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

TL;DR

This paper introduces a space-time non-local multi-continua method for efficiently upscaling parabolic equations with moving, heterogeneous media, capturing complex space-time heterogeneities with multiscale basis functions.

Contribution

It extends existing non-local multi-continua approaches to space-time heterogeneities, providing a systematic and efficient way to construct multiscale basis functions for time-dependent problems.

Findings

The method accurately approximates solutions in numerical experiments.

It offers computational savings over traditional space-only approaches.

Multiscale basis functions decay exponentially outside the oversampled domain.

Abstract

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.