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On the numerical structure preservation of nonlinear damped stochastic oscillators

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Abstract

The paper is focused on analyzing the conservation issues of stochastic 𝜃-methods when applied to nonlinear damped stochastic oscillators. In particular, we are interested in reproducing the long-term properties of the continuous problem over its discretization through stochastic 𝜃-methods, by preserving the correlation matrix. This evidence is equivalent to accurately maintaining the stationary density of the position and the velocity of a particle driven by a nonlinear deterministic forcing term and an additive noise as a stochastic forcing term. The provided analysis relies on a linearization of the nonlinear problem, whose effectiveness is proved theoretically and numerically confirmed.

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Acknowledgments

The authors thank the anonymous referee, for valuable remarks, in particular, for suggesting further comparisons in Section 9. This work is supported by the GNCS-INDAM project and by the PRIN2017-MIUR project. The authors are member of the INdAM Research group GNCS.

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Authors and Affiliations

  1. Department of Information Engineering and Computer Science and Mathematics, University of L’Aquila Via Vetoio, Loc. Coppito, 67100, L’Aquila, Italy

    Raffaele D’Ambrosio & Carmela Scalone

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  1. Raffaele D’Ambrosio
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  2. Carmela Scalone
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Correspondence to Raffaele D’Ambrosio.

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D’Ambrosio, R., Scalone, C. On the numerical structure preservation of nonlinear damped stochastic oscillators. Numer Algor 86, 933–952 (2021). https://doi.org/10.1007/s11075-020-00918-5

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  • DOI: https://doi.org/10.1007/s11075-020-00918-5

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Mathematics Subject Classification (2010)