四阶Runge-Kutta格式及五阶Runge-Kutta格式的证明

A Proof for the Fourth Order and Fifth Order of Runge-Kutta Forms

基于Taylor展开方法,在二阶和三阶Runge-Kutta格式的基础上证明了具有四阶精度的四阶Runge-Kutta格式所满足的条件,同时推导出了五阶Runge-Kutta格式,并证明了具有五阶精度的五阶Runge-Kutta格式所满足的条件。数值例子的计算结果表明五阶Runge-Kutta格式具有较高精度。

Based upon the method of Taylor series expansion,the conditions which the forth order of Runge-Kutta form with fourth precision satisfies is proved upon the second and the third order of Runge-Kutta forms. At the same time,the fifth order of Runge-Kutta form is deduced,and it is also proved that the conditions which the fifth order of Runge-Kutta form with fifth precision satisfies. The calculating results of numerical examples show that the fifth order Runge-Kutta form is highly accurate.