Well-balanced methods for the shallow water equations in spherical coordinates

The goal of this work is to obtain a family of explicit high order well-balanced methods for the shallow water equations in spherical coordinates. Application of shallow water models to large scale problems requires the use of spherical coordinates: this is the case, for instance, of the simulation of the propagation of a Tsunami wave through the ocean. Although the PDE system is similar to the shallow water equations in Cartesian coordinates, new source terms appear. As a consequence, the derivation of high order numerical methods that preserve water at rest solutions is not as straightforward as in that case. Finite Volume methods are considered based on a first order path-conservative scheme and high order reconstruction operators. Numerical methods based on these ingredients have been successfully applied previously to the nonlinear SWEs in Cartesian coordinates. Some numerical tests to check the well-balancing and high order properties of the scheme, as well as its ability to simulate planetary waves or tsunami waves over realistic bathymetries are presented.