Motivated by weak convergence results in the paper of Takahashi and Yoshida (2005), we show strong convergence for an accelerated Euler- aruyama scheme applied to perturbed stochastic differential equations. The Milstein scheme with the same acceleration is also discussed as an extended result. The theoretical results can be applied to analyzing the multi-level Monte Carlo method originally developed by M.B. Giles. Several numerical experiments for the SABR stochastic volatility model are presented in order to confirm the efficiency of the schemes.