Abstract:
I will first give a brief review of Godunov-type central-upwind schemes for hyperbolic systems of balance laws. These finite volume methods offer a simple, robust and yet highly accurate alternative to more complicated and problem oriented upwind schemes.
The second part of the talk will be devoted to applications of central-upwind schemes to the Saint-Venant system of shallow water equations and related problems. I will show how to construct the well-balanced positivity preserving central-upwind scheme, which accurately balances the flux and the source terms and at the same time preserves positivity of the water depth -- both properties are absolutely required for practical applications of the method. The scheme, derived in both one- and two-dimensional cases (in the latter case we use both Cartesian and triangular grids), is applied to a variety of test problems.
Finally, I will discuss extensions of the central-upwind schemes to a more complicated shallow water model -- the Ripa system, in which the water temperature fluctuations are taken into account. This system admits substantially more complicated (than the original Saint-Venant system) steady-state solution and thus designing a well-balanced scheme for it is a very challenging task which will be addressed in the final part of the talk. |