Computational uncertainty principle in nonlinear ordinary differential equations (I) - Numerical results

In a majority of cases of long-time numerical integration for initial-value problems, roundoff error has received little attention. Using twenty-nine numerical methods, the influence of round-off error on numerical solutions is generally studied through a large number of numerical experiments. Here we find that there exists a strong dependence on machine precision (which i s a new kind of dependence different from the sensitive dependence on initi al conditions), maximally effective computation time (MECT) and optimal ste psize (OS) in solving nonlinear ordinary differential equations (ODEs) in f inite machine precision. And an optimal searching method for evaluating MEC T and OS under finite machine precision is presented. The relationships bet ween MECT, OS, the order of numerical method and machine precision are foun d. Numerical results show that round-off error plays a significant role in the above phenomena. Moreover, we find two universal relations which are in dependent of the types of ODEs, initial values and numerical schemes. Based on the results of numerical experiments, we present a computational uncert ainty principle, which is a great challenge to the reliability of long-time numerical integration for nonlinear ODEs.