A shallow water equation based on displacement and pressure and its numerical solution

Abstract

The primary purpose of this paper is to develop an efficient numerical scheme for solving the shallow water wave problem with a sloping water bottom and wet-dry interface. For this purpose, the Lagrange method and the constrained Hamilton variational principle are used to solve the shallow water wave problem. According to the constrained Hamilton variational principle, a shallow water equation based on the displacement and pressure (SWE-DP) is derived. Based on the discretized constrained Hamilton variational principle, a numerical scheme is developed for solving the SWE-DP. The proposed scheme combines the finite element method for spatial discretization and the simplectic Zu-class method for time integration. The correctness of the SWE-DP and the effectiveness of the proposed scheme are verified by three classical numerical examples. Numerical examples show that the proposed method performs well in the simulation of the shallow water problem with a sloping water bottom and wet-dry interface.