Multiscale simulation of advection-diffusion problems has been the subject of many studies in recent years. Although methods like MsFEM and MsFVM have worked well for elliptic and some parabolic problems, there are challenges in extension to hyperbolic problems in general. We developed a theoretical framework, “the global-local analysis of PDE” to rectify this. Based on the framework, we equip the well-established Generalized FEM(GFEM) with transient local problems. Some highlights of our resulting framework include: 1) resolves multiple space and time scales; 2) computationally efficient; 3) recovers fine scale field on the coarse global mesh