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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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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D’Ambrosio, R., Scalone, C. On the numerical structure preservation of nonlinear damped stochastic oscillators. Numer Algor (2020). https://doi.org/10.1007/s11075-020-00918-5
- Stochastic differential equations
- Stochastic 𝜃-methods
- Nonlinear damped stochastic oscillators
- Numerical structure preservation.
Mathematics Subject Classification (2010)