Central-Upwind Schemes for the Saint-Venant System
Department of Mathematics, University of Michigan,
Ann Arbor, MI 48109-1109 and Mathematics Department, Tulane
University, New Orleans, LA 70118, USA. firstname.lastname@example.org.
2 Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA. email@example.com.
Revised: 17 February 2002
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.
Mathematics Subject Classification: 65M06 / 35L65
Key words: Saint-Venant system / shallow water equations / high-order central-upwind schemes / balance laws conservation laws / source terms.
© EDP Sciences, SMAI, 2002