Ohmic decay and the visco-resistive Alfvén wave

ID	Group	Milestone	Backends	Figure	Regression
VV-5	Dissipative terms (M4)	M4 (#65)	CPU, FNL	yes	alfven-resistive

Case and scope

A transverse field $b_t=A\sin (\mathbf{k}\cdot \mathbf{x})$ on a gas at rest decays by Ohmic diffusion at the rate $\eta k^2$; with a normal field $B_n$ the same mode is an Alfvén wave damped by viscosity and resistivity together. The oracle (viscous/waves.py) integrates the $2\times 2$ linear system for $(v_t,b_t)$ exactly; a fourth leg reruns the acoustic wave of VV-2 on the MHD model with $\mathbf{B}=0$.

What the case catches. The Ohmic flux has its own kernels, a curl form whose cross terms an axis-aligned wave never reads, and a sign that the plan in #65 printed wrong (an anti-diffusion; see numerics). Each defect is a number the exact linear mode pins:

• a sign error grows the field;
• a missing cross term changes the diagonal rate;
• a coefficient error changes every rate;
• a wrong temperature on MHD (a wrong slot, or a wrong $T$ read from the MHD auxiliary state by the viscous and heat fluxes) shows on the acoustic leg on MHD and nowhere else: the constant laws of the Alfvén leg never read $T$.

Models and numerics

Item	Choice	Code
Equations	ideal MHD (mhd-ideal), $\gamma =1.4$, with resistivity $\eta$ and, on two legs, viscosity and conductivity	models
Divergence control	none (mhd-none) on the Ohmic and acoustic legs; GLM, $c_h=3$, $\alpha =0$ (mhd-glm) on the Alfvén leg	GLM; adam_flume_cpu_mhd_glm_kernels.F90
Ohmic flux	curl form, dissipative_order 4 or 2	add_resistive_fluxes_o4, add_resistive_fluxes_o2, adam_flume_cpu_dissipation_kernels.F90; compute_resistive_flux, adam_flume_dissipation_library.F90; numerics
Viscous and heat fluxes	as in VV-1, the temperature from the MHD auxiliary state	add_dissipative_fluxes_o4, adam_flume_cpu_dissipation_kernels.F90
Inviscid fluxes	WENO5 flux splitting, characteristic variables	compute_residuals_weno, adam_flume_cpu_object.F90
Time	runge-kutta-ssp-54, CFL 0.2	adam_rk_object.F90
FNL backend	the same fluxes as device kernels	add_resistive_fluxes_o4_dev, adam_flume_fnl_dissipation_kernels.F90

Initial and boundary conditions

Item	Value
Domain	$[0,1{]}^2$, $z$ null
Grid	quadtree, $4\times 4$ blocks, $N/4$ cells per block along the active axes; an axis-aligned magnetic wave keeps the transverse axis active and thin (4 blocks of 6 cells), the acoustic wave on MHD a null transverse axis (1 cell)
Ranks	2
Initial condition	sine-wave in its magnetic mode ($b_t=A\sin (\mathbf{k}\cdot \mathbf{x})$, $A={10}^{-5}$) or acoustic mode, ${\rho }_0=p_0=1$, $u=0$; wavelength 1 along an axis, $1/\sqrt{2}$ along the diagonal
Boundaries	periodic on the $x$ and $y$ faces, extrapolation on the $z$ faces

Leg	Model	Direction	dissipative_order	Coefficients	$\mathbf{B}$ background	$N$	$t$
Ohmic x	mhd-none	$x$	4 and 2	$\eta =0.01$	0	32 / 64 / 128	0.5
Ohmic diagonal	mhd-none	45° (the cross terms ${\partial }_i{B}_d$ of the curl form)	4	$\eta =0.01$	0	24 / 48 / 96	0.1
Alfvén x	mhd-glm	$x$	4	$\mu =0.01$, $\eta =0.005$	$B_n=1$	32 / 64 / 128	0.5
acoustic on MHD	mhd-none	$x$	4	$\mu =0.01$, $k=0.02$; odd part of the $\pm A$ twins	0	32 / 64 / 128	0.5

The rates the oracle holds: Ohmic $\eta k^2$; Alfvén damping $(\nu +\eta )k^2/2$.

Input files

File	Role
src/tests/flume/verification/viscous/waves.py	make --mode magnetic writes each input; oracle integrates the linear $(v_t,b_t)$ system exactly
src/tests/flume/verification/viscous/check.sh --legs vv5	runs the five ladders (--np N; FLUME_EXE selects the backend)
src/tests/flume/regression/alfven-resistive/input.ini	the frozen visco-resistive Alfvén wave at 45°, MHD-GLM, $B_n=1$, $\mu =0.01$, $\eta =0.005$, ${32}^2$, $t=0.05$
make_doc_figures.py (fig_viscous, fig_viscous_ladders)	the figures, from the CPU and FNL work directories of viscous/check.sh

cd src/tests/flume/verification/viscous && ./check.sh --legs vv5             # CPU
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl ./check.sh --legs vv5       # FNL

Results

Middle panel: the five VV-5 ladders, each normalised by its coarsest error, against the order-2 and order-4 slopes.

Centre: the VV-5 ladders as absolute RMS errors, CPU (filled, solid) and FNL (open, dotted), which coincide. Right: the decay against the exact rate, $|A_{\mathrm{measured}}/A_{\mathrm{exact}}-1|$, converging at order 4 (order 2 for the order-2 flux).

Oracle: as VV-1, the observed order of the finest pair $\ge 3.8$ at order 4, $\ge 1.9$ at order 2.

Leg	N	RMS error (CPU)	Order (CPU)	FNL
Ohmic x, order 4	32 / 64 / 128	$3.64\cdot {10}^{-11}$ / $2.29\cdot {10}^{-12}$ / $1.43\cdot {10}^{-13}$	+3.99, +4.00	the same
Ohmic x, order 2	32 / 64 / 128	$3.68\cdot {10}^{-9}$ / $9.20\cdot {10}^{-10}$ / $2.30\cdot {10}^{-10}$	+2.00, +2.00	the same
Ohmic diagonal, order 4	24 / 48 / 96	$1.14\cdot {10}^{-10}$ / $7.19\cdot {10}^{-12}$ / $4.50\cdot {10}^{-13}$	+3.99, +4.00	the same
Alfvén x, order 4	32 / 64 / 128	$6.37\cdot {10}^{-11}$ / $1.10\cdot {10}^{-12}$ / $2.18\cdot {10}^{-14}$	+5.85, +5.66	the same
acoustic on MHD, order 4	32 / 64 / 128	$8.20\cdot {10}^{-11}$ / $1.69\cdot {10}^{-12}$ / $2.19\cdot {10}^{-14}$	+5.60, +6.27	$2.21\cdot {10}^{-14}$ at 128 cells, +6.26

CPU and FNL on 2 ranks, identical to the printed digits except the round-off of the acoustic leg's odd part.

Quantity	Measured	Reference
Ohmic decay	$0.3947841$	$\eta k^2=0.3947842$
Alfvén damping	$0.29610$	$(\nu +\eta )k^2/2=0.29609$
acoustic amplitude on MHD, 32 cells (CPU)	$8.6252132315\cdot {10}^{-6}$	VV-2 on Euler, CPU: $8.6252132316\cdot {10}^{-6}$ (FNL: $8.6252132321\cdot {10}^{-6}$)

Related cases: the same oracle on Euler is VV-1 and VV-2; the energy budget and the divergence of the same resistive flux with GLM and EGLM are in VV-6; the diagonal Ohmic wave crosses 2:1 seams in VV-8.

Findings

• The measured Ohmic rate is the derived one: the sign printed in the #65 plan was wrong, the code's is right.
• Along $x$ the Ohmic decay of $b_y$ is the same equation as the shear wave's $u_y$ in VV-1, and its errors are the VV-1 errors scaled by the amplitude, to three digits: two kernels, one stencil.
• The Alfvén wave's error is dominated by the inviscid WENO part, as on the acoustic wave, hence the order above four.
• The acoustic wave on MHD reproduces VV-2 on Euler: the measured amplitude agrees to ten digits at 32 cells; the RMS errors agree to three, the inviscid parts of the two models being different kernels.
• The diagonal runs without divergence control. With GLM or EGLM the same ladder measured +7.65 then +1.72, and +3.32 from 96 to 192 cells. The cause is the ideal scheme: run with $\eta =0$, where the exact field is static, GLM's upwinding ($c_h=3$) damps it at fifth order ($8.5\cdot {10}^{-12}$ at 48 cells, $2.7\cdot {10}^{-13}$ at 96), with the opposite sign to the Ohmic error and as large, so the two cancel near 48 cells and a 24/48/96 ladder is not asymptotic. Without divergence control the $\eta =0$ field stays at round-off. A smaller time step (CFL 0.05) changed no printed digit, so neither error is temporal. GLM and EGLM with resistivity are covered by VV-6.
• The magnetic amplitude is ${10}^{-5}$: the wave's magnetic pressure drives an $O(A^2)$ flow that advects it at $O(A^3)$, odd in $A$ (no twin cancels it), relative $A^2={10}^{-10}$ here.
• #85 asks for the decay against the exact rate as a figure; the published figure shows the error ladders only.