Abstract: We consider the Saint-Venant system for shallow water flows with non-flat bottom. This is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, and granular flows when completed with friction. Numerical approximate solutions to this system may be generated using conservative finite volume methods, which are known to properly handle any shocks and contact discontinuities. Yet, these schemes prove to be problematic for near steady states, as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions. This difficulty can be overcome by using so called well-balanced schemes.
We describe a general strategy based on a local hydrostatic reconstruction that allows us to derive a well-balanced scheme from any solver for the homogeneous problem (Godunov, Roe, kinetic...). Whenever the initial solver satisfies some classical stability properties, it yields a simple and fast well-balanced scheme that preserves the nonnegativity of the water height and satisfies a semi-discrete entropy inequality.
Conservation Laws Preprint Server <firstname.lastname@example.org> Last modified: Tue Jul 8 10:01:19 MEST 2003