不可分哈密顿问题的显式高阶对称法构造与应用

A symplectic integrator, which conserves the symplectic structure and gives no secular change in energy errors, is an ideal tool for studing the long-term dynamical evolution of Hamiltonian system. For an inseparable Hamiltonians, the application of explicit integrators becomes inconvenient and implicit algorithms are widely employed. Although the second order midpoint method is the most commonly used implicit symplectic method, its computational efficiency is low. Following the extended phase space second order explicit symmetric leapfrog methods of Pihajoki (2015) for an inseparable Hamiltonian system, The leapfrogs combined with coordinate mixing transformations are symmetric non-symplectic in the original dimension although they are symplectic in the extended phase space. This makes the leapfrog-like methods display good long term stability and error behaviour similar to symplectic ones. we construct extended phase space explicit symmetric integration schemes. we survey a critical problem on how to mix the variables in the extended phase space and how to project a solution in the extended phase space back to one in the original dimension. It is demonstrated via numerical tests that sequent permutations of coordinates and momenta can make the leapfrog-like methods yield the best result in both accuracy and long term stabilized error behaviour. We will present a method how to construct many fourth order extended phase space explicit symmetric integration schemes. In a similar way, extended phase space sixth, eighth and other higher order explicit algorithms can be available. With a help of invariant chaos indicators, the extended phase space explicit symplectic-like methods are well suited for various inseparable Hamiltonian problems, including post-Newtonian restricted three-body problem and post-Newtonian Hamiltonian formulations of non-spinning or spinning compact objects. In short, the present proposal may have a breakthrough on research of chaos in spinning compact binaries. It will be helpful to develop the interdiscipline of numerical mathematics and nonlinear celestial mechanics and general relativity in our country.

辛算法因保辛结构并使能量误差无长期变化是研究哈密顿系统长期定性演化的最佳积分方法。针对坐标和动量不可分的哈密顿问题，一般不适合显辛算法而适合如二阶隐式中点法那样的隐辛算法，会降低计算效率。2015年Pihajoki引入扩展相空间及坐标与动量置换方法构造二阶显式蛙跳对称法，取得类似辛算法那样好的长期能量误差效果。在此基础上，本项目将找到坐标与动量最佳置换方法并进一步构造扩展相空间的高阶显式对称法，最后借助这些显式对称法并利用相对论不变混沌指标去讨论后牛顿限制性致密星三体问题以及后牛顿非自旋或自旋致密哈密顿系统的动力学性质。这项研究有望在天体力学数值方法上取得重要进展，构造出高精度扩展相空间的显式对称法，以便为正确探讨动力学服务，促进计算数学、非线性天体力学和相对论等交叉领域发展。

辛算法因保辛结构保能量是研究哈密顿系统长期定性演化的最佳几何积分方法。针对坐标和动量不可分的哈密顿问题，适合计算效率低的隐辛算法而不适合显辛算法。2015年Pihajoki引入扩展相空间和坐标与动量置换方法构造出类似辛算法的二阶显式蛙跳对称法。.在此基础上，我们提出了一个新的方法用来构建许多四阶显式相空间扩充的类辛算法，每种算法都是由六个常用的不经过任何排列的蛙跳算法构成的.同理，在扩充的相空间中六阶，八阶，甚至更高阶显式类辛算法都是可以得到的；我们改进了为不可分离哈密顿量设计的四阶扩充相空间显式类辛算法，其中利用了Yoshita三重混合积分和中点置换---原变量和它对应扩展变量之间的中点，每一步积分都调整原变量及其对应扩展变量的值为他们的中点值。这中点置换的四阶扩充相空间显式类辛算法由一个三重积构建并明显比以前的两个三重积构建的方法有更高的计算效率；针对4自由度8维哈密顿系统，我们还构造了具有二阶精度、不含任何截断误差的8维哈密顿系统保能量算法。.最后借助这些显式对称法并利用相对论不变混沌指标去讨论带有耗散力的后牛顿限三体问题以及后牛顿非自旋或自旋致密哈密顿系统的动力学性质。这项研究在天体力学数值方法上取得一些进展，构造出高精度扩展相空间的显式对称法，以便为正确探讨动力学服务，促进计算数学、非线性天体力学和相对论等交叉领域发展。