Space-time multiscale model reduction for transport equations

TL;DR

This paper introduces a space-time GMsFEM approach for transport equations, enabling efficient multiscale modeling by coupling space and time basis functions, resulting in better dimension reduction and coarser time steps.

Contribution

It develops a novel space-time multiscale model reduction method for transport equations, combining snapshot spaces and spectral problems for improved accuracy and efficiency.

Findings

Achieves good accuracy with few basis functions.

Allows coarser time stepping compared to spatial-only methods.

Demonstrates effectiveness through numerical examples.

Abstract

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.