Exponential BGK integrator for multiscale flow simulation

The simulation of gas flows covering a large range of Knudsen number is still challenging, as the coupling of solvers based on the Navier-Stokes equations, which become inaccurate for rarefied flows, with the DSMC method, computationally costly in dense regions, is not straightforward. However, a multiscale method capable of handling both regimes would be beneficial to numerous applications, ranging from aerospace engineering to microelectromechanical systems. The use of the BGK collision operator in the Boltzmann equation has proven efficient for both discrete velocity methods and particle-based simulations, for gas flows in a range that goes from rarefied to lower Knudsen number regimes. The latter type of solvers also has the advantage to be easily coupled with the widely used DSMC, but corresponds however to schemes that are usually limited to first order and can therefore require fine resolutions. We present here a method that achieves second order in time by integrating the BGK equation using exponential time differencing. The idea is similar to DUGKS, which reaches second order with a Crank-Nicolson integration, but in our method all prefactors remain positive regardless of the timestep. This allows its extension from a discrete velocity-type method to a particle-based scheme without introducing any negative weighted particle, hereby keeping the implementation analogous to the standard stochastic particle BGK method. The resulting exponential differencing scheme is implemented with both stochastic particle and discrete velocity approaches, and its accuracy and efficiency are compared to preexisting BGK methods on several test cases.