High order finite volume WENO schemes for the shallow water flows through channels with irregular geometry

Abstract

The shallow water equations are widely used to model flows in rivers and coastal areas. In this paper, we consider the shallow water flows in open channels with irregular geometry and a non-flat bottom topography, and design high order finite volume weighted essentially non-oscillatory (WENO) methods. A special source term approximation is introduced so that the proposed methods can preserve the still water steady state exactly. We also employ a simple positivity-preserving limiter to provide efficient and robust simulations near the wetting and drying front. The proposed methods are well-balanced for the still water steady state solutions, preserve the non-negativity of the wet cross section, and are genuinely high order accurate in smooth regions for general solutions and essentially non-oscillatory for general solutions with discontinuities. Numerical examples are performed at the end to verify these properties.