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

## TL;DR

This paper develops a numerical method combining central discontinuous Galerkin and finite element techniques to solve the two-dimensional Green-Naghdi model with enhanced dispersive properties, ensuring accuracy, efficiency, and physical fidelity.

## Contribution

It introduces a novel CDG-FE scheme for the 2D Green-Naghdi model that improves dispersive effects and maintains well-balanced and positivity-preserving features.

## Key findings

- The proposed method accurately captures dispersive wave behavior.
- Numerical tests demonstrate the scheme's efficiency and stability.
- The approach effectively preserves physical properties like positivity.

## Abstract

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 the purpose of 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 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.