Study On Numerical Model Of Shallow Water Equations Based On Quadrature-Free Nodal Discontinuous Galerkin Method

The numerical shallow water model(SWE)is one of the most widely used hydrodynamic models,and has been applied for various flow regimes in rivers,coastal and ocean problems.When applying numerical models to real applications,the numerical schemes used by the model have a great effect on the simulated results.It becomes an important discipline of hydrodynamics to use new high-order accuracy schemes to develop numerical model of shallow water equations.The shallow water equations are discretized with high-order quadrature-free nodal discontinuous Galerkin(DG)methods in this dissertation,and a two-dimensional model suitable for complex realistic applications and a three-dimensional linear model with high-order accuracy are developed.The main contents and results of this dissertation are given as follows.(1)In order to improve the computational efficiency of the model,a quadrature-free nodal DG scheme with arbitrary quadrilateral elements is proposed.Compared with the scheme of traditional triangular meshes,the new method of quadrilateral meshes of similar resolutions could effectively improve computational efficiency without destroying the accuracy.(2)To avoid generating numerical oscillations near the discontinuities,a new vertexbased slope limiter for high-dimensional problems is proposed by modifying traditional vertex-based slope limiter.This method effectively reduces the excessive numerical diffusions in the traditional slope limiter,and can be applied to arbitrary polygons such as triangles and quadrilaterals.(3)The quadrature-free nodal DG method is applied to solve the two-dimensional SWE model.With analyzing the process of numerical discretization,a pre-balanced discretizing approach for the topography source term is proposed to ensure the equilibrium with the flux terms in the quadrature-free volume discretization.In surface discretization,the hydrostatic reconstruction method is used to balance the numerical and normal flux terms,where the well-balanced property is maintained with complex stair-like topography.(4)In the wetting/drying treatment,a special reconstruction method is proposed for the variables of the partially wet elements,which can provide accurate results and preserve numerical stability simultaneously.In addition,a restricted time step is proposed to maintain the positivity of averaged water depth in the quadrature-free nodal DG method.(5)To solve the primitive continuity equation and obtain high-order accuracy in the three-dimensional model,an efficient and accurate approach is proposed for calculating the vertical-averaged velocities.A vectorization approach is proposed to calculating the vertical velocities,which avoids the numerical integrations and is suitable for the quadrature-free nodal DG methods.