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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.

##### MSC:
 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
HE-E1GODF; HLLE
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