Volume 25, Issue 6
Non-Existence of Conjugate-Symplectic Multi-Step Methods of Odd Order

J. Comp. Math., 25 (2007), pp. 690-696.

Published online: 2007-12

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• Abstract

We prove that any linear multi-step method $G_1^{\tau}$ of the form $$\sum^{m}_{k=0}\alpha _k Z_{k}=\tau \sum^{m}_{k=0}\beta _k J^{-1}\nabla H(Z_k)$$ with odd order $u$ ($u\ge 3$) cannot be conjugate to a symplectic method $G_2^{\tau}$ of order $w$ ($w\ge u$) via any generalized linear multi-step method $G_3^{\tau}$ of the form $$\sum^m_{k=0} \alpha_k Z_k = \tau\sum^m_{k=0} \beta_k J^{-1}\nabla H(\sum^m_{l=0}\gamma_{kl}Z_l).$$ We also give a necessary condition for this kind of generalized linear multi-step methods to be conjugate-symplectic. We also demonstrate that these results can be easily extended to the case when $G_3^{\tau}$ is a more general operator.

• Keywords

Linear multi-step method, Generalized linear multi-step method, Step-transition operator, Infinitesimally symplectic, Conjugate-symplectic.

65L06.

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@Article{JCM-25-690, author = {}, title = {Non-Existence of Conjugate-Symplectic Multi-Step Methods of Odd Order}, journal = {Journal of Computational Mathematics}, year = {2007}, volume = {25}, number = {6}, pages = {690--696}, abstract = {

We prove that any linear multi-step method $G_1^{\tau}$ of the form $$\sum^{m}_{k=0}\alpha _k Z_{k}=\tau \sum^{m}_{k=0}\beta _k J^{-1}\nabla H(Z_k)$$ with odd order $u$ ($u\ge 3$) cannot be conjugate to a symplectic method $G_2^{\tau}$ of order $w$ ($w\ge u$) via any generalized linear multi-step method $G_3^{\tau}$ of the form $$\sum^m_{k=0} \alpha_k Z_k = \tau\sum^m_{k=0} \beta_k J^{-1}\nabla H(\sum^m_{l=0}\gamma_{kl}Z_l).$$ We also give a necessary condition for this kind of generalized linear multi-step methods to be conjugate-symplectic. We also demonstrate that these results can be easily extended to the case when $G_3^{\tau}$ is a more general operator.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8722.html} }
TY - JOUR T1 - Non-Existence of Conjugate-Symplectic Multi-Step Methods of Odd Order JO - Journal of Computational Mathematics VL - 6 SP - 690 EP - 696 PY - 2007 DA - 2007/12 SN - 25 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8722.html KW - Linear multi-step method, Generalized linear multi-step method, Step-transition operator, Infinitesimally symplectic, Conjugate-symplectic. AB -

We prove that any linear multi-step method $G_1^{\tau}$ of the form $$\sum^{m}_{k=0}\alpha _k Z_{k}=\tau \sum^{m}_{k=0}\beta _k J^{-1}\nabla H(Z_k)$$ with odd order $u$ ($u\ge 3$) cannot be conjugate to a symplectic method $G_2^{\tau}$ of order $w$ ($w\ge u$) via any generalized linear multi-step method $G_3^{\tau}$ of the form $$\sum^m_{k=0} \alpha_k Z_k = \tau\sum^m_{k=0} \beta_k J^{-1}\nabla H(\sum^m_{l=0}\gamma_{kl}Z_l).$$ We also give a necessary condition for this kind of generalized linear multi-step methods to be conjugate-symplectic. We also demonstrate that these results can be easily extended to the case when $G_3^{\tau}$ is a more general operator.

Yandong Jiao, Guidong Dai, Quandong Feng & Yifa Tang. (1970). Non-Existence of Conjugate-Symplectic Multi-Step Methods of Odd Order. Journal of Computational Mathematics. 25 (6). 690-696. doi:
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