Viscous waves

ID	Group	Milestone	Backends	Figure	Regression
VV-1, VV-2	Dissipative terms (M4)	M4 (#65)	CPU, FNL	yes	shear-viscous

Case and scope

A small sine wave on a gas at rest is one Fourier mode of the linearised compressible Navier–Stokes equations. The oracle (viscous/waves.py) integrates that mode exactly, $\exp (Mt)$ of the $3\times 3$ system for $({\rho }^{\prime },u_n,T^{\prime })$ plus the decoupled transverse velocity, from the Fourier coefficients of the initial state, so it holds every mode the start excites (the acoustic pair and the entropy mode), not only a decay rate. The error is the RMS over the cells of the point values (FLUME is a finite-difference scheme).

• VV-1, the shear wave: the transverse velocity $u_t=A\sin (k\mathbf{n}\cdot \mathbf{x})e^{-\nu k^{2}t}$; the error is measured on $u_t$. It pins the order of the viscous flux.
• VV-2, the right-running acoustic wave with a conductivity as well, so $\mu$ and $k$ act together; the error is measured on ${\rho }^{\prime }$.

What the case catches: a dissipative flux that is consistent and still wrong in its order (a missing correction term), its coefficient (a missing 4/3, a conductivity without $R$) or its direction (a cross derivative read with the wrong stencil, exercised by the diagonal leg).

Models and numerics

Item	Choice	Code
Equations	compressible Navier–Stokes, ideal gas, $\gamma =1.4$, $R=1$, constant $\mu$ and $k$	models
Dissipative fluxes	dissipative_order = 4 (conservative fourth-order flux with the Shu–Osher correction) or 2 (compact central)	add_dissipative_fluxes, adam_flume_cpu_object.F90; add_dissipative_fluxes_o4, add_dissipative_fluxes_o2, adam_flume_cpu_dissipation_kernels.F90; compute_dissipative_flux, adam_flume_dissipation_library.F90; numerics
Inviscid fluxes	WENO5 (weno-u-5) flux splitting on characteristic variables (scheme_space = weno)	compute_residuals_weno, adam_flume_cpu_object.F90; numerics
Time	SSP Runge–Kutta, 5 stages, 4th order (runge-kutta-ssp-54), CFL 0.2, the diffusive limit included	adam_rk_object.F90; compute_lambda_max_dissipative, adam_flume_cpu_euler_kernels.F90
Boundaries	periodic in $x$ and $y$, extrapolation on the null $z$	set_boundary_conditions, adam_flume_cpu_object.F90
FNL backend	the same fluxes as device kernels	add_dissipative_fluxes_dev, adam_flume_fnl_object.F90; add_dissipative_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, $N/4$ cells per block along the active axes, ngc = 3. A wave along an axis keeps the other in-plane axis: active and thin for the shear wave (4 blocks of 6 cells across it, because FLUME freezes the momentum normal to a null direction, which would freeze the shear), null (1 cell) for the acoustic wave
Ranks	2
Initial condition	sine-wave, ${\rho }_0=p_0=1$, $u=0$; wavelength 1 along an axis, $1/\sqrt{2}$ along the diagonal
Boundaries	periodic on the $x$ and $y$ faces, extrapolation on the $z$ faces

Leg	Mode	Direction	dissipative_order	$A$	$\mu$, $k$	$N$	$t$
VV-1 shear x	shear	$x$	4	${10}^{-4}$	0.01, 0	32 / 64 / 128	0.5
VV-1 shear y	shear	$y$	4	${10}^{-4}$	0.01, 0	32 / 64 / 128	0.5
VV-1 shear x, order 2	shear	$x$	2	${10}^{-4}$	0.01, 0	32 / 64 / 128	0.5
VV-1 shear diagonal	shear	45° (both directions active: the cross terms ${\partial }_i{u}_d$ and the tangential derivatives)	4	${10}^{-4}$	0.01, 0	24 / 48 / 96	0.1
VV-2 acoustic	acoustic, and its $-A$ twin	$x$	4	${10}^{-5}$	0.01, 0.02	32 / 64 / 128	0.5

A sound wave steepens at $O(A^2)$: at $A={10}^{-4}$ that residue (${10}^{-8}$) flattened the error at every resolution. VV-2 therefore runs the twin with amplitude $-A$ and judges the odd part $(f(+A)-f(-A))/2$, whose residue is $O(A^3)$, at $A={10}^{-5}$.

Input files

File	Role
src/tests/flume/verification/viscous/waves.py	make writes each input; oracle integrates the linear mode exactly and judges the ladder
src/tests/flume/verification/viscous/check.sh --legs vv1,vv2	runs the ladders (--np N; FLUME_EXE selects the backend)
src/tests/flume/regression/shear-viscous/input.ini	the frozen diagonal shear wave, ${48}^2$, $t=0.05$
make_doc_figures.py (fig_viscous, fig_viscous_fields, fig_viscous_ladders)	the figures, from the CPU (and, for the ladders, FNL) work directories of viscous/check.sh

cd src/tests/flume/verification/viscous && ./check.sh --legs vv1,vv2             # CPU
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl ./check.sh --legs vv1,vv2       # FNL

Results

Left: the Navier–Stokes ladders (VV-1 to VV-4), each normalised by its coarsest error, against the order-2 and order-4 slopes; middle: the Ohmic ladders (VV-5); right: Becker's shock (VV-4).

Absolute RMS errors of the VV-1 and VV-2 legs (left) and of the VV-5 legs (centre), CPU (filled, solid) and FNL (open, dotted), which coincide; right, the Ohmic decay: the measured amplitude of the mode over the exact one.

Top row: the diagonal shear wave on the uniform ${96}^2$ grid at $t=0.1$ and its pointwise error, smooth and spread over the domain. The bottom row belongs to VV-8.

Oracle: the observed order of the finest pair $\ge 3.8$ at order 4, $\ge 1.9$ at order 2.

Leg	N	RMS error (CPU)	Order (CPU)	FNL
shear x, order 4	32 / 64 / 128	$3.64\cdot {10}^{-10}$ / $2.29\cdot {10}^{-11}$ / $1.43\cdot {10}^{-12}$	+3.99, +4.00	the same to the printed digits
shear y, order 4	32 / 64 / 128	identical to x, bit for bit	+3.99, +4.00	the same
shear x, order 2	32 / 64 / 128	$3.68\cdot {10}^{-8}$ / $9.20\cdot {10}^{-9}$ / $2.30\cdot {10}^{-9}$	+2.00, +2.00	the same
shear diagonal, order 4	24 / 48 / 96	$6.72\cdot {10}^{-10}$ / $1.16\cdot {10}^{-11}$ / $2.43\cdot {10}^{-13}$	+5.86, +5.57	the same
acoustic, order 4	32 / 64 / 128	$8.20\cdot {10}^{-11}$ / $1.69\cdot {10}^{-12}$ / $2.18\cdot {10}^{-14}$	+5.60, +6.28	$8.20\cdot {10}^{-11}$ / $1.69\cdot {10}^{-12}$ / $2.21\cdot {10}^{-14}$, +5.60, +6.26 (at 32 cells $8.196853\cdot {10}^{-11}$ against the CPU's $8.196891\cdot {10}^{-11}$)

Every leg passes on both backends (2 ranks). On FNL the numbers are the same to the printed digits except where the round-off of the run itself shows: the acoustic odd part, whose $\pm A$ difference amplifies the round-off of a ${10}^{-5}$ wave.

Related cases: the same oracle drives Ohmic decay and the Alfvén wave; the diagonal waves cross 2:1 seams in VV-7 and VV-8; the run-by-run backend comparison is in CPU and FNL agree.

Findings

• The measured decay rate of the shear wave is $\nu k^2$ to $2\cdot {10}^{-8}$ relative at 128 cells.
• The diagonal wave converges faster than its design order: on the diagonal the leading truncation errors of the $x$ and $y$ fluxes appear to cancel.
• On the acoustic wave the error is dominated by the inviscid WENO part, not by the viscous one. VV-2 therefore checks that $\mu$ and $k$ enter with the right coefficients (the measured amplitude equals the exact one to $6\cdot {10}^{-10}$ relative), and VV-1 pins the order of the dissipative flux.
• The measured acoustic rate, $0.29576$, differs by 4% from the first-order Kirchhoff–Stokes rate $k^2/(2{\rho }_0)(\frac{4}{3}\mu +(\gamma -1)k/c_p)=0.30831$: here $\nu k/a=0.053$, and the exact linear oracle carries that correction.
• #85 lists absolute shear and acoustic ladders (RMS error against $N$ with reference slopes) as a figure still to extend fig_viscous with; the published figure shows the ladders normalised by their coarsest error.