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

作　　者：ZHANG RuiLi TANG YiFa ZHU BeiBei TU XiongBiao ZHAO Yue 

机构地区：[1]lsec,icmsec,academy of mathematics and systems science,chinese academy of sciences,Beijing 100190,China

出　　处：《Science China Mathematics》2016年第2期379-396,共18页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant No.11371357);the Marine Public Welfare Project of China(Grant No.201105032)

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 ∞; in the opposite direction, the coefficient sequence of the bushy trees of height m(m ≥ 2), whose subtrees are all tall trees, approaches ∞ at large speed as the number of branches goes to +∞. 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.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 ∞; in the opposite direction, the coefficient sequence of the bushy trees of height m（m ≥ 2）, whose subtrees are all tall trees, approaches ∞ at large speed as the number of branches goes to ＋∞. 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 

分 类 号：O241.8[理学—计算数学]