Some properties of the explicit symplectic scheme $\phi^{\frac{\tau}{2}}_1\circ\phi^{\frac{\tau}{2}}_2\circ\phi^{\tau}_3 \circ\phi^{\frac{\tau}{2}}_2\circ\phi^{\frac{\tau}{2}}_1$ for splitable Hamiltonian $H=H^{(1)}+H^{(2)}+H^{(3)}$

Quandong Feng, Yifa Tang

LSEC, ICMSEC, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100080, P.R. China

fqd@lsec.cc.ac.cn

Poster

For a splitable hamiltonian $H(p,q)=H^{(1)}(p,q)+H^{(2)}(p,q)+H^{(3)}(p,q),\quad p, q \in \mathbb{R}^n$, if $\phi^t_1$, $\phi^t_2$, $\phi^t_3$ are the phase flows of $H^{(1)}$, $H^{(2)}$, $H^{(3)}$ respectively, then $\widetilde{Z}=\Psi^t(Z) =\phi^{\frac{t}{2}}_1\circ\phi^{\frac{t}{2}}_2\circ \phi^t_3\circ\phi^{\frac{t}{2}}_2\circ\phi^{\frac{t}{2}}_1(Z)$ is a time-reversible symplectic scheme ( RESS) with order $2$. We do two things for $\Psi^t$: It is known that $\Psi^{\beta t}\circ\Psi_2^{\alpha t}\circ\Psi_3^{\beta t} = \phi_1^{\frac{\bet... ...hi_3^{\beta t}\circ \phi_2^{\frac{\beta t}{2}}\circ\phi_1^{\frac{\beta t}{2}}$ ( $\alpha=\frac{-^3\sqrt{2}}{2-^3\sqrt{2}}$, $\beta=\frac{1}{2-^3\sqrt{2}}$, and $\gamma=\alpha+\beta=\frac{1-^3\sqrt{2}}{2-^3\sqrt{2}}$) is a $4$th-order RESS for Hamiltonian $H=H^{(1)}+H^{(2)}+H^{(3)}$ [Forest & Ruth (1990), Suzuki (1990), Yoshida (1990)].
But how about the converse proposition? For this we establish the following: Theorem If $13$-fold composition
$\widetilde{Z}=\Theta^t(Z)= \phi^{\delta t}_{n_7}\circ\phi^{\alpha t}_{n_6}\ci... ... \phi^{\beta t}_{n_5}\circ\phi^{\alpha t}_{n_6}\circ\phi^{\delta t}_{n_7} (Z)$ where $n_1,\cdots,n_7\in \{1,2,3\}$ and any two neighbouring numbers are different, is a 4th-order RESS for Hamiltonian $H=H^{(1)}+H^{(2)}+H^{(3)}$, then $\Theta^t$ is exactly the $3$-fold composition of $\Psi^t$: $\Theta^t=\Psi^{\kappa_2t}\circ\Psi^{\kappa_1t}\circ\Psi^{\kappa_2t}$ with $\kappa_1=\frac{-^3\sqrt{2}}{2-^3\sqrt{2}}$ and $\kappa_2=\frac{1}{2-^3\sqrt{2}}$. For a fixed step size $t$, any symplectic scheme has a formal energy (or perturbed hamiltonian) which is the `` exact hamiltonian" for the symplectic scheme to be the `` formal phase flow" when evaluated at the discrete temporal points (Benettin & Giorgilli, Feng, Hairer, Yoshida). Using $B$-series and the technique mainly due to Hairer, we write the formal energy of $\Psi^t$ as $\widetilde{H}=H+t^2H_2+t^4H_4+\cdots$, the expasions of $H_2$ and $H_4$ consist of $15$ terms and $333$ terms corresponding to the free unlabeled trees of vertices $3$ and $5$ respectively.