Article Contents
Article Contents

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

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.

Mathematics Subject Classification: 65C30.

 Citation:

• 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).

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).

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).

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).

•  E. Buckwar and R. D'Ambrosio, Exponential mean-square stability properties of stochastic multistep methods, submitted. 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. 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. 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. 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. 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. 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). P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, 1962. D. J. Higham , An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001) , 525-546.  doi: 10.1137/S0036144500378302. P. E. Kloeden and E. Platen, The Numerical Solution of Stochastic Differential Equations, Springer-Verlag, 1992. 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. 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.

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