On the Asymptotic Preserving property of the Unified Gas Kinetic Scheme for the diffusion limit of linear kinetic models

TL;DR

This paper proves that the unified gas kinetic scheme (UGKS) remains asymptotic preserving in the diffusion limit of linear kinetic models, and introduces modifications for better boundary layer capture using a standard finite volume approach.

Contribution

The paper demonstrates the asymptotic preserving property of UGKS for diffusion limits and modifies it for improved boundary layer accuracy without complex decompositions.

Findings

UGKS is asymptotic preserving in diffusive regimes

Modified scheme captures boundary layers accurately

Numerical tests confirm scheme's effectiveness

Abstract

The unified gas kinetic scheme (UGKS) of K. Xu et al. [K. Xu and J.-C. Huang, J. Comput. Phys., 229, pp. 7747--7764, 2010], originally developed for multiscale gas dynamics problems, is applied in this paper to a linear kinetic model of radiative transfer theory. While such problems exhibit purely diffusive behavior in the optically thick (or small Knudsen) regime, we prove that UGKS is still asymptotic preserving (AP) in this regime, but for the free transport regime as well. Moreover, this scheme is modified to include a time implicit discretization of the limit diffusion equation, and to correctly capture the solution in case of boundary layers. Contrary to many AP schemes, this method is based on a standard finite volume approach, it does neither use any decomposition of the solution, nor staggered grids. Several numerical tests demonstrate the properties of the scheme.