On the Asymptotic Preserving property of the Unified Gas Kinetic Scheme for the diffusion limit of linear kinetic models
This work provides a theoretical foundation for UGKS in radiative transfer, addressing a known bottleneck in asymptotic preserving schemes for multiscale kinetic problems.
The paper proves that the Unified Gas Kinetic Scheme (UGKS) is asymptotic preserving for linear kinetic models in both the diffusive and free transport regimes, and modifies it to handle boundary layers with implicit time discretization. Numerical tests confirm the scheme's accuracy and stability.
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.