# American Institute of Mathematical Sciences

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

## 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

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.

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
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##### References:
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).
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).
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).
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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