On the Spectral Properties of Van Leer and AUSM Flux-Vector Splitting Schemes

TL;DR

This paper rigorously analyzes the eigenvalues of Van Leer and AUSM flux-splitting schemes for Euler equations, confirming Van Leer's eigenvalue sign condition and exploring the spectral properties of AUSM variants.

Contribution

Provides a detailed eigenvalue analysis of Van Leer and AUSM flux-splitting schemes, extending Van Leer's original proof and examining AUSM's spectral characteristics with different pressure splittings.

Findings

Van Leer's scheme has one zero and two positive eigenvalues within certain parameters.

AUSM with linear pressure splitting has eigenvalues of mixed signs.

AUSM with second-order pressure splitting has all positive eigenvalues.

Abstract

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 $1 \le \gamma \le 3$, $|M|<1$, 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.