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Volume 10, Issue 3
On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws

Michael Dumbser & Eleuterio F. Toro

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

Published online: 2011-10

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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 PNPM schemes on unstructured meshes recently proposed in [9].

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@Article{CiCP-10-635, author = {Michael Dumbser and Eleuterio F. Toro}, title = {On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws}, journal = {Communications in Computational Physics}, year = {2011}, volume = {10}, number = {3}, pages = {635--671}, 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 PNPM schemes on unstructured meshes recently proposed in [9].

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.170610.021210a}, url = {http://global-sci.org/intro/article_detail/cicp/7455.html} }
TY - JOUR T1 - On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws AU - Michael Dumbser & Eleuterio F. Toro JO - Communications in Computational Physics VL - 3 SP - 635 EP - 671 PY - 2011 DA - 2011/10 SN - 10 DO - http://doi.org/10.4208/cicp.170610.021210a UR - https://global-sci.org/intro/article_detail/cicp/7455.html KW - AB -

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 PNPM schemes on unstructured meshes recently proposed in [9].

Michael Dumbser and Eleuterio F. Toro. (2011). On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws. Communications in Computational Physics. 10 (3). 635-671. doi:10.4208/cicp.170610.021210a
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