Multidimensional High Order Method of Transport for the Shallow Water Equations

Abstract

The Method of Transport was originally developed for the Euler equation in 1993 by M. Fey. He introduced the physical property of infinitely many propagation directions into the numerical method. Here, we present the extension of this method to equations with inhomogeneous fluxes, such as the shallow water equations. For efficiency reasons and to reach higher order accuracy certain modifications had to be made to the method, whereby the multidimensional character will be kept. The resulting scheme can then be interpreted as a decomposition of the nonlinear equations into a system of linear advection equations with variable coefficients in conservative form. We present a multidimensional high order resolution scheme for the advection equation and for the shallow water equations. A special limiting technique is used for these methods to keep the multidimensional properties.