  The BGK model of the Boltzmann equation allows for efficient flow
simulations, especially in the transition regime between continuum and high
rarefaction. However, ensuring efficient performances for multiscale flows, in
which the Knudsen number varies by several orders of magnitude, is never
straightforward. Discrete velocity methods as well as particle-based solvers
can each reveal advantageous in different conditions, but not without
compromises in specific regimes. This article presents a second-order
asymptotic preserving discrete velocity method to solve the BGK equation, with
the particularity of thoroughly maintaining positivity when operations are
conducted on the distribution function. With this procedure based on
exponential differencing, it is therefore also possible to construct an adapted
version of this second-order method using the stochastic particle approach, as
previously presented. The deterministic variant is detailed here and its
performances are evaluated on several test cases. Combined to the probabilistic
solver and with the possibility of a future coupling, our exponential
differencing DVM provides a robust toolbox that would be useful in efficiently
simulating multiscale gas phenomena.
