High order exactly well-balanced numerical methods for shallow water systems

This paper is concerned with the development of high-order methods for the shallow water system that preserve every stationary solution of the system and not only the water-at-rest ones. The pursued strategy is based on the generalized hydrostatic reconstruction introduced in [10], and on the technique proposed in [8] to obtain high-order well-balanced reconstruction operators. The more difficult step in the reconstruction procedure is the first one: given a cell value, a stationary solution of the system has to be found whose average at the cell equals the given value. The implementation of this step is discussed in detail, as well as, the similarities and differences with some previous approaches to obtain well-balanced methods for the shallow water system. Finally, the well-balanced properties of the developed numerical methods are checked in some test cases.