Balanced Schemes for the Shallow Water Equations

The Saint-Venant (SV) system is commonly used to model flows in rivers or coastal areas. This system describes the flow as a conservation law with an additional source term due to bottom topography. Similarly to other balance laws, the SV system admits steady-state solutions in which nonzero flux gradients are exactly balanced by the source terms. Such steady-states as well as their perturbations, are difficult to capture numerically. Standard numerical schemes for conservation laws will, in general, fail to preserve the delicate balance between the fluxes and the source terms.

In this talk we will show how to derive semi-discrete Godunov-type central schemes that preserve stationary steady-state solutions of the SV system. The main idea is to combine modern methods for approximating solutions of multidimensional systems of hyperbolic conservation laws with a careful discretization of the source terms. The schemes will be constructed both on Cartesian meshes and on unstructured grids. Along the way, we will comment on some recent developments in the areas of non-oscillatory approximations and high-order schemes for conservation laws.

Refreshments will be served at 3:45 p.m. (MATX 1115, Math Lounge).