Use of Jordan forms for convection-pressure split Euler solvers
This work provides a method to handle weakly hyperbolic systems in computational fluid dynamics, which is a specific technical challenge for researchers developing numerical schemes for Euler equations.
The authors addressed the problem of constructing upwind solvers for convection-pressure split Euler flux functions with genuine weakly hyperbolic subsystems. They used Jordan canonical forms to generate generalized eigenvectors and developed two numerical schemes, which were tested on 1-D and 2-D benchmark problems, showing effectiveness in handling shock instabilities.
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 Vázquez 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.