KMS Of Academy of mathematics and systems sciences, CAS
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 Publication | SCIENCE CHINA-MATHEMATICS |
ISSN | 1674-7283 |
Volume | 59Issue:2Pages:379-396 |
Abstract | 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. |
Keyword | convergence analysis formal energy symplectic method Hamiltonian system bushy tree |
DOI | 10.1007/s11425-015-5003-7 |
Language | 英语 |
Funding Project | National Natural Science Foundation of China[11371357] ; Marine Public Welfare Project of China[201105032] |
WOS Research Area | Mathematics |
WOS Subject | Mathematics, Applied ; Mathematics |
WOS ID | WOS:000369949400009 |
Publisher | SCIENCE PRESS |
Citation statistics | |
Document Type | 期刊论文 |
Identifier | http://ir.amss.ac.cn/handle/2S8OKBNM/22043 |
Collection | 计算数学与科学工程计算研究所 |
Corresponding Author | Tang YiFa |
Affiliation | Chinese 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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