Well‐balanced central schemes for the Saint‐Venant systems

Abstract

In this paper we present a new well‐balanced unstaggered central finite volume scheme for the numerical solution of the one‐dimensional Saint‐Venant systems on flat or variable bottom topographies. The Saint‐Venant equations also known as the shallow water equations (SWE) are useful when modeling the hydrodynamics of coastal oceans and lakes and the simulation of tsunami waves and inundation floods. The Saint‐Venant system is a hyperbolic system with a nonzero geometrical source term when flows over variable bottom topographies are considered. In this work we develop a new one‐dimensional well‐balanced central scheme for the numerical solution of SWE systems: The numerical base scheme is an unstaggered extension of the 1D Nessyahu and Tadmor central scheme that evolves the numerical solution on a unique grid and avoids the resolution of the Riemann problems arising at the cell interfaces thanks to a layer of ghost staggered cells. As is the case with general balance laws, the SWE system admits steady‐state solutions in which the nonzero divergence of the flux is exactly balanced by the source term. This balance is very difficult to maintain at the discrete level, and in general most numerical schemes for hyperbolic systems fail to preserve it, and thus generate nonphysical waves. In this work we propose a special discretization of the source term according to the discretization of the divergence of the flux by the numerical base scheme, and then adapt the surface gradient method to the case of one‐dimensional unstaggered central schemes. The resulting scheme ensures the well‐balanced constraint at the discrete level and is consistent with the SWE system.