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# 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).

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