中点置换的相空间扩充显式类辛算法

发布时间：2018-04-18 03:37

本文选题：辛积分器 + 相空间扩充法　；参考：《南昌大学》2017年硕士论文

【摘要】：辛算法提出以后在天体力学领域内得到长足的发展和广泛的应用,由于它运算过程中能够保持哈密顿系统的辛结构(表现在长期误差的积累没有发生在能量和角动量中)。辛算法通常可分为两类,一种是显式辛算法,另外一种是隐式辛算法,前者常用于坐标动量可分的哈密顿系统,但在不可分的哈密顿系统中的应用却比较困难,这时候隐式辛算法就开始发挥重要的作用,但是隐式辛算法需要大量计算时间重复迭代,效率底下。人们开始考虑把显式算法的好处带到不可分哈密顿系统。在本篇论文中我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。总的来说,这新的中点置换比起目前已知的坐标动量置换能更有效限制原变量和对应扩充变量在每一个积分步数相互相等,作为结果,这种新的构建方法能把辛积分器的好处带给二阶后牛顿自旋紧密双星哈密顿量,特别是在混沌轨道,这新的方法有很好的表现,但以前的置换方法却不行。然后我推广中点置换的相空间扩充显式类辛算法到非保守系统,例如带引力耗散效应的自旋紧密双星、阻尼谐振子和尘埃粒子在Poynting-Robertson阻力下的轨道运动。在数值模拟中,这中点置换的显式类辛积分器明显比坐标动量相继置换以及隐式辛算法更高级更精确更有效率。这新的方法在研究长期演变的不可分哈密顿问题特别有效。
[Abstract]:The symplectic algorithm has been developed and widely applied in the field of celestial mechanics since it can maintain the symplectic structure of Hamiltonian system (as shown by the accumulation of long-term errors not occurring in energy and angular momentum).The symplectic algorithm is usually divided into two categories, one is explicit symplectic algorithm, the other is implicit symplectic algorithm.At this point the implicit symplectic algorithm begins to play an important role, but the implicit symplectic algorithm needs a lot of computation time to iterate over and over again, which is inefficient.People are beginning to consider bringing the benefits of explicit algorithms to indivisible Hamiltonian systems.In this paper, I improve the explicit symplectic algorithm of the fourth order extended phase space for inseparably Hamiltonian, in which we use the Yoshita triple mixed integral and the midpoint permutation --primitive variable and the midpoint between its corresponding extended variables.Each step integral adjusts the value of the original variable and its corresponding extended variable to their midpoint value.The fourth-order extended phase space explicit symplectic algorithm for midpoint permutation is constructed from a triple product and is obviously more efficient than the previous method.In general, this new midpoint permutation is more effective than the known coordinate momentum permutation in limiting the equal number of integral steps between the original variable and the corresponding extended variable as a result.This new construction method can bring the advantage of symplectic integrator to the second-order post-Newtonian spin-tight binary star Hamiltonian, especially in chaotic orbits. This new method has a good performance, but the previous replacement method is not.Then I extend the extended explicit symplectic algorithm of midpoint permutation to non-conservative systems, such as spin compact binary stars with gravitational dissipation effect, damped harmonic oscillator and dust particle orbital motion under Poynting-Robertson resistance.In numerical simulation, the explicit symplectic integrator of midpoint permutation is more efficient than coordinate momentum successive permutation and implicit symplectic algorithm.This new method is particularly effective in studying the long-term evolution of the indivisible Hamiltonian problem.
【学位授予单位】：南昌大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：P13

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