泰勒有限差分方法数值求解常微分方程

摘要

本文提出一种新的更高精度的泰勒有限差分公式并且应用于求解常微分方程。这种应用泰勒有限差分公式来求解常微分方程的方法称为泰勒有限差分方法。此外，出于比较的目的，使用欧拉方法求解常微分方程的算法流程也被提及，并且在MATLAB软件平台进行了两组对比性的数值实验。两组对比性的数值实验结果均表明，使用泰勒有限差分方法求解常微分方程的精度要比使用欧拉方法求解常微分方程的精度更高。后续可以应用该方法去开发常微分方程数值求解器的软件。%In this paper, a new finite difference formula with higher accuracy, termed Taylor finite difference formula, is presented and applied to solving ordinary differential equations (ODE). Such a method is thus considered as Taylor fi-nite difference method. For comparison, Euler method is also presented. In addition, numerical experiments are carried out, of which the results substantiate that the accuracy of Taylor finite difference method is indeed higher than that of Euler method for solving ODE. Finally, Taylor finite difference method can be used to develop ODE numerical solver software.