Ohmic heating and the divergence constraint

ID	Group	Milestone	Backends	Figure	Regression
VV-6	Dissipative terms (M4)	M4 (#65)	CPU, FNL	no	—

Case and scope

The diagonal Ohmic wave of VV-5 with GLM and with EGLM, judged by its energy budget and by its divergence instead of its error.

What the case catches:

• a wrong Ohmic energy flux: $G_d^E=\sum _i{B}_i{G}_d^{B_i}$ must return to the gas, as heat, exactly the energy the field loses. A wrong (or wrong-signed) energy flux leaves the field equations right and shows only in this budget; the total energy, conserved by the flux form whatever the fluxes, would not see it either;
• a resistive flux that feeds $\mathrm{\nabla }\cdot \mathbf{B}$: the curl form must not raise the constraint GLM and EGLM are there to keep.

Models and numerics

Item	Choice	Code
Equations	ideal MHD (mhd-ideal), $\gamma =1.4$, $\eta =0.01$, $\mu =0$	models
Divergence control	GLM (mhd-glm) and EGLM (mhd-eglm), $c_h=3$, $\alpha =0$	GLM, EGLM; adam_flume_cpu_mhd_glm_kernels.F90, adam_flume_cpu_mhd_eglm_kernels.F90
Ohmic flux	curl form with its energy flux, dissipative_order = 4	add_resistive_fluxes_o4, adam_flume_cpu_dissipation_kernels.F90; compute_resistive_flux, adam_flume_dissipation_library.F90; numerics
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 of ${12}^2$ cells (${48}^2$), uniform
Ranks	2
Initial condition	sine-wave, magnetic mode at 45°, $A={10}^{-4}$ (the budget compares energies of $O(A^2)$ with differences of the internal energy, so it needs the signal), ${\rho }_0=p_0=1$, wavelength $1/\sqrt{2}$
Boundaries	periodic on the $x$ and $y$ faces, extrapolation on the $z$ faces
Final time	$t=0.2$
Runs	mhd-glm and mhd-eglm, each with its ideal twin ($\eta =0$)

Input files

File	Role
src/tests/flume/verification/viscous/waves.py	make writes the inputs; budget judges the energy budget and $\mathrm{\nabla }\cdot \mathbf{B}$ against the ideal twin
src/tests/flume/verification/viscous/check.sh --legs vv6	runs the four runs (--np N; FLUME_EXE selects the backend)

cd src/tests/flume/verification/viscous && ./check.sh --legs vv6             # CPU
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl ./check.sh --legs vv6       # FNL

No figure (#85 lists none for VV-6).

Results

Oracle (waves.py budget):

• heat gained over field and kinetic energy lost within ${10}^{-3}$ of 1;
• the drift of the total energy at most ${10}^{-13}$ of the total;
• $max|\mathrm{\nabla }\cdot \mathbf{B}|$ (central differences) at most 1.05 times the ideal twin's plus ${10}^{-14}$.

Check	Bound	CPU	FNL
field energy lost (of $2.5\cdot {10}^{-9}$)	—	$6.770374\cdot {10}^{-10}$	the same
internal energy gained	—	$6.770375\cdot {10}^{-10}$	$6.770384\cdot {10}^{-10}$
heat / (field + kinetic loss)	$1\pm {10}^{-3}$	1.0000002	1.0000015
total energy drift, relative	${10}^{-13}$	$1.8\cdot {10}^{-16}$	$1.8\cdot {10}^{-16}$
$max|\mathrm{\nabla }\cdot \mathbf{B}|$, resistive / ideal (GLM; EGLM alike)	$1.05\times$ ideal $+{10}^{-14}$	$2.1\cdot {10}^{-18}$ / $2.6\cdot {10}^{-18}$	$1.6\cdot {10}^{-18}$ / $2.3\cdot {10}^{-18}$ (GLM)

Both models give the same budget.

Related cases: the wave itself is the diagonal Ohmic leg of VV-5; the divergence at 2:1 seams is exercised by VV-7 and VV-8.

Findings

• The departures of the heat ratio from 1 are round-off: the internal energy is the total minus the kinetic and magnetic parts, and a difference of $6.8\cdot {10}^{-10}$ out of a total of 2.5 keeps six digits.
• What the divergence leg does not show. On a uniform periodic grid the diagonal field is discretely solenoidal from the start, and the curl-form flux keeps it so; both runs sit at round-off, so the leg checks only that the resistive flux adds no divergence there. The test with teeth is the 2:1 AMR seam, where the discrete operators no longer commute (PRISM's #29 floor): it belongs to M4 P4 (VV-7, VV-8).