Interval of effective time-step size for the numerical computation of nonlinear ordinary differential equations

由于满足计算的不确定性原理,需适当选取时间步长以保证非线性常微分方程组数值解的可靠性,目前尚未见关于有效步长区间的理论结果。本文对于给定的误差限,将方法截断误差与机器舍入误差的相关曲线分别进行平移,从而得到一种确定有效步长近似区间的方法,并推导出近似区间相比于原区间的相对误差公式。另外,研究了有效步长区间随积分时间的变化规律,并对已有的数值结果给出解释。本文所得结论可为数值求解常微分方程组选取有效步长并得到可靠的数值解提供理论支持。

The computational uncertainty principle states that the numerical computation of nonlinear ordinary differential equations（ODEs） should use appropriately sized time steps to obtain reliable solutions.However,the interval of effective step size（IES） has not been thoroughly explored theoretically.In this paper,by using a general estimation for the total error of the numerical solutions of ODEs,a method is proposed for determining an approximate IES by translating the functions for truncation and rounding errors.It also illustrates this process with an example.Moreover,the relationship between the IES and its approximation is found,and the relative error of the approximation with respect to the IES is given.In addition,variation in the IES with increasing integration time is studied,which can provide an explanation for the observed numerical results.The findings contribute to computational step-size choice for reliable numerical solutions.