A reconstructed central discontinuous Galerkin-finite element method for the fully nonlinear weakly dispersive Green–Naghdi model

Abstract

In this paper, we present a class of high order reconstructed central discontinuous Galerkin-finite element methods for the fully nonlinear weakly dispersive Green–Naghdi model, which describes a large spectrum of shallow water waves. In the proposed methods, we first reformulate the Green–Naghdi model into conservation laws coupled with an elliptic equation, and then discretize the conservation laws with reconstructed central discontinuous Galerkin methods and the elliptic equation with continuous FE methods. The reconstructed central discontinuous Galerkin methods can be viewed as a class of fast central discontinuous Galerkin methods, in which we replace the standard formula for the numerical solution defined on the dual mesh in the central discontinuous Galerkin method with a projection equation in the L2 sense. The proposed methods reduce the computational cost of the traditional methods by nearly half but still maintain the formal high order accuracy. We study the L2 stability and an L2a priori error estimate for smooth solutions of the reconstructed central discontinuous Galerkin method for linear hyperbolic equation. Numerical tests are presented to illustrate the accuracy and computational efficiency of the proposed method.