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

Abstract

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. 2010 Mathematics Subject Classification. 65C30. Key words and phrases. Stochastic differential equations, stochastic damped oscillators, stochastic two-step methods, long-term numerical dynamics. The work is supported by GNCS-Indam project.

Stability Issues for Selected Stochastic Evolutionary Problems: A Review

“…This issue will be analyzed in detail in future contributions. The tool introduced in [1][2][3] gives the possibility to provide a priori analysis relying on simple computations on 2 by 2 matrices. The structure-preserving approach to SDEs will next be devoted to stochastic Hamiltonian problems [7,8] and the stochastic extension of existing deterministic approaches [32][33][34].…”

“…where X n and V n are numerical solutions of Equation (2) generated by the ITS method given by Equation (5). As proved in [3], Σ satisfies the matrix equation…”

“…In [1][2][3], the authors have analyzed the ability of some of the most common numerical methods for SDEs in preserving the correlation matrix over long times. The analysis given in [1,2] only involves one-step methods, while that provided in [3] is focused on the larger class of indirect two-step methods (ITS methods), i.e., the family of stochastic two-step methods applied to the first order representation given by Equation (2) of the second order SDE in Equation (1). ITS methods assume the form…”

“…The first highlighted aspect, i.e., stability as numerical preservation of qualitative properties, is here framed in the context of stochastic differential equations (SDEs), with special emphasis on problems describing stochastic oscillators [1][2][3][4].The perspective we follow consists of providing a long-term analysis of numerical methods for SDEs in terms of preserving invariance laws that characterize the dynamics provided by the exact solution. A first contribution in this sense was given in [5], where the author analyzed the invariance of asymptotic laws characterizing linear stochastic systems under given discretizations.…”

“…A first contribution in this sense was given in [5], where the author analyzed the invariance of asymptotic laws characterizing linear stochastic systems under given discretizations. The analysis of partitioned methods for linear oscillators in the presence of additive noise has been an object of [6], while the analysis of linear second order SDEs describing damped stochastic oscillators has been provided by Burrage and Lythe in [1,2] and inspired the paper [3] giving a more general two-step framework. More recent contributions have regarded stochastic Hamiltonian problems [7,8] and stochastic oscillators with multiplicative noise [9].…”