An evolution Galerkin scheme for the shallow water magnetohydrodynamic equations in two space dimensions

In this paper we propose a new finite volume evolution Galerkin (FVEG) scheme for the shallow water magnetohydrodynamic (SMHD) equations. We apply the exact integral equations already used in our earlier publications to the SMHD system. Then, we approximate these integral equation in a general way which does not exploit any particular property of the SMHD equations and should thus be applicable to arbitrary systems of hyperbolic conservation laws in two space dimensions. In particular, we investigate more deeply the approximation of the spatial derivatives which appear in the integral equations. The divergence free condition is satisfied discretely, i.e. at each vertex. First numerical results confirm reliability of the numerical scheme.