TY - JOUR
T1 - Expansions of Step-Transition Operators of Multi-Step Methods and Order Barriers for Dahlquist Pairs
JO - Journal of Computational Mathematics
VL - 1
SP - 45
EP - 58
PY - 2006
DA - 2006/02
SN - 24
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jcm/8733.html
KW - Linear Multi-Step Method
KW - Step-Transition Operator
KW - $B$-series
KW - Dahlquist (Conjugate) pair
KW - Symplecticity
AB - Using least parameters, we expand the step-transition operator of any linear multi-step method ({LMSM}) up to $O(\tau ^{s+5})$ 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 $G_3^{\lambda\tau} \circ G_1^{\tau}=G_2^{\tau}\circ G_3^{\lambda\tau}$ 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.