High-order IMAX-RK finite volume methods for multidimensional hyperbolic systems

In this paper we present a high-order accurate cell-centered finite volume method for the semi-implicit discretization of multidimensional hyperbolic systems in conservative form on unstructured grids. This method is based on a special splitting of the physical flux function into a convective and a non-convective part. The convective contribution to the global flux is treated implicitly by mimicking the upwinding of a scalar linear flux function while the rest of the flux is discretized in an explicit way. The spatial accuracy is ensured by allowing non-oscillatory polynomial reconstruction procedures, while the time accuracy is attained by adopting a Runge-Kutta stepping scheme. The method can be naturally considered in the framework of the IMplicit-EXplicit (IMEX) schemes and the properties of the resulting operators are analysed using the properties of M-matrices.