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Long-Term Stability of Multi-Value Methods for Ordinary Differential Equations

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Abstract

Much effort is put into the construction of general linear methods with the aim of achieving an excellent long-time behavior for the integration of Hamiltonian systems. In this article, a backward error analysis is presented, which permits to get sharp estimates for the parasitic solution components and for the error in the Hamiltonian. For carefully constructed methods (symmetric and zero growth parameters) the error in the parasitic components typically grows like \(h^{p+4}\exp (h^2Lt)\), where \(p\) is the order of the method, and \(L\) depends on the problem and on the coefficients of the method. This is confirmed by numerical experiments.

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Notes

  1. All the manipulations with B-series are done with the excellent Mathematica package by Ander Murua.

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Acknowledgments

This work was partially supported by the Fonds National Suisse, project No. 200020-144313/1. A large part of this work was carried out while the first author visited the University of Geneva.

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Authors and Affiliations

  1. Dipartimento di Matematica, Univ. degli Studi di Salerno, 84084 , Fisciano, SA, Italy

    R. D’Ambrosio

  2. Section de Mathématiques, Univ. de Genève, 1211 , Genève 4, Switzerland

    E. Hairer

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  1. R. D’Ambrosio
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  2. E. Hairer
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Correspondence to R. D’Ambrosio.

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D’Ambrosio, R., Hairer, E. Long-Term Stability of Multi-Value Methods for Ordinary Differential Equations. J Sci Comput 60, 627–640 (2014). https://doi.org/10.1007/s10915-013-9812-y

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  • DOI: https://doi.org/10.1007/s10915-013-9812-y

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