Commun. Comput. Phys., 10 (2011), pp. 635-671.


On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws

Michael Dumbser 1*, Eleuterio F. Toro 1

1 Laboratory of Applied Mathematics, University of Trento, I-38100 Trento, Italy.

Received 17 June 2010; Accepted (in revised version) 2 December 2010
Available online 1 June 2011
doi:10.4208/cicp.170610.021210a

Abstract

This paper is concerned with a new version of the Osher-Solomon Riemann solver and is based on a numerical integration of the path-dependent dissipation matrix. The resulting scheme is much simpler than the original one and is applicable to general hyperbolic conservation laws, while retaining the attractive features of the original solver: the method is entropy-satisfying, differentiable and complete in the sense that it attributes a different numerical viscosity to each characteristic field, in particular to the intermediate ones, since the full eigenstructure of the underlying hyperbolic system is used. To illustrate the potential of the proposed scheme we show applications to the following hyperbolic conservation laws: Euler equations of compressible gasdynamics with ideal gas and real gas equation of state, classical and relativistic MHD equations as well as the equations of nonlinear elasticity. To the knowledge of the authors, apart from the Euler equations with ideal gas, an Osher-type scheme has never been devised before for any of these complicated PDE systems. Since our new general Riemann solver can be directly used as a building block of high order finite volume and discontinuous Galerkin schemes we also show the extension to higher order of accuracy and multiple space dimensions in the new framework of $P_NP_M$ schemes on unstructured meshes recently proposed in [9].

AMS subject classifications: 35L65, 65M08, 76M12, 76L05

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Key words: Universal Osher-Solomon flux, universal Roe flux, high resolution shock-capturing finite volume schemes, WENO schemes, reconstructed discontinuous Galerkin methods, $P_N P_M$ schemes, Euler equations, gas dynamics, ideal gas and real gas equation of state, MHD equations, relativistic MHD equations, nonlinear elasticity.

*Corresponding author.
Email: michael.dumbser@ing.unitn.it (M. Dumbser), toroe@ing.unitn.it (E. F. Toro)
 

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