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

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.