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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
Source PublicationSCIENCE CHINA-MATHEMATICS
ISSN1674-7283
Volume59Issue:2Pages:379-396
AbstractBased 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.
Keywordconvergence analysis formal energy symplectic method Hamiltonian system bushy tree
DOI10.1007/s11425-015-5003-7
Language英语
Funding ProjectNational Natural Science Foundation of China[11371357] ; Marine Public Welfare Project of China[201105032]
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000369949400009
PublisherSCIENCE PRESS
Citation statistics
Cited Times:2[WOS]   [WOS Record]     [Related Records in WOS]
Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/22043
Collection计算数学与科学工程计算研究所
AffiliationChinese Acad Sci, Acad Math & Syst Sci, ICMSEC, LSEC, Beijing 100190, Peoples R China
Recommended Citation
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.
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