离散空间分数阶非线性薛定谔方程的预处理迭代方法

发布时间：2018-05-29 23:47

本文选题：分数阶 + 非线性　；参考：《华东师范大学》2017年硕士论文

【摘要】：众所周知,非线性Schr(?)dinger方程在量子力学中占有十分重要的地位.近些年来,作为传统Schr(?)dinger方程的推广,分数阶Schr(?)dinger方程受到越来越多的关注,特别是关于其数值求解方法的研究.在本文中,我们主要讨论空间分数阶非线性Schr(?)dinger方程的预处理方法,包括耦合和非耦合情形.经过Crank-Nicolson差分离散后,得到一组复的非线性方程组.在用Newton方法求解时,每个迭代步都需要求解一个Jacobi矩阵线性方程组.本论文主要讨论的就是这些线性方程组的预处理Krylov子空间迭代方法,具体工作如下:(1)对于非耦合的分数阶非线性Schr(?)dinger方程,我们首先将原问题转化为具有2 × 2分块结构的实线性方程组,然后利用此系数矩阵的特殊结构,提出了一类基于交替方向和循环矩阵的预处理方法,并做了理论分析.数值算例表明,这类预处理子具有很好的数值表现.(2)针对耦合的分数阶非线性Schr(?)dinger方程,我们采用类似的方法,将原问题转化为4 × 4分块结构的实线性方程组.通过适当的矩阵分裂,将其中具有Toeplitz结构的矩阵与其它矩阵分离开来,提出了相应的交替方向预处理方法,并通过数值算例验证了预处理方法的有效性.
[Abstract]:As we all know, nonlinear Schr(?)dinger equation plays an important role in quantum mechanics. In recent years, as a generalization of the traditional Schr(?)dinger equation, the fractional Schr(?)dinger equation has attracted more and more attention, especially the research on its numerical solution. In this paper, we mainly discuss the preprocessing methods of fractional order nonlinear Schr(?)dinger equations, including coupled and uncoupled cases. After Crank-Nicolson difference discretization, a set of complex nonlinear equations are obtained. When solving by Newton method, each iteration step needs to solve a Jacobi matrix linear equation system. In this paper, we mainly discuss the preprocessing Krylov subspace iterative method for these linear equations. The main work is as follows: 1) for the uncoupled fractional nonlinear Schr(?)dinger equations, We first transform the original problem into a real linear equation system with 2 脳 2 block structure. Then, by using the special structure of the coefficient matrix, we propose a kind of preprocessing method based on alternating direction and cyclic matrix, and do theoretical analysis. Numerical examples show that this kind of preprocessor has good numerical performance. (2) for the coupled fractional nonlinear Schr(?)dinger equation, we use a similar method to transform the original problem into a real linear system with 4 脳 4 block structure. The matrix with Toeplitz structure is separated from other matrices by proper matrix splitting, and the corresponding alternating direction preprocessing method is proposed, and the validity of the preprocessing method is verified by numerical examples.
【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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