Asymptotic Preserving numerical schemes for multiscale parabolic problems

TL;DR

This paper introduces an Asymptotic Preserving numerical scheme for multiscale parabolic problems that remains accurate across different regimes of oscillation, bridging the gap between highly oscillatory and non-oscillatory cases.

Contribution

The paper presents a novel micro-macro decomposition-based scheme that is consistent for both small and large scale oscillations in multiscale parabolic problems.

Findings

Scheme accurately captures asymptotic behaviour as  ightarrow 0

Method remains stable and consistent across regimes

Improves upon existing homogenization methods

Abstract

We consider a class of multiscale parabolic problems with diffusion coefficients oscillating in space at a possibly small scale $\varepsilon$. Numerical homogenization methods are popular for such problems, because they capture efficiently the asymptotic behaviour as $\varepsilon \rightarrow 0$, without using a dramatically fine spatial discretization at the scale of the fast oscillations. However, known such homogenization schemes are in general not accurate for both the highly oscillatory regime $\varepsilon \rightarrow 0$ and the non oscillatory regime $\varepsilon \sim 1$. In this paper, we introduce an Asymptotic Preserving method based on an exact micro-macro decomposition of the solution which remains consistent for both regimes.