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

Abstract

We study 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 Itô stochastic differential equation. We propose and analyse temporal discretization schemes for strong and weak effective approximation of the slow component. Such schemes satisfy an asymptotic preserving property. We first prove strong (mean-square) error estimates with rate of convergence 1{2 with respect to the time-step size, which are uniform with respect to the time scale separation parameter. This strong convergence result is illustrated by some numerical experiments. We also prove uniform weak error estimates which exhibit an intriguing order reduction phenomenon: the obtained rate of convergence is also 1{2. We also prove weak error estimates with rate of convergence 1 with respect to both ∆t and .