Asymptotic-Preserving Schemes for Multiscale Physical Problems

TL;DR

This paper reviews Asymptotic-Preserving schemes that enable efficient multiscale simulations by bridging microscopic and macroscopic physics without resolving all scales explicitly.

Contribution

It introduces AP strategies for various asymptotic transitions in physical systems, highlighting their ability to handle multiscale problems across different regimes.

Findings

AP schemes effectively capture macroscopic behavior without fine-scale resolution

The paper demonstrates AP methods for quantum to classical, classical to kinetic, and kinetic to hydrodynamics transitions

AP strategies improve computational efficiency in multiscale physical simulations

Abstract

We present the asymptotic transitions from microscopic to macroscopic physics, their computational challenges and the Asymptotic-Preserving (AP) strategies to efficiently compute multiscale physical problems. Specifically, we will first study the asymptotic transition from quantum to classical mechanics, from classical mechanics to kinetic theory, and then from kinetic theory to hydrodynamics. We then review some representative AP schemes that mimic, at the discrete level, these asymptotic transitions, hence can be used crossing scales and, in particular, capture the macroscopic behavior without resolving numerically the microscopic physical scale.