High Order Well-balanced Central Local Discontinuous Galerkin-Finite Element Methods For Solving The Green-Naghdi Model

Green-Naghdi model is a kind of the fully nonlinear weakly dispersive shallow water equations. Due to the dispersive effect, it has solitary wave solutions and can simulate the long-time propagation of a single solitary wave. Furthermore, such model can be applied into large amplitude problem. Therefore, it is very important to study this model.In this paper, we consider a one-dimensional Green–Naghdi model for shallow water waves over flat and variable bottom topographies. Since the Green-Naghdi model over variable bottom topographies has still water stationary solutions, the numerical methods should preserve these solutions in numerical simulations, and also need to maintain the non-negativity of the water depth. In addition, this model contains mixed spatial and time derivatives of the unknowns in the source term and flux term, which is an issue for designing robust numerical methods. To remove the mixed derivatives, we first rewrite such model into balance laws coupled with an elliptic equation in terms of new variables adapted for numerical studies. Then a hybrid numerical method, combining the well-balanced central local discontinuous Galerkin method with continuous finite element method, is proposed to solve the Green-Naghdi model with non-flat bottom. The balance laws are computed by the well-balanced central local discontinuous Galerkin methods, and the elliptic equation is solved by the continuous finite element methods.Since the source term is zero in the Green-Naghdi model with flat bottom, we do not need to balance the source term and the flux term. For the convenience of calculation, we first reformulate the Green-Naghdi model into conservation laws coupled with an elliptic equation in terms of new variables adapted for numerical studies. Then we propose a hybrid numerical method, combining the central local discontinuous Galerkin method with continuous finite element method. The conservation laws are solved by the central local discontinuous Galerkin methods, and the elliptic equation is computed by the continuous finite element methods.Finally, numerical tests are presented to illustrate the performance of the proposed schemes.