Becker's viscous shock

ID	Group	Milestone	Backends	Figure	Regression
VV-4	Dissipative terms (M4)	M4 (#65)	CPU, FNL	yes	becker-shock

Case and scope

Becker's (1922) exact profile of a shock resolved by the viscosity, with constant $\mu$ and $Pr=3/4$, in the shock frame. It is an exact steady solution of the compressible Navier–Stokes equations, so, like Couette, the run starts from the exact solution and the departure at the final time is the error of the scheme, with no fit of a shock position.

What the case catches: the strongest nonlinear viscous flux of the suite, with the energy flux carrying both the viscous work and the heat flux. A wrong coefficient, a wrong viscous work or a wrong heat flux moves the profile.

Models and numerics

Item	Choice	Code
Equations	compressible Navier–Stokes, $\gamma =1.4$, $R=1$, constant $\mu$, $Pr=0.75$	models
Dissipative fluxes	dissipative_order = 4	add_dissipative_fluxes_o4, adam_flume_cpu_dissipation_kernels.F90; compute_dissipative_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
Initial profile	Becker's implicit solution, solved cell by cell	becker_shock, adam_flume_ic_object.F90
Boundaries	supersonic inflow upstream, extrapolation elsewhere	set_boundary_conditions, adam_flume_cpu_object.F90; boundary conditions
FNL backend	the same kernels on the device	add_dissipative_fluxes_o4_dev, adam_flume_fnl_dissipation_kernels.F90

Initial and boundary conditions

Item	Value
Domain	$x\in [-0.25,0.25]$, the shock at $x=0$; $y$ and $z$ null
Grid	quadtree, $4\times 4$ blocks of $N/4\times 1$ cells, $N$ = 64 / 128 / 256 along $x$
Ranks	2
Initial condition	becker-shock: upstream state ${\rho }_1=1$, $p_1=1/\gamma$ (so $a_1=1$), $u_1=M$
bc_x_min	inflow with the upstream state
Other faces	extrapolation
Final time	$t=0.05$

Leg	Mach	$\mu$	Cells across the shock at $N=256$
becker-m2-o4	2	0.005	about 8
becker-m3-o4	3	0.01	about 8

At Mach 3 with $\mu =0.005$ the shock spans 4 cells, WENO's nonlinear weights still act, and the measured order was +2.25; the leg therefore takes $\mu =0.01$ at Mach 3.

Input files

File	Role
src/tests/flume/verification/viscous/profiles.py	make --case becker writes each input; oracle measures $(u-u_{\mathrm{exact}})/(u_1-u_2)$
src/tests/flume/verification/viscous/check.sh --legs vv4	runs the two ladders (--np N; FLUME_EXE selects the backend)
src/tests/flume/regression/becker-shock/input.ini	the frozen case: Mach 2, 64 cells, $t=0.01$
make_doc_figures.py (fig_viscous)	the figure, from the CPU work directories of viscous/check.sh

cd src/tests/flume/verification/viscous && ./check.sh --legs vv4             # CPU
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl ./check.sh --legs vv4       # FNL

Results

Right panel: $(u-u_2)/(u_1-u_2)$ at 64 and 256 cells against Becker's profile, Mach 2 and 3; the left panel holds the two Becker ladders.

Oracle: order $\ge 3$ on the RMS and on the maximum of $(u-u_{\mathrm{exact}})/(u_1-u_2)$ (finest pair).

Mach	64	128	256	Orders (RMS)
2	RMS $1.05\cdot {10}^{-2}$, max $6.2\cdot {10}^{-2}$	$1.92\cdot {10}^{-3}$, $1.15\cdot {10}^{-2}$	$1.30\cdot {10}^{-4}$, $8.3\cdot {10}^{-4}$	+2.45, +3.88
3	RMS $1.47\cdot {10}^{-2}$, max $8.9\cdot {10}^{-2}$	$3.09\cdot {10}^{-3}$, $1.89\cdot {10}^{-2}$	$2.77\cdot {10}^{-4}$, $1.7\cdot {10}^{-3}$	+2.25, +3.48

CPU, 2 ranks. Every leg passes on FNL (2 ranks) with the same numbers to the printed digits.

Related cases: the other exact steady viscous profile is Couette; the backend agreement of these runs, the largest of the suite, is in CPU and FNL agree.

Findings

• The coarse pair converges at about second order and the finest pair above third: the shock width is fixed by $\mu$, so it spans about 2 cells at 64 cells and about 8 at 256. With 4 cells across it the order was still pre-asymptotic (+2.25).
• The Becker runs hold the largest CPU–FNL differences of the suite ($4.3\cdot {10}^{-13}$ of the momentum scale), being its steepest gradients.