Non-Existence of Conjugate-Symplectic Multi-Step Methods of Odd Order (Journal of Computational Mathematics, 2021)
We prove that any linear multi-step method $G_1^{τ}$ of the form $$\sum^{m}_{k=0}α _k Z_{k}=τ \sum^{m}_{k=0}β _k J^{-1} abla H(Z_k)$$ with odd order $u$ ($u\ge 3$) cannot be conjugate to a symplectic method $G_2^{τ}$ of order $w$ ($w\ge u$) via any generalized linear multi-step method $G_3^{τ}$ of the form $$\sum^m_{k=0} α_k Z_k = τ\sum^m_{k=0} β_k J^{-1} abla H(\sum^m_{l=0}γ_{kl}Z_l).$$ 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 $G_3^{τ}$ is a more general operator.