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Central-Upwind Schemes for the Saint-Venant System

Published online by Cambridge University Press:  15 August 2002

Alexander Kurganov  and
Doron Levy
Alexander Kurganov
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109 and Mathematics Department, Tulane University, New Orleans, LA 70118, USA. kurganov@math.tulane.edu.
Doron Levy
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA. dlevy@math.stanford.edu.
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Abstract

We present one- and two-dimensional central-upwind schemes for approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutions in which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preserve this delicate balance with numerical schemes. Small perturbations of these states are also very difficult to compute. Our approach is based on extending semi-discrete central schemes for systems of hyperbolic conservation laws to balance laws. Special attention is paid to the discretization of the source term such as to preserve stationary steady-state solutions. We also prove that the second-order version of our schemes preserves the nonnegativity of the height of the water. This important feature allows one to compute solutions for problems that include dry areas.

Keywords

Saint-Venant systemshallow water equationshigh-order central-upwind schemesbalance laws conservation lawssource terms.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 3 , May 2002 , pp. 397 - 425
Copyright
© EDP Sciences, SMAI, 2002

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