Conservative Multidimensional Upwinding for the Steady Two-Dimensional Shallow Water Equations

In recent years upwind differencing has gained acceptance as a robust and accurate technique for the numerical approximation of the one-dimensional shallow water equations. In two dimensions the benefits have been less marked due to the reliance of the methods on standard operator splitting techniques. Two conservative genuinely multidimensional upwind schemes are presented which have been adapted from flux balance distribution methods recently proposed for the approximation of steady state solutions of the Euler equations on unstructured triangular grids. A method for dealing with source terms, such as those introduced by modelling bed slope and friction, is also suggested and results are presented for two-dimensional steady state channel flows to illustrate the accuracy and robustness of the new algorithms.