【作者】 王艳丽；

【导师】 赵平福；

【作者基本信息】 北京交通大学 ， 计算数学， 2006， 硕士

【摘要】 本文以特殊的线性振荡方程y″+g(t)y=0(其中(?) g(t)=+∞)为例讨论了高振荡常微分方程数值解问题。高振荡微分方程是指其解含有高振荡函数的一类微分方程，它在分子动力学、天体力学、量子化学以及原子物理等方面有着广泛的应用。对于高振荡微分方程给出一种好的数值解法是一件非常困难的事情。例如，对于形如y″+g(t)y=0的线性高振荡方程，用经典的方法，如Runge-Kutta法、线性多步法等方法在处理该类问题时均会产生较大的误差。近来，Iserles利用Magnus展开方法详细研究了该类方程数值解法问题，给出了计算结果较好的数值算法。在这篇论文中，我们首先介绍一些基本概念和基本知识，为后面的内容做准备工作。然后用梯形方法对线性振荡方程y″+g(t)y=0数值求解，理论分析及数值实验均显示，用梯形方法求解会产生较大的误差。我们就梯形格式作了几种修改，误差分析及数值结果均显示，修改后格式的数值解都要优于梯形格式的数值解。我们还系统地介绍了Magnus展开方法及修正的Magnus展开方法，从修正的Magnus展开方法出发，我们考虑利用Cayley变换构造线性高振荡微分方程的数值解法。这样构造的解法涉及到高振荡函数的积分，我们采用Filon方法计算，数值结果显示，该数值解法具有好的长时间数值跟踪能力。另外，我们也利用数值实验比较了用Gauss方法及Filon方法计算高振荡函数积分所给出的数值解法，实验显示，Filon方法相应的数值解法优于Gaoss方法所相应的数值解法。更多还原

【Abstract】 This paper deals with the numerical solutions of highly-oscillatory ordinary differential equations,with a special reference to the linear oscillatory" + g(t)y = 0,where lim g(t) = +∞.Highly-oscillatory ordinary differential equations are a kind of equations whose solutions include highly-oscillating functions. They are extensively applied in molecular dynamics、 celestial mechanics、 quantum chemistry、 atomic physics and so on. It is very difficult to give a good numerical method for highly-oscillatory ordinary differential equations. For example, when handling with the linear oscillator y" + g(t)y = 0, with classical methods Runge-Kutta, multi-step methods, these methods may generate bigger error. Recently,Iserles has studied specifically numerical solution question of this kinds of equations using Magnus expansion and given good results.In this paper, we first introduce some basic concepts and knowledge preparing for the following content,and then give a numerical solution for the linear oscillator y" + g(t)y = 0 with trapezoidal method. Both theoretical analysis and numerical experiment indicate that this method generates bigger error. We modify the trapezoidal scheme in several ways. Both error analysis and numerical results indicate that the numerical solutions of modified trapezoidal schemes are better than the trapezoidal scheme.Also, we introduce systematically Magnus methods and modified Magnus methods. Beginning with modified Magnus methods,We consider numerical method of the linear highly-oscillatory ordinary differential based on the Cayley map.The method refers to the highly-oscillating integral.We adopt Filon method ,numerical result indicates that the method has good ability of long time intervals tracking computation.In addition,we compare the numerical results of highly-oscillating integral with Filon and Gauss method separately by numerical experiment.The experiment indicates that Filon method is better than Gauss method.更多还原