On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws (Communications in Computational Physics, 2011)
This paper is concerned with a new version of the Osher-Solomon Riemannsolver 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 togeneral hyperbolic conservation laws, while retaining the attractive features of the originalsolver: the method is entropy-satisfying, differentiable and complete in the sensethat it attributes a different numerical viscosity to each characteristic field, in particularto the intermediate ones, since the full eigenstructure of the underlying hyperbolic systemis used. To illustrate the potential of the proposed scheme we show applicationsto the following hyperbolic conservation laws: Euler equations of compressible gasdynamicswith ideal gas and real gas equation of state, classical and relativistic MHDequations 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 neverbeen devised before for any of these complicated PDE systems. Since our new generalRiemann solver can be directly used as a building block of high order finite volumeand discontinuous Galerkin schemes we also show the extension to higher order ofaccuracy and multiple space dimensions in the new framework of PNPM schemes onunstructured meshes recently proposed in [9].