Hybrid Multiscale Methods II. Kinetic Equations

In this work we consider the development of a new family of hybrid numerical methods for the solution of kinetic equations which involves different scales. The basic idea is to couple macroscopic and microscopic model in all cases in which the macroscopic model does not provide correct results. The key aspect in the development of the algorithms is the choice of a suitable hybrid representation of the solution and a merging of Monte Carlo methods in non-equilibrium regimes with deterministic methods in equilibrium ones. This approach permits to treat e±ciently both the microscopic and the macroscopic scales. Applications to the Boltzmann-BGK equation are presented to show the performance of the new methods