High order well-balanced finite difference WENO schemes for shallow water flows along channels with irregular geometry

Abstract

In this paper, we present high order finite difference weighted essentially non-oscillatory (WENO) schemes for the shallow water flows along open channels with irregular geometry and over a non-flat bottom topography. The proposed schemes maintain the well-balanced property for the still water steady state solutions, namely preserve steady state at the discrete level, when there is a exact balance between the flux gradient and the source term. Compared with the traditional shallow water equations with constant cross section, the construction of the well-balanced schemes is not a trivial work due to the effect induced by the irregular geometry of the channels. To preserve the well-balanced property, we first reformulate the source term, then propose to construct the numerical fluxes by means of a flux modification technique, and finally discrete the source term with the help of a novel source term approximation. Benchmark numerical examples are applied to validate the good performances of the resulting schemes: well-balanced property, high order accuracy, and high resolution for the discontinuous solutions.