Sensor-gated reflux prototype

ID	Group	Milestone	Backends	Figure	Regression
B2	Seam studies	M6 (#68)	NumPy prototype (no ADAM run)	yes	—

Case and scope

B1 recovers order 5 only without the reflux, and A3 showed that a shock needs it. Phase E would apply the reflux only where a WENO sensor flags the face (reflux_policy = troubled); D-M6-4 builds it only if a prototype shows the order recovered with the shock positions intact. B2 is that go/no-go: does a sensor-gated reflux recover the smooth order, keep the shocks where the full reflux puts them, and how much conservation does it give up?

Models and numerics

Item	Choice	Code
Composite grid	the B1 grid: the box $[0.25,0.75{]}^2$ refined 2:1 on the periodic unit square	Composite, proto2d.py
Space	WENO5-JS with Lax–Friedrichs splitting, the weights of adam_weno_object ($ϵ={10}^{-6}$, exponent 3)	weno5, weno_flux, proto2d.py; adam_weno_object.F90
Time	classical RK4, $\mathrm{\Delta }t=0.4h(64h{)}^{1/4}/\text{speed}$	advance, proto2d.py
Reflux policies	always (today), never, troubled: a coarse box face takes the mean of its fine fluxes only where the sensor $r=\underset{k}{min}{w}_k/d_k$ of the coarse face or of either fine face is below $\tau =0.2$ (the weno-riemann threshold)	Composite.rhs, proto2d.py
Arms	today's mean/cubic/always, and lag6/quintic under each policy	b2, proto2d.py
FLUME counterpart	reflux_policy = troubled (E2)	flag_troubled_seam_cells, adam_flume_cpu_object.F90

Initial and boundary conditions

Leg	Law	Initial condition	Crossing	Final time
smooth	$u_t+u_x+u_y=0$	the B1 advection, $\sin 2\pi x\sin 2\pi y$	—	$t=1$
c2f	Burgers along $x$	$u=1$ on $[0,0.15]$, 0 elsewhere	$x=0.25$, coarse to fine, at $t=0.2$	$t=0.5$
f2c	Burgers along $x$	$u=1$ on $[0.45,0.6]$, 0 elsewhere	$x=0.75$, fine to coarse, at $t=0.3$	$t=0.6$
weak	Burgers along $x$	a ${10}^{-3}$ pulse on $[0,0.15]$ riding on $u=1$	$x=0.25$	$t=0.5$

The rarefaction catches the shock on the way. The weak pulse's smoothness indicators sit near $ϵ$, so the JS sensor misses it: the missed-detection control. The pulses start from exact cell averages, so the discrete mass is the exact one (sampling the step at the centres offsets it by up to a cell, which made the first version of the shock metric erratic). Grids: smooth leg $N$ = 32 / 64 / 128; shock legs $N=64$ (--long: 64 and 128).

Input files

File	Role
src/tests/flume/verification/amr-proto/proto2d.py	b2 [--cells ...] [--shock-cells ...] [--leg L ...] [--tau 0.2] [--check], L one of smooth, c2f, f2c, weak
src/tests/flume/verification/amr-proto/check.sh	runs B1, B2 and B3 (--legs b2 for B2 alone)
src/tests/flume/verification/make_doc_figures.py --only seam-proto-b2	draws the figure (fig_seam_proto_b2; NumPy only, about 3 min)

cd src/tests/flume/verification/amr-proto && ./check.sh --legs b2  # B2: about 3 min
./check.sh --legs b2 --long                                        # shock legs at N = 64 and 128, about 5 min

Metrics against the exact solution: the mass defect over the pulse's own mass, and the mass-equivalent shock position of every row (the cells within $6H$ of the exact shock), in local cells. Bounds (--check): the smooth L1 and L∞ orders of troubled at least 4.7; on c2f and f2c, troubled within 0.05 cells of always and its mass defect at most ${10}^{-3}$, while never's is at least ${10}^{-3}$ (the negative control: the metrics see a missing reflux).

Results

Left: the smooth leg, $L_1$ against $N$ per arm (64 → 128 orders in the legend); always is capped near order 3, troubled keeps the uniform order. Centre and right: the Burgers pulses at $N=64$, the mean shock position error and the mass defect per policy. troubled places the strong shocks exactly where always does and keeps the mass defect below ${10}^{-3}$ of the pulse; never misplaces the c2f shock by a cell and loses $1.8\cdot {10}^{-2}$; on the weak pulse the sensor flags nothing, so troubled equals never (the missed-detection control).

Smooth leg, orders of $N$ = 64 → 128:

Arm	L1	L∞	L∞ corner / edge	Faces flagged
uniform	5.01	4.95	–	–
mean, cubic, always (today)	3.03	2.22	2.22 / 2.05	–
lag6, quintic, always	2.96	2.10	2.10 / 1.96	–
lag6, quintic, troubled	5.01	4.95	5.10 / 5.02	0 %
lag6, quintic, never	5.01	4.95	5.10 / 5.02	–

Shock legs (lag6/quintic; mean shock position in local cells, mass defect over the pulse's mass, the uniform control in brackets):

Case, N	always	troubled	never	Faces flagged
c2f, 64	−0.53, $0$	−0.53, $5.0\cdot {10}^{-5}$	−1.12, $1.8\cdot {10}^{-2}$	15 %
c2f, 128 (−0.30)	−0.50, $2\cdot {10}^{-16}$	−0.50, $3.6\cdot {10}^{-5}$	−1.07, $8.6\cdot {10}^{-3}$	8 %
f2c, 64	−0.25, $2\cdot {10}^{-16}$	−0.25, $6.5\cdot {10}^{-5}$	+0.05, $1.7\cdot {10}^{-2}$	15 %
f2c, 128 (−0.37)	−0.25, $4\cdot {10}^{-16}$	−0.25, $6.9\cdot {10}^{-6}$	+0.06, $8.4\cdot {10}^{-3}$	7 %
weak, 64 (+0.02)	−0.003, ${10}^{-12}$	−0.014, $3.9\cdot {10}^{-4}$	−0.014, $3.9\cdot {10}^{-4}$	0 %
weak, 128 (+0.01)	−0.015, ${10}^{-12}$	−0.022, $3.2\cdot {10}^{-5}$	−0.022, $3.2\cdot {10}^{-5}$	0 %

Related cases: B1 is the linear prototype; E2 measures the policy in FLUME; AL measured the sensor's false positives on the Euler vortex.

Findings

• troubled recovers the order. On the smooth leg the sensor flags no seam face, so troubled is never: order 5 in L1 and L∞, as the uniform control, against about 2 with the reflux always on.
• troubled keeps the shock where always puts it. The two differ by at most 0.003 cells; never moves the shock by 0.3 to 0.6 cells and loses ${10}^{-2}$ of the pulse's mass. The reflux at the flagged faces (7 to 15 % of the seam-face evaluations) is what holds the shock.
• troubled is near-conservative, not conservative. Its mass defect, $7\cdot {10}^{-6}$ to $6.5\cdot {10}^{-5}$ of the pulse, comes from the unflagged smooth faces (the rarefaction crossing the seam) and from the steps before the shock reaches the sensor's stencil; it is two to three orders below never's.
• A missed detection is a small error here. The weak shock is never flagged, so troubled runs as never: its shock moves by 0.007 to 0.011 cells against always and it loses $3\cdot {10}^{-5}$ to $4\cdot {10}^{-4}$ of its own mass. What the JS sensor misses is what the reflux matters least for, but not nothing.
• The corners are again not the L∞ cause. Under the nonlinear weights corner and edge maxima still converge at the same rate; A1's Euler L∞ order 1.46 is not reproduced by a scalar model, and stays open for the system (the characteristic projection).
• Verdict (D-M6-4): go for Phase E, with three conditions on record. The order is recovered only together with the high-order transfers of Phases C and D: with today's mean restriction the order stays at 2 whatever the reflux does (B1). The policy is near-conservative (${10}^{-5}$ of the shocked mass), not exact. And A4 (AL) measured 5 % false positives on the Euler vortex with the sensor read on primitive variables, where this scalar model flags none; E must measure its own false-positive rate, since each flagged smooth face brings the order cap back.