oracle.py	$⟨|B_z|⟩/A_0$, its rate, $E_B(T)/E_B(0)$, the CPU–FNL agreement
src/tests/flume/verification/mhd/seam_divb_oracle.py	the seam div(B) decay (--amr)
src/tests/flume/verification/make_doc_figures.py fig_field_loop	draws field-loop.png from <runs>/field-loop (--only field-loop)

cd src/tests/flume/verification/mhd
field-loop/check.sh                                                     # CPU (about 10 min)
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl field-loop/check.sh    # FNL
field-loop/check.sh --agree                                             # CPU against FNL, after both runs
field-loop/check.sh --amr                                               # 2:1 seams (about 28 min)

make_doc_runs.sh <runs> mhd writes the figure run (<runs>/field-loop: $128\times 64$); make_doc_figures.py --runs <runs> draws it.

Results

Magnetic pressure $|B{|}^2/2$ of the loop at the final time.

Check	Bound	CPU	FNL
$⟨|B_z|⟩/A_0$, $N=64/128$	$1.040\cdot {10}^{-3}$ / $6.003\cdot {10}^{-4}$ (baseline + 2%)	$1.02\cdot {10}^{-3}$ / $5.89\cdot {10}^{-4}$	pass
its rate under refinement	$>0$	0.79 (0.20 from 128 to 256)	pass
$E_B(T)/E_B(0)$, $N=64/128$	$\ge 0.87011$ / $0.93544$ (baseline − 0.2%)	0.872 / 0.937	pass
--amr: $⟨|B_z|⟩/A_0$, $N=32/64$	$2.277\cdot {10}^{-3}$ / $1.118\cdot {10}^{-3}$	$2.23\cdot {10}^{-3}$ / $1.10\cdot {10}^{-3}$, rate +1.03	pass
--amr: $E_B(T)/E_B(0)$, $N=32/64$	$\ge 0.80066$ / $0.89888$	0.802 / 0.901	pass
--amr: final seam div(B) / its peak	0.1	0.033 / 0.011	pass
--agree: CPU against FNL, field by field ($\mathbf{B}$ relative to $A_0$)	${10}^{-10}$	—	$\sim {10}^{-14}$ at step 4

The uniform $N=256$ run (26 more minutes, not in the ladder) gives $5.13\cdot {10}^{-4}$ and $E_B$ 0.967. The uniform $32/64$ runs that the --amr ladder is compared with give $1.99\cdot {10}^{-3}$, $1.11\cdot {10}^{-3}$ and $E_B$ 0.759, 0.884. The weno-riemann energies are in MHD on weno-riemann.

Related cases: glm cleaning measures the GLM transport and damping on an exact solution; runtime regrids and accuracy of runtime AMR track this loop with a moving refinement (regression field-loop-regrid); the M6 seam studies G3a and G3b run it across a 2:1 seam under GLM and EGLM.

Findings

• The loop field jumps at $r=R$, so the discrete div(B) there is $\sim A_0/h$ over a ring $\sim h$ wide: the $B_z$ source is $O(1)$ in $h$ and the decrease is slow. No convergence order is expected; the measured level matches the GLM level of Mignone & Tzeferacos ($\sim {10}^{-3}$).
• On AMR the refined half dissipates less ($E_B$ 0.802, 0.901 against 0.759, 0.884 uniform); the seams add $B_z$ at $N=32$ and none at $N=64$. The seam div(B) peaks at the first step (the initial-data error of the loop edge) and GLM removes it; the collocated PRISM seam runs away instead (#29).
• The backends agree to round-off on $\mathbf{B}$ as on the fluid. Until #79 they differed by $\sim {10}^{-6}$ on $\mathbf{B}$: the slow-wave normalisation of the eigenvectors cancelled catastrophically; it is now cancellation-free (numerics).