关于Van Leer和AUSM通量-向量分裂方案的谱性质
On the Spectral Properties of Van Leer and AUSM Flux-Vector Splitting Schemes
AI总结：
本文研究了Van Leer和AUSM分裂方案的谱性质，证明了其雅可比矩阵的特征值特性，并通过数值实验验证了AUSM二阶压力分裂方案的特征值为实且正。
摘要：
Van Leer的通量-向量分裂方案是计算流体力学的基石，但其原始证明特征值符号条件的证明形式较为简略。本文对Van Leer分裂方案在一维欧拉方程中所关联的雅可比矩阵的特征值进行了详尽且严谨的分析。通过构造判别式的斯特姆序列，我们证明，在允许的参数范围内$1 \le γ\le 3$，$|M|<1$，且$a>0$时，雅可比$\partial F^+/\partial U$具有一个零特征值和两个正实特征值，证实了Van Leer的原始主张。此外，我们还将分析扩展到Liou和Steffen提出的原始AUSM方案的两个变种，考虑了线性和二阶压力分裂。对于线性压力分裂，我们显示特征值并非全部同号；而对于二阶压力分裂，我们证明了特征方程的所有系数均为正。附录中报告的数值实验证实了AUSM二阶压力分裂的判别式非负性，暗示其特征值为实且正。
英文摘要：
The flux-vector splitting scheme of Van Leer is a cornerstone of computational fluid dynamics, yet its original proof of the eigenvalue sign condition was presented in a condensed form. In this work, we provide a detailed and rigorous analysis of the eigenvalues of the Jacobian matrices associated with the Van Leer splitting for the one-dimensional Euler equations. By constructing the Sturm sequence of the discriminant, we prove that for the admissible parameter range $1 \le γ\le 3$, $|M|<1$, and $a>0$, the Jacobian $\partial F^+/\partial U$ has one zero eigenvalue and two positive real eigenvalues, confirming Van Leer's original claim. Furthermore, we extend our analysis to two variants of the original AUSM scheme (Advection Upstream Splitting Method) proposed by Liou and Steffen, considering both linear and second-order pressure splittings. For the linear pressure splitting we show that the eigenvalues are not all of the same sign, while for the second-order pressure splitting we prove that all coefficients of the characteristic equation are positive. Numerical experiments reported in the appendix confirm the non-negativity of the discriminant for the AUSM with the second-order pressure splitting, implying that its eigenvalues are real and positive.
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