A Two-Dimensional Version of the Godunov Scheme for Scalar Balance Laws

A Godunov scheme is derived for two-dimensional scalar conservation laws without or with source terms following ideas originally proposed by Boukadida and LeRoux [9] in the context of a staggered Lax-Friedrichs scheme. In both situations, the numerical fluxes are obtained at each interface of a uniform Cartesian computational grid just by means of the “external waves” involved in the entropy solution of the elementary 2d Riemann problems; in particular, all the wave-interaction phenomena is overlooked. This restriction of the wave pattern suffices for deriving the exact numerical fluxes of the staggered LxF scheme, but it furnishes only an approximation for the Godunov scheme: we show that under convenient assumptions, these flux functions are smooth and the resulting discretization process is stable under nearly-optimal CFL restriction. A well-balanced extension is presented, relying on the Curl-free component of the Helmholtz decomposition of the source term. Several numerical tests against exact 2D solutions are performed for convex, non-convex and inhomogeneous equations and the time evolution of the L truncation error is displayed.