I will first 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 shallow water equations. This can be done in a straightforward manner, but then the resulting scheme may suer 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 diculties, we have developed a special technique, which guarantees that the designed second-order central-upwind scheme is both well-balanced and positivity preserving, [1,3].

Finally, I will show how the scheme can be extended to the two-layer shallow water equations [4] and to the Savage-Hutter type model of submarine landslides and generated tsunami waves [2], which, in addition to the geometric source term, contain nonconservative interlayer exchange terms. It is well-known that such terms, which arise in many dierent multiphase models, are extremely sensitive to a particular choice their numerical discretization. To circumvent this diculty, we rewrite the studied systems 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 numerical examples.

REFERENCES [1] A. Kurganov and D. Levy, Central-upwind schemes for the Saint-Venant system, M2AN Math. Model. Numer. Anal., 36 (2002), pp. 397425. [2] A. Kurganov and J. Miller, Central-upwind scheme for Savage-Hutter type model of submarine landslides and generated tsunami waves, submitted. [3] A. Kurganov and G. Petrova, A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system, Commun. Math. Sci., 5 (2007), pp. 133160. [4] A. Kurganov and G. Petrova, Central-upwind schemes for two-layer shallow equations, SIAM J. Sci. Comput., 31 (2009), pp. 17421773.