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September  2018, 23(7): 2763-2773. doi: 10.3934/dcdsb.2018105

Numerical preservation of long-term dynamics by stochastic two-step methods

Raffaele D'Ambrosio 1, , Martina Moccaldi 2, and  Beatrice Paternoster 2,

1. 

Department of Engineering and Computer Science and Mathematics, University of L'Aquila, L'Aquila (AQ), Italy
2. 

Department of Mathematics, University of Salerno, Fisciano (SA), Italy

Received  March 2017 Revised  January 2018 Published  September 2018 Early access  March 2018

Fund Project: The work is supported by GNCS-Indam project.

The paper aims to explore the long-term behaviour of stochastic two-step methods applied to a class of second order stochastic differential equations. In particular, the treatment focuses on preserving long-term statistics related to the dynamics of a linear stochastic damped oscillator whose velocity, in the stationary regime, is distributed as a Gaussian variable and uncorrelated with the position. By computing the solution of a very simple matrix equality, we a-priori determine the long-term statistics characterizing the numerical dynamics and analyze the behaviour of a selection of methods.

Keywords: Stochastic differential equations, stochastic damped oscillators, stochastic two-step methods, long-term numerical dynamics.
Mathematics Subject Classification: 65C30.
Citation: Raffaele D'Ambrosio, Martina Moccaldi, Beatrice Paternoster. Numerical preservation of long-term dynamics by stochastic two-step methods. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2763-2773. doi: 10.3934/dcdsb.2018105
References:
[1]

E. Buckwar and R. D'Ambrosio, Exponential mean-square stability properties of stochastic multistep methods, submitted.

[2]

E. Buckwar, R. Horvath-Bokor and R. Winkler, Asymptotic mean-square stability of two-step methods for stochastic ordinary differential equations, BIT Numer. Math., 46 (2006), 261-282.  doi: 10.1007/s10543-006-0060-5.

[3]

P. M. Burrage and K. Burrage, Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise, Numer. Algor., 65 (2014), 519-532.  doi: 10.1007/s11075-013-9796-6.

[4]

P. M. Burrage and K. Burrage, Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise, J. Comput. Appl. Math., 236 (2012), 3920-3930.  doi: 10.1016/j.cam.2012.03.007.

[5]

K. Burrage, I. Lenane and G. Lythe, Numerical methods for second-order stochastic differential equations, SIAM J. Sci. Comput., 29 (2007), 245-264.  doi: 10.1137/050646032.

[6]

K. Burrage and G. Lythe, Accurate stationary densities with partitioned numerical methods for stochastic differential equations, SIAM J. Numer. Anal., 47 (2009), 1601-1618.  doi: 10.1137/060677148.

[7]

D. Conte, R. D'Ambrosio and B. Paternoster, On the stability of ϑ-methods for stochastic Volterra integral equations, Discr. Cont. Dyn. Sys. - B, accepted for publication, (2017).

[8]

P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, 1962.

[9]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.

[10]

P. E. Kloeden and E. Platen, The Numerical Solution of Stochastic Differential Equations, Springer-Verlag, 1992.

[11]

A. H. Strömmen and D. J. Melbö Higham, Numerical simulation of a linear stochastic oscillator with additive noise, Appl. Numer. Math., 51 (2004), 89-99.  doi: 10.1016/j.apnum.2004.02.003.

[12]

G. Vilmart, Weak second order multi-revolution composition methods for highly oscillatory stochastic differential equations with additive or multiplicative noise, SIAM J. Sci. Comput., 36 (2014), 1770-1796.  doi: 10.1137/130935331.

show all references

References:
[1]

E. Buckwar and R. D'Ambrosio, Exponential mean-square stability properties of stochastic multistep methods, submitted.

[2]

E. Buckwar, R. Horvath-Bokor and R. Winkler, Asymptotic mean-square stability of two-step methods for stochastic ordinary differential equations, BIT Numer. Math., 46 (2006), 261-282.  doi: 10.1007/s10543-006-0060-5.

[3]

P. M. Burrage and K. Burrage, Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise, Numer. Algor., 65 (2014), 519-532.  doi: 10.1007/s11075-013-9796-6.

[4]

P. M. Burrage and K. Burrage, Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise, J. Comput. Appl. Math., 236 (2012), 3920-3930.  doi: 10.1016/j.cam.2012.03.007.

[5]

K. Burrage, I. Lenane and G. Lythe, Numerical methods for second-order stochastic differential equations, SIAM J. Sci. Comput., 29 (2007), 245-264.  doi: 10.1137/050646032.

[6]

K. Burrage and G. Lythe, Accurate stationary densities with partitioned numerical methods for stochastic differential equations, SIAM J. Numer. Anal., 47 (2009), 1601-1618.  doi: 10.1137/060677148.

[7]

D. Conte, R. D'Ambrosio and B. Paternoster, On the stability of ϑ-methods for stochastic Volterra integral equations, Discr. Cont. Dyn. Sys. - B, accepted for publication, (2017).

[8]

P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, 1962.

[9]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.

[10]

P. E. Kloeden and E. Platen, The Numerical Solution of Stochastic Differential Equations, Springer-Verlag, 1992.

[11]

A. H. Strömmen and D. J. Melbö Higham, Numerical simulation of a linear stochastic oscillator with additive noise, Appl. Numer. Math., 51 (2004), 89-99.  doi: 10.1016/j.apnum.2004.02.003.

[12]

G. Vilmart, Weak second order multi-revolution composition methods for highly oscillatory stochastic differential equations with additive or multiplicative noise, SIAM J. Sci. Comput., 36 (2014), 1770-1796.  doi: 10.1137/130935331.

Figure 1.  Patterns over $\eta$ of $|\widetilde{\sigma}_{x,EM}^2-\sigma_x^2|$ (continuous line), $|\widetilde{\sigma}_{v,EM}^2-\sigma_v^2|$ (dashed line) and $|\widetilde{\mu}_{EM}-\mu|$ (dashed-dotted line), for $g = 1$, $\Delta t = 10^{-2}$, $\varepsilon = 1$ for the Eulero-Maruyama method (16) applied to the stochastic problem (4).
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Figure 2.  Patterns over $\eta$ of $|\widetilde{\sigma}_{x,TRAP}^2-\sigma_x^2|$ (continuous line), $|\widetilde{\sigma}_{v,TRAP}^2-\sigma_v^2|$ (dashed line, almost overlapping the continuous line) and $|\widetilde{\mu}_{TRAP}-\mu|$ (dashed-dotted line), for $g = 1$, $\Delta t = 10^{-2}$, $\varepsilon = 1$ for the trapezoidal method (17) applied to the stochastic problem (4).
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Figure 3.  Patterns over $\eta$ of $|\widetilde{\sigma}_{x,AM}^2-\sigma_x^2|$ (continuous line), $|\widetilde{\sigma}_{v,AM}^2-\sigma_v^2|$ (dashed line) and $|\widetilde{\mu}_{AM}-\mu|$ (dashed-dotted line), for $g = 1$, $\Delta t = 10^{-2}$, $\varepsilon = 1$ for the Adams-Moulton method (19) applied to the stocastic problem (4).
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Figure 4.  Patterns over $\eta$ of $|\widetilde{\sigma}_{x,BDF}^2-\sigma_x^2|$ (continuous line), $|\widetilde{\sigma}_{v,BDF}^2-\sigma_v^2|$ (dashed line) and $|\widetilde{\mu}_{BDF}-\mu|$ (dashed-dotted line), for $g = 1$, $\Delta t = 10^{-2}$, $\varepsilon = 1$ for the BDF method (21) applied to the stochastic problem (4).
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