A New Well-Balanced Finite Volume CWENO Scheme for Shallow Water Equations over Bottom Topography

Abstract

In this article, we develop a new well-balanced finite volume central weighted essentially non-oscillatory (CWENO) scheme for one- and two-dimensional shallow water equations over uneven bottom. The well-balanced property is of paramount importance in practical applications, where many studied phenomena can be regarded as small perturbations to the steady state. To achieve the well-balanced property, we construct numerical fluxes by means of a decomposition algorithm based on a novel equilibrium preserving reconstruction procedure and we avoid applying the traditional hydrostatic reconstruction technique accordingly. This decomposition algorithm also helps us realize a simple source term discretization. Both rigorous theoretical analysis and extensive numerical examples all verify that the proposed scheme maintains the well-balanced property exactly. Furthermore, extensive numerical results strongly suggest that the resulting scheme can accurately capture small perturbations to the steady state and keep the genuine high-order accuracy for smooth solutions at the same time.