Numerical methods for nonconservative hyperbolic systems: a theoretical framework

The goal of this paper is to provide a theoretical framework allowing to extend some general concepts related to the numerical approximation of 1d 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 the 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 we 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.