  We consider a kinetic description of a multi-species gas mixture modeled with
Bhatnagar-Gross-Krook (BGK) collision operators, in which the collision
frequency varies not only in time and space but also with the microscopic
velocity. In this model, the Maxwellians typically used in standard BGK
operators are replaced by a generalization of such target functions, which are
defined by a variational procedure \cite{arXiv:2101.09047}. In this paper we
present a numerical method for simulating this model, which uses an
Implicit-Explicit (IMEX) scheme to minimize a certain potential function,
mimicking the Lagrange functional that appears in the theoretical derivation.
We show that theoretical properties such as conservation of mass, total
momentum and total energy as well as positivity of the distribution functions
are preserved by the numerical method, and illustrate its usefulness and
effectiveness with numerical examples.
