Space–time non-local multi-continua multiscale method for channelized-media parabolic equations

Abstract

In this paper, we consider a parabolic problem with time-dependent heterogeneous coefficients. Many applied problems have coupled spatial and temporal 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 to handle space–time heterogeneities, by identifying macroscopic parameters in space–time regions. Our proposed method space–time Non-local multi-continua (space–time NLMC) offers an efficient numerical solver to deal with time-dependent heterogeneous coefficients. It provides a flexible and systematic approach to construct multiscale basis functions, enabling accurate approximation of the solution. The construction of these multiscale basis functions involves solving local energy minimization problems within the oversampled space–time regions. The resulting basis functions exhibit exponential decay outside the selected domain. Unlike the classical time-stepping methods combined with full-discretization technique, our space–time NLMC efficiently constructs the multiscale basis functions in the space–time domain, resulting in computational savings compared to space-only approaches as discussed in the paper. We present two numerical experiments that demonstrate the accuracy and effectiveness of the proposed approach.