On time integrators for Generalized Multiscale Finite Element Methods applied to advection–diffusion in high-contrast multiscale media

Despite recent progress in dealing with advection–diffusion problems in high-contrast multiscale settings, there is still a need for methods that speed up calculations without compromising accuracy. In this paper, we consider the challenges of unsteady diffusion–advection problems in the presence of multiscale high-contrast media. We use the Generalized Multiscale Method (GMsFEM) as the space discretization and pay extra attention to the time solver. Traditional finite-difference methods’ accuracy and stability deteriorate in the presence of high contrast and also with an advection term. Following Contreras et al. (2023), we use exponential integrators to handle the time dependence, fully utilizing the advantages of the generalized multiscale method. For situations dominated by diffusion, our approach aligns with previous work. However, in cases where advection starts to dominate, we introduce a different local generalized eigenvalue problem to build the multiscale basis functions. This adjustment makes things more efficient since the basis functions retain more information related to the advection term. We present experiments to demonstrate the effectiveness of our proposed method.