A Second-Order Well-Balanced Positivity Preserving Central-Upwind Scheme for the Saint-Venant System

Abstract

A family of Godunov-type central-upwind schemes for the Saint-Venant system of shallow water equations has been first introduced in A. Kurganov and D. Levy, Central-upwind schemes for the Saint-Venant system. Depending on the reconstruction step, the second-order versions of the schemes there could be made either well-balanced or positivity preserving, but fail to satisfy both properties simultaneously. Here, we introduce an improved second-order central-upwind scheme which, unlike its forerunners, is capable to both preserve stationary steady states (lake at rest) and to guarantee the positivity of the computed fluid depth. Another novel property of the proposed scheme is its applicability to models with discontinuous bottom topography. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of one- and two-dimensional examples.