常微分方程数值积分的计算稳定性

§1.引言 1.问题的提出在工程技术上,很多问题归结为解常微分方程组的初值问题,而求得具有分析表达式的解,通常是不可能的,必须借助数值积分法求其近似解。在数值积分常微分方程,特别是小参数微分方程(最高阶导数项含有小参数的微分方程)时,步长的选择是一个复杂的问题:步长大了,就会引起计算的不稳定;而步长取得过小,又会花费大量的机器时间。通常所谓的“小参数问题”就是由于这个原因而著称的。而在自动控制系统

THE STABILITY FOR NUMERICAL INTEGRATION OF SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

In this paper the following linear system of ordinary differential equations are considered, where A is a(n×n)-dimensional constant matrix, X and F are n-dimensional vectors. In view of stability, the essential connection between step of integration, numerical integ-rative formula and property of the differential equations are studied, general criterion of choice of step of integration is given.