Reflux policy: troubled against always

ID	Group	Milestone	Backends	Figure	Regression
E2 (E1)	Seam studies	M6 (#68)	FNL (flags checked equal to the CPU's); AO euler-tq: CPU, FNL	yes	—

Case and scope

The Berger–Colella reflux keeps a 2:1 seam conservative, and caps the composite at second order (AO, B1): it replaces the coarse face flux by the restricted fine one everywhere. B2 showed in a prototype that a sensor-gated reflux recovers the order and keeps the shocks in place (D-M6-4: go). E1 builds [numerics] reflux_policy = troubled; E2 measures it against always on the four checks of the plan: the smooth order (A1), the shock crossings (A3), the integral drift (A4) and the seam divergence (A2).

Models and numerics

Item	Choice	Code
reflux_policy = always	default: every seam face cell takes the restricted fine flux	apply_reflux_to_stage_forest, adam_flume_cpu_object.F90
reflux_policy = troubled	the reflux applies only at the seam face cells flagged in a stage of the step: those the positivity limiter blended, and those where the WENO-5 sensor $\underset{k}{min}{w}_k/d_k$ of the density, the normal velocity or the pressure across the face is below 0.2; a cell flagged at any stage is refluxed	flag_troubled_seam_cells, adam_flume_cpu_object.F90 and adam_flume_fnl_object.F90; seam_weno5_sensor, adam_flume_seam_sync_object.F90; FLUX_CORRECTION_SENSOR_TAU, adam_flume_parameters.F90
Key	[numerics] reflux_policy, with reflux = .true.	adam_flume_numerics_object.F90
Seam operators	default (mean + tricubic) or lagrange6 + quintic	C1, D1
Scheme, time	as in each baseline case	numerics

Initial and boundary conditions

Check	Setup
A1	the AO euler and mhd ladders, $N$ = 64 / 128 / 256; euler also with lagrange6 + quintic (euler-tq)
A3	the AS crossings, Sod and RJ2a, c2f and f2c
A4	the AL vortex ($N=64$ to $t=1$) and Sod c2f
A2	the AD cpaw, $N$ = 64 / 128

FNL, 2 ranks.

Input files

File	Role
src/tests/flume/verification/amr-order/check.sh --legs euler-tq	the AO leg under troubled with lagrange6 + quintic: orders $\ge 5.5$, errors at most the euler-q6 baseline plus 2 %
src/tests/flume/verification/amr-order/make_amr_order.py --reflux-policy troubled	the other AO runs
src/tests/flume/verification/amr-shock/make_amr_shock.py --reflux-policy troubled	the crossings
src/tests/flume/verification/make_doc_runs.sh <runs> seams, make_doc_figures.py --only reflux-policy	re-runs the crossings e2-troubled-{c2f,f2c} and d-quintic-{c2f,f2c} (CPU); the AO ladders from the kept work directories

cd src/tests/flume/verification/amr-order && ./check.sh --legs euler-tq
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl ./check.sh --legs euler-tq

The A2 and A4 runs under troubled: amr-divb/make_amr_divb.py --reflux-policy troubled and amr-log/check.sh --reflux-policy troubled (issue #85 R2b; before, the policy was added to the inputs by hand).

Results

CPU. Left: the AO ladder, troubled with lagrange6 + quintic at the order of the run without reflux (6.11), today's operators at 2. Centre and right: the Sod shock after the crossing, lagrange6 + quintic under always and troubled: the same profile (the shock faces are flagged, so they are refluxed).

Check	always	troubled
AO euler, default operators, N = 256: L1, order	$4.05\cdot {10}^{-6}$, 2.01	$1.80\cdot {10}^{-6}$, 2.34
AO euler, lagrange6 + quintic, N = 256: L1, order	$4.05\cdot {10}^{-6}$, 2.01	$6.03\cdot {10}^{-8}$, 6.11
AO mhd, default operators, N = 256: L1, order	$1.83\cdot {10}^{-6}$, 2.07	$1.10\cdot {10}^{-6}$, 2.04
AO face cells flagged, N = 64 / 128 / 256	—	3–5 % / 0–0.6 % / 0 %
A3 shock position (Sod, RJ2a, c2f and f2c)	reference	equal within $2\cdot {10}^{-4}$ cells
A3 plateau and band defects	reference	within 1 % (RJ2a c2f band +44 %)
A3 mass drift	$\le 3\cdot {10}^{-10}$	${10}^{-11}$ to $1.5\cdot {10}^{-7}$
A4 drift, vortex N = 64 to $t=1$ (r, ru, rv, rE)	$\le 3\cdot {10}^{-16}$	$1.8$–$4.4\cdot {10}^{-5}$, growth 1.3–1.6 (rv 8.8)
A4 drift, Sod c2f	$3\cdot {10}^{-16}$	$1.9\cdot {10}^{-7}$
CPAW D_flux seam floor, N = 64 / 128 (growth)	$5.39\cdot {10}^{-4}$ / $5.43\cdot {10}^{-5}$ (0.95 / 0.72)	$5.52\cdot {10}^{-4}$ / $3.80\cdot {10}^{-5}$ (0.76 / 0.68)

On FNL the flags equal the CPU's to the cell.

Related cases: B2 is the prototype; AL measured the sensor's false positives; G3a uses troubled as the #78 workaround.

Findings

• The reflux was what capped the high-order seam operators. With lagrange6 and quintic the reflux on (always) holds the vortex at second order, $4.05\cdot {10}^{-6}$; under troubled no smooth face is flagged once resolved and the run is sixth order, $6.03\cdot {10}^{-8}$, the error of the uniform coarse run (67 times smaller). AO leg euler-tq asserts it.
• Shocks are unchanged: positions equal to $2\cdot {10}^{-4}$ cells, defects within 1 % but one band value; the flags cover 35 to 42 % of the face-cell steps of a crossing.
• Conservation is the price: an unflagged face keeps its flux mismatch, so the integrals drift at the reflux-off rate on smooth flows (${10}^{-5}$ per period at $N=64$, bounded but one component accumulating) and by ${10}^{-7}$ across a shock.
• MHD: the D_flux seam floor is bounded and 1.4 times lower at $N=128$ (83 against 119 times the uniform floor).
• Decision (D-M6-2, owner): the default stays always, exactly conservative; troubled is the opt-in that makes the high-order seam operators worth their cost with the reflux on.