Asymptotic Preserving and Multiscale Methods for Kinetic and Hyperbolic Problems

An asymptotic preserving unified gas kinetic scheme for grey radiative transfer equations

Song Jiang

Beijing Instutute of Applied Physics and Computational Mathematics


The solutions of radiative transport equations can cover both optical thin and optical thick regimes due to the large variation of photon’s mean-free path and its interaction with the material. In the small mean free path limit, the nonlinear time-dependent radiative transfer equations can converge to an equilibrium diffusion equation due to the intensive interaction among radiation and material. In the optical thin limit, the photon free transport mechanism will emerge. In this paper, we are going to develop an accurate and robust asymptotic preserving unified gas kinetic scheme (AP-UGKS) for the grey radiative transfer equations, where the radiation transport equation is coupled with the material thermal energy equation. The current work is based on the UGKS framework for the rarefied gas dynamics [K. Xu and J.C. Huang, J. Comput. Phys. 229 (2010), 7747-7764], and is an extension of a recent work [L. Mieussens, J. Comput. Phys. 253(2013), 138-156] from a one-dimensional linear radiation transport equation to a nonlinear two-dimensional grey radiative system. The newly developed scheme has the asymptotic preserving (AP) property in the optically thick regime in the capturing of diffusive solution without using a cell size being smaller than the photon’s mean free path and time step being less than the photon collision time. Besides the diffusion limit, the scheme can capture the exact solution in the optical thin regime as well. The current scheme is a finite volume method. Due to the direct modeling for the time evolution solution of the interface radiative intensity, a smooth transition of the transport physics from optical thin to optical thick can be accurately recovered. Many numerical examples are included to validate the current approach. Finally, an extension of the proposed AP UGKS scheme to the frequency-dependent radiative transfer system will be discussed. (joint work with Wenjun Sun and Kun Xu)