Flux difference splitting and the balancing of source terms and flux gradients

Abstract Flux difference splitting methods are widely used for the numerical approximation of homogeneous conservation laws where the flux depends only on the conservative variables. However, in many practical situations this is not the case. Not only are source terms commonly part of the mathematical model, but also the flux can vary spatially even when the conservative variables do not. It is the discretisation of the additional terms arising from these two situations which is addressed in this work, given that a specific flux difference splitting method has been used to approximate the underlying conservation law. The discretisation is constructed in a manner which retains an exact balance between the flux gradients and the source terms when this is appropriate. The effectiveness of these new techniques, in both one and two dimensions, is illustrated using the shallow water equations, in which the additional terms arise from the modelling of bed slope and, in one dimension, breadth variation. Roe's scheme is chosen for the approximation of the conservation laws and appropriate discrete forms are constructed for the additional terms, not only in the first-order case but also in the presence of flux- and slope-limited high-resolution corrections. The method is then extended to two-dimensional flow where it can be applied on both quadrilateral and triangular grids.