Integration-point stress history grows an inlet instability when dt is halved (Wi 5 cylinder)

[lmoresi]

On the viscoelastic DFG cylinder (pure Maxwell, Wi 5, 1/20 mesh, DEVSS eta_a = nu/4, EulerianSUPG momentum) the integration_point stress history is stable at dt = 1/120 for 240 steps and grows an instability at dt = 1/240 that starts at the inlet, not at the cylinder. The eulerian (grid) history at the same two time steps is unchanged to 0.7% in drag.

Wi 5, 1/20, drag window t = 0.67 to 1, near-cylinder stress L2, max stress peak:

history	dt	cd	near L2	peak max
grid	1/120	1.112	0.0034	0.109
grid	1/240	1.120	0.0036	0.116
IP	1/120	1.193	0.0033	0.090
IP	1/240	0.932	0.0347	2.281

Stress peak on the IP dt = 1/240 run by step (dt = 1/240, so step 240 is t = 1): 140: 0.15, 160: 0.27, 180: 0.49, 200: 1.04, 220: 1.60, 240: 2.28. Doubling every ~25 steps, still growing at the end of the run. The stress invariant frames show it as a faceted, cell-scale stress structure attached to the inlet boundary and spreading downstream to the cylinder; the grid run is clean there.

Candidate mechanism, not yet tested. The trace-back flavours do not use inflow_value; a departure point that lands outside the domain is restored to the boundary and takes the transported field's value there (docstring on inflow_value). At the inlet that is an exact carry of the boundary value from the committed flux, with no relaxation applied on the way, so a boundary node whose stress is above the relaxed inflow stress feeds itself through the memory term. The same exact-carry-at-a-boundary mechanism is the candidate for #735 at the no-slip wall. The grid history is clean because it has a real inflow condition (-uw_stress_inflow 1 in the benchmark script, the relaxed stress of the incoming flow).

Predictions being tested: the nodal trace-back at the same dt and Wi should show the same inlet growth (same boundary carry); the IP history at Wi 1 and dt = 1/240 should not, or much more slowly (relaxation per step is five times stronger).

Runs: ~/+Simulations/stress_transport/cylinder/{w5_grid,w5dt_grid,w5_ip,w5dt_ip}, script run_wi5_dt.sh; frames frames_w5dt_tauII (steps 120, 168, 232). Worktree branch feature/levelset-supg at b5bfc29d.

[lmoresi]

Discriminators run (1/20, dt = 1/240, DEVSS eta_a = nu/4, 240 steps to t = 1). Stress peak by step:

run	100	140	180	220	240
IP, Wi 5	0.087	0.146	0.493	1.602	2.281
nodal, Wi 5	0.178	0.268	0.330	0.365	0.371
IP, Wi 1	0.175	0.242	0.252	0.254	0.254
grid, Wi 5	0.088	0.099	0.112	0.116	0.114

The nodal trace-back at the same dt and Wi does not grow, so the boundary-carry candidate in the report above is not the mechanism: the clamp at the inlet is shared by both trace-back flavours and only the integration-point one grows. The IP history at Wi 1 does not grow either. The growth needs all three of: the integration-point sampling, the halved time step (Courant about 0.3 at the inlet centre instead of 0.6), and the long relaxation time (per-step retention 0.992 at Wi 5 against 0.958 at Wi 1).

That is the low-Courant growing mode of the integration-point scheme (#703 analysis: inter-cell aliasing of the shifted sampled field, a creep of ~1e-3 per step on the scalar problem) with the viscoelastic memory term as its amplifier: whatever the transport gains at grid scale is re-injected almost in full at high Wi, which turns a creep into a doubling every ~25 steps. It shows at the inlet first, where the flow is unidirectional and the stress signal is smallest, so the mode is not masked.

The damping the #703 analysis calls for, kappa_stab = eps h^2 / (pi^2 dt), has no diffusion term to attach to in a stress history; the analogue here is a smoothing weight on the per-step commit projection (SNES_MultiComponent_Projection, currently smoothing 0). A run with that weight set from the formula is the confirming test, and if it holds the option belongs with the stabilisation contributions being collected on the planning list, off by default.

[lmoresi]

Ruling (Louis): the integration-point history is operated at Courant ~1, as settled for the scalar case, and the damping term is not the remedy — it was not liked then and any use of it needs a careful look first. Below Courant 1 the scheme is to be treated with care, more so with a memory term. For convergence checks on this history, refine h and dt together at fixed Courant rather than halving dt alone. The grid history is unaffected by dt and remains the reference for a dt sweep.

[lmoresi]

Integrator check. ETD-1 on the integration-point history reproduces BDF-1 step for step: flat at dt 1/120 for 240 steps at Wi 2, 5 and 10, and at Wi 5, dt 1/240 the same growth with the same doubling time (stress peak 0.14 to 2.10 over steps 140 to 240, against 0.15 to 2.28 for BDF-1). Consistent with the mechanism as stated: the amplifier is the per-step retention of the carried stress, and e^{-dt/lambda} and lambda/(lambda+dt) agree to four digits there. The integrator is an accuracy lever on this history, not a stability one.

[lmoresi]

Order 2 on the integration-point history, revisited. BDF-2 on IP at Wi 5, dt 1/120 doubles every ~10 steps from step 60 (peak 5.06 by step 120) where BDF-1 and ETD-1 are flat. ETD-2 with a traced forcing history (5e1266e3: the strain rate sampled at the same departure point as the stress) is flat for 240 steps at the same settings, peak 0.09 throughout. So the second level itself is not the problem; extrapolating the stress across two levels with opposite-sign weights is. The Courant floor of this history is unchanged by the integrator (ETD-1 = BDF-1 step for step at dt 1/240), and order 2 is now available on it without lowering that floor.

[lmoresi]

Cross-reference #745: on a plain box channel with no cylinder the integration-point history grows at the inlet from t ~ 0.8 at Wi 1 and Wi 5, at dt 1/60 and 1/120 alike and with DEVSS off, while nodal and grid are stable. That inlet growth is dt-independent there, so it is a different mechanism from the low-Courant mode discussed here, though the dt 1/240 cylinder case above also grew at the inlet first.