The structure of the stationary kinetic boundary layer for the linear BGK equation

Abstract

Recently, very accurate numerical methods for solving kinetic boundary layer problems for linear kinetic equations have been developed. To test such methods, detailed information about exactly solvable models can be very helpful. To this purpose the authors present results for the stationary, 1D linear BGK equation, obtained by the singular eigenfunction method, that extend results already known in the literature, both in scope and in precision. Special attention is paid to the nature of the singularity in the distribution function for small velocities near the boundary; this singularity cannot be reproduced exactly by the numerical methods available. Explicit expressions are presented for the Milne problem and for the albedo problem with input particles of a single velocity. They compare the results with those of a recently developed variant of the two-stream moment method. They find excellent agreement, except close to the singularity; for many quantities of interest, the accuracy obtainable by the novel two-stream moment method cannot be reproduced with comparable numerical effort by evaluation of the exact solution.