Uniform strong and weak error estimates for numerical schemes applied to multiscale SDEs in a Smoluchowski-Kramers diffusion approximation regime

TL;DR

This paper develops and analyzes numerical schemes for multiscale stochastic differential equations, providing uniform strong and weak error estimates in the Smoluchowski-Kramers diffusion approximation regime, ensuring accurate approximation as the scale separation vanishes.

Contribution

The paper introduces and rigorously analyzes schemes that are asymptotic preserving and provide uniform error estimates for multiscale SDEs in the diffusion approximation regime.

Findings

Established uniform strong error estimates.

Established uniform weak error estimates.

Schemes are asymptotic preserving.

Abstract

We study a family of numerical schemes applied to a class of multiscale systems of stochastic differential equations. When the time scale separation parameter vanishes, a well-known Smoluchowski--Kramers diffusion approximation result states that the slow component of the considered system converges to the solution of a standard It\^o stochastic differential equation. We propose and analyse schemes for strong and weak effective approximation of the slow component. Such schemes satisfy an asymptotic preserving property and generalize the methods proposed in a recent article. We fill a gap in the analysis of these schemes and prove strong and weak error estimates, which are uniform with respect to the time scale separation parameter.