十维相空间守恒Hamilton系统的能量守恒积分器
Energy-conserving integrator for conservative Hamiltonian systems with ten-dimensional phase space
摘要：
本文针对十维五自由度相空间守恒哈密顿系统，设计了一种隐式非辛精确保能积分器。它是基于一个合适的离散平均的哈密顿梯度，与二阶精度的数值解。本文以一维无序离散非线性薛定谔方程和自旋紧双星的后牛顿哈密顿系统为例，讨论了它们的解。我们数值证明，该算法具有良好的长期性能，在能量的保存，如果忽略roundeness错误。该结果与时间步长、初始轨道偏心率、规则和混沌轨道动力学行为无关。特别是，适当大的时间步长的新算法的应用，有助于减少耗时和循环误差。这种新的方法，结合快速李雅普诺夫指标，是非常适合相关的混沌在两个例子的问题。结果表明，混沌是前一个系统的主要原因之一。在后一个问题中，小的初始间距和高的初始偏心率的组合可以容易地诱导混沌。
英文摘要：
In this paper, an implicit nonsymplectic exact energy-preserving integrator is specifically designed for a ten-dimensional phase-space conservative Hamiltonian system with five degrees of freedom. It is based on a suitable discretization-averaging of the Hamiltonian gradient, with a second-order accuracy to numerical solutions. A one-dimensional disordered discrete nonlinear Schrödinger equation and a post-Newtonian Hamiltonian system of spinning compact binaries are taken as our two examples. We demonstrate numerically that the proposed algorithm exhibits good long-term performance in the preservation of energy, if roundoff errors are neglected. This result is independent of time steps, initial orbital eccentricities, and regular and chaotic orbital dynamical behavior. In particular, the application of appropriately large time steps to the new algorithm is helpful in reducing time-consuming and roundoff errors. This new method, combined with fast Lyapunov indicators, is well suited related to chaos in the two example problems. It is found that chaos in the former system is mainly responsible for one of the parameters. In the latter problem, a combination of small initial separations and high initial eccentricities can easily induce chaos.
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