Loss of positive definiteness is a symptom, not the cause, of high-Weissenberg-number breakdown

Abstract

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 $λ_{\min}(A)<-β/(1-β)$; stress diffusion supplies a closed-form cutoff, while the classical $σ\propto k$ 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 $\mathrm{Wi}=50$ while carrying $\det A\approx-8.5\times10^5$; the divergence-coupled variant fails at $t^*=47$. The surviving scheme matches published benchmarks within 0.05% and 0.18% at $\mathrm{Wi}=10$ 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.