Relaxed micro–macro schemes for kinetic equations

Abstract

In Bennoune et al. (2008) [1], Lemou and Mieussens (2008) [6] we recently developed a general approach to design asymptotic preserving (AP) schemes for kinetic equations which are able to solve both macroscopic (small Knudsen number ε) and kinetic scales using the same model and the same numerical parameters. The strategy is based on micro/macro decompositions of the distribution function and can be applied to a large class of kinetic models (Boltzmann, Landau, etc.). However, the so-obtained schemes are shown to be AP for only close-to-equilibrium initial data in the non-linear case and, in general, they require costly inversions of non-local collision operators. In the present work, we introduce a new formulation of this strategy with the following properties: i) Initial data does not need to be well prepared, and may be independent of ε. ii) No inversion (even linear) of collision operators is needed and time-implicit schemes are obtained for the asymptotic models in the limit ε → 0 , making the schemes free from the usual diffusive CFL constraint. iii) The numerical schemes are consistent with the models for all values of ε > 0 , and degenerate into consistent discretizations of the asymptotic model (Euler, Navier–Stokes, diffusion) when ε goes to 0, the numerical parameters (time–space–velocity steps) being fixed. Preliminary numerical validations of this approach are done on the non-local linear transport equation and its diffusion limit.