IMEX finite volume methods for multi-dimensional hyperbolic systems

A general framework for the semi-implicit discretization of multidimensional conservative hyperbolic systems is proposed. The discretization approach is based on the method-of-line strategy. The spatial discretization uses an unstructured Finite Volume (FV) technique, and a non-oscillatory reconstruction procedure to provide a spatial accuracy of order higher than one. The time derivative is discretized by an Implicit-Explicit Runge-Kutta (IMEX-RK) stepping scheme. The resulting matrix operators are analized within the framework of the M-matrix theory. Sufficient conditions for positive-in-the-mean discrete solution are derived. It is also proved that the non-linear implicit problem, whose solution is needed by the IMEX-RK approach, always admits a unique solution under general hypothesis. Several numerical examples illustrates the behavior of the method