  In this paper, we consider a parabolic problem with time-dependent
heterogeneous coefficients. Many applied problems have coupled space and time
heterogeneities. Their homogenization or upscaling requires cell problems that
are formulated in space-time representative volumes for problems with scale
separation. In problems without scale separation, local problems include
multiple macroscopic variables and oversampled local problems, where these
macroscopic parameters are computed. These approaches, called Non-local
multi-continua, are proposed for problems with complex spatial heterogeneities
in a number of previous papers. In this paper, we extend this approach for
space-time heterogeneities, by identifying macroscopic parameters in space-time
regions. Our proposed method space-time Non-local multi-continua (space-time
NLMC) is an efficient numerical solver to deal with time-dependent
heterogeneous coefficients. It provides a flexible and systematic way to
construct multiscale basis functions to approximate the solution. These
multiscale basis functions are constructed by solving a local energy
minimization problems in the oversampled space-time regions such that these
multiscale basis functions decay exponentially outside the oversampled domain.
Unlike the classical time-stepping methods combined with full-discretization
technique, our space-time NLMC efficiently constructs the multiscale basis
functions in a space-time domain and can provide a computational savings
compared to space-only approaches as we discuss in the paper. We present two
numerical experiments, which show that the proposed approach can provide a good
accuracy.
