A boundary matching micro/macro decomposition for kinetic equations

Abstract

We introduce a new micro/macro decomposition of collisional kinetic equations which naturally incorporates the exact space boundary conditions. The idea is to write the distribution function f in all its domain as the sum of a Maxwellian adapted to the boundary (which is not the usual Maxwellian associated with f) and a reminder kinetic part. This Maxwellian is defined such that its ‘incoming’ velocity moments coincide with the ‘incoming’ velocity moments of the distribution function. Important consequences of this strategy are the following: (i) 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; (ii) It provides a new class of the so-called ‘Asymptotic Preserving’ (AP) numerical schemes: such schemes are consistent with the original kinetic equation for all fixed positive value of the Knudsen number ε, and if ε → 0 with fixed numerical parameters then these schemes degenerate into consistent numerical schemes for the various corresponding asymptotic fluid or diffusive models. Here, the strategy provides AP schemes not only inside the physical domain but also in the space boundary layers. We provide a numerical test in the case of a diffusion limit of the one-group transport equation, and show that our AP scheme recovers the boundary layer and a good approximation of the theoretical boundary value, which is usually computed from to the so-called Chandrasekhar function.