水波问题的高阶数值方法

In this project, we design high order numerical methods for the water wave problems. We first consider the Green-Naghdi models in one and two dimensional spaces, which are fully nonlinear weakly dispersive shallow water equations. Most of traditional numerical methods cannot maintain the still water stationary solution and the non-negativity of the water depth of the Green-Naghdi model, thus we will construct a high order numerical method which can maintain the still water stationary solution and the non-negativity of the water depth in the framework of discontinuous Galerkin methods. In order to improve the computational efficiency, we will implement the proposed methods on the structured adaptive mesh for solving two-dimensional Green-Naghdi model. Then we consider the free surface water wave equations. For this problem, the velocity potential of the fluid satisfies the Laplace equation and linear or nonlinear boundary conditions on the free surface and the bottom of the fluid. We will develop a coupling method combining central discontinuous Galerkin methods and boundary element methods for solving this problem. Due to the advantage that boundary element methods can reduce the dimension of the problem by one, we solve the Laplace equation in the fluid domain by using Galerkin boundary element methods in the proposed coupling method, we then solve equations defined on the free surface by using central discontinuous Galerkin methods.

本项目将研究水波问题的高阶数值方法。首先，我们考虑一维和二维完全非线性弱色散的浅水波方程-Green-Naghdi模型。很多传统的数值方法不能保持该模型的静水稳定解和水深的非负性，因此本项目将在间断Galerkin法的框架下，构造一个保持静水稳定解和水深非负的高阶数值方法。为了提高计算效率，我们将在结构化自适应网格上执行所提出的方法，并用来求解二维Green-Naghdi模型。然后，我们考虑自由表面水波方程。对该问题，流体的速度势满足Laplace方程，且在自由表面和流体底部满足线性或非线性边界条件。我们将利用中心间断Galerkin法和边界元法的耦合算法来求解该问题。由于边界元法具有降维的优点， 因此在该耦合算法中，我们使用Galerkin边界元法来求解流体区域上的Laplace方程，然后利用中心间断Galerkin法来求解自由表面上的方程。

本项目研究了水波方程的高阶数值方法。具体来说，我们首先对非线性浅水波方程设计了一个保持静水稳定解和水深非负的中心间断伽辽金法；然后将该方法与有限元法结合求解了非线性弱色散浅水波方程-Green-Naghdi模型；第三，我们改进了Green-Naghdi模型的色散性，并耦合中心间断伽辽金法和有限元法设计了保持静水稳定解和水深非负的高阶数值方法；第四，为了提高中心间断伽辽金法的计算效率，我们设计了一个重构的中心间断伽辽金法，与原方法相比，新方法节约了一半的计算时间，该方法首先被用来求解一些守恒律方程，然后与有限元法结合求解了Green-Naghdi模型；第五，我们也研究了基于结构化自适应网格加密的间断伽辽金法，其进一步提高了间断伽辽金法计算效率，该方法也被使用来解决浅水波方法；第六，为了能够解决复制区域上的水波问题，我们设计了一个基于非结构重叠网格的中心间断伽辽金法，该方法首先被用来求解一些守恒律方程，然后被用来求解浅水波问题；最后我们结合中心间断伽辽金法和边界元法来求解了线性和非线性的自由表面水波方法。