Foundations Of Computational Mathematics (FoCM 2005)

Workshop on Foundations of Numerical PDEs

July 7 - July 9, 2005

Universidad de Cantabria in Santander, Spain

Two-sided Generalized Riemann solvers:
Advantages and Shortcomings

Rosa Donat

Department of Applied Mathematics at Universitat de València

Abstract (*):   Many modern shock capturing schemes can be classified as characteristic based schemes. These rely on a local diagonalization of the hyperbolic system that involves the Jacobian matrix of the flux function at each cell interface. It is now relatively well known that the interface state used for the computation of the Jacobian can have a significant effect on the fine features of the numerical solution [1]. A new numerical flux function is developed in [2] that takes into account the two physically relevant states at the interface, without ever constructing an artificially mixed interface state. This new flux function reduces to the well known Shu-Osher scheme [3] for scalar conservation laws, but becomes fundamentally different in the case of systems. The resulting numerical scheme can be combined with the ENO-reconstruction technology to produce state of the art HRSC schemes that have been applied in various scenarios. This talk will review recent results that derive from the use of these new schemes, with special emphasis in the advantages and shortcomings found in the simulations.

[1] R. P. Fedkiw, B. Merriman, R. Donat, S. Osher, The Penultimate Scheme for Systems of
Conservation Laws: Finite Difference ENO with Marquina's Flux Splitting, Innovative methods for Numerical solutions of PDEs, Ed. M. M. Hafez, J. J. Chattot 49, 1998.
[2] R. Donat and A. Marquina, Capturing shock reflections: An improved flux formula, J. Comp. Phys., 125, 1996.
[3] C. W Shu and S. J. Osher, Efficient implementation of essentially non-oscillatory, shock-capturing schemes II.  J. Comp. Phys., 83, 1989.

*With G. Chiavassa and G. Haro