Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. (English) Zbl 0931.76055

Summary: 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 \(Q\)-scheme of van Leer and Roe are proposed which generate natural upwind discretizations of the source terms. We analyze the consistency of the proposed schemes. 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. We also obtain a low-order asymptotic unsteady analytical solution for small Froude numbers. The numerical results confirm the improved properties of the proposed schemes for a transient test problem. \(\copyright\) Academic Press.


76M12 Finite volume methods applied to problems in fluid mechanics
76M20 Finite difference methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
86A05 Hydrology, hydrography, oceanography


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