Isentropic vortex: order of accuracy

ID	Group	Milestone	Backends	Figure	Regression
V2, RV-2	Euler	M1 (#35), M3 (#47)	CPU (all schemes), FNL (weno, weno-riemann LLF and HLLC)	yes	vortex-periodic

Case and scope

An exact smooth solution of the Euler equations: an isentropic vortex (Shu 1998, ICASE 97-65, §5.1) convected diagonally by the free stream through a doubly periodic box. Every run is compared cell by cell with the exact solution, and the observed order between successive resolutions, $p={\log }_2(E_N/E_{2N})$, must reach the design order.

What the case catches: a loss of design order on smooth flows, which Sod cannot see (WENO is first order at discontinuities):

• a wrong stencil weight or reconstruction table;
• a wrong ghost fill across periodic block faces, across blocks and across ranks;
• a first-order boundary treatment.

Models and numerics

Item	Choice	Code
Equations	compressible Euler, ideal gas, $\gamma =1.4$	models
Space, weno	WENO5 (weno-u-5) flux splitting on characteristic variables	compute_residuals_weno, adam_flume_cpu_object.F90; compute_face_split_fluxes, adam_flume_euler_library.F90
Space, weno-riemann (RV-2)	WENO5 interpolation (characteristic), LLF, HLL or HLLC, the 6th-order flux correction with the WENO-weight sensor	compute_residuals_riemann, adam_flume_cpu_object.F90; initialize_interpolation, adam_weno_object.F90
Time	SSP Runge–Kutta, 5 stages, 4th order (runge-kutta-ssp-54), CFL 0.1 (small, so the space error dominates the time error)	adam_rk_object.F90
Initial condition	the isentropic-vortex kind	isentropic_vortex, adam_flume_ic_object.F90
Boundaries	library periodicity in $x$ and $y$ (across blocks and ranks)	update_ghost, adam_flume_cpu_object.F90; boundary conditions
FNL backend	the same kernels on the device	adam_flume_fnl_object.F90

Initial and boundary conditions

Item	Value
Domain	$[0,1{]}^2$, $z$ null
Grid	quadtree (ratio = 4, nk = 1), 16 blocks (4 per side) of ${16}^2$, ${32}^2$, ${64}^2$ cells: ${64}^2$, ${128}^2$, ${256}^2$ cells, ngc = 3
Ranks	2 (8/8 blocks)
Free stream	$(\rho ,u,v,w,p)=(1,1,1,0,1)$
Vortex	centre $(0.5,0.5)$, radius 0.07 (tails $e^{-25}$ at the box edge), strength 5
Initial condition	isentropic-vortex
Boundaries	periodic on $x$ and $y$; extrapolation on the null $z$ faces
Final time	$t=0.1$ (the vortex centre moves by $(0.1,0.1)$)

Input files

File	Role
src/tests/flume/verification/vortex/vortex-n{064,128,256}.ini	the ladder
src/tests/flume/verification/vortex/check.sh	V2: runs the ladder (--np N, --legs v2; FLUME_EXE selects the backend), order bound 4.5
src/tests/flume/verification/vortex/vortex_oracle.py	the exact density, $L_1$ and $L_{\mathrm{\infty }}$ errors, the finest-pair order, the null-$z$ copies identical
src/tests/flume/verification/riemann-flux/check.sh --legs vortex	RV-2: the ladder on weno-riemann, order bound 4.8
src/tests/flume/verification/make_doc_figures.py (fig_vortex)	the figure, from the work directories of the two scripts

cd src/tests/flume/verification/vortex && ./check.sh                    # V2, CPU
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl ./check.sh             # V2, FNL
cd ../riemann-flux && ./check.sh --legs vortex --solver hllc            # RV-2 (also llf, hll)

Results

Left: density at $t=0.1$, ${128}^2$ cells, weno-riemann HLLC. Right: $L_1(\rho )$ against the cells per side for the four schemes, with the order-5 slope.

Scheme	Bound (finest-pair order)	$L_1(\rho )$, ${256}^2$	Order 128 → 256	FNL
weno (split)	4.5 (V2)	$8.38\cdot {10}^{-8}$	5.79	the same
weno-riemann LLF	4.8 (RV-2)	$1.42\cdot {10}^{-7}$	5.64	the same
weno-riemann HLL	4.8 (RV-2)	$8.98\cdot {10}^{-8}$	6.26	not run
weno-riemann HLLC	4.8 (RV-2)	$8.04\cdot {10}^{-8}$	6.49	the same

The full ladders on the CPU:

Scheme	$L_1(\rho )$, ${64}^2$ / ${128}^2$ / ${256}^2$
weno (split)	$8.218\cdot {10}^{-5}$ / $4.621\cdot {10}^{-6}$ / $8.379\cdot {10}^{-8}$
weno-riemann LLF, WENO-weight sensor	$1.182\cdot {10}^{-4}$ / $7.073\cdot {10}^{-6}$ / $1.415\cdot {10}^{-7}$
weno-riemann LLF, sensor none	$9.113\cdot {10}^{-5}$ / $4.748\cdot {10}^{-6}$ / $1.415\cdot {10}^{-7}$

Related cases: Sod checks the same schemes at discontinuities; the MHD orders are in Order of accuracy.

Findings

• The order is asserted on the finest pair only. From ${64}^2$ to ${128}^2$ it is lower (4.15 for the splitting scheme): at ${64}^2$ the vortex radius spans 4.5 cells and the WENO weights are still far from the linear ones.
• At ${256}^2$ the WENO-weight sensor of weno-riemann is inactive (the same digits with and without it); at ${128}^2$ it trips at the vortex extrema and costs a factor 1.5. The library weights use the exponent of $S=3$, the NumPy prototype 2.
• LLF is 1.7× more dissipative than the splitting scheme on the vortex; HLL and HLLC close the gap, and HLLC is the most accurate of the four.
• The quadtree layout is bitwise identical to the octree one with a null $z$ axis, and 16 times cheaper.
• The earlier verification page and the figure's panel title called the run "one period"; the input runs to $t=0.1$, one tenth of the period of the diagonal convection through the unit box (the title is corrected since #85 R1).