  This work is devoted to the numerical simulation of the \BGK equation for two
species in the fluid limit using a particle method. Thus, we are interested in
a gas mixture consisting of two species without chemical reactions assuming
that the number of particles of each species remains constant. We consider the
kinetic two species model proposed by Klingenberg, Pirner and Puppo in 2017,
which separates the intra and interspecies collisions. We want to study
numerically the influence of the two relaxation term, one corresponding to
intra, the other to interspecies collisions. For this, we use the method of
micro-macro decomposition. First, we derive an equivalent model based on the
micro-macro decomposition (see Bennoune, Lemou and Mieussens, 2007 and
Crestetto, Crouseilles and Lemou, 2013). The kinetic micro part is solved by a
particle method, whereas the fluid macro part is discretized by a standard
finite volume scheme. Main advantages of this approach are: (i) the noise
inherent to the particle method is reduced compared to a standard (without
micro-macro decomposition) particle method, (ii) the computational cost of the
method is reduced in the fluid limit since a small number of particles is then
sufficient.
