MHD rotor

ID	Group	Milestone	Backends	Figure	Regression
MV-13	Ideal MHD	M2 (#41)	CPU, FNL	yes	rotor

Case and scope

A dense disc spins in a magnetised ambient gas (Balsara & Spicer 1999; the first rotor of Tóth 2000) and launches torsional Alfvén waves that brake it, compresses the field and drives strong shocks: a stringent 2-D test of positivity and of the multi-dimensional field evolution. There is no exact solution; the oracle checks:

• symmetry: the 180° rotation composed with $\mathbf{B}\to -\mathbf{B}$ (the ambient field is uniform) maps the solution to itself. A symmetry defect of the scheme is at truncation level (${10}^{-3}$) from the first steps; a correct scheme leaves a round-off seed amplified by the flow;
• positivity: the floors are disabled, so a non-positive density or pressure stops the run.

There is no conservation check: the outflow boundaries do not conserve.

Models and numerics

Item	Choice	Code
Equations	ideal MHD, $\gamma =1.4$ ($c_p=3.5$, $c_v=2.5$)	models
Space	WENO5 flux splitting on characteristic variables; weno-riemann under --numerics	compute_residuals_weno, compute_residuals_riemann, adam_flume_cpu_object.F90
Divergence control	GLM, $c_h=6$ (glm_ch_check = error), glm_alpha = 0.18, glm_damping_length = 1.0	mhd_glm_face_split_fluxes, adam_flume_mhd_library.F90; models
Floors	disabled	models
Time	SSP Runge–Kutta, 5 stages (runge-kutta-ssp-54), CFL 0.4	adam_rk_object.F90
Boundaries	zeroth-order extrapolation (outflow)	set_boundary_conditions, adam_flume_cpu_object.F90; boundary conditions
FNL backend	the same kernels on the device	adam_flume_fnl_object.F90

Initial and boundary conditions

Item	Value
Domain	$[0,1{]}^2$, $z$ null
Grid	the vortex grid of V2 (quadtree, $4\times 4$ blocks), ${256}^2$ cells (${64}^2$ per block); a power of two, so the cell centres are exact binary fractions and the initial state is bitwise symmetric (Tóth uses ${200}^2$)
Ranks	2
Initial condition	mhd-rotor: ambient $\rho =1$ at rest, $p=1$, $\mathbf{B}=(5/\sqrt{4\pi },0,0)$; disc $\rho =10$ for $r<0.1$ around $(0.5,0.5)$, rigid rotation at speed 2 at $r=0.1$, linear taper to $r=0.115$
Boundaries	extrapolation on $x$ and $y$
Final time	$t=0.15$ (1152 steps)

Input files

File	Role
src/tests/flume/verification/mhd/rotor/make_rotor.py	writes the input from vortex/vortex-n064.ini (--cells, --cfl, --time-max, --glm-ch, --it-save)
src/tests/flume/verification/mhd/rotor/check.sh	runs the case (--np N, --numerics SPEC, --legs mv13)
src/tests/flume/verification/mhd/orszag-tang/orszag_tang_oracle.py	the symmetry check, --field-parity even (the $\mathbf{B}\to -\mathbf{B}$ composition)
src/tests/flume/verification/make_doc_figures.py fig_rotor	draws rotor.png from <runs>/rotor (--only rotor)

cd src/tests/flume/verification/mhd
rotor/check.sh                                                     # CPU (about 10 min)
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl rotor/check.sh    # FNL
rotor/check.sh --numerics hlld:primitive                           # weno-riemann, HLLD, primitive interpolation

make_doc_runs.sh <runs> mhd writes the figure run (<runs>/rotor: ${256}^2$); make_doc_figures.py --runs <runs> draws it.

Results

Density and magnetic pressure $|B{|}^2/2$ of the rotor.

Check	Bound	CPU	FNL
symmetry (180° rotation with $\mathbf{B}\to -\mathbf{B}$), splitting, $t=0.15$	${10}^{-8}$	$1.8\cdot {10}^{-9}$ (${10}^{-11}$ at step 100)	pass
positivity (floors disabled)	run completes	pass	pass
--numerics: symmetry at step 100	${10}^{-8}$	$3.9\cdot {10}^{-12}$ HLLD primitive, $4.9\cdot {10}^{-10}$ HLLD characteristic	$2.1\cdot {10}^{-12}$, $7.9\cdot {10}^{-10}$
--numerics: symmetry at the end	${10}^{-4}$ (gross failures only)	$3.4\cdot {10}^{-11}$ HLL; $5.4\cdot {10}^{-9}$ HLLD primitive; $7.4\cdot {10}^{-8}$ HLLD primitive without the sensor; $1.1\cdot {10}^{-7}$ HLLD characteristic	$2.3\cdot {10}^{-7}$ and $6.1\cdot {10}^{-6}$ for the two HLLD

$w$ and $B_z$ stay exactly 0 on the CPU.

Related cases: Orszag–Tang shares the symmetry oracle; MHD on weno-riemann explains the two-point symmetry check; multi-realm runs split this rotor on a quadtree.

Findings

• The final symmetry defect grows smoothly from round-off (the time series has no jump, so no switch flips) and orders by the dissipation of the solver and by the backend's round-off: the least dissipative interpolations amplify the seed most, and FNL contracts differently into FMAs.
• A single end-of-run tolerance cannot separate a round-off seed from a defect across solvers, so under --numerics the symmetry is asserted twice: at step 100, before the flow amplifies the seed, within ${10}^{-8}$, and at the end within ${10}^{-4}$.