Long-term adaptive symplectic numerical integration of linear stochastic oscillators driven by additive white noise
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In this paper, we present an adaptive variable step size numerical scheme for the integration of linear stochastic oscillator equations driven by additive Brownian white noise. We first show that traditional adaptive schemes based on local error estimation destroy the long-time behavior of the underlying method. As a remedy, we extend the idea presented in Hairer and Söderlind (SIAM J. Sci. Comput. 26(6), 1838–1851 2005) to the stochastic setting and show that using step density control mechanisms based on time regularization and local error tracking, we are able to obtain numerical schemes which preserve the important qualitative features of the solution process such as symmetry, time reversibility, symplecticity, linear growth rate of the second moment, and infinite oscillation. Numerical experiments confirm the theoretical findings of the paper.
KeywordsStochastic Hamiltonian systems Linear stochastic oscillator Additive white noise Variable step size Time regularization Symplecticity
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The authors would like to thank the editor and two anonymous referees for providing valuable and helpful suggestions and comments on the paper. The second author also gratefully acknowledges support by the Institute for Advanced Studies in Basic Sciences (IASBS) Research Council under grant No. G2017IASBS22614.
- 2.Arnold, L.: Random Dynamical Systems. Springer (2013)Google Scholar
- 10.De la Cruz, H., Jimenez, J., Zubelli, J.: Locally linearized methods for the simulation of stochastic oscillators driven by random forces. BIT Numer. Math., 1–29 (2016)Google Scholar
- 11.De la Cruz, H., Zubelli, J.: Numerical schemes for the long-term simulation of SDEs with additive noise and their effectiveness in the integration of a stochastic oscillator. Preprint (2016)Google Scholar
- 13.Dormand, J.R.: Numerical Methods for Differential Equations: a Computational Approach. CRC Press (1996)Google Scholar
- 16.Gitterman, M.: The Noisy Oscillator: the First Hundred Years, from Einstein until Now. World Scientific (2005)Google Scholar
- 30.Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press (1990)Google Scholar
- 35.Mao, X.: Stochastic Differential Equations and their Applications. Woodhead Publishing (1997)Google Scholar
- 41.Misawa, T.: Symplectic integrators to stochastic Hamiltonian dynamical systems derived from composition methods. Math. Probl. Eng., 2010 (2010)Google Scholar
- 42.Øksendal, B.: Stochastic Differential Equations: an Introduction with Applications, 6th edn. Springer (2003)Google Scholar
- 43.Preto, M., Tremaine, S.: A class of symplectic integrators with adaptive time step for separable Hamiltonian systems. Astronom. J. 118(5), 2532 (1999)Google Scholar
- 45.Seeßelberg, M., Breuer, H., Mais, H., Petruccione, F., Honerkamp, J.: Simulation of one-dimensional noisy Hamiltonian systems and their application to particle storage rings. Zeitschrift für Physik C Particles Fields 62(1), 63–73 (1994)Google Scholar
- 51.Soong, T.T.: Random Differential Equations in Science and Engineering. Elsevier (1973)Google Scholar
- 52.Sotiropoulos, V., Kaznessis, Y.N.: An adaptive time step scheme for a system of stochastic differential equations with multiple multiplicative noise: chemical Langevin equation, a proof of concept. J. Chem. Phys. 128(1), 014,103 (2008)Google Scholar
- 59.Wang, L.: Variational Integrators and Generating Functions for Stochastic Hamiltonian Systems. KIT Scientific Publishing (2009)Google Scholar
- 61.Wang, L., Hong, J., Scherer, R.: Symplectic numerical methods for a linear stochastic oscillator with two additive noises. In: Proceedings of the World Congress on Engineering, vol. 1 (2011)Google Scholar
- 63.Zhou, W., Zhang, L., Hong, J., Song, S.: Projection methods for stochastic differential equations with conserved quantities. BIT Numer. Math., 1–22 (2016)Google Scholar