A high-order well-balanced central scheme for the shallow water equations in channels with irregular geometry

Abstract

This paper presents a new numerical scheme based on the finite volume method to solve the shallow water equations in channels with rectangular section and variable width. Time integration is carried out by means of a Runge-Kutta scheme with a natural continuous extension, using a new temporary forward flow at the midpoint of each cell which considers the physical flow and the source term primitive of the shallow water model. That term takes into account the gradient of bed height, channel width and friction energy loss model. Spatial integration is based on a central scheme in which flows only have to be evaluated on the midpoint of the cells where the solution is reconstructed. In this way, it is not necessary to know the structure of the partial differential equations to be solved. A centered three degree reconstruction polynomial is applied, using a slope correction to the midpoint of each cell to prevent the occurrence of spurious numerical oscillations. Some benchmark examples show the non-oscillatory behavior of numerical solutions in channels with a variable width. A comparison between numerical results and those obtained experimentally on a laboratory flume is also carried out.