Hamilton方程保能量数值方法的研究
发布时间：2018-08-11 09:58
【摘要】：Hamilton系统在天体力学、量子力学等领域有着广泛的应用.一切真实的、耗散效应可忽略不计的物理过程都可以表达为这样或那样的Hamilton形式.但是,对于许多Hamilton方程,求其精确解常常是非常困难的.因此系统地研究Hamilton方程的数值解法具有重要的理论和实际意义.我们希望数值解法尽可能保持Hamilton系统的重要性质.离散梯度在构造Hamilton方程保能量数值方法中起着非常重要的作用.可利用离散梯度构造保能量数值解法,离散梯度包括坐标增量离散梯度、平均离散梯度等.本文介绍了 Hamilton方程的一些性质及辛几何算法的相关知识,主要研究了Hamilton方程保能量数值方法.我们研究了 Hamilton函数fg,fgh的一些离散梯度之间的联系,给出了它们的关系式.在此基础上,给出了由一阶保能量的数值解法构造二阶保能量的数值解法的方法.最后通过具体的例子进行了数值模拟,数值结果显示新的数值解法是保能量数值方法.
[Abstract]:Hamilton system is widely used in celestial mechanics, quantum mechanics and other fields. All real, dissipative physical processes can be expressed in one form or another in Hamilton form. However, for many Hamilton equations, it is often very difficult to find its exact solution. Therefore, it is of great theoretical and practical significance to study the numerical solution of Hamilton equation systematically. We hope that the numerical solution can preserve the important properties of the Hamilton system as much as possible. Discrete gradient plays an important role in constructing the energy preserving numerical method of Hamilton equation. The energy preserving numerical solution can be constructed by using discrete gradient. The discrete gradient includes coordinate increment discrete gradient, average discrete gradient and so on. In this paper, some properties of Hamilton equation and the knowledge of symplectic geometric algorithm are introduced, and the energy preserving numerical method of Hamilton equation is mainly studied. In this paper, we study the relations between some discrete gradients of Hamilton function fgfgh, and give their relations. On this basis, the method of constructing the second order energy preserving numerical solution from the first order energy preserving numerical solution is given. Finally, numerical simulation is carried out through a concrete example. The numerical results show that the new numerical method is energy preserving numerical method.
【学位授予单位】：北京交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
本文编号：2176657
[Abstract]:Hamilton system is widely used in celestial mechanics, quantum mechanics and other fields. All real, dissipative physical processes can be expressed in one form or another in Hamilton form. However, for many Hamilton equations, it is often very difficult to find its exact solution. Therefore, it is of great theoretical and practical significance to study the numerical solution of Hamilton equation systematically. We hope that the numerical solution can preserve the important properties of the Hamilton system as much as possible. Discrete gradient plays an important role in constructing the energy preserving numerical method of Hamilton equation. The energy preserving numerical solution can be constructed by using discrete gradient. The discrete gradient includes coordinate increment discrete gradient, average discrete gradient and so on. In this paper, some properties of Hamilton equation and the knowledge of symplectic geometric algorithm are introduced, and the energy preserving numerical method of Hamilton equation is mainly studied. In this paper, we study the relations between some discrete gradients of Hamilton function fgfgh, and give their relations. On this basis, the method of constructing the second order energy preserving numerical solution from the first order energy preserving numerical solution is given. Finally, numerical simulation is carried out through a concrete example. The numerical results show that the new numerical method is energy preserving numerical method.
【学位授予单位】：北京交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
【参考文献】
相关期刊论文 前5条
1 蒋朝龙;黄荣芳;孙建强;;耦合非线性薛定谔方程的平均离散梯度法[J];工程数学学报;2014年05期
2 骆思宇;蒋朝龙;孙建强;;高阶非线性薛定谔方程的离散梯度法[J];海南大学学报(自然科学版);2013年02期
3 刘小强;聂淑媛;;基于Padé逼近的辛算法及其稳定性分析[J];三门峡职业技术学院学报;2007年02期
4 张素英,邓子辰;广义Hamilton(控制)系统的离散梯度积分法[J];计算力学学报;2004年05期
5 冯康;秦孟兆;;HAMILTONIAN ALGORITHMS FOR HAMILTONIAN DYNAMICAL SYSTEMS[J];Communication of State Key Laboratories of China;1991年02期
相关硕士学位论文 前3条
1 孙非凡;Hamilton系统保能量算法的研究[D];北京交通大学;2015年
2 刘小菊;保能量计算方法[D];北京交通大学;2012年
3 陈钊;高振荡微分方程的对称数值解法[D];北京交通大学;2008年
，本文编号：2176657
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