Brio–Wu and Ryu–Jones 4d

ID	Group	Milestone	Backends	Figure	Regression
— (mhd/riemann, M2-P3 exit)	Ideal MHD	M2 (#41)	CPU, FNL	yes	brio-wu

Case and scope

The Brio–Wu shock tube ($\gamma =2$, $B_x=0.75$, $B_y=\pm 1$) develops a compound wave; Ryu–Jones 4d a switch-on/switch-off structure. Both stress the MHD eigensystem near its degeneracies, where a scheme that is not clean needs the positivity floors to survive. The leg also exercises the floors themselves on a state no clean solver produces: a negative pressure from the first stage.

What the case catches:

• an unclean scheme: with the floors armed at ${10}^{-10}$ the runs must reach the final time with zero floored cells in every stage, and a positive, finite final density and pressure;
• floors that do not engage: with p_floor = 1e-3 the negative-pressure state must be floored and the run must complete;
• floors that hide a failure: with the floors disabled the same state must stop the run with its message.

Models and numerics

Item	Choice	Code
Equations	ideal MHD; Brio–Wu and the negative state $\gamma =2$, Ryu–Jones 4d $\gamma =5/3$ ($c_v=1$, $c_p=\gamma$)	models
Space	WENO5 flux splitting on characteristic variables	compute_residuals_weno, adam_flume_cpu_object.F90; mhd_face_split_fluxes, adam_flume_mhd_library.F90
Divergence control	none; Brio–Wu also with GLM (glm_ch = 4.0, glm_alpha = 0.18, glm_damping_length = 1.0, glm_ch_check = warning)	mhd_glm_face_split_fluxes, adam_flume_mhd_library.F90; models
Floors	armed at ${10}^{-10}$ (clean legs); p_floor = 1e-3 (floored leg); disabled (stop leg)	apply_floors, adam_flume_cpu_object.F90 and adam_flume_cpu_mhd_kernels_agnostic.INC; models
Time	SSP Runge–Kutta, 3 stages, CFL 0.5	adam_rk_object.F90
Boundaries	zeroth-order extrapolation	set_boundary_conditions, adam_flume_cpu_object.F90; boundary conditions
FNL backend	the same kernels on the device	compute_residuals_weno_dev, apply_floors, adam_flume_fnl_object.F90

Initial and boundary conditions

Item	Value
Domain	$[0,1]$ along $x$; $y$ and $z$ null
Grid	the Sod grid of V1, $N=512$: 64 blocks of $128\times 4\times 4$ cells, 4 along $x$
Ranks	2
Initial condition	riemann-problem, discontinuity at $x=0.5$
Boundaries	extrapolation on all six faces

States $(\rho ,u,v,w,p,B_x,B_y,B_z)$ (code units; Brio–Wu from Athena++ athinput.bw, Ryu–Jones 4d from the Athena C test athinput.rj4d):

Leg	$\gamma$	Left	Right	$t$
Brio–Wu, none and GLM	2	$(1,0,0,0,1,0.75,1,0)$	$(0.125,0,0,0,0.1,0.75,-1,0)$	0.1
Ryu–Jones 4d, none	5/3	$(1,0,0,0,1,0.7,0,0)$	$(0.3,0,0,1,0.2,0.7,1,0)$	0.16
negative pressure (floored; stop)	2	the Brio–Wu left state	$(0.125,0,0,0,-0.1,0.75,-1,0)$	0.02

Input files

File	Role
src/tests/flume/verification/mhd/riemann/make_riemann.py	writes the input from sod/sod-x.ini (--case bw|rj4d|negative, --cells, --divergence-control, --rho-floor, --p-floor)
src/tests/flume/verification/mhd/riemann/check.sh	runs the legs and checks the logs and the final checkpoint (--np N, --numerics SPEC, --legs LIST: clean, floored, stop)
src/tests/flume/verification/make_doc_figures.py fig_mhd_riemann	draws brio-wu.png and rj4d.png from <runs>/brio-wu and <runs>/rj4d (--only mhd-riemann)

cd src/tests/flume/verification/mhd
riemann/check.sh                                                     # CPU
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl riemann/check.sh    # FNL
riemann/check.sh --numerics hlld                                     # weno-riemann, HLLD

make_doc_runs.sh <runs> mhd writes the figure runs (<runs>/brio-wu: 512 cells, GLM; <runs>/rj4d: 512 cells, no cleaning); make_doc_figures.py --runs <runs> draws it.

Results

Brio–Wu shock tube with GLM: density, normal velocity, pressure and the tangential fields at $t=0.1$.

Ryu–Jones 4d at $t=0.16$: the switch-on and switch-off structure.

Check	Bound	CPU	FNL
Brio–Wu (none, GLM), Ryu–Jones 4d: stages with floored cells (no MHD floors: log line)	0	0	0
the same: final $\rho$, $p$ positive and finite	$>0$	pass	pass
negative pressure, p_floor = 1e-3: floors engaged, run completes	engaged	pass	pass
negative pressure, floors disabled: the run stops with "cells with a non-positive density or pressure"	stop	pass	pass

On weno-riemann (HLLD, either interpolation) the floored-cell count and the HLLD fallbacks to HLL are both 0 (MHD on weno-riemann); the fallbacks are reported, not asserted.

Related cases: Ryu–Jones 2a has an exact solution on the same grid; positivity limiter removes the floors from the hard problems.

Findings

• Both problems run clean on the splitting scheme: the eigensystem holds near its degeneracies without floors.
• A negative-pressure state is floored when the floors are armed and stops the run when they are not: the floors never act silently on a run that does not ask for them.
• The figures have no exact solution overlaid; the leg asserts cleanliness and positivity, not accuracy.