Development of innovative numerical integrators which preserving all of the conserved quantities

开发保留所有守恒量的创新数值积分器

基本信息

批准号：
15340030
负责人：
NAKAMURA Yoshimasa
金额：
$ 10.37万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

项目摘要

Symplectic integrators and energy preserving difference methods have been widely used for Hamiltonian dynamical systems. However, since such numerical integrators do not preserve all of the conserved quantities, their behavior may be rather different from the trajectory of Hamiltonian systems. Indeed, these integrators can not describe a long-time behavior of the gravitational 2-body motion of Kepler. The purpose of the research project is to develop a new numerical integrator named TCI, totally conservative integrator, which preserving all of the conserved quantities of Hamiltonian systems. It was shown in a paper by Minesaki and Nakamura that a long-time behavior of the Kepler motion is completely preserved by the TCI.In 2006, the last year of the project, Minesaki and Inoue developed a new integrator which preserving all the conserved quantities of the gravitational 3-body motion. Since the general 3-body motion is a chaotic dynamical system, a candidate of such an integrator is formulated for the case of Langrage's equilateral triangle solution on a plain. The basic design of the integrator is to divide the equations of motion into 2-body parts and interaction parts of 3-body and then discretize them, individually. A key idea is a use of a gap in a time variable which appears in the discrete 2-body motion to keep the sum of relative coordinates zero. Consequently, it is shown that the resulting numerical integrator preserves Langrage's equilateral triangle solution exactly. Such a remarkable property is viewed in numerical simulation except for a round-off error in computer. The new integrator is shown to be superior than Stormer-Verlet's symplectic integrator for 3-body motion. It is also verified that the new integrator behaves well for the letter "8" solution of the 3-body motion and the corresponding energy is kept constant for a long period.

符号积分器和能量保留差异方法已被广泛用于哈密顿动力系统。但是，由于这种数值集成剂并不能保留所有保守量，因此它们的行为可能与汉密尔顿系统的轨迹有很大不同。确实，这些集成剂无法描述开普勒重力2体运动的长期行为。研究项目的目的是开发一个名为TCI，完全保守的集成商TCI的新数值集成商，该集成商保留了所有保守数量的汉密尔顿系统。 Minesaki和Nakamura的一篇论文中显示，开普勒运动的长期行为由TCI.in 2006（该项目的最后一年）完全保存，Minesaki和Inoue开发了一个新的集成仪，该集成蛋白保留了所有保守量的重力3体运动。由于一般的三体运动是一种混乱的动力学系统，因此为Langrage在平原上的等边三角形溶液的情况下制定了这种积分器的候选。集成商的基本设计是将运动方程分为2体零件和3体的相互作用，然后单独离散。一个关键的想法是在离散2体运动中出现的时间变量中使用差距，以保持相对坐标的总和为零。因此，结果表明，所得的数值积分器准确地保留了Langrage的等边三角形解决方案。除了计算机中的圆形错误外，在数值模拟中查看了这种非凡的属性。对于三体运动，新的积分器比Stormer-Verlet的符号积分器优越。还可以验证，新的集成器对于三体运动的字母“ 8”解决方案的行为很好，并且相应的能量长期保持恒定。