Dynamical Chaos in the Wisdom-Holman Integrator: Origins and Solutions
Dynamical Chaos in the Wisdom-Holman Integrator: Origins and Solutions
复制标题
Wisdom-Holman 积分器中的动态混沌：起源和解决方案
DOI：
10.1086/300720
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发表时间：
1998
期刊：
影响因子：
--
通讯作者：
M. Holman
中科院分区：
文献类型：
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作者：
K. Rauch;M. Holman
我们研究了Wisdom-Holman（WH）辛映射应用于扰动，高偏心（e = 0.9）两体轨道的积分的非线性稳定性。我们发现，该方法是不稳定的，并引入人工混沌到这类问题的计算轨迹，除非选择的步长足够小，periapse总是解决，在这种情况下，该方法是一般稳定的。这种“径向轨道不稳定性”甚至在弱扰动系统中也存在。使用斯塔克问题作为基准测试案例，我们调查这种不稳定性的动力学起源，并认为，数值混乱的结果从重叠的步长共振;有趣的是，斯塔克问题的许多这些共振似乎是绝对稳定的。我们同样研究了几种替代积分方法的鲁棒性：Mikkola提出的WH映射的时间正则化版本;邓肯，Levison和Lee的势分裂（PS）方法;以及两种基于Stark运动而不是Keplerian运动的近似方法（比较纽曼等人）。两个不动点问题和一个相关的，更一般的问题是用来进行比较测试的各种方法的几种类型的运动。在测试的算法中，时间变换WH映射显然是最有效和最稳定的方法，在没有近距离接触的情况下，将偏心，近开普勒轨道。对于测试粒子受到高偏心率和非常接近的遭遇，我们发现一个增强版本的PS方法，将时间正则化，力中心切换，和一个改进的内核功能，是既经济和高度通用。我们得出结论，斯塔克为基础的方法是边际效用的N体类型的集成。额外的影响辛积分的N体系统进行了讨论。
We examine the nonlinear stability of the Wisdom-Holman (WH) symplectic mapping applied to the integration of perturbed, highly eccentric (e ≳ 0.9) two-body orbits. We find that the method is unstable and introduces artificial chaos into the computed trajectories for this class of problems, unless the step size chosen is small enough that periapse is always resolved, in which case the method is generically stable. This "radial orbit instability" persists even for weakly perturbed systems. Using the Stark problem as a fiducial test case, we investigate the dynamical origin of this instability and argue that the numerical chaos results from the overlap of step-size resonances; interestingly, for the Stark problem many of these resonances appear to be absolutely stable. We similarly examine the robustness of several alternative integration methods: a time-regularized version of the WH mapping suggested by Mikkola; the potential-splitting (PS) method of Duncan, Levison, & Lee; and two original methods incorporating approximations based on Stark motion instead of Keplerian motion (compare Newman et al.). The two fixed point problem and a related, more general problem are used to conduct a comparative test of the various methods for several types of motion. Among the algorithms tested, the time-transformed WH mapping is clearly the most efficient and stable method of integrating eccentric, nearly Keplerian orbits in the absence of close encounters. For test particles subject to both high eccentricities and very close encounters, we find an enhanced version of the PS method—incorporating time regularization, force-center switching, and an improved kernel function—to be both economical and highly versatile. We conclude that Stark-based methods are of marginal utility in N-body type integrations. Additional implications for the symplectic integration of N-body systems are discussed.