非线性对流扩散方程LDG方法的误差估计

发布时间：2018-05-16 12:46

本文选题：局部间断有限元方法 + 非线性对流扩散方程　；参考：《哈尔滨工业大学》2016年硕士论文

【摘要】：自然界中,如水污染,热传播等自然现象以及油田勘探等工程上的实际问题,几乎都可以用对流扩散方程来表示,这类方程在求解时一般无法得到精确解,因此,对其数值解法的研究以及误差的估计具有实际应用价值。本文主要研究的内容是利用局部间断有限元法(Local Discontinuous Galerkin Method,以下简记为LDG)分别在一维网格剖分和多维Cartesian网格剖分情况下对非线性对流扩散方程进行误差估计,定义了数值流通量和投影,同时给出了单元熵不等式和稳定性分析。本文首先利用LDG方法将非线性对流扩散方程改写成一阶方程组的形式,其次利用间断有限元法(Discontinuous Galerkin Method,以下简记为DG)的思想得到方程组的离散格式,讨论其数值流通量的取值。为了保证数值解的误差不随解过程增加,即方程的解能够受初值控制,本文利用单元熵不等式分别讨论了一维网格剖分和多维Cartesian网格剖分下解的稳定性。进一步,本文最重要的研究内容是误差估计。一维网格剖分下,对于Kd V方程一般采用能量方程进行误差估计,而在本文中,我们利用能量范数和Taylor展开对其进行估计,通过选取合适的投影,消去许多边界项,从而得到误差估计的结果。上述思想也被应用于多维Cartesian网格剖分情况下,唯一的不同之处在于投影的定义和性质。最后,本文通过上述方法证明了,在一维网格剖分和多维Cartesian网格剖分下,误差估计的结果均为O (h~(k+1/2)).
[Abstract]:In nature, such natural phenomena as water pollution, heat propagation and engineering problems such as oil field exploration can almost be expressed by convection-diffusion equations, which are generally unable to obtain exact solutions when solving these equations. The study of its numerical solution and the estimation of its error have practical application value. In this paper, the local discontinuous finite element method is used to estimate the error of nonlinear convection-diffusion equations in the case of one-dimensional mesh generation and multidimensional Cartesian mesh generation, respectively. The numerical flux and projection are defined, and the element entropy inequality and stability analysis are given. In this paper, the nonlinear convection-diffusion equation is firstly rewritten into the first order equations by using LDG method, and then the discrete scheme of the equations is obtained by using the idea of discontinuous Galerkin method of discontinuous finite element method, and the numerical flux of the equations is discussed. In order to ensure that the error of the numerical solution does not increase with the solution process, that is, the solution of the equation can be controlled by the initial value, the stability of the solution under one-dimensional mesh generation and multidimensional Cartesian mesh generation is discussed by using the element entropy inequality, respectively. Furthermore, the most important research content of this paper is error estimation. In this paper, the energy equation is used to estimate the error of the KDV equation. In this paper, the energy norm and the Taylor expansion are used to estimate the KDV equation. By selecting the appropriate projection, many boundary terms are eliminated. The result of error estimation is obtained. The above idea is also applied in the case of multi-dimensional Cartesian mesh generation. The only difference lies in the definition and properties of projection. Finally, through the above methods, it is proved that the results of error estimation for one-dimensional mesh generation and multi-dimensional Cartesian mesh generation are O ~ (1 / 2) ~ (2 / 2).
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O241.82

【相似文献】