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Well-balanced schemes for the shallow water equations with Coriolis forces. (English) Zbl 1448.65097

Summary: 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.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N08 Finite volume methods for boundary value problems involving PDEs
35L45 Initial value problems for first-order hyperbolic systems
35L65 Hyperbolic conservation laws
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76U60 Geophysical flows
76U65 Rossby waves
35Q86 PDEs in connection with geophysics
86A05 Hydrology, hydrography, oceanography
65J10 Numerical solutions to equations with linear operators

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