American Institute of Mathematical Sciences

Advanced
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  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 & 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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[8]

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

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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).
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Angelamaria Cardone, Dajana Conte, Beatrice Paternoster. Two-step collocation methods for fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2709-2725. doi: 10.3934/dcdsb.2018088
[2]

William F. Thompson, Rachel Kuske, Yue-Xian Li. Stochastic phase dynamics of noise driven synchronization of two conditional coherent oscillators. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2971-2995. doi: 10.3934/dcds.2012.32.2971
[3]

Igor Chueshov, Peter E. Kloeden, Meihua Yang. Long term dynamics of second order-in-time stochastic evolution equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 991-1009. doi: 10.3934/dcdsb.2018139
[4]

Jiaohui Xu, Tomás Caraballo. Long time behavior of fractional impulsive stochastic differential equations with infinite delay. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2719-2743. doi: 10.3934/dcdsb.2018272
[5]

Tingting Wu, Yufei Yang, Huichao Jing. Two-step methods for image zooming using duality strategies. Numerical Algebra, Control & Optimization, 2014, 4 (3) : 209-225. doi: 10.3934/naco.2014.4.209
[6]

Nguyen Dinh Cong, Nguyen Thi Thuy Quynh. Coincidence of Lyapunov exponents and central exponents of linear Ito stochastic differential equations with nondegenerate stochastic term. Conference Publications, 2011, 2011 (Special) : 332-342. doi: 10.3934/proc.2011.2011.332
[7]

Nikolaos Halidias. Construction of positivity preserving numerical schemes for some multidimensional stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 153-160. doi: 10.3934/dcdsb.2015.20.153
[8]

Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061
[9]

Yulan Lu, Minghui Song, Mingzhu Liu. Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 695-717. doi: 10.3934/dcdsb.2018203
[10]

Dingshi Li, Lin Shi, Xiaohu Wang. Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5121-5148. doi: 10.3934/dcdsb.2019046
[11]

Ludwig Arnold, Igor Chueshov. Cooperative random and stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 1-33. doi: 10.3934/dcds.2001.7.1
[12]

Jianhua Huang, Tianlong Shen, Yuhong Li. Dynamics of stochastic fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2051-2067. doi: 10.3934/dcdsb.2015.20.2051
[13]

Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459
[14]

Philippe Angot, Pierre Fabrie. Convergence results for the vector penalty-projection and two-step artificial compressibility methods. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1383-1405. doi: 10.3934/dcdsb.2012.17.1383
[15]

Andrew Vlasic. Long-run analysis of the stochastic replicator dynamics in the presence of random jumps. Journal of Dynamics & Games, 2018, 5 (4) : 283-309. doi: 10.3934/jdg.2018018
[16]

Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115
[17]

Christopher Rackauckas, Qing Nie. Adaptive methods for stochastic differential equations via natural embeddings and rejection sampling with memory. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2731-2761. doi: 10.3934/dcdsb.2017133
[18]

Lifeng Chen, Jifa Jiang. Stochastic epidemic models driven by stochastic algorithms with constant step. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 721-736. doi: 10.3934/dcdsb.2016.21.721
[19]

Chuchu Chen, Jialin Hong. Mean-square convergence of numerical approximations for a class of backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2051-2067. doi: 10.3934/dcdsb.2013.18.2051
[20]

Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (73)
  • HTML views (345)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences