Conservative arbitrary order finite difference schemes for shallow-water flows

The classical nonlinear shallow-water model (SWM) of an ideal fluid is considered. For the model, a new method for the construction of mass and total energy conserving finite difference schemes is suggested. In fact, it produces an infinite family of finite difference schemes, which are either linear or nonlinear depending on the choice of certain parameters. The developed schemes can be applied in a variety of domains on the plane and on the sphere. The method essentially involves splitting of the model operator by geometric coordinates and by physical processes, which provides substantial benefits in the computational cost of solution. Besides, in case of the whole sphere it allows applying the same algorithms as in a doubly periodic domain on the plane and constructing finite difference schemes of arbitrary approximation order in space. Results of numerical experiments illustrate the skillfulness of the schemes in describing the shallow-water dynamics.