Well-balanced schemes for the shallow water equations with Coriolis forces

In the present paper we study shallow water equations with bottom topography and Coriolis forces. The latter yield non-local potential operators that need to be taken into account in order to derive a well-balanced numerical scheme. In order to construct a higher order approximation a crucial step is a well-balanced reconstruction which has to be combined with a well-balanced update in time. We implement our newly developed second-order reconstruction in the context of well-balanced central-upwind and finite-volume evolution Galerkin schemes. Theoretical proofs and numerical experiments clearly demonstrate that the resulting finite-volume methods preserve exactly the so-called jets in the rotational frame. For general two-dimensional geostrophic equilibria the well-balanced methods, while not preserving the equilibria exactly, yield better resolution than their non-well-balanced counterparts.