A Self-Adaptive Algorithm of the Clean Numerical Simulation (CNS) for Chaos (Advances in Applied Mathematics and Mechanics, 2023)
The background numerical noise $ε_0$ is determined by the maximum of truncation error and round-off error. For a chaotic system, the numerical error $ε(t)$ growsexponentially, say, $ε(t) = ε_0 {\rm exp}(κt),$ where $κ > 0$ is the so-called noise-growing exponent. This is the reason why one can not gain a convergent simulation of chaotic systems in a long enough interval of time by means of traditional algorithms in doubleprecision, since the background numerical noise $ε_0$ might stop decreasing because ofthe use of double precision. This restriction can be overcome by means of the clean numerical simulation (CNS), which can decrease the background numerical noise $ε_0$ toany required tiny level. A lot of successful applications show the novelty and validityof the CNS. In this paper, we further propose some strategies to greatly increase thecomputational efficiency of the CNS algorithms for chaotic dynamical systems. It ishighly suggested to keep a balance between truncation error and round-off error andbesides to progressively enlarge the background numerical noise $ε_0,$ since the exponentially increasing numerical noise $ε(t)$ is much larger than it. Some examples aregiven to illustrate the validity of our strategies for the CNS.