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

Quandong Feng Yifa Tang

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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COPYRIGHT: © Global Science Press

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@Article{JCM-24-45, author = {Quandong Feng , and Yifa Tang , }, 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 AU - Quandong Feng , AU - Yifa Tang , 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, Step-Transition Operator, $B$-series, Dahlquist (Conjugate) pair, 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.

Quandong Feng & Yifa 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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