Order of accuracy: linear waves, CPAW, magnetised vortex

ID	Group	Milestone	Backends	Figure	Regression
MV-5, MV-6, MV-7	Ideal MHD	M2 (#41)	CPU, FNL	yes	—

Case and scope

Three smooth problems whose exact solution after one period (or one crossing) is the initial state, so every smooth-flow ingredient (eigen-decomposition, WENO5 characteristic reconstruction, fluxes, periodic ghost fill, time integration) is measured against an exact answer:

• MV-5, linear waves (Stone et al. 2008, section 8.2): a small-amplitude eigenmode per family (fast, Alfvén, slow, entropy), 1-D along $x$ and 2-D inclined. It catches a wrong eigenvector or a family-specific loss of order.
• MV-6, circularly polarised Alfvén wave (Tóth 2000): an exact nonlinear solution at any amplitude. At $B_{\perp }=0.1$ it exercises the WENO nonlinear weights, which the ${10}^{-7}$ linear waves leave at their linear values. Left and right polarisations are mirror images under $z\to -z$, so their errors must be equal bitwise (rounding is symmetric under a sign change).
• MV-7, magnetised vortex (Balsara 2004): a steady 2-D equilibrium of pressure, centrifugal force, magnetic tension and magnetic pressure convected by the free stream, with every variable varying in both directions and a divergence-free field. It checks the $L_1$ and $L_{\mathrm{\infty }}$ order and that $max|\mathrm{\nabla }\cdot \mathbf{B}|$ stays at truncation level.

Models and numerics

Item	Choice	Code
Equations	ideal MHD, $\gamma =5/3$	models
Space	WENO5 flux splitting on characteristic variables (all three); weno-riemann HLLD under --numerics (MV-5, MV-6)	mhd_face_split_fluxes, mhd_eigenvectors, adam_flume_mhd_library.F90; numerics
Divergence control	MV-5, MV-6: none (MV-5 also GLM or EGLM with --divergence-control, $c_h=1.1(|u_n|+c_f)$, no damping); MV-7: GLM, $c_h=3$, glm_alpha = 0.18, glm_damping_length = 1.0	models
div(B) operator (MV-7)	second-order centred difference (fdv_order = 2)	compute_divb_history, adam_flume_cpu_object.F90
Time	SSP Runge–Kutta, 5 stages order 4 (runge-kutta-ssp-54); CFL 0.1 (MV-5, MV-6: the spatial error dominates), 0.4 (MV-7: errors equal to 4–5 digits at 0.1)	adam_rk_object.F90; numerics
Boundaries	periodic	boundary conditions
FNL backend	the same kernels on the device	adam_flume_fnl_object.F90

Initial and boundary conditions

Leg	Domain and grid	Initial condition	Final time
MV-5, 1-D	$[0,1]$ along $x$, the Sod grid of V1 (4 blocks along $x$), $N=16,32$ cells per wavelength (the figure adds 64)	mhd-linear-wave: $\rho =1$, $p=3/5$, $\mathbf{B}=(1,\sqrt{2},1/2)$ in the wave frame ($c_f=2$, $c_a=1$, $c_s=1/2$), $u_n=1$ for the entropy wave, amplitude ${10}^{-7}$, wavelength 1	one period, $1/|u_n+c|$
MV-5, 2-D	$[0,\sqrt{5}]\times [0,\sqrt{5}/2]$, the vortex grid of V2 (quadtree, $4\times 4$ blocks), $N\times N/2$ square cells, $N=32,64$; wave at $\tan a=2$ ($a=63.43°$), one wavelength along each axis	the same	one period
MV-6	the MV-5 2-D grid, $N=32,64,128$	mhd-cpaw: $\rho =1$, $p=0.1$, $B_{\parallel }=1$, $v_{\parallel }=0$, $B_{\perp }=0.1$, wavelength 1 (Alfvén speed and period 1); right and left polarisations	$t=1$
MV-7	$[-7,7{]}^2$ (tails $e^{-24}$ at the box edge), the V2 grid, $N^2$ cells, $N=64,128,256$	mhd-vortex: $\rho =1$, $p=1$, $\mathbf{v}=(1,1,0)$, $\mathbf{B}=0$ background, $\kappa =\mu =1$, radius 1	one crossing, $t=14$

All legs: 2 ranks, periodic on the active axes.

Input files

File	Role
src/tests/flume/verification/mhd/linear-wave/make_linear_wave.py	writes the MV-5 input (--wave, --geometry 1d|2d, --cells, --amplitude, --divergence-control)
src/tests/flume/verification/mhd/linear-wave/check.sh, linear_wave_oracle.py	MV-5: the error $ϵ$ of Stone et al. (last against first checkpoint), the bounds and the order (--numerics, --divergence-control, --legs LIST: mv5-1d, mv5-2d)
src/tests/flume/verification/mhd/cpaw/make_cpaw.py	writes the MV-6 input (--polarisation, --geometry, --cells)
src/tests/flume/verification/mhd/cpaw/check.sh, cpaw_oracle.py	MV-6: bounds, order, left = right (--numerics, --legs mv6)
src/tests/flume/verification/mhd/vortex/make_mhd_vortex.py	writes the MV-7 input (--cells, --half-width, --cfl)
src/tests/flume/verification/mhd/vortex/check.sh, mhd_vortex_oracle.py	MV-7: $L_1$, $L_{\mathrm{\infty }}$, orders, div(B), initial checkpoint against the exact vortex (--legs mv7)
src/tests/flume/verification/make_doc_runs.sh group order	the 1-D ladders 16/32/64 on the two schemes (24 runs)
src/tests/flume/verification/make_doc_figures.py fig_order	draws linear-wave-order.png (--only order)

cd src/tests/flume/verification/mhd
linear-wave/check.sh                       # MV-5
cpaw/check.sh                              # MV-6
vortex/check.sh                            # MV-7 (about 40 min on the CPU, 31 of them at N = 256)
linear-wave/check.sh --numerics hlld       # MV-5 on weno-riemann

Results

The 1-D linear waves on 16, 32 and 64 cells per wavelength, splitting scheme and weno-riemann HLLD; the legend gives the orders of the two pairs.

Test	Bound	CPU	FNL
MV-5, every family, observed order	$\ge 4.7$ (design 5 minus 0.3); 4.8 under hlld:primitive/hlld:characteristic	about 5: 4.96–4.97 from 16 to 32 cells (1-D); 4.96–5.02 on the asserted ladders	pass
MV-5, error of every run	per run, CPU baseline + 2% (e.g. 1-D fast $1.512\cdot {10}^{-10}$ / $4.843\cdot {10}^{-12}$ at 16 / 32)	below	pass
MV-6, observed order on 32/64/128	$\ge 4.7$ (4.8 under the HLLD specs)	4.89 and 4.99	pass
MV-6, $ϵ$ at $N=32/64/128$	$1.049\cdot {10}^{-3}$ / $3.537\cdot {10}^{-5}$ / $1.112\cdot {10}^{-6}$	below	pass
MV-6, left against right polarisation	bitwise (LR_TOL = 0)	bitwise	bitwise
MV-7, $L_1$ at $N=64/128/256$	$4.327\cdot {10}^{-4}$ / $2.224\cdot {10}^{-5}$ / $6.900\cdot {10}^{-7}$	below	pass
MV-7, $L_{\mathrm{\infty }}$ at $N=64/128/256$	$7.812\cdot {10}^{-3}$ / $5.194\cdot {10}^{-4}$ / $2.439\cdot {10}^{-5}$	below	pass
MV-7, $L_1$ order (finest pair)	$\ge 4.7$	5.01	pass
MV-7, $L_{\mathrm{\infty }}$ order (finest pair)	$\ge 4.0$	4.41 (pre-asymptotic: 3.41, 3.91, 4.41 on 32/64, 64/128, 128/256)	pass
MV-7, $max|\mathrm{\nabla }\cdot \mathbf{B}|$ convergence	order $\ge 1.7$ (second-order operator)	truncation level	pass
MV-7, initial checkpoint against the exact vortex	${10}^{-14}$	pass	pass

FNL ran every ladder (work directories of both backends); the tables record the CPU values, and the backends agree to the printed digits.

Related cases: isentropic vortex is the Euler order test on the same grid; MHD on weno-riemann holds the orders of the HLLD runs; EGLM cleaning repeats MV-5 under EGLM; order across seams measures the order across 2:1 seams.

Findings

• Every linear-wave family converges at 4.96–4.97 from 16 to 32 cells. At 64 cells the error is ${10}^{-13}$, six decades below the amplitude, and the round-off floor lowers the slow and entropy orders to 4.85: MV-5 asserts on the 16/32 pair (1-D) and on 32/64 (2-D) for that reason.
• The amplitude is ${10}^{-7}$, not the ${10}^{-6}$ of Stone et al.: at ${10}^{-6}$ the $O(A^2)$ nonlinear error of the mode ($ϵ$ scales as $A^2$ at $N=128$: $3.26\cdot {10}^{-10}$ at $A={10}^{-5}$, $3.26\cdot {10}^{-12}$ at ${10}^{-6}$) floors a fifth-order scheme already at $N=64$–128, and below about ${10}^{-8}$ the accumulated round-off floors it ($1.5\cdot {10}^{-14}$ at $A={10}^{-8}$). At 2-D $N=128$ the slow and Alfvén waves reach the floor (orders 2.6 and 4.3 on 64/128).
• With GLM or EGLM on MV-5, $c_h$ sets the time step, so the errors differ from the none baseline (2-D slow wave 2.4×).
• MV-7 runs GLM because with none the discrete div(B) of the vortex grows exponentially at a rate proportional to $1/h$: the run stops on a negative pressure at $t=13.4$, 8.2, 4.3 for $N=64$, 128, 256, whatever the CFL.
• The MV-7 $L_{\mathrm{\infty }}$ order is still pre-asymptotic at $N=256$ ($N=512$ costs 2.5 h on the CPU); 32 cells is pre-asymptotic in $L_1$ too (order 3.65), so the ladder starts at 64.