  In this paper, we study the strong and weak convergence rates for multi-scale
one-dimensional stochastic Burgers equation. Based on the techniques of
Galerkin approximation, Kolmogorov equation and Poisson equation, we obtain the
slow component strongly and weakly converges to the solution of the
corresponding averaged equation with optimal orders 1/2 and 1 respectively. The
highly nonlinear term in system brings us huge difficulties, we develop new
technique to overcome these difficulties. To the best of our knowledge, this
work seems to be the first result in which the optimal convergence orders in
strong and weak sense for multi-scale stochastic partial differential equations
with highly nonlinear term.
