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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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References

  1. Anton, R., Cohen, D.: Exponential integrators for stochastic schrödinger equations driven by Ito noise. J. Comput. Math. 36(2), 276–309 (2019)

    MATH  Google Scholar 

  2. Buckwar, E., D’Ambrosio, R.: Exponential mean-square stability properties of stochastic multistep methods, submitted

  3. Buckwar, E., Sickenberger, T.: A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods. Math. Comput. Simul. 81, 1110–1127 (2011)

    Article  MathSciNet  Google Scholar 

  4. Bryden, A., Higham, D. J.: On the boundedness of asymptotic stability regions for the stochastic theta method. BIT 43, 1–6 (2003)

    Article  MathSciNet  Google Scholar 

  5. Burrage, P. M., Burrage, K.: Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise. Numer. Algor. 65, 519–532 (2012)

    Article  MathSciNet  Google Scholar 

  6. Burrage, P. M., Burrage, K.: Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise. J. Comput. Appl. Math. 236, 3920–3930 (2014)

    Article  MathSciNet  Google Scholar 

  7. Burrage, K., Lenane, I., Lythe, G.: Numerical methods for second-order stochastic differential equations. SIAM. J. Sci. Comput. 29(1), 245–264 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Burrage, K., Lythe, G.: Accurate stationary densities with partitioned numerical methods for stochastic differential equations. SIAM. J. Numer. Anal. 47, 1601–1618 (2009)

    Article  MathSciNet  Google Scholar 

  9. Burrage, K., Lythe, G.: Accurate stationary densities with partitioned numerical methods for stochastic partial differential equations. Stochastic Partial Differential Equations: Analysis and Computations. 2(2), 262–280 (2014)

    Article  MathSciNet  Google Scholar 

  10. Chen, C., Cohen, D., D’Ambrosio, R., Lang, A.: Drift-preserving numerical integrators for stochastic Hamiltonian systems. Adv. Comput. Math. 46(2), 27 (2020)

    Article  MathSciNet  Google Scholar 

  11. Citro, V., D’Ambrosio, R.: Long-term analysis of stochastic 𝜃-methods for damped stochastic oscillators, Appl. Numer. Math. 18–26. https://doi.org/10.1016/j.apnum.2019.08.011 (2019)

  12. Citro, V., D’Ambrosio, R., Di Giovacchino, S.: A-stability preserving perturbation of Runge–Kutta methods for stochastic differential equations, Appl. Math. Lett. 102, 106098 (2020)

    MathSciNet  MATH  Google Scholar 

  13. Cohen, D., Gauckler, L., Hairer, E., Lubich, C.: Long-term analysis of numerical integrators for oscillatory Hamiltonian systems under minimal non-resonance conditions, BIT. Numer. Math. 55(3), 705–732 (2015)

    Article  Google Scholar 

  14. Conte, D., D’Ambrosio, R., Paternoster, B.: On the stability of 𝜃-methods for stochastic Volterra integral equations. Discret. Cont. Dyn. Syst. B 23, 2695–2708 (2018)

    MathSciNet  MATH  Google Scholar 

  15. D’Ambrosio, D., Moccaldi, M., Paternoster, B.: Numerical preservation of long-term dynamics by stochastic two-step methods. Discrete and Continuous Dynamical Systems Series B. 23(7), 2763–2773 (2018)

    Article  MathSciNet  Google Scholar 

  16. D’Ambrosio, R., Di Giovacchino, S.: Mean-square contractivity of stochastic 𝜃-methods, submitted.

  17. Gardiner, C. W.: Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, 3rd ed. Springer-Verlag, Berlin (2004)

    Book  Google Scholar 

  18. Higham, D. J.: Mean-square asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal. 38, 753–769 (2000)

    Article  MathSciNet  Google Scholar 

  19. Schurz, H.: The invariance of asymptotic laws of linear stochastic systems under discretization. Z. Angew. Math. Mech. 6, 375–382 (1999)

    Article  MathSciNet  Google Scholar 

  20. Strömmen Melbö, A. H., Higham, D. J.: Numerical simulation of a linear stochastic oscillator with additive noise. Appl. Numer. Math. 51, 89–99 (2004)

    Article  MathSciNet  Google Scholar 

  21. Vilmart, G.: Weak second order multirevolution composition methods for highly oscillatory stochastic differential equations with additive or multiplicative noise. SIAM J. Sci. Comput. 36(4), A1770–A1796 (2014)

    Article  MathSciNet  Google Scholar 

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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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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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