Scientific Computing Group Seminars - Detail View

Speaker: A. Kurganov

Affiliation: Tulane University

Talk Title: Central-Upwind Schemes for Two-Layer Shallow Water Equations

Invited by: David Gottlieb

Time: Sept. 19 2008 11 a.m.

Location: 182 George Street, Room 110

Abstract:

I will first give a brief review on simple and robust central-upwind schemes for hyperbolic conservation laws. I will then discuss their application to the Saint-Venant system of (single layer) shallow water equations. This can be done in a straightforward manner, but then the resulting scheme may suffer from the lack of balance between the fluxes and (possibly singular) geometric source term, which may lead to a so-called numerical storm, and from appearance of negative values of the water height, which may destroy the entire computed solution. To circumvent these difficulties, we have developed a special technique, which guarantees that the designed second-order central-upwind scheme is both well-balanced and positivity preserving. Finally, I will show how the scheme can be extended to a more complicated case of a two-layer shallow water equations, which, in addition to the geometric source term, contains nonconservative interlayer exchange terms. It is well-known that such terms, which arise in many different multiphase models, are extremely sensitive to a particular choice their numerical discretization. To circumvent this difficulty, we rewrite the system in a different way so that the nonconservative terms are multiplied by a quantity, which is, in all practically meaningful cases, very small. We then apply the central-upwind scheme to the rewritten system and demonstrate robustness and superb performance of the proposed method on a number of one- and two-dimensional examples. The talk is based on a series of joint works with Guergana Petrova, Texas A&M University.