Orszag–Tang vortex

ID	Group	Milestone	Backends	Figure	Regression
MV-12	Ideal MHD	M2 (#41)	CPU, FNL	yes	orszag-tang

Case and scope

The standard 2-D test of MHD turbulence and shock interactions (Stone et al. 2008, section 8.4): interacting shocks, a current sheet and low-$\beta$ regions from a smooth, divergence-free, doubly periodic start. It has no exact solution, so the oracle checks what the equations guarantee:

• symmetry: the 180° rotation about $(1/2,1/2)$ maps the initial state to itself (scalars even, vectors odd), so the discrete solution must keep that symmetry. A wrong ghost fill or a sign error gives a truncation-level defect ($\sim {10}^{-3}$); a correct scheme leaves a round-off seed amplified by the flow;
• conservation: the periodic box conserves the integrals of $\rho$, $\rho \mathbf{u}$, $E$ and $\mathbf{B}$;
• positivity: the floors are disabled, so a non-positive density or pressure stops the run; a completed run has zero floored cells.

With --amr the centre box is refined once, so 2:1 seams are crossed by the shocks: symmetry and conservation must hold there too, and the seam must not be a larger div(B) source than the flow features themselves.

Models and numerics

Item	Choice	Code
Equations	ideal MHD, $\gamma =5/3$	models
Space	WENO5 flux splitting on characteristic variables	compute_residuals_weno, adam_flume_cpu_object.F90; mhd_glm_face_split_fluxes, adam_flume_mhd_library.F90
Divergence control	GLM, $c_h=4$ (above the fastest signal, glm_ch_check = error), glm_alpha = 0.18, glm_damping_length = 1.0	models
Floors	disabled	models
Time	SSP Runge–Kutta, 5 stages (runge-kutta-ssp-54), CFL 0.4	adam_rk_object.F90
AMR (--amr)	geometric primitive-box marker at initialisation, no regrid; 2:1 restriction, tricubic ghost fill, reflux	interp_seam_ghost, adam_field_object.F90; apply_reflux_corrections, adam_forest_object.F90; numerics
Diagnostics	conservation history every step; div(B) history (seam-local column on AMR)	compute_conservation, 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	$[0,1{]}^2$, $z$ null
Grid, uniform	the vortex grid of V2 (quadtree, $4\times 4$ blocks), ${128}^2$ cells (${32}^2$ per block)
Grid, --amr	${32}^2$ base ($8^2$ per block), the blocks whose centroid lies in $[0.25,0.75{]}^2$ refined once (2:1); octree with null $z$ and nk = 4 (mhd/amr_box.py); plus the uniform ${32}^2$ run as reference
Ranks	2
Initial condition	orszag-tang, evaluated as odd functions of $2x-1$, $2y-1$ (bitwise symmetric)
Boundaries	periodic on $x$ and $y$
Final time	$t=0.5$ (1281 steps at ${128}^2$)

Input files

File	Role
src/tests/flume/verification/mhd/orszag-tang/make_orszag_tang.py	writes the input from vortex/vortex-n064.ini (--cells, --refine-box, --ratio, --nk, --divergence-control, --it-save)
src/tests/flume/verification/mhd/amr_box.py	refines a box by one 2:1 level at initialisation
src/tests/flume/verification/mhd/orszag-tang/check.sh	runs the uniform leg, or --amr, or --divergence-control eglm (--numerics SPEC, --legs mv12)
src/tests/flume/verification/mhd/orszag-tang/orszag_tang_oracle.py	the 180° symmetry, the conservation, the positivity
src/tests/flume/verification/mhd/seam_divb_oracle.py	the seam div(B) against the uniform reference (--amr)
src/tests/flume/verification/make_doc_figures.py fig_orszag_tang	draws orszag-tang.png and orszag-tang-amr.png from <runs>/orszag-tang and <runs>/orszag-tang-amr (--only orszag-tang)

cd src/tests/flume/verification/mhd
orszag-tang/check.sh                                                     # uniform 128^2, CPU (about 4.5 min)
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl orszag-tang/check.sh    # FNL
orszag-tang/check.sh --amr                                               # 2:1 seams (about 6 min)

make_doc_runs.sh <runs> mhd writes the figure runs (<runs>/orszag-tang: ${128}^2$; <runs>/orszag-tang-amr: ${32}^2$ with $[0.25,0.75{]}^2$ refined); make_doc_figures.py --runs <runs> draws it.

Results

Density and pressure at $t=0.5$, ${128}^2$.

Density on the ${32}^2$ base with the centre box refined; the block outlines show the 2:1 seams the shocks cross.

Check	Bound	CPU	FNL
180° rotational symmetry at $t=0.5$, relative defect	${10}^{-8}$	$1.3\cdot {10}^{-9}$	pass
integrals of $\rho$, $\rho \mathbf{u}$, $E$, $\mathbf{B}$, relative drift	${10}^{-13}$	$1.2\cdot {10}^{-14}$	pass
positivity (floors disabled)	run completes	pass	pass
--amr: symmetry	${10}^{-8}$	$9.1\cdot {10}^{-13}$	pass
--amr: conservation	${10}^{-13}$	$1.1\cdot {10}^{-16}$	pass
--amr: peak seam $max|\mathrm{\nabla }\cdot \mathbf{B}|$ / peak $max|\mathrm{\nabla }\cdot \mathbf{B}|$ of the uniform ${32}^2$ run	1.0	0.76 ($0.887/1.170$)	pass

The symmetry defect of the uniform run grows from $1.7\cdot {10}^{-13}$ at step 100 to $3\cdot {10}^{-11}$ at step 300, when the shocks form, and $1.3\cdot {10}^{-9}$ at $t=0.5$ (CPU, M2-P6a). $w$ and $B_z$ stay exactly 0 on the CPU; on FNL they carry round-off noise ($\sim {10}^{-12}$ of the in-plane field), which the oracle scales by the vector magnitude. FNL ran every leg; the tables record the CPU values.

Related cases: rotor uses the same symmetry oracle; EGLM cleaning runs this problem under EGLM (EV-4); quadtree AMR runs the --amr problem on three trees; MHD on weno-riemann runs it on HLLD.

Findings

• The initial state is bitwise symmetric and the scheme is symmetric in exact arithmetic, but its left- and right-biased reconstructions round differently: the defect is a round-off seed amplified by the flow, four to five decades below what a wrong fill or a sign error produces.
• On AMR the refined box is symmetric about the centre and the reflux keeps the integrals: symmetry ($9.1\cdot {10}^{-13}$) and conservation ($1.1\cdot {10}^{-16}$) hold across the seams.
• The div(B) beside the seams stays below the uniform-grid peak: the shocks, not the seams, set the div(B) maximum.
• The AMR variant runs on an octree with null $z$ and nk = 4 because, before #46, a quadtree seam produced a spurious $z$ dependence; the comparison with a quadtree is MV-15.