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Title of Journal: Sci China Math

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Abbravation: Science China Mathematics

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Science China Press

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1869-1862

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Convergence analysis of the formal energies of sym

Authors: RuiLi Zhang YiFa Tang BeiBei Zhu XiongBiao Tu Yue Zhao
Publish Date: 2015/03/20
Volume: 59, Issue: 2, Pages: 379-396
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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 Bseries and Bernoulli functions we prove that in the formal energy of the midpoint 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 RungeKutta 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

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