A large time step upwind scheme for the shallow water equations with source terms

Abstract

Numerical methods to predict the water profile and discharge variations in the case of steady and unsteady flows in hydraulic systems modelling have become a common tool. Upwind methods in particular have proved a suitable way to discretize the shallow water equations. Finite difference schemes for time dependent equations are traditionally divided in two main groups, according to the way of discretization used for the time derivative, as explicit and implicit. Implicit schemes offer unconditional numerical stability at the extra cost of having to deal with the resolution of an algebraic, and often nonlinear, system with as many unknowns as grid points at every time step. On the other hand, conceptual simplicity is the most valuable characteristic of the former in which variables at a future time can be independently evaluated at every single point. The allowable time step size is nevertheless restricted by stability reasons to fulfil the Corant-Friedrichs-Lewy (CFL) condition. Is is possible to relax the condition over the time step size when using explicit schemes. A generalization of the first order explicit upwind and Roe method (see Roe 1981), modified to allow large time steps, was explored by Leveque (see Leveque 1981, Leveque 1985) first in the scalar non-linear case and then adapted to systems of equations. It becomes stable for CFL’s larger than 1 and provides an accurate and correct solution of shocks. The technique is devised to cope well with flow transients and even discontinuities far from the boundaries, being able to give a resolution of the shocks even sharper for CFL>1. When rarefactions containing a sonic point are present it needs some adjustments and the preferable procedure is not clear. A way to overcome that situation will be proposed and explored. The extension of the explicit method considers situations with or without source terms. In the first case, they are discretized according to the upwind formulation (see Murillo et al., 2010). Also, the use of this technique at open and closed boundaries is considered. The outline is as follows: the one dimensional discretization is described first, for a scalar equation. After, the performance of the scheme is evaluated to solve the shallow water equations and the way to deal with bed slope and friction source terms was incorporated into the proposed procedure. Steady open channel flow problems with analytical solutions and where bed slope and friction source terms play an important role are used as validation test cases. Also this method has been tested for the ideal dam break problem and other Riemann problems with analytical solutions.