Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry

This paper deals with the numerical solution of the shallow water equations in channels with irregular geometry but with a locally rectangular cross section. This type of channel leads to the presence of source terms involving the gradient of the depth and the breadth of the channel. Extensions of the e-scheme of van Leer and Roe are proposed which generate natural upwind discretizations of the source terms. The consistency of the proposed schemes is analyzed. A stationary solution that emphasizes the source terms considered is obtained which is used to test the proposed extensions in terms of a "conservation" property. A low-order asymptotic unsteady analytical solution for a small Froude number is also obtained. The numerical results presented confirm the improved properties of the proposed schemes for a transient test problem. (C) 1999 Academic Press.