Time-symmetric integration in astrophysics
Time-symmetric integration in astrophysics
复制标题
天体物理学中的时间对称积分
DOI：
10.1093/mnras/sty184
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发表时间：
2017
影响因子：
4.8
通讯作者：
E. Bertschinger
中科院分区：
文献类型：
--
作者：
David M. Hernandez;E. Bertschinger
计算常微分方程组的长期解，如N$天体问题的解，对于理解从星系形成到行星混沌的天体物理学中的广泛动力学是至关重要的。由于这些方程一般不存在解析解，研究人员依赖于容易产生各种误差的数值方法。为了减少这些误差，采用了强大的辛积分器。但是，辛积分器可能会受到严重的限制，因为它们与自适应步长不兼容，因此它们难以适应不断变化的时间和长度尺度。一种很有前途的替代方案是时间可逆积分，它可以处理自适应时间步长，但由于天体物理学中的时间可逆积分而产生的误差却鲜为人知。这项工作的目的是解析和数值研究时间可逆积分在有和没有自适应步长的情况下所引起的误差。为了进行误差分析，我们推导了这些积分器的修正的微分方程组。作为一个例子，我们考虑了可逆非辛积分器--梯形规则，并证明了它对一个摆问题和一个Henon-Heiles轨道都有长期的能量误差增加。我们的结论是，使用可逆积分并不能保证良好的能量守恒，在可能的情况下，最好使用辛积分器。我们还证明了时间对称性和时间可逆性对于积分器来说是截然不同的性质。
Calculating the long term solution of ordinary differential equations, such as those of the $N$-body problem, is central to understanding a wide range of dynamics in astrophysics, from galaxy formation to planetary chaos. Because generally no analytic solution exists to these equations, researchers rely on numerical methods which are prone to various errors. In an effort to mitigate these errors, powerful symplectic integrators have been employed. But symplectic integrators can be severely limited because they are not compatible with adaptive stepping and thus they have difficulty accommodating changing time and length scales. A promising alternative is time-reversible integration, which can handle adaptive time stepping, but the errors due to time-reversible integration in astrophysics are less understood. The goal of this work is to study analytically and numerically the errors caused by time-reversible integration, with and without adaptive stepping. We derive the modified differential equations of these integrators to perform the error analysis. As an example, we consider the trapezoidal rule, a reversible non-symplectic integrator, and show it gives secular energy error increase for a pendulum problem and for a Henon---Heiles orbit. We conclude that using reversible integration does not guarantee good energy conservation and that, when possible, use of symplectic integrators is favored. We also show that time-symmetry and time-reversibility are properties that are distinct for an integrator.