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In this paper, we consider a parabolic problem with time-dependentheterogeneous coefficients. Many applied problems have coupled space and timeheterogeneities. Their homogenization or upscaling requires cell problems thatare formulated in space-time representative volumes for problems with scaleseparation. In problems without scale separation, local problems includemultiple macroscopic variables and oversampled local problems, where thesemacroscopic parameters are computed. These approaches, called Non-localmulti-continua, are proposed for problems with complex spatial heterogeneitiesin a number of previous papers. In this paper, we extend this approach forspace-time heterogeneities, by identifying macroscopic parameters in space-timeregions. Our proposed method space-time Non-local multi-continua (space-timeNLMC) is an efficient numerical solver to deal with time-dependentheterogeneous coefficients. It provides a flexible and systematic way toconstruct multiscale basis functions to approximate the solution. Thesemultiscale basis functions are constructed by solving a local energyminimization problems in the oversampled space-time regions such that thesemultiscale basis functions decay exponentially outside the oversampled domain.Unlike the classical time-stepping methods combined with full-discretizationtechnique, our space-time NLMC efficiently constructs the multiscale basisfunctions in a space-time domain and can provide a computational savingscompared to space-only approaches as we discuss in the paper. We present twonumerical experiments, which show that the proposed approach can provide a goodaccuracy.

Space-time Non-local multi-continua upscaling for parabolic equations with moving channelized media