高阶显式类辛法的构建和应用

发布时间：2018-06-14 08:26

本文选题：混沌-重力-方法:数值-天体力学-行星和卫星:动力学演化和稳定 + 核-壳时空　；参考：《南昌大学》2017年硕士论文

【摘要】：Pihajoki提出了一种适用于不可分离哈密顿系统的二阶显式相空间扩充的的对称蛙跳积分方法。在这个基础上,我们在扩充的相空间中如何组合这些变量的方法上提出了一些见解。数据测试表明,连续的置换坐标和动量可以使类似的蛙跳算法得到更加准确的计算结果和最佳的长时间稳定性,我们也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。这种构建模式由四部分组成:置换坐标,没有置换的三个二阶蛙跳格式的积分,置换动量,没有置换的三个二阶蛙跳格式的积分。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。我们用一些常见的不可分离的哈密顿系统为例,就像有后牛顿项的无旋转的致密双星系统来证明我们提出的四阶算法要优于目前已知的算法,也包括了在Chin系统中的四阶辛算法和四阶显式和隐式中点法混合的辛算法。当给予适当的初始值时,我们可以发现显式相空间扩充的类辛算法更适合用于不可分离的哈密顿模型中,这些模型包括太阳系系统中常用的带微小扰动的重力系统和具有后牛顿项的自旋的致密双星系统。之后我们将之用于更加复杂的哈密顿系统中,并讨论二阶,四阶,八阶的性质,讨论相对论核-壳系统的测地线方程的一些物理学性质。最稳定的圆形轨道的半径在赤道平面只取决于四极矩。给定的扁圆的四极矩会导致两个最稳定的圆形轨道存在,它们的半径大于史瓦西半径。然而,一个扁长的四极矩只对应一个最稳定的圆轨道,其半径小于史瓦西半径。在一般的测地线轨道中,上述的四阶显式相空间空充的对称算法能有效的运用到这个哈密顿系统中,尽管它们相对论中核与壳不能分离开来。借助这样的积分方法,可以快速的计算这个轨道并且不会伴随能量误差的增长。这些核壳之间极矩对测地线方程的影响是由数值估计的。
[Abstract]:Pihajoki proposes a symmetric leapfrog integral method for second order explicit phase space expansion for inseparable Hamiltonian systems. On this basis, we put forward some ideas on how to combine these variables in the extended phase space. The data test shows that continuous permutation coordinates and momentum can make the similar leapfrog algorithm more accurate calculation results and the best long-term stability. We also propose a new method to construct many symplectic algorithms with fourth-order explicit phase space expansion. Each algorithm is composed of six commonly used leapfrog algorithms without any arrangement. The construction model consists of four parts: permutation coordinate, integration of three second-order leapfrog schemes without permutation, permutation momentum, and integration of three second-order leapfrog schemes without permutation. Similarly, explicit symplectic algorithms of order 6, order 8, and even higher order can be obtained in extended phase space. We take some common inseparability Hamiltonian systems as examples, such as dense binary systems with post-Newtonian terms that do not rotate to prove that our fourth order algorithm is superior to the current known algorithm. It also includes the fourth order symplectic algorithm in Chin system and the hybrid symplectic algorithm of four order explicit and implicit intermediate point method. When given a proper initial value, we can find that the symplectic algorithm with explicit phase space expansion is more suitable for the inseparability Hamiltonian model. These models include gravity systems with small perturbations commonly used in the solar system and dense binary systems with post-Newtonian spin. Then we apply it to more complex Hamiltonian systems and discuss the properties of second order, fourth order and eighth order, and discuss some physical properties of geodesic equations of relativistic core-shell systems. The radius of the most stable circular orbit in the equatorial plane depends only on the quadrupole moment. The quadrupole moment of a given flat circle results in the existence of two most stable circular orbits with a radius larger than that of Schwarzie. However, a flat and long quadrupole moment only corresponds to the most stable circular orbit and its radius is smaller than that of Schwarzie. In a general geodesic orbit, the above four order explicit phase space space-filled symmetry algorithm can be effectively applied to this Hamiltonian system, even though the core and shell can not be separated in their relativistic theory. With the help of this integral method, the orbit can be calculated quickly without increasing the energy error. The effect of polar moments between core-shell on geodesic equations is estimated numerically.
【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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