High order well-balanced central local discontinuous Galerkin-finite element methods for solving the Green-Naghdi model

In this paper, a hybrid numerical method, combining the central local discontinuous Galerkin method with continuous finite element method, is proposed to solve the fully nonlinear weakly dispersive Green-Naghdi model describing a large spectrum of shallow water waves. In our numerical approach, the Green-Naghdi model is first rewritten as balance laws coupled with an elliptic equation in terms of new variables adapted for numerical studies. Then we discretize the balance laws with well-balanced central local discontinuous Galerkin methods and the elliptic part with continuous finite element methods. Numerical tests are presented to illustrate the performance of the proposed schemes.