Expansions of Step-Transition Operators of Multi-Step Methods and Order Barriers for Dahlquist Pairs (Journal of Computational Mathematics, 2018)
Using least parameters, we expand the step-transition operator of any linear multi-step method (LMSM) up to $O(τ ^{s+5})$ with order $s=1$ and rewrite the expansion of the step-transition operator for $s=2$ (obtained by the second author in a former paper). We prove that in the conjugate relation $G_3^{λτ} ° G_1^{τ}=G_2^{τ}° G_3^{λτ}$ with $G_1$ being an LMSM, (1) the order of $G_2$ can not be higher than that of $G_1$; (2) if $G_3$ is also an LMSM and $G_2$ is a symplectic $B$-series, then the orders of $G_1$, $G_2$ and $G_3$ must be $2$, $2$ and $1$ respectively.