等譜流方法及其應(yīng)用
發(fā)布時(shí)間：2018-07-15 22:03
【摘要】：等譜流近些年應(yīng)用廣泛,從分子動(dòng)力學(xué)到微磁學(xué),再到線(xiàn)性代數(shù),都與之有著密切的關(guān)系,因此近年來(lái)引起許多人的興趣.等譜流的一般形式是由矩陣微分方程表示L=[B(L),L],L(0)=L0,其中Lo是給定的對(duì)稱(chēng)矩陣,B(L)對(duì)所有的L都是反對(duì)稱(chēng)的,且[B(L),L]= B(L)L-LB(L)矩陣B(L)的不同選擇分別刻畫(huà)了解L(t)的不同動(dòng)態(tài)特征,本文針對(duì)Toda流,給出了一種新的等譜流方法,并從能量守恒的角度給出了各種等譜流方法的比較.主要工作如下：1.介紹了等譜流的相關(guān)定義及其主要特性,即等譜性,并且簡(jiǎn)單介紹了它與QR算法的聯(lián)系,以及幾種常見(jiàn)的等譜流.2.研究了等譜流方法中的修正的Gauss-Legendre方法(MGLRK)和半顯式的Taylor方法,并從能量守恒的角度給出了他們的數(shù)值比較.3.基于李群算法的基本概念和RKMK方法的基本原理,構(gòu)造一種新的等譜流方法,即Cayley變換下的RK方法.利用數(shù)值實(shí)驗(yàn)驗(yàn)證新算法較其他等譜流方法更有效,同樣也考察了新方法在能量守恒上的數(shù)值表現(xiàn).
[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.
【學(xué)位授予單位】：南京師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類(lèi)號(hào)】：O241.81
本文編號(hào)：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.
【學(xué)位授予單位】：南京師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類(lèi)號(hào)】：O241.81
【參考文獻(xiàn)】
相關(guān)碩士學(xué)位論文 前1條
1 胡勛鋒;李群算法及其應(yīng)用[D];海南大學(xué);2012年
，本文編號(hào)：2125517
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