Abstract
This paper presents a new method for the solution of multiscale stochastic differential equations at the diffusive time scale. In contrast to averaging-based methods, e.g., the heterogeneous multiscale method (HMM) or the equation-free method, which rely on Monte Carlo simulations, in this paper we introduce a new numerical methodology that is based on a spectral method. In particular, we use an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients of the homogenized equation. Spectral convergence is proved under suitable assumptions. Numerical experiments corroborate the theory and illustrate the performance of the method. A comparison with the HMM and an application to singularly perturbed stochastic PDEs are also presented.
Highlights
Multiscale stochastic systems arise frequently in applications
Several numerical methods for multiscale stochastic systems that are based on scale separation and on the existence of a coarse-grained equation for the slow variables have been proposed in the literature
In this paper we develop further the methodology introduced in [9] and we apply it to the numerical solution of fast/slow systems of stochastic differential equations (SDEs), including singularly perturbed stochastic partial differential equations (SPDEs) in bounded domains
Summary
Multiscale stochastic systems arise frequently in applications. Examples include atmosphere/ocean science [45] and materials science [19]. The calculation of the drift and diffusion coefficients that appear in this effective (coarse-grained) equation requires appropriate averaging over the fast scales. Several numerical methods for multiscale stochastic systems that are based on scale separation and on the existence of a coarse-grained equation for the slow variables have been proposed in the literature. Examples include the heterogeneous multiscale method (HMM) [62, 21, 1] and the equation-free approach [36]. These techniques are based on evolving the coarse-grained dynamics while calculating the drift and diffusion coefficients “on-the-fly” using short simulation bursts of the fast dynamics
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