Use of Jordan forms for convection-pressure split Euler solvers

TL;DR

This paper introduces a novel approach using Jordan forms to construct upwind Euler solvers for weakly hyperbolic systems, enabling better handling of convection-pressure splits with genuine weak hyperbolicity.

Contribution

The paper develops new upwind solvers based on Jordan canonical forms to address weak hyperbolicity in convection-pressure split Euler equations, ensuring complete eigenvector sets.

Findings

Successfully constructed two new numerical schemes using Jordan forms.

Schemes tested on 1-D and 2-D benchmark problems, including shock instabilities.

Jordan form approach effectively handles weakly hyperbolic parts of Euler systems.

Abstract

In this study, we analyze convection-pressure split Euler flux functions which contain genuine weakly hyperbolic convection subsystems. A system is said to be a genuine weakly hyperbolic if all eigenvalues are real with no complete set of linearly independent (LI) eigenvectors. To construct an upwind solver based on flux difference splitting (FDS) framework, we require to generate complete set of LI eigenvectors. This can be done through addition of generalized eigenvectors which can be computed from theory of Jordan canonical forms. Once we have complete set of LI generalized eigenvectors, we construct upwind solvers in convection-pressure splitting framework. Since generalized eigenvectors are not unique, we take extra care to ensure no direct contribution of generalized eigenvectors in the final formulation of both the newly developed numerical schemes. First scheme is based on Zha and Bilgen type splitting approach, while second is based on Toro and V\'azquez splitting. Both the schemes are tested on several bench-mark test problems on 1-D and one of them is tested on some typical 2-D test problems which involve shock instabilities. The concept of generalized eigenvector based on Jordan forms is found to be useful in dealing with the genuine weakly hyperbolic parts of the considered Euler systems.