TY - JOUR
T1 - A Self-Adaptive Algorithm of the Clean Numerical Simulation (CNS) for Chaos
AU - Qin , Shijie
AU - Liao , Shijun
JO - Advances in Applied Mathematics and Mechanics
VL - 5
SP - 1191
EP - 1215
PY - 2023
DA - 2023/06
SN - 15
DO - http://doi.org/10.4208/aamm.OA-2022-0340
UR - https://global-sci.org/intro/article_detail/aamm/21773.html
KW - Chaos, Clean Numerical Simulation (CNS), self-adaptive algorithm, computational efficiency.
AB - <p style="text-align: justify;">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 $ε(t)$ grows
exponentially, say, $ε(t) = ε_0 {\rm exp}(κt),$ where $κ &gt; 0$ 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 double
precision, since the background numerical noise $ε_0$ might stop decreasing because of
the use of double precision. This restriction can be overcome by means of the clean numerical simulation (CNS), which can decrease the background numerical noise $ε_0$ to
any required tiny level. A lot of successful applications show the novelty and validity
of the CNS. In this paper, we further propose some strategies to greatly increase the
computational efficiency of the CNS algorithms for chaotic dynamical systems. It is
highly suggested to keep a balance between truncation error and round-off error and
besides to progressively enlarge the background numerical noise $ε_0,$ since the exponentially increasing numerical noise $ε(t)$ is much larger than it. Some examples are
given to illustrate the validity of our strategies for the CNS.</p>