Expansions of Step-Transition Operators of Multi-Step Methods and Order Barriers for Dahlquist Pairs

Expansions of Step-Transition Operators of Multi-Step Methods and Order Barriers for Dahlquist Pairs

Year:    2006

Author:    Quandong Feng, Yifa Tang

Journal of Computational Mathematics, Vol. 24 (2006), Iss. 1 : pp. 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/2006-JCM-8733

Journal of Computational Mathematics, Vol. 24 (2006), Iss. 1 : pp. 45–58

Published online:    2006-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    14

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

Author Details

Quandong Feng

Yifa Tang