题╲t目常微分方程的数值解法及其实现.doc

题 目 常微分方程的数值解法及其实现 英文题目 The numerical solution of ordinary differential equations and its implementation 学生姓名： 学 号： 专 业： 信息与计算科学 学 院： 指导教师： 二 0 年 月 日 摘 要 随着计算机的迅速发展和广泛运用，在众多领域内，我们越来越认识到科学计算是科学研究的重要方法。 虽然求解微分方程有许多解析方法,但解析方法只能够求解一些特殊类型的方程,从实际意义上来讲我们更关心的是某些 特定的自变量在某一个定义范围内的一系列离散点上的近似值.把这样一组近似解称为微分方程在该范围内的数值解. 常微分方程的数值解法常用来求近似解。数值解法得到的近似解(含误差)是一个离散的函数表. 常微分方程的数值有很多，解法精确度不高的是欧拉方法，也就是一阶数值方法。也有其改进的方法，称为改进Euler法也叫预测-校正法，可以证明该算法具有 2 阶精度，同时可以看到它是个单步递推格式，比隐式公式的迭代求解过程简单。后面将看到，它的稳定性高于显式欧拉法。其他的主要就是龙格库塔法，有二阶和四阶之分.现在计算机中使用的是RK4，也就是4阶龙格库塔方法来计算常微分方程的初值问题。 关键词：欧拉算法；龙格库塔法； ABSTRACT With the rapid development of the computer and used widely, in many fields, we have become increasingly aware of the scientific calculation is an important method of scientific research. Although there are many analytical method for solving differential equations, but the analytical method can only solve some special types of equations, from a practical sense, we are more concerned about the specific variables in one of a series of discrete point approximation to define the range of values. To such a group of approximate solution is called numerical differential equations in the scope of the solution. The numerical solution of ordinary differential equations are used to find approximate solutions. The numerical solution of the approximate solution obtained (error) is a discrete function table. There are many numerical ordinary differential equations, solution accuracy is not high for the Euler method, also is the first-order numerical method. Also improved the method, called the improved Euler method is also called the predictor-corrector method, the algorithm can be proved with 2 order accuracy, at the same time we can see it is a one-step recursion scheme, simple process than the iterative method for solving the implicit equations. Youll see, it is more s