正定的丧失是高魏森贝格数崩溃的症状，而非原因
Loss of positive definiteness is a symptom, not the cause, of high-Weissenberg-number breakdown
AI总结：
研究高魏森贝格数下数值崩溃问题，推导不定状态定量理论，通过多种测试和分析得出正定丧失非崩溃必要充分条件，离散耦合路径起决定作用，违反是分辨率指标并提供监测器。
AI中文摘要：
高魏森贝格数下的数值崩溃常被归因于构象张量对称正定（SPD）的丧失，这一结论源于无溶剂粘度的麦克斯韦型模型。当溶剂分数β>0时，任意对称应力的初值问题在局部是适定的。我们推导了不定状态缺失的定量理论并测试其计算结果。冻结系数分析给出了在波数上一致有界的增长率和方向分辨不稳定性阈值λmin(A)<-β/(1 - β)；应力扩散提供了一个封闭形式的截止值，当溶剂粘度消失时恢复经典的σ∝k突变。一个行列式恒等式表明违反情况在时间尺度λ/2上自我修复，所以持续的违反衡量了重新产生它们的截断误差。光谱和格子玻尔兹曼测试再现了阈值、溶剂分数反转和分辨率独立性。在四辊轧机干预中，强制SPD将爆炸延迟15个对流时间，但使驻点魏森贝格数降低30%。在五种耦合方案中，局部二阶矩应力源在Wi = 50时通过完整预算保持稳定，同时携带det A≈ - 8.5×10^5；散度耦合变体在t* = 47时失败。存活的方案在Wi = 10和20时与已发表的基准在0.05%和0.18%内匹配。因此，正定的丧失对于崩溃既非必要也非充分条件：离散耦合路径起决定作用，违反是一种分辨率指标，我们提供了运行时监测器。
英文摘要：
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.
