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

Abstract

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.