On splitting‐based mass and total energy conserving arbitrary order shallow‐water schemes

A new method for numerical solution to the shallow‐water equations is suggested. The method allows constructing a family of finite difference schemes of different approximation order that conserve the mass and the total energy. Our approach is based on the method of splitting, and unlike others it permits to derive conservative numerical schemes after discretizing all the partial derivatives, both spatial and temporal. The schemes thus appear to be fully discrete, both in time and in space. Besides, due to a simple structure of the matrices appeared therewith, the method provides essential benefits in the computational cost of solution and is easy‐to‐implement in the Cartesian and spherical geometries. Numerical results confirm functionality and efficiency of the developed method. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007