Asymptotic-Preserving Schemes for Multiscale Physical Problems

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.

Time complexity analysis of quantum difference methods for linear high dimensional and multiscale partial differential equations

“…The AP schemes have been developed for a wide range of time-dependent kinetic and hyperbolic equations. The fundamental idea is to design numerical methods that preserve the asymptotic limits from the microscopic to the macroscopic models in the discrete setting [16,17]. We first consider two typical AP schemes presented in [15,16] for solving (6.1) or (6.2).…”

“…Among various multiscale methods, the Asymptotic-Preserving (AP) scheme, which preserves the asymptotic transition from the micro models to the macro ones at the discrete level, has the merit of using one solver that works across scales naturally, thus has been widely used for multiscale hyperbolic and kinetic problems [16,17].…”