A two-dimensional version of the Godunov scheme for convex, 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 [Math. Comput., 63 (1994), pp. 541-553] 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 two-dimensional (2D) Riemann problems; in particular, all the wave-interaction phenomena are overlooked. This restriction of the wave pattern suffices for deriving the exact numerical fluxes of the staggered Lax-Friedrichs 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, nonconvex, and inhomogeneous equations and the time-evolution of the L1 truncation error is displayed.