Numerical methods for nonconservative hyperbolic systems: a theoretical framework.

2006/01/01 by Carlos Parés · 16 citations
Engineering · Mathematics · #Computational Fluid Dynamics and Aerodynamics #Navier-Stokes equation solutions #Fluid Dynamics and Turbulent Flows

paper · doi:10.1137/050628052

Abstract

The goal of this paper is to provide a theoretical framework allowing one to extend some general concepts related to the numerical approximation of 1-d conservation laws to the more general case of first order quasi-linear hyperbolic systems. In particular this framework is intended to be useful for the design and analysis of well-balanced numerical schemes for solving balance laws or coupled systems of conservation laws. First, the concept of path-conservative numerical schemes is introduced, which is a generalization of the concept of conservative schemes for systems of conservation laws. Then, we introduce the general definition of approximate Riemann solvers and give the general expression of some well-known families of schemes based on these solvers: Godunov, Roe, and relaxation methods. Finally, the general form of a high order scheme based on a first order path-conservative scheme and a reconstruction operator is presented.

Citations

• A Well-Balanced Scheme for the Numerical Processing of Source Terms in Hyperbolic Equations
• Why nonconservative schemes converge to wrong solutions: error analysis

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