We introduce a new micro-macro decomposition of collisional kinetic equations in the specific case of the diffusion limit, which naturally incorporates the incoming boundary conditions. The idea is to write the distribution function $f$ in all its domain as the sum of an equilibrium adapted to the boundary (which is not the usual equilibrium associated with $f$) and a remaining kinetic part. This equilibrium is defined such that its incoming velocity moments coincide with the incoming velocity moments of the distribution function. A consequence of this strategy is that no artificial boundary condition is needed in the micro-macro models and the exact boundary condition on $f$ is naturally transposed to the macro part of the model. This method provides an 'Asymptotic preserving' numerical scheme which generates a very good approximation of the space boundary values at the diffusive limit, without any mesh refinement in the boundary layers. Our numerical results are in very good agreement with the exact so-called Chandrasekhar value, which is explicitely known in some simple cases.
A particle micro-macro decomposition based numerical scheme for collisional kinetic equations in the diffusion scaling
INRIA Rennes - Bretagne Atlantique（法国国家信息与自动化研究所雷恩-大西洋布列塔尼中心）；Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, Université de Nantes（南特大学让·勒雷数学实验室，CNRS UMR 6629）；Institut de Recherche Mathématiques de Rennes, CNRS UMR 6625, Université de Rennes 1（雷恩大学1号数学研究中心，CNRS UMR 6625）；ENS Rennes（雷恩高等师范学院）
An Asymptotic-Preserving Micro--Macro Scheme for Plasma Simulations in Quasi-Neutral and Low-Mach-Number Regimes with Kinetic Upgrades
Harbin Institute of Technology（哈尔滨工业大学）；Université de Toulouse（图卢兹大学）；UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse, CNRS, Institut de Mathématiques de Toulouse UMR 5219
