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




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