Convergence analysis of the formal energies of symplectic methods for Hamiltonian systems

doi:10.1007/s11425-015-5003-7

	Convergence analysis of the formal energies of symplectic methods for Hamiltonian systems
	Zhang RuiLi; Tang YiFa; Zhu BeiBei; Tu XiongBiao; Zhao Yue
	2016-02-01
发表期刊	SCIENCE CHINA-MATHEMATICS
ISSN	1674-7283
卷号	59 期号:2 页码:379-396
摘要	Based on Feng's theory of formal vector fields and formal flows, we study the convergence problem of the formal energies of symplectic methods for Hamiltonian systems and give the clear growth of the coefficients in the formal energies. With the help of B-series and Bernoulli functions, we prove that in the formal energy of the mid-point rule, the coefficient sequence of the merging products of an arbitrarily given rooted tree and the bushy trees of height 1 (whose subtrees are vertices), approaches 0 as the number of branches goes to infinity; in the opposite direction, the coefficient sequence of the bushy trees of height m (m >= 2), whose subtrees are all tall trees, approaches infinity at large speed as the number of branches goes to + infinity. The conclusion extends successfully to the modified differential equations of other Runge-Kutta methods. This disproves a conjecture given by Tang et al. (2002), and implies: (1) in the inequality of estimate given by Benettin and Giorgilli (1994) for the terms of the modified formal vector fields, the high order of the upper bound is reached in numerous cases; (2) the formal energies/formal vector fields are nonconvergent in general case.
关键词	convergence analysis formal energy symplectic method Hamiltonian system bushy tree
DOI	10.1007/s11425-015-5003-7
语种	英语
资助项目	National Natural Science Foundation of China[11371357] ; Marine Public Welfare Project of China[201105032]
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied ; Mathematics
WOS记录号	WOS:000369949400009
出版者	SCIENCE PRESS
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/22043
专题	计算数学与科学工程计算研究所
通讯作者	Tang YiFa
作者单位	Chinese Acad Sci, Acad Math & Syst Sci, ICMSEC, LSEC, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Zhang RuiLi,Tang YiFa,Zhu BeiBei,et al. Convergence analysis of the formal energies of symplectic methods for Hamiltonian systems[J]. SCIENCE CHINA-MATHEMATICS,2016,59(2):379-396.
APA	Zhang RuiLi,Tang YiFa,Zhu BeiBei,Tu XiongBiao,&Zhao Yue.(2016).Convergence analysis of the formal energies of symplectic methods for Hamiltonian systems.SCIENCE CHINA-MATHEMATICS,59(2),379-396.
MLA	Zhang RuiLi,et al."Convergence analysis of the formal energies of symplectic methods for Hamiltonian systems".SCIENCE CHINA-MATHEMATICS 59.2(2016):379-396.