### Central-Upwind Schemes for Shallow Water Models

- Series
- Applied and Computational Mathematics Seminar
- Time
- Monday, April 15, 2013 - 14:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Alexander Kurganov – Tulane University

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 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 the two-layer shallow
water equations and to the Savage-Hutter type model of submarine landslides
and generated tsunami waves, which, in addition to the geometric source
term, contain 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 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.