等谱流方法及其应用
发布时间：2018-07-15 22:03
【摘要】：等谱流近些年应用广泛,从分子动力学到微磁学,再到线性代数,都与之有着密切的关系,因此近年来引起许多人的兴趣.等谱流的一般形式是由矩阵微分方程表示L=[B(L),L],L(0)=L0,其中Lo是给定的对称矩阵,B(L)对所有的L都是反对称的,且[B(L),L]= B(L)L-LB(L)矩阵B(L)的不同选择分别刻画了解L(t)的不同动态特征,本文针对Toda流,给出了一种新的等谱流方法,并从能量守恒的角度给出了各种等谱流方法的比较.主要工作如下：1.介绍了等谱流的相关定义及其主要特性,即等谱性,并且简单介绍了它与QR算法的联系,以及几种常见的等谱流.2.研究了等谱流方法中的修正的Gauss-Legendre方法(MGLRK)和半显式的Taylor方法,并从能量守恒的角度给出了他们的数值比较.3.基于李群算法的基本概念和RKMK方法的基本原理,构造一种新的等谱流方法,即Cayley变换下的RK方法.利用数值实验验证新算法较其他等谱流方法更有效,同样也考察了新方法在能量守恒上的数值表现.
[Abstract]:Isospectral flows have been widely used in recent years, ranging from molecular dynamics to micromagnetics to linear algebras, which have attracted many people's interest in recent years. The general form of isospectral flow is expressed by a matrix differential equation L = [B (L) L] L (0) L _ 0, where Lo is a given symmetric matrix B (L) is antisymmetric to all L, and [B (L) L] = B (L) L-LB (L) matrix B (L) characterizes the different dynamic characteristics of L (t), respectively. In this paper, a new isospectral flow method is presented, and the comparison of various isospectral flow methods from the point of view of energy conservation is given. The main work is as follows: 1. This paper introduces the definition of isospectral flow and its main characteristics, that is, isospectral property, and briefly introduces its relation with QR algorithm, and several common isospectral flows. The modified Gauss-Legendre method (MGLRK) and the semi-explicit Taylor method in the isospectral flow method are studied, and their numerical comparisons are given from the point of view of energy conservation. Based on the basic concept of Li Qun algorithm and the basic principle of RKMK method, a new isospectral flow method, RK method under Cayley transform, is constructed. Numerical experiments show that the new method is more effective than other isospectral flow methods, and the numerical performance of the new method in energy conservation is also investigated.
【学位授予单位】：南京师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.81
本文编号：2125517
[Abstract]:Isospectral flows have been widely used in recent years, ranging from molecular dynamics to micromagnetics to linear algebras, which have attracted many people's interest in recent years. The general form of isospectral flow is expressed by a matrix differential equation L = [B (L) L] L (0) L _ 0, where Lo is a given symmetric matrix B (L) is antisymmetric to all L, and [B (L) L] = B (L) L-LB (L) matrix B (L) characterizes the different dynamic characteristics of L (t), respectively. In this paper, a new isospectral flow method is presented, and the comparison of various isospectral flow methods from the point of view of energy conservation is given. The main work is as follows: 1. This paper introduces the definition of isospectral flow and its main characteristics, that is, isospectral property, and briefly introduces its relation with QR algorithm, and several common isospectral flows. The modified Gauss-Legendre method (MGLRK) and the semi-explicit Taylor method in the isospectral flow method are studied, and their numerical comparisons are given from the point of view of energy conservation. Based on the basic concept of Li Qun algorithm and the basic principle of RKMK method, a new isospectral flow method, RK method under Cayley transform, is constructed. Numerical experiments show that the new method is more effective than other isospectral flow methods, and the numerical performance of the new method in energy conservation is also investigated.
【学位授予单位】：南京师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.81
【参考文献】
相关硕士学位论文 前1条
1 胡勋锋;李群算法及其应用[D];海南大学;2012年
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