Volume 24, Issue 1
Expansions of Step-Transition Operators of Multi-Step Methods and Order Barriers for Dahlquist Pairs

Quan-dong Feng & Yi-fa Tang

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J. Comp. Math., 24 (2006), pp. 45-58

Published online: 2006-02

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  • Abstract

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.

  • Keywords

Linear Multi-Step Method Step-Transition Operator $B$-series Dahlquist (Conjugate) pair Symplecticity

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@Article{JCM-24-45, author = {}, title = {Expansions of Step-Transition Operators of Multi-Step Methods and Order Barriers for Dahlquist Pairs}, journal = {Journal of Computational Mathematics}, year = {2006}, volume = {24}, number = {1}, pages = {45--58}, abstract = { 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. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8733.html} }
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.
Quan-dong Feng & Yi-fa Tang. (1970). Expansions of Step-Transition Operators of Multi-Step Methods and Order Barriers for Dahlquist Pairs. Journal of Computational Mathematics. 24 (1). 45-58. doi:
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