基于局部细化网格的离散分数阶微分方程的预处理迭代方法

发布时间：2018-05-03 10:26

本文选题：分数阶微分方程 + 边界层　；参考：《华东师范大学》2017年硕士论文

【摘要】：分数阶微分算子具有典型的非局部特性,当前被广泛应用于科学和工程计算的各个领域.但也正因为分数阶微分算子的非局部性,使得离散后的代数方程组往往是稠密的,这大大增加了数值求解的难度和复杂度.同时,有些分数阶微分方程的解还存在边界层现象和弱正则性,即使它的扩散系数和右端项都是光滑的.本论文主要研究的是解带边界层的分数阶微分方程的预处理方法.我们讨论了一类带Dirichlet边界条件的非齐次变系数分数阶微分方程.由于存在边界层现象,需要采用局部加密的网格剖分方法.但此时离散后的代数方程组不再具有整体Toeplitz结构,这给设计高效的预处理方法带来了一定的困难.我们通过对系数矩阵的细致分析和研究,提出了基于循环矩阵的块对角预处理子和块下三角预处理子,并在理论上证明了这两类预处理方法的有效性.数值实验也表明,这两种预处理方法都具有很好的数值表现.我们也考虑了网格局部细化的方法.为了尽可能地减少运算量,我们提出了一种网格局部细化方法,使得数值解在达到同等精度时所需的网格加密次数尽可能地少.同时,我们还提出了一种均衡局部细化的网格剖分方法,使得在不改变均匀网格的前提下得到更高精度的数值解.我们对这两种网格加密方法都进行了数值测试.
[Abstract]:Fractional differential operators have typical nonlocal properties and are widely used in various fields of science and engineering. However, because of the nonlocality of fractional differential operators, the discrete algebraic equations are often dense, which greatly increases the difficulty and complexity of numerical solution. At the same time, some fractional differential equations have boundary layer phenomena and weak regularity, even if their diffusion coefficient and right end term are smooth. In this paper, the preprocessing method of solving fractional differential equations with boundary layer is studied. In this paper, we discuss a class of nonhomogeneous fractional differential equations with Dirichlet boundary conditions. Due to the phenomenon of boundary layer, it is necessary to use the method of local encryption mesh generation. However, the discrete algebraic equations no longer have a global Toeplitz structure, which makes it difficult to design efficient preprocessing methods. Based on the detailed analysis and study of the coefficient matrix, we propose a block diagonal preprocessor and a block lower triangular preprocessor based on the cyclic matrix, and theoretically prove the effectiveness of these two preprocessing methods. Numerical experiments also show that the two preprocessing methods have good numerical performance. We also consider the method of local mesh refinement. In order to reduce computational complexity as much as possible, we propose a local mesh thinning method to minimize the number of mesh encryption required for numerical solutions with the same accuracy. At the same time, we propose a balanced local thinning mesh generation method, which can obtain a more accurate numerical solution without changing the uniform mesh. We have carried on the numerical test to these two kinds of mesh encryption methods.
【学位授予单位】：华东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.6

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