Micro-macro schemes for kinetic equations including boundary layers

TL;DR

This paper presents a novel micro-macro decomposition method for kinetic equations that naturally incorporates boundary conditions, leading to asymptotic preserving schemes that accurately approximate boundary values in the diffusion limit without mesh refinement.

Contribution

It introduces a new boundary-aware micro-macro decomposition for kinetic equations, eliminating artificial boundary conditions and improving boundary value approximations in diffusive regimes.

Findings

The method accurately reproduces the Chandrasekhar boundary value in simple cases.

Numerical results show excellent agreement with theoretical boundary values.

The scheme avoids mesh refinement in boundary layers, enhancing computational efficiency.

Abstract

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.