2026/07/16 by Yuan Yu, Lanjin Lian, Feiyang Chu
#physics.comp-ph
Numerical breakdown at high Weissenberg number is often attributed to loss of symmetric positive definiteness (SPD) of the conformation tensor. That conclusion follows from Maxwell-type models without solvent viscosity. With solvent fraction β>0, the initial-value problem is locally well posed for arbitrary symmetric stress. We derive the missing quantitative theory for indefinite states and test its computational consequences. Frozen-coefficient analysis gives a growth rate uniformly bounded in wavenumber and the direction-resolved instability threshold λmin(A)<-β/(1-β); stress diffusion supplies a closed-form cutoff, while the classical σ∝ k catastrophe is recovered as solvent viscosity vanishes. A determinant identity shows that violations self-heal on the timescale λ/2, so persistent violations measure the truncation error that recreates them. Spectral and lattice Boltzmann tests reproduce the threshold, solvent-fraction reversal, and resolution independence. In four-roll-mill interventions, enforcing SPD delays blow-up by 15 convective times but reduces the stagnation-point Weissenberg number by 30%. Across five coupling schemes, a local second-moment stress source remains stable through the full budget at Wi=50 while carrying det A≈-8.5×105; the divergence-coupled variant fails at t^*=47. The surviving scheme matches published benchmarks within 0.05% and 0.18% at Wi=10 and 20. Thus loss of positive definiteness is neither necessary nor sufficient for breakdown: the discrete coupling route decides, and the violation is a resolution gauge for which we provide run-time monitors.