Godunov method for nonconservative hyperbolic systems

This paper is concerned with the numerical approximation of Cauchy problems for one-dimensional nonconservative hyperbolic systems. The theory developed by Dal Maso et al. [J. Math. Pures Appl. 74 (1995) 483–548] is used in order to define the weak solutions of the system: an interpretation of the nonconservative products as Borel measures is given, based on the choice of a family of paths drawn in the phase space. Even if the family of paths can be chosen arbitrarily, it is natural to require this family to satisfy some hypotheses concerning the relation of the paths with the integral curves of the characteristic fields. The first goal of this paper is to investigate the implications of three basic hypotheses of this nature. Next, we show that, when the family of paths satisfies these hypotheses, Godunov methods can be written in a natural form that generalizes their classical expression for systems of conservation laws. We also study the well-balance properties of these methods. Finally, we prove the consistency of the numerical scheme with the definition of weak solutions: we prove that, under hypothesis of bounded total variation, if the approximations provided by a Godunov method based on a family of paths converge uniformly to some function as the mesh is refined, then this function is a weak solution (related to that family of paths) of the nonconservative system. We extend this result to a family of numerical schemes based on approximate Riemann solvers.