Before revision:  In this paper  we provide a general framework for understanding the weak error of numerical approximation techniques in the multi-scale setting. We quantify how the error of three different methods depends upon both the natural scalings within a given system , and with the step-size of the numerical method.
After revision:  In this paper we provide a general framework for understanding the weak error of numerical approximation techniques . The analysis takes into account both the natural scalings within a given system , and with the step-size of the numerical method.