High order well-balanced CDG–FE methods for shallow water waves by a Green–Naghdi model

In this paper, we consider a one-dimensional fully nonlinear weakly dispersive Green–Naghdi model for shallow water waves over variable bottom topographies. Such model describes a large spectrum of shallow water waves, and it is thus of great importance to design accurate and robust numerical methods for solving it. The governing equations contain mixed spatial and temporal derivatives of the unknowns. They also have still-water stationary solutions which should be preserved in stable numerical simulations. In our numerical approach, we first reformulate the Green–Naghdi equations into balance laws coupled with an elliptic equation. We then propose a family of high order numerical methods which discretize the balance laws with well-balanced central discontinuous Galerkin methods and the elliptic part with continuous finite element methods. Linear dispersion analysis for both the (reformulated) Green–Naghdi system and versions of the proposed numerical scheme is performed when the bottom topography is flat. Numerical tests are presented to illustrate the accuracy and stability of the proposed schemes as well as the capability of the Green–Naghdi model to describe a wide range of shallow water wave phenomena.