A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit

We propose a new numerical scheme for linear transport equations. It is based on a decomposition of the distribution function into equilibrium and non-equilibrium parts. We also use a projection technique that allows to reformulate the kinetic equation into a coupled system of an evolution equation for the macroscopic density and a kinetic equation for the non-equilibrium part. By using a suitable time semi-implicit discretization, our scheme is able to accurately approximate the solution in both kinetic and diffusion regimes. It is asymptotic preserving in the following sense: when the mean free path of the particles is small, our scheme is asymptotically equivalent to a standard numerical scheme for the limit diffusion model. A uniform stability property is proved for the simple telegraph model. Various boundary conditions are studied. Our method is validated in one-dimensional cases by several numerical tests and comparisons with previous asymptotic preserving schemes.