A novel flux splitting scheme for the Euler equations with general equation of state (Eleuterio Toro, (Université de Trente, Italie)

This talk is concerned with a recently proposed flux vector splitting method for the Euler equations [1] which, compared to existing splitting methods, has some distinctive advantages, such as exact recognition of stationary isolated contact waves, simplicity, robustness and efficiency. Two features of the new splitting are: complete separation of pressure from advection terms and identification of a reduced pressure system that furnishes all required information for constructing the full numerical flux in a simple manner. The first-order method was originally proposed for the 1D Euler equations for ideal gases. Here we present the following extensions: (a) general equation of state, (2) multiple space dimensions and (c) high-order of accuracy in both space and time, on unstructured meshes, using the ADER approach [2]-[4], both in its finite volume form [4] and its discontinuous Galerkin form [5], [6]. Computational results for a carefully selected suit of test problems will be presented.

References

[1] E F Toro and M E Vázquez-Cendón. Flux splitting schemes for the Euler equations. Computers and Fluids. Vol. 70, Pages 1-12, 2012.

[2] E F Toro, R C Millington and L A M Nejad. Towards very high-order Godunov schemes. In Godunov Methods: Theory and Applications. Edited Review. E F Toro (Editor). Kluwer Academic/Plenum Publishers. Conference in Honour of S K Godunov. Vol. 1, pages 897- 902. New York, Boston and London, 2001. 


[3] E F Toro and V A Titarev. Solution of the generalised Riemann problem for advection-reaction equations. Proceedings of the Royal Society of London. Series A. Vol. 458, pages 271-281, 2002.

[4] V A Titarev and E F Toro. ADER: arbitrary high order Godunov approach. Journal of Scientific Computing. Vol. 17, pages 609-618, 2002.

[5] M Dumbser, D Balsara, E F Toro and C D Munz. A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes on unstructured meshes. Journal of Computational Physics. Vol. 227, pages 8209-8253, 2008.

[6] C E Castro, E F Toro and M Kaeser. ADER scheme on unstructured meshes for shallow water: simulation of tsunami waves. Geophysics Journal International. Vol. 189, pages 1505-1520, 2012. doi: 10.1111/j.1365-246X.2012.05471.x