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.

翻译：我们研究一组适用于一组多尺度的随机差分方程式的数值方法。当时间尺度分离参数消失时,一个众所周知的Smoluchowski-Kramers扩散近似结果显示,被考虑的系统缓慢部分与标准的Itçóo随机差分方程式的解决方案相融合。我们提出并分析对慢度方程式的强弱有效近差的系统方法。这种系统满足了一种无防护特性,并概括了最近一篇文章中建议的方法。我们在分析这些计划时填补了一个空白,并证明对时间尺度分离参数的误差估计数是强弱的,与时间尺度分离参数一致。