Contrast-independent partially explicit time discretizations for multiscale flow problems

TL;DR

This paper introduces a contrast-independent time discretization method for multiscale flow problems, enabling larger time steps without stability issues by combining implicit and explicit treatments of multiscale modes.

Contribution

It proposes a novel splitting algorithm with specially designed multiscale spaces that achieves contrast-independent stability in time-dependent multiscale problems.

Findings

Unconditional stability achieved regardless of contrast.

Numerical results demonstrate computational savings.

Effective treatment of dominant multiscale modes in implicit fashion.

Abstract

Many multiscale problems have a high contrast, which is expressed as a very large ratio between the media properties. The contrast is known to introduce many challenges in the design of multiscale methods and domain decomposition approaches. These issues to some extend are analyzed in the design of spatial multiscale and domain decomposition approaches. However, some of these issues remain open for time dependent problems as the contrast affects the time scales, particularly, for explicit methods. For example, in parabolic equations, the time step is $dt=H^2/\kappa_{max}$, where $\kappa_{max}$ is the largest diffusivity. In this paper, we address this issue in the context of parabolic equation by designing a splitting algorithm. The proposed splitting algorithm treats dominant multiscale modes in the implicit fashion, while the rest in the explicit fashion. The unconditional stability of these algorithms require a special multiscale space design, which is the main purpose of the paper. We show that with an appropriate choice of multiscale spaces we can achieve an unconditional stability with respect to the contrast. This could provide computational savings as the time step in explicit methods is adversely affected by the contrast. We discuss some theoretical aspects of the proposed algorithms. Numerical results are presented.