A well balanced scheme for the shallow water wave equations in open channels with (discontinuous) varying width and bed

Abstract

Finite volume methods have proven themselves a powerful tool for finding solutions to the shallow water wave equations. They are based on the conservation laws for the mass and momentum, integrated over discrete finite volumes. These methods tend to do well at the diffcult problem of capturing solutions involving shocks. However, one area that causes problems is the approximation of steady or near steady states when there is a sloping bed elevation. The problem arises due to a poor balance between the discretisation of the flux terms across the edge of a finite volume and the pressure terms due to the sloping bed. Methods that overcome these diffculties and reproduce the still lake steady state solution, are called well balanced. In this work we are interested in a well balanced scheme for the one dimensional shallow water wave equations but with a modification that allows for varying width in the transverse direction. Here a well balanced method developed by Audusse et al. for the constant width case is extended to the case of varying (possibly discontinuous) width. Numerical validation of this new method is provided.