An hybrid method for the Boltzmann equation

In this work, we discuss the generalization of the hybrid Monte Carlo schemes proposed in [1, 2] to the challenging case of the Boltzmann equation. The proposed schemes are designed in such a way that the number of particles used to describe the solution decreases when the solution approaches the equilibrium state and consequently the statistical error decreases as the system approaches this limit. Moreover, as opposite to standard Monte Carlo methods which computational cost increases with the number of collisions, here the time step and thus also the computational cost is independent from the collisional scale. Thanks to the local coupling of Monte Carlo techniques for the solution of the Boltzmann equation with macroscopic numerical methods for the compressible Euler equations, the scheme degenerates to a finite volume scheme for the compressible Euler equations in the limit of an infinite number of collisions without introducing any artificial transition. A simple applications to one-dimensional Boltzmann equation is presented to show the performance of the new method.