EGLM cleaning

ID	Group	Milestone	Backends	Figure	Regression
EV-2, EV-3, EV-4	Ideal MHD	M3 (#47, M3-P4)	CPU, FNL	yes	orszag-tang-eglm

Case and scope

The energy-consistent cleaning (divergence_control = eglm, Derigs et al. 2018) puts ${\psi }^2/2$ in the total energy and adds nonconservative sources,

$$-(\mathrm{\nabla }\cdot \mathbf{B})(0,\mathbf{B},\mathbf{u}\cdot \mathbf{B},\mathbf{u},0)-(\mathbf{u}\cdot \mathrm{\nabla }\psi )(0,0,\psi ,0,1)$$

in the order $(\rho ,\rho \mathbf{u},E,\mathbf{B},\psi )$. On a periodic box only the density integral is then conserved; the momentum, $\mathbf{B}$ and energy integrals move by the time integral of their sources. The legs run through the --divergence-control eglm option of the glm-pulse, Orszag–Tang and linear-wave scripts, and rj2a/check.sh compares EGLM with GLM:

• EV-2: the GLM pulse against the telegraph solution, which EGLM shares with $\psi$ in field units (the oracle compares $c_h\psi$). It catches a wrong EGLM cleaning block or damping;
• EV-3: Ryu–Jones 2a, EGLM against GLM. With $\psi =0$ the EGLM energy coupling and sources vanish, so the runs must agree bitwise on the 8 shared variables: it catches a source or an energy term that does not vanish;
• EV-4: Orszag–Tang, the drifts bounded by their sources, and the 180° symmetry. It catches a source with a wrong magnitude or sign;
• MV-5 under EGLM: the linear-wave orders;
• the Balsara–Spicer blast without limiter and floors, the case EGLM was expected to help.

Models and numerics

Item	Choice	Code
Equations	ideal MHD with EGLM, $\psi$ in field units, ${\psi }^2/2$ in $E$	models; mhd_eglm_conservative_to_auxiliary, mhd_eglm_eigenvectors, mhd_eglm_flux, adam_flume_mhd_library.F90
Space, splitting	WENO5 flux splitting on characteristic variables	mhd_eglm_face_split_fluxes, adam_flume_mhd_library.F90
Space, weno-riemann	HLLD, characteristic interpolation (and primitive on the blast)	mhd_eglm_riemann_hlld, adam_flume_mhd_riemann_library.F90
Sources and damping	the nonconservative sources above; the damped cleaning energy becomes heat	add_eglm_sources, add_glm_damping, adam_flume_cpu_mhd_eglm_kernels.F90
Time, boundaries	as in each verification (SSP-33 for the pulse and Ryu–Jones 2a, SSP-54 for Orszag–Tang, the waves and the blast)	adam_rk_object.F90
FNL backend	the same kernels on the device	adam_flume_fnl_mhd_eglm_kernels.F90, adam_flume_fnl_object.F90

Initial and boundary conditions

Leg	Setup
EV-2	the MV-3 legs of GLM cleaning (d'Alembert and damped at $N=64,128,256$, min-cell, walls) at pulse amplitude ${10}^{-3}$ instead of 0.1
EV-3	Ryu–Jones 2a, $N=256$, $x$, glm_ch = 2.0, glm_alpha = 0.18
EV-4	Orszag–Tang, ${128}^2$, $t=0.5$, a checkpoint every 25 steps (the oracle integrates the $\psi$ source from them)
MV-5 under EGLM	the MV-5 ladders of order of accuracy, $c_h=1.1(|u_n|+c_f)$, no damping
Blast	Balsara–Spicer, $\beta =2.5\cdot {10}^{-4}$, ${128}^2$ (the positivity limiter PV-1 setup), no limiter, no floors

Input files

File	Role
src/tests/flume/verification/mhd/glm-pulse/check.sh --divergence-control eglm	EV-2
src/tests/flume/verification/mhd/rj2a/check.sh	EV-3 (the EGLM leg ev3, which needs mv4, is part of the default run)
src/tests/flume/verification/mhd/orszag-tang/check.sh --divergence-control eglm	EV-4 and the symmetry
src/tests/flume/verification/mhd/eglm_conservation_oracle.py	the EV-4 drift bounds
src/tests/flume/verification/mhd/linear-wave/check.sh --divergence-control eglm	MV-5 under EGLM
src/tests/flume/verification/mhd/positivity-probe/make_blast.py --eglm	the blast input
src/tests/flume/verification/make_doc_figures.py fig_eglm_energy	draws eglm-energy.png from the conservation histories of the orszag-tang work directories (--only eglm-energy)

cd src/tests/flume/verification/mhd
glm-pulse/check.sh --divergence-control eglm                                  # EV-2
orszag-tang/check.sh --divergence-control eglm [--numerics hlld]              # EV-4, symmetry
linear-wave/check.sh --divergence-control eglm [--numerics hlld]              # MV-5 under EGLM
rj2a/check.sh [--numerics hlld]                                               # EV-3 (EGLM leg)

Results

The total energy of the periodic Orszag–Tang box, ${128}^2$: GLM keeps it to round-off; EGLM changes it through its nonconservative sources, by $3.0\cdot {10}^{-4}$ (splitting) and $4.4\cdot {10}^{-4}$ (HLLD) at $t=0.5$, within the EV-4 bound.

Measured in issue #47, M3-P4; CPU and FNL agree to the printed digits unless a cell says otherwise:

Check	Bound	Splitting	HLLD characteristic
EV-2, pulse against the telegraph solution (amplitude ${10}^{-3}$): $L_1$ at 256 cells, d'Alembert / damped	$6.484\cdot {10}^{-9}$ / $1.406\cdot {10}^{-9}$ (CPU + 2%)	$6.36\cdot {10}^{-9}$ / $1.38\cdot {10}^{-9}$	—
EV-2, $\int B_x$ drift	${10}^{-14}$	$3.7\cdot {10}^{-16}$ ($N=128$)	—
EV-3, Ryu–Jones 2a, EGLM against GLM	bitwise (splitting); ${10}^{-11}$ (--numerics)	bitwise, $\psi =0$	$6.3\cdot {10}^{-13}$ (FNL: bitwise)
EV-4, Orszag–Tang: $\int \rho$ drift; $\int E$ drift / its source bound	round-off; $\le 1$	$3\cdot {10}^{-17}$; $2.8\cdot {10}^{-3}$	0; $7.0\cdot {10}^{-3}$
Orszag–Tang symmetry at step 100 / at $t=0.5$	${10}^{-8}$ / ${10}^{-4}$	$5.4\cdot {10}^{-13}$ / $4.3\cdot {10}^{-7}$	$1.2\cdot {10}^{-12}$ / $1.1\cdot {10}^{-11}$
linear waves, observed order	$\ge 4.7$	4.96–4.98	4.96–4.97
Balsara–Spicer blast ($\beta =2.5\cdot {10}^{-4}$, ${128}^2$, no limiter, no floors)	—	non-positive pressure at step 3	reaches $t=0.01$ (396 steps)

The EV-2 legs at 64 and 128 cells, min-cell and walls have their own bounds (CPU M3-P4c + 2%: $6.686\cdot {10}^{-7}$, $5.789\cdot {10}^{-8}$; damped $1.487\cdot {10}^{-7}$, $1.266\cdot {10}^{-8}$; min-cell$1.201\cdot {10}^{-9}$; walls $9.021\cdot {10}^{-8}$). The linear-wave bounds under EGLM are per run, for the splitting scheme and for hlld:characteristic.

Related cases: GLM cleaning is the GLM version of EV-2; Orszag–Tang the GLM version of EV-4; positivity limiter makes the splitting scheme run the blast; G3b handles the EGLM reflux at 2:1 seams.

Findings

• EV-2 runs at amplitude ${10}^{-3}$ because EGLM couples the $(B_x,\psi )$ pair to the fluid: the damped cleaning energy becomes heat, the heat drives a flow, and the source $-(\mathrm{\nabla }\cdot \mathbf{B})\mathbf{u}$ moves $B_x$. At the GLM amplitude 0.1 this floors the damped legs at a relative $L_1$ of $8\cdot {10}^{-5}$ (against $7\cdot {10}^{-6}$ for GLM); at ${10}^{-3}$ EGLM equals GLM to 3 digits (relative $6.6\cdot {10}^{-6}$ both, on the damped leg at 256), so the telegraph oracle is exact for EGLM there. The $\int B_x$ drift scales as the amplitude cubed.
• EV-4 asserts $\int \rho$ to round-off and bounds every other drift, at every step, by twice the time integral of its source magnitude: $max|\mathbf{B}|\int \parallel \mathrm{\nabla }\cdot \mathbf{B}{\parallel }_1$ (momentum), $max|\mathbf{u}|\int \parallel \mathrm{\nabla }\cdot \mathbf{B}{\parallel }_1$ ($\mathbf{B}$), $max|\mathbf{u}\cdot \mathbf{B}|\int \parallel \mathrm{\nabla }\cdot \mathbf{B}{\parallel }_1+\int \parallel \psi \mathbf{u}\cdot \mathrm{\nabla }\psi {\parallel }_1$ (energy). It is a necessary bound, not a tight one: the signed sources largely cancel (energy drift $2.8\cdot {10}^{-3}$ of its bound; $\rho \mathbf{u}$ and $\mathbf{B}$ $\sim {10}^{-11}$, where the 180° symmetry cancels their sources).
• The symmetry defect is round-off amplified by the flow, more by the EGLM sources on the splitting scheme ($4.3\cdot {10}^{-7}$ against $5.9\cdot {10}^{-10}$ for GLM at $t=0.5$), so it is checked at step 100 within ${10}^{-8}$ and at the end within ${10}^{-4}$, as for the rotor.
• On the blast HLLD with primitive interpolation also reaches $t=0.01$; the splitting scheme needs the positivity limiter (M3-P5).
• --amr is not combined with EGLM in orszag-tang/check.sh: the EGLM sources at 2:1 seams are out of scope there.