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The necessary condition for a Runge-Kutta scheme to be symplectic for Hamiltonian systems

✍ Scribed by Y.-F. Tang

Publisher: Elsevier Science
Year: 1993
Tongue: English
Weight: 411 KB
Volume: 26
Category: Article
ISSN: 0898-1221
DOI: 10.1016/0898-1221(93)90082-7

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✦ Synopsis

In

this paper, beginning with the expansion of a Runge-Kutta scheme (Lemma l), the author introduces the so-called Y-T-type array for each irreducible one (Lemmss 2-3). As a study of the properties of Y-T-type arrays, the author establishes Lemmas 4-5, and then the key Theorem 1. The interesting Theorem 2 is concluded from Lemma 1. The last result (Theorem 3) is obtained from Theorem 2 and Theorem 1; thus the author solves the hard problem on the necessary condition for a Runge-Kutta scheme to be symplectic for Hamiltonian systems. Hence, a sufficient and necessary condition is given. Finally, Remark 1 is given as a necessary supplement of Theorem 3.

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