常微分方程基于GAUSS型积分的数值解法

提出用Gauss-Legendre求积公式构造常微分方程初值问题的离散化格式,以给出一种求解此类问题的数值方法。文中根据两点与三点Gauss-Legendre求积公式及逼近Gauss点处函数值的不同方法,列出十余种计算格式,并说明它们的收敛性和稳定性。各种格式是针对一阶常微分方程提出的,但同样也适用于一阶常微分方程组和高阶常微分方程的初值问题。

NUMERICAL METHODS BASED ON GAUSS QUADRATURE FORMULA FOR ORDINARY DIFFERFNTIAL EQUATIONS

With the aid of Gauss-Legendre quadrature formulas, this paper suggests some discrete schemes of initial-value problems of ordinary differential equations to obtain a numerical method to solve these problems. According to Gauss-Legendre formulas with two or three arguments and various methods to approach to the function values at Gauss points we enumerate more than ten computation schemes and explain their convergence and stability.Those schemes is aimed to first-order equations, but they can also use to systems of equations and high-order equations.