一种用于混沌的洁净数值模拟（CNS）自适应算法
A Self-Adaptive Algorithm of the Clean Numerical Simulation (CNS) for Chaos
AI总结：
针对混沌系统长期模拟中数值噪声指数增长的问题，本文提出平衡截断与舍入误差并自适应增大背景噪声的 CNS 策略，显著提升计算效率。
摘要：
背景数值噪声 $\varepsilon_{0}$ 由截断误差和舍入误差的最大值决定。对于混沌系统，数值误差 $\varepsilon(t)$ 呈指数增长，即 $\varepsilon(t) = \varepsilon_{0} \exp(κ\\,t)$，其中 $κ>0$ 是所谓的噪声增长指数。这就是为什么人们无法借助双精度传统算法在足够长的时间区间内获得混沌系统的收敛模拟，因为由于使用双精度，背景数值噪声 $\varepsilon_{0}$ 可能停止减小。这一限制可通过洁净数值模拟（CNS）克服，该方法能够将背景数值噪声 $\varepsilon_{0}$ 减小到任何所需的微小水平。大量成功应用展示了 CNS 的新颖性和有效性。本文中，我们进一步提出若干策略，以大幅提高用于混沌动力系统的 CNS 算法的计算效率。强烈建议在截断误差与舍入误差之间保持平衡，此外，由于指数增长的数值噪声 $\varepsilon(t)$ 远大于背景数值噪声 $\varepsilon_{0}$，应逐步增大 $\varepsilon_{0}$。文中给出若干示例，以说明我们这些 CNS 策略的有效性。
英文摘要：
The background numerical noise $\varepsilon_{0} $ is determined by the maximum of truncation error and round-off error. For a chaotic system, the numerical error $\varepsilon(t)$ grows exponentially, say, $\varepsilon(t) = \varepsilon_{0} \exp(κ\,t)$, 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 double precision, since the background numerical noise $\varepsilon_{0}$ 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 $\varepsilon_{0}$ 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 $\varepsilon_{0}$, since the exponentially increasing numerical noise $\varepsilon(t)$ is much larger than it. Some examples are given to illustrate the validity of our strategies for the CNS.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
