Alexander Kurganov, Department of Mathematics,Tulane University
601 Pao Yue-Kong Library
In the first part of the talk, I will describe a general framework for designing finite-volume methods (both upwind and central) for hyperbolic systems of conservation laws. I will focus on Riemann-problem-solver-free non-oscillatory central schemes and, in particular, on central-upwind schemes that belong to the class of central schemes, but has some upwind features that help to reduce the amount of numerical diffusion typically present in staggered central schemes such as, for example, the first-order Lax-Friedrichs and second-order Nessyahu-Tadmor scheme.
In the second part of the talk, I will discuss how central-upwind schemes can be extended to hyperbolic systems of balance laws, such as the Saint-Venant system and related shallow water models. The main difficulty in this extension is preserving a delicate balance between the flux and source terms. This is especially important in many practical situations, in which the solutions to be captured are (relatively) small perturbations of steady-state solutions. The other crucial point is preserving positivity of the computed water depth (and/or other quantities, which are supposed to remain nonnegative). I will present a general approach of designing well-balanced positivity preserving central-upwind schemes and illustrate their performance on a number of shallow water models.