  Despite the development of an extensive toolbox of multi-scale rarefied flow
simulators, such simulations remain challenging due to the significant
disparity of collisional and macroscopic spatio-temporal scales. Our study
offers a novel and consistent numerical scheme for a coupled treatment of
particles advection and collision governed by the BGK evolution, honouring
positivity of the velocity distribution. Our method shares its framework, in
spirit, with the unified gas kinetic class of multi-scale schemes. Yet it
provides attractive features for particle-based stochastic simulations, readily
implementable to existing direct simulation Monte-Carlo codes. We demonstrate
accuracy and performance of the devised scheme for prototypic gas flows, over a
wide-range of rarefaction parameters. Due to the resulting robustness and
flexibility of the devised exponential BGK integrator, the scheme paves the way
towards more affordable simulations of large- and multi-scale rarefied gas
phenomena.
