目录
目录
摘要Ⅱ
关键词Ⅱ
AbstractⅢ
引言
引言1
1数值方法概述1
1.1研究现状 1
1.2基本的线性多步法1
1.2.1线性多步法1
1.2.2混合法2
1.3指数拟合的线性多步法2
1.4两导数RungeKutta方法2
2数值方法分析3
2.1指数多步法3
2.2对称多步法的相延迟分析3
3 一类新的三角拟合对称线性多步法5
3.1一个两步方法5
3.2一个四步方法5
4数值实验5
5进一步的应用问题及其数值求解方法 7
5.1 Schrödinger方程—共振问题7
5.2 N体问题8
5.3非齐次方程问题9
5.4非线性方程问题9
5.5二体问题10
6 总结与展望 10
致谢12
参考文献12
二阶振荡微分方程的高效线性多步法
摘 要
二阶振荡微分方程广泛存在于经典力学、量子力学、天文学、生物工程学等领域。基于很多微分方程的解析解是很难得到的客观事实，数值方法的优化和研究在各级各类领域的应用中展现了相当的必要性和重要性。与传统的一些数值方法相比，线性多步法展示出了非常多的优势，方法更为简捷、数值解更加精确、并且拥有更高的计算效率。
因为线性多步法所显示出的优越性，本文旨在对二阶振荡微分方程的线性多步法进行深入的了解，在前人的基础上展开探究，熟练掌握并灵活运用线性多步法，并用于讨论量子力学中的典型问题。
本文的结构将主要分为如下六个部分：第一节简要介绍二阶振荡微分方程数值解的基础知识和研究背景，概述国内外专家学者对于二阶振荡微分方程的研究现状；第二节，我们将分别介绍基础的线性多步法和指数拟合型线性多步法，并对对称多步法进行相延迟分析；第三节构造了实用的两个双频三角拟合对称线性多步法；第四节通过数值求解两个经典的测试问题，证明新方法相对于文献中几个高效方法的优越 *51今日免费论文网|www.51jrft.com +Q: &351916072& 
性。第五节进一步讨论了应用科学中几个典型问题的数值求解策略；最后，我们简要地总结了本文讨论的主要问题，并展望了未来研究的若干课题。
EFFICIENT LINEAR MULTISTEP METHOD FOR SECOND ORDER OSCILLATORY DIFFERENTIAL EQUATIONS
ABSTRACT
Second order oscillatory differential equations are widely used in classical mechanics, quantum mechanics, astronomy, bioengineering and other fields. Based on the fact that analytical solutions of many differential equations are difficult to obtain, the optimization and research of numerical methods have shown considerable necessity and importance in various fields. Compared with some traditional numerical methods, linear multistep method has many advantages, such as simpler method, more accurate numerical solution and higher computational efficiency.
Because of the advantages of linear multistep method, this paper aims to deeply understand the linear multistep method of secondorder oscillatory differential equation, to explore on the basis of predecessors, to master and flexibly use linear multistep method, and to discuss the typical problems in quantum mechanics.
The structure of this paper is divided into four parts as follows: In Section 1, the basic knowledge and research background of the numerical solution of the secondorder oscillatory differential equation are briefly introduced, and the research status of the secondorder oscillatory differential equation by domestic and foreign experts is summarized. In Section, we introduce the basic linear multistep and exponential fitting linear multistep respectively, and analyze the phase delay of the symmetric multistep. In Section 3 we present two practical twofrequency trigonometrically fitted linear multistep methods. In Section 4, by numerically solving two classical test problems, our new methods are shown to be superior to some highly effective codes from recent literature. In Section 5, we discuss numerical strategies for some classical problems in applied sciences. Finally, we summarize the main problems of this paper and outlook some promising topics for future research.

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