MHD on weno-riemann

ID	Group	Milestone	Backends	Figure	Regression
RV-5, RV-6, RV-7	Ideal MHD	M3 (#47, M3-P3c)	CPU, FNL	yes	rj2a-hlld

Case and scope

The MHD verifications of M2 were built on the WENO flux splitting. The weno-riemann scheme (WENO interpolation of the face states, an LLF, HLL or HLLD Riemann solver, a high-order flux correction) reuses their inputs and oracles through the shared --numerics option, which rewrites the [numerics] block of the generated inputs:

--numerics SOLVER[:RECON[:CORRECTION[:SENSOR]]]

with SOLVER = llf | hll | hlld, RECON = primitive | characteristic (default characteristic, the MHD default), CORRECTION = none | 4th | 6th (default 6th), SENSOR = weno | none (default weno). Seven scripts take it: linear-wave, cpaw, riemann, rj2a, orszag-tang, rotor and field-loop.

What the case catches:

• a Riemann-solver path that loses accuracy: the specs hlld:primitive and hlld:characteristic (6th, weno) assert bounds measured on them in rj2a, linear-wave, cpaw and field-loop; HLLD must be at least as accurate as HLL on Ryu–Jones 2a (RV-6);
• a scheme-dependent defect: every spec runs the scheme-independent checks (order, bitwise x/y/z and left/right, symmetry, conservation, positivity); the other specs are measurements.

Models and numerics

Item	Choice	Code
Equations	ideal MHD, no cleaning, GLM or EGLM, as in each verification	models
Space	scheme_space = weno-riemann: WENO interpolation of the face states (primitive or characteristic variables)	compute_residuals_riemann, adam_flume_cpu_object.F90; mhd_face_interpolation_fields, mhd_face_states, adam_flume_mhd_riemann_library.F90; numerics
Riemann solvers	LLF, HLL, HLLD (with a fallback to HLL, reported in the log); GLM and EGLM variants	mhd_riemann_llf, mhd_riemann_hll, mhd_riemann_hlld, mhd_glm_riemann_hlld, mhd_eglm_riemann_hlld, adam_flume_mhd_riemann_library.F90; adam_flume_cpu_mhd_hlld_kernels.F90 (and the _llf, _hll, _glm_*, _eglm_* twins)
Flux correction	6th with the WENO sensor (default)	numerics
Time, boundaries, grids	as in each verification	—
FNL backend	the same kernels on the device	compute_residuals_riemann_dev, adam_flume_fnl_object.F90

Initial and boundary conditions

The inputs are those of the verifications, unchanged except for the [numerics] block:

Problem	Setup
Ryu–Jones 2a	256 and 512 cells, $t=0.2$; RV-6 adds an HLL run at 256
linear waves, CPAW	the MV-5 and MV-6 ladders
Brio–Wu, Ryu–Jones 4d	512 cells, floors armed at ${10}^{-10}$
field loop	the uniform 64/128 ladder
Orszag–Tang, rotor	as in MV-12, MV-13

Input files

File	Role
src/tests/flume/verification/mhd/numerics.sh	numerics_check, numerics_tag, numerics_apply: validates the spec and rewrites [numerics]
src/tests/flume/verification/make_doc_runs.sh <runs> rj2a mhd, make_doc_figures.py --only weno-riemann	rj2a re-runs the splitting and HLLD-characteristic runs, mhd the HLLD-primitive run rj2a-hlld-prim
src/tests/flume/verification/mhd/{rj2a,linear-wave,cpaw,riemann,orszag-tang,rotor,field-loop}/check.sh	the seven scripts that take --numerics

cd src/tests/flume/verification/mhd
rj2a/check.sh --numerics hlld                     # HLLD, characteristic interpolation
rotor/check.sh --numerics hlld:primitive          # HLLD, primitive interpolation

Results

Ryu–Jones 2a at 256 cells, CPU, over the exact solution: the splitting scheme and weno-riemann with HLLD on characteristic and on primitive interpolation. The three agree at this scale; the primitive interpolation has the largest $L_1$ sum of the three (table below).

Measured in issue #47, M3-P3c; CPU and FNL agree to the printed digits:

Problem	Splitting	HLLD primitive	HLLD characteristic
Ryu–Jones 2a, $L_1$ sum at 256 / 512 cells	$3.84/2.12\cdot {10}^{-2}$	$5.13/2.93\cdot {10}^{-2}$	$3.76/2.12\cdot {10}^{-2}$
linear waves, CPAW: observed order	4.96–5.02	4.96–5.00	4.96–4.99
Brio–Wu, Ryu–Jones 4d: floored cells, HLLD fallbacks	0	0, 0	0, 0
field loop, magnetic energy kept at 64 / 128 cells	0.872 / 0.937	0.869 / 0.936	0.887 / 0.944

Bounds asserted under the two HLLD specs (CPU M3-P3c + 2%; energies − 0.2%):

Check	HLLD primitive	HLLD characteristic
Ryu–Jones 2a, $L_1$ sum at 256 / 512	$5.235/2.985\cdot {10}^{-2}$	$3.840/2.166\cdot {10}^{-2}$
CPAW $ϵ$ at 32 / 64 / 128	$1.124\cdot {10}^{-3}$ / $3.774\cdot {10}^{-5}$ / $1.176\cdot {10}^{-6}$	$1.098\cdot {10}^{-3}$ / $3.726\cdot {10}^{-5}$ / $1.174\cdot {10}^{-6}$
linear waves, CPAW: observed order	$\ge 4.8$	$\ge 4.8$
field loop $⟨|B_z|⟩/A_0$ at 64 / 128	$9.950\cdot {10}^{-4}$ / $5.546\cdot {10}^{-4}$	$1.083\cdot {10}^{-3}$ / $5.923\cdot {10}^{-4}$
field loop $E_B(T)/E_B(0)$ at 64 / 128	$\ge 0.86757$ / $0.93417$	$\ge 0.88559$ / $0.94189$
Ryu–Jones 2a, GLM against no cleaning	${10}^{-11}$	${10}^{-11}$
rotor symmetry at step 100 / at the end	${10}^{-8}$ / ${10}^{-4}$	${10}^{-8}$ / ${10}^{-4}$

The linear-wave bounds are per run, the larger of the two interpolations (linear-wave/check.sh).

Related cases: Sod, Lax and Shu–Osher compare the Euler Riemann solvers; EGLM cleaning runs on HLLD characteristic; positivity limiter runs HLLD and HLLC with the limiter.

Findings

• On Ryu–Jones 2a HLLD is more accurate than HLL ($5.98$ and $4.32\cdot {10}^{-2}$): RV-6 asserts the HLLD $L_1$ sum at most the HLL one.
• HLLD with characteristic interpolation matches the splitting scheme on Ryu–Jones 2a; primitive interpolation is about 35% less accurate there. On the smooth waves the two interpolations agree to 3 digits (linear WENO weights).
• The GLM run differs from the run without cleaning by round-off (${10}^{-12}$; bitwise with the splitting), because the interpolated uniform $B_n$ seeds $\psi$.
• The rotor's symmetry defect is round-off amplified by the flow and grows as the dissipation decreases (final $5\cdot {10}^{-9}$ to $6\cdot {10}^{-6}$ depending on the interpolation and the backend), so under --numerics it is checked at step 100 within ${10}^{-8}$ and at the end within ${10}^{-4}$ (rotor).
• The scripts glm-pulse, divb-peak, vortex and rotated-shock-tube do not take --numerics: MV-3, MV-7, MV-8 and MV-10 run on the splitting scheme only.