Well-balanced central schemes for systems of shallow water equations with wet and dry states

Abstract

In this paper we propose a new well-balanced unstaggered central finite volume scheme for the shallow water equations on variable bottom topographies, with wet and dry states. Based on a special piecewise linear reconstruction of the cell-centered numerical solution and a careful discretization of the system of partial differential equations, the proposed numerical scheme ensures both a well-balanced discretization and the positivity requirement of the water height component. More precisely, the well-balanced requirement is fulfilled by following the surface gradient method, while the positivity requirement of the computed water height component is ensured by following a new technique specially designed for the unstaggered central schemes. The developed scheme is then validated and classical shallow water equation problems on variable bottom topographies with wet and dry states are successfully solved. The reported numerical results confirm the potential and efficiency of the proposed method.