  In this paper, we propose a space-time GMsFEM for transport equations.
Multiscale transport equations occur in many geoscientific applications, which
include subsurface transport, atmospheric pollution transport, and so on. Most
of existing multiscale approaches use spatial multiscale basis functions or
upscaling, and there are very few works that design space-time multiscale
functions to solve the transport equation on a coarse grid. For the time
dependent problems, the use of space-time multiscale basis functions offers
several advantages as the spatial and temporal scales are intrinsically
coupled. By using the GMsFEM idea with a space-time framework, one obtains a
better dimension reduction taking into account features of the solutions in
both space and time. In addition, the time-stepping can be performed using much
coarser time step sizes compared to the case when spatial multiscale basis are
used. Our scheme is based on space-time snapshot spaces and model reduction
using space-time spectral problems derived from the analysis. We give the
analysis for the well-posedness and the spectral convergence of our method. We
also present some numerical examples to demonstrate the performance of the
method. In all examples, we observe a good accuracy with a few basis functions.
