常微分方程初值问题的基本数值解法分析

Analysis of Basic Numerical Solutions for the Initial Value Problem of Ordinary Differential Equations

微分方程的数值解法在科学技术及生产实践等多方面应用广泛.文章分析了构造常微分方程初值问题数值解法的三种常用基本方法,差商代替导数法,数值积分法及待定系数法,推导出了Euler系列公式及三阶龙格-库塔公式,指出了各公式的优劣性及适用条件,并对Euler公式的收敛性、稳定性进行了分析.

The numerical solution of differential equations is widely used in science, technology, production practices and many other fields. This paper analyzed three kinds of basic methods for constructing numerical solutions for initial value problem of ordinary differential equations ： difference quotient instead of derivative method, numerical integral method and undetermined coefficients method. At the same time, the paper deduces the Euler series formula and the classical third order Runge-Kutta formula. In addition, the paper pointed out the advantages and disadvantages of each formula and application condition, it also analyzed the convergence and stability of the Euler formula.