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

Published online by Cambridge University Press:  20 August 2015

Michael Dumbser  and
Eleuterio F. Toro
Michael Dumbser*
Affiliation:
Laboratory of Applied Mathematics, University of Trento, I-38100 Trento, Italy
Eleuterio F. Toro*
Affiliation:
Laboratory of Applied Mathematics, University of Trento, I-38100 Trento, Italy
*
Corresponding author.Email:michael.dumbser@ing.unitn.it
Email:toroe@ing.unitn.it
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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 gas-dynamics 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 PNPM schemes on unstructured meshes recently proposed in [9].

Keywords

35L6565M0876M1276L05Universal Osher-Solomon fluxuniversal Roe fluxhigh resolution shock-capturing finite volume schemesWENO schemesreconstructed discontinuous Galerkin methodsPNPM schemesEuler equationsgas dynamicsideal gas and real gas equation of stateMHD equationsrelativistic MHD equationsnonlinear elasticity
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 3 , September 2011 , pp. 635 - 671
Copyright
Copyright © Global Science Press Limited 2011

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