Development Of Two-Dimensional Hydrodynamic Model Based On Quadrature-free Nodal Discontinuous Galerkin Method

The numerical simulation of two-dimensional hydrodynamics and large-scale storm surges in estuarine and coastal areas is of great significance to port design and construction,ocean hydrological research,beach protection and the safety of the people and structures along the coastline.At present,most of numerical methods solving shallow water equations have one or more shortcomings in stability,local conservation preserving,describing discontinuous,calculation accuracy or efficiency.The discontinuous Galerkin method combines the advantages of finite element method and finite volume method.It performs well in stability,high accuracy,self-adaptive mesh generation,parallel algorithm implementation and local conservation preserving.In this paper,the shallow water equations are discretized with the quadrature-free nodal discontinuous Galerkin method.After adding the source terms to the governing equations in the existing shallow water model,a two-dimensional hydrodynamic model for tide simulation in estuarine and coastal areas is preliminarily established.The main results of this study are as follows.(1)Source terms including Coriolis force term,wind stress term,bottom friction term and horizontal diffusion term are added into the shallow water equations and discretized with the quadrature-free nodal discontinuous Galerkin method.Corresponding benchmarks are selected respectively to verify the calculation of these source terms in the established two-dimensional hydrodynamic model.After verification,the model is applied to simulate the tidal flow of Hongtang bay,Sanya,and the simulation results are in good agreement with the observed data in field.Therefore,the established model has the basic functions of simulating the two-dimensional hydrodynamics in estuarine and coastal areas(2)Considering the interaction of waves and current,the radiation stress term,surface roller term,bottom friction term and diffusion term due to combined waves and current are added into the governing equations.Numerical examples are designed based on the experiments conducted to study the wave induced current with and without structures.The simulation results are in good agreement with the experimental data.So far,a proper wave-current coupling module is added into the model.(3)Considering the situation of simulating large-scale tidal current and storm surges,a stereographic projection is used to project two-dimensional shallow water equations with the spherical coordinates into the plane coordinate system.The quadrature-free nodal discontinuous Galerkin method is again used to solve the governing equations.The model established is applied to the simulation of tidal flow in field.The simulation results are in good agreement with the observed data and the simulation results of the Cartesian coordinate model established in this paper.It indicates that the established preliminary two-dimensional spherical hydrodynamic model for large-scale calculation is feasible.