非线性动力方程精细积分法的自适应步长研究

Study on adaptive step size of precise intrgration method for nonlinear dynamic equations

基于Adams显式和隐式预估公式实现对时间步长的自适应选择,利用当前时刻v(tk),采用预估公式的两种形式(显式与隐式),对v(tk+1)进行两次预估,利用两公式局部截断误差关系,得出误差估计值ξ(tk+1),并根据其大小自适应调节时间步长。将该思想应用于预估型(求解过程需要用到预估公式)精细积分算法中,使精细积分算法的时间步长依赖于给定的每步误差限值,提高计算精度,且使算法具有很好的稳定性,对刚度硬化和软化问题均有很好的效果。数值算例验证了本文思想的有效性与适用性。

In this paper,the adaptive selection of time steps is realized based on Adams explicit and implicit estimation formulas:For the state equation used for a nonlinear dynamic system,given v(tk) at the current time instant,two forms of the prediction formula (explicit and implicit) were used to respectively estimate the truncation errors,and the obtain ξ(tk+1) was taken as the error of this time step,and the value of the time step was adjusted according to the value of the result.This idea is applied to the precise integration algorithm of the predictive type (the solution process needs to use the prediction formula),making the time step of the precise integration algorithm depend on the given error limit of each step,improving the calculation accuracy,and making the algorithm have good stability,and have good effect on the stiffness hardening and softening problems.Numerical examples demonstrate the validity and applicability of the proposed ideas.