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A technique to construct symmetric variable-stepsize linear multistep methods for second-order systems

Authors: B. Cano and A. Durán
Journal: Math. Comp. 72 (2003), 1803-1816
MSC (2000): Primary 65L06, 70F05, 70H33
Published electronically: May 29, 2003
MathSciNet review: 1986805
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Abstract: Some previous works show that symmetric fixed- and variable-stepsize linear multistep methods for second-order systems which do not have any parasitic root in their first characteristic polynomial give rise to a slow error growth with time when integrating reversible systems. In this paper, we give a technique to construct variable-stepsize symmetric methods from their fixed-stepsize counterparts, in such a way that the former have the same order as the latter. The order and symmetry of the integrators obtained is proved independently of the order of the underlying fixed-stepsize integrators. As this technique looks for efficiency, we concentrate on explicit linear multistep methods, which just make one function evaluation per step, and we offer some numerical comparisons with other one-step adaptive methods which also show a good long-term behaviour.

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Additional Information

B. Cano
Affiliation: Departamento de Matemática Aplicada y Computación, Facultad de Ciencias, Universidad de Valladolid, Valladolid, Spain

A. Durán
Affiliation: Departamento de Matemática Aplicada y Computación, Facultad de Ciencias. Universidad de Valladolid, Valladolid, Spain

Keywords: Explicit linear multistep methods, variable stepsizes, error growth, reversible second-order systems, symmetric integrators, efficiency, high-order methods
Received by editor(s): January 1, 2026
Received by editor(s) in revised form: April 30, 2002, and January 1, 2000
Published electronically: May 29, 2003
Additional Notes: This work was supported by DGICYT PB95–705 and JCL VA36/98
Article copyright: © Copyright 2003 American Mathematical Society