On displacement shallow water wave equation and symplectic solution

In this paper, the shallow water wave problem is discussed in the Lagrangian description. By using the Hamilton variational principle in analytical mechanics, a displacement shallow water wave equation (DSWWE) is developed for the shallow water wave problem with a sloping water bottom and wet-dry interface. A numerical scheme based on the discretized Hamilton principle is constructed for solving the proposed displacement shallow water wave equation. The proposed numerical scheme is symplectic and explicit, and can preserve the total energy and mass of the shallow water system in the discrete sense. The correctness of the DSWWE and the effectiveness of the proposed numerical scheme are verified by using four classical numerical examples. Numerical examples show that the proposed method performs well with respect to the simulation of the shallow water problem with a sloping water bottom and wet-dry interface.