Numerical Methods Of Shallow Water Wave Problems And Stokes Problems

This dissertation consists of two parts. The first part concerns the numerical methods of two classes of shallow water models:central discontinuous Galerkin methods for the nonlinear shallow water equations and the Green-Naghdi model. The nonlinear shallow water equations properly describe the propagation of the fully nonlinear non-dispersive waves, while the Green-Naghdi model is used to simulate the propagation of the fully nonlinear weekly dispersive waves. The second part is devoted to the numerical solutions of the Stokes problem by using multiwavelet Galerkin boundary element methods, the Stokes problem has been usually applied to model incompressible viscous flows where the fluid Reynolds number is very low.To solve the nonlinear shallow water equations numerically, we usually encounter two issues. The first one is that the model is a balance law which admits still water stationary solutions, for the solutions the flux term is not zero but balanced by the source term, however many standard numerical methods cannot balance the source term and the flux term, so that they may introduce the numerical oscillations when condidering the problems related to the stationary solutions. The second one is that during the simulation of the waves, the numerical methods may produce negative water depth if the problems contain dry or near dry area. To overcome these issues, we develop a well-balanced central discontinuous Galerkin method and a positivity-preserving (preserving the non-negativity of water depth) central discontinuous Galerkin method for the one dimensional nonlinear shallow water equations, respectively. The former can balance the source term and the flux term and the latter can maintain the non-negativity of the water depth, respectively. The well-balanced and positivity-preserving properties are proved in this thesis. Based on these methods, we then present a positivity-preserving well-balanced central discontinous Galerkin method for the model, which can maintain the n on-negativity of the water depth and balance the source term and the flux term simultaneously. These methods are also extended to two dimensional nonlinear shallow water equations.As the nonlinear shallow water equations, the Green-Naghdi model also has still water stationary solutions which are desired to be preserved in numerical simulations and the numerical methods also need to maintain the non-negativity of the water depth. Besides, the model contains mixed spatial and time derivatives of the unknowns in the source term and flux term, which is also an issue for developing robust numerical methods. To design the robust numerical methods, we first reformulate the Green-Naghdi model into coupled balance laws and an elliptic equation which removes the mixed derivatives. 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, therefore we develop a central discontinuous Galerkin-finite element method for the simulation of the Green-Naghdi model with flat bottom. Then a well-balanced central discontinuous Galerkin-finite element method is presented to solve the Green-Naghdi model with non-flat bottom. In addition, a positivity-preserving well-balanced central discontinuous Galerkin-finite element method is proposed for the problems which contain dry or near dry area. In these mehods, the balance laws are solved by the central discontinuous Galerkin methods with the corresponding properties proposed for the nonlinear shallow water equations, and the elliptic equation is computed by the continuous finite element methods. We also proved the well-balanced and positivity-preserving properties of the proposed methods.To solve the Stokes problems by the boundary element methods, the countiuous equations or the impressiblility conditions of the fluids can be included in the boundary integral equations, moreover the velocity and pressure of the fluids can be computed separately, and therefore, there has been growing interest in applying boundary element method for the solution of Stokes problem. However, the associated system matrix is computed densely due to the nonlocal nature of the boundary integral operators. Hence the computational complexity of the system matrixis O(N2)(where Nis the number of unknowns). To overcome the drawback, a multiwavelet Galerkin boundary element method is developed for the simulation of two dimensional Stokes problems in this thesis. This method combines a boundary integral formulation for the Stokes equation with the Alpert multiwavelets for construction of trial and test functions for Galerkin variational formulation. Because of the use of multiwavelets, the present method reduces the computational complexity of boundary element matrix from O(N2)to almost O(N) by using two compression strategies. To evaluate logarithmic singular double integrals more efficiently, the analytical formulae are presented to calculate the inner integrals and then the Gaussian quadrature is used for the outer integrals.Finally, numerical examples are presented to show the accuracy, reliability and efficiency of the proposed methods.