三级Runge-Kutta方法阶条件的推导

Derivation of the Order Conditions for Three Stages Runge-Kutta Methods

Runge-Kutta(RK)方法是数值求解常微分方程的基本方法,也是构造高阶单步法的重要途径.实际计算中具有构造思路简单、计算高效等优点.然而,这类方法在构造时涉及到多变量复合函数的高阶微分运算,运算起来非常复杂.几乎所有的计算方法教材、专著都只给出方法的构造思想,同时给出几个常用的RK方法,很少讨论高阶方法的构造过程和相关细节,初学者学起来非常吃力,不能彻底理解RK方法.基于此,本文给出三级RK方法的构造过程,从而彻底理解方法的构造思想.最后通过一些例子来检验.

Runge-Kutta(RK)method is a fundamental method to numerically solve ODEs.It is also an important approach to design high order one-step methods.It feathers the advantages of simplicity and efficiency in practical computing.However,it is very troublesome since it involves high order differential manipulation for multi-variate compound functions.Almost all the computational methods textbook,monographs only present the idea of RK methods and give some commonly used RK methods.It seldom discusses how to construct RK methods and the details.It is very difficult for beginner to completely understand RK methods.This paper is devoting to presenting the constructing process of RK method of three stages.Thus,beginners can completely understand the idea.Some examples are presented to verify the methods.