A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems

We propose a simple extension of the well-known Riemann solver of Osher and Solomon (Math. Comput. 38:339-374, 1982) to a certain class of hyperbolic systems in non-conservative form, in particular to shallow-water-type and multi-phase flow models. To this end we apply the formalism of path-conservative schemes introduced by Parés (SIAM J. Numer. Anal. 44:300-321, 2006) and Castro et al. (Math. Comput. 75:1103-1134, 2006). For the sake of generality and simplicity, we suggest to compute the inherent path integral numerically using a Gaussian quadrature rule of sufficient accuracy. Published path-conservative schemes to date are based on either the Roe upwind method or on centered approaches. In comparison to these, the proposed new path-conservative Osher-type scheme has several advantages. First, it does not need an entropy fix, in contrast to Roe-type path-conservative schemes. Second, our proposed non-conservative Osher scheme is very simple to implement and nonetheless constitutes a complete Riemann solver in the sense that it attributes a different numerical viscosity to each characteristic field present in the relevant Riemann problem; this is in contrast to centered methods or incomplete Riemann solvers that usually neglect intermediate characteristic fields, hence leading to excessive numerical diffusion. Finally, the interface jump term is differentiable with respect to its arguments, which is useful for steady-state computations in implicit schemes. We also indicate how to extend the method to general unstructured meshes in multiple space dimensions. We show applications of the first order version of the proposed path-conservative Osher-type scheme to the shallow water equations with variable bottom topography and to the two-fluid debris flow model of Pitman & Le. Then, we apply the higher-order multi-dimensional version of the method to the Baer-Nunziato model of compressible multi-phase flow. We also clearly emphasize the limitations of our approach in a special chapter at the end of this article.