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Analysis of variable-stepsize linear multistep methods with special emphasis on symmetric ones


Authors: B. Cano and A. Durán
Journal: Math. Comp. 72 (2003), 1769-1801
MSC (2000): Primary 65L06, 70F05, 70H33
DOI: https://doi.org/10.1090/S0025-5718-03-01538-2
Published electronically: May 29, 2003
MathSciNet review: 1986804
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Abstract: In this paper we deal with several issues concerning variable-stepsize linear multistep methods. First, we prove their stability when their fixed-stepsize counterparts are stable and under mild conditions on the stepsizes and the variable coefficients. Then we prove asymptotic expansions on the considered tolerance for the global error committed. Using them, we study the growth of error with time when integrating periodic orbits. We consider strongly and weakly stable linear multistep methods for the integration of first-order differential systems as well as those designed to integrate special second-order ones. We place special emphasis on the latter which are also symmetric because of their suitability when integrating moderately eccentric orbits of reversible systems. For these types of methods, we give a characterization for symmetry of the coefficients, which allows their construction, and provide some numerical results for them.


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Additional Information

B. Cano
Affiliation: Departamento de Matemática Aplicada y Computación, Facultad de Ciencias, Universidad de Valladolid, Spain
Email: bego@mac.mac.uva.es

A. Durán
Affiliation: Departamento de Matemática Aplicada y Computación, Facultad de Ciencias, Universidad de Valladolid, Spain
Email: angel@mac.mac.uva.es

DOI: https://doi.org/10.1090/S0025-5718-03-01538-2
Keywords: Linear multistep methods, variable stepsizes, stability, periodic orbits, error growth, reversible systems, symmetric integrators, asymptotic expansion of the error
Received by editor(s): December 26, 2000
Received by editor(s) in revised form: April 30, 2002
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

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