二阶动力学方程数值积分的一种新格式

A Numerical Integration Schema of Second Order Equations in Dynamics

许多动力学问题归结为求解一个二阶常微分方程组.本文提出了适用于二阶方程组的一种包含不同阶次公式的自动调节步长数值积分格式,对方法的稳定性进行了分析,并给出了数值计算例子,说明了在同样的精度要求下,与传统的Runge-Kutta 法相比,这种新格式可以节省计算机时.

Many problems in dynamics can be reduced to second order ordinarydifferential equation systems in the form of A(y,y,t)y=b(y,y,t),wherey ∈(?)~n,A being an n×n matrix and b an n-dimensional vector.These systems areusually high nonlinear and in most cases can only be treated with the numericalintegration method.However,the available numerical integration schemata aremostly for the first order ordinary differential equation system being constru-cted.If we apply these schemata to deal with second order equation systems,wehave to inverse the matrices many times and therefore the efficiency is low.A numerical integration schema with adaptive step-length by comparing theresults from the formulas of different orders is suggested in this paper.Itespecially suits the above-mentioned second order equation systems.Thestability of the method is analysed with numerical examples given.It is shownthat the new schema is cheaper than the conventional Runge-Kutta method.