A CDG-FE method for the two-dimensional Green-Naghdi model with the enhanced dispersive property

In this work, we investigate numerical solutions of the two-dimensional shallow water wave using a fully nonlinear Green-Naghdi model with an improved dispersive effect. For numerics, the Green-Naghdi model is rewritten into a formulation coupling a pseudo-conservative system and a set of pseudo-elliptic equations. Since the pseudo-conservative system is no longer hyperbolic and its Riemann problem can only be approximately solved, we consider the utilization of the central discontinuous Galerkin method, which possesses an important feature of the needlessness of Riemann solvers. Meanwhile, the stationary elliptic part will be solved using the finite element method. Both the well-balanced and the positivity-preserving features, which are highly desirable in the simulation of the shallow water wave, will be embedded into the proposed numerical scheme. The accuracy and efficiency of the numerical model and method will be illustrated through numerical tests. •A 2D modified Green-Naghdi model with enhanced dispersive property is derived.•A 2D numerical model is presented to reformulate the modified Green-Naghdi model.•A positivity-preserving well balanced CDG-FE method is presented for the model.•The presented method is a high order Galerkin scheme and free of Riemann solver.•A wide range of numerical tests demonstrate the numerical model and scheme.