Hybrid multiscale methods for hyperbolic and kinetic problems

In these notes we present some recent results on the development of hybrid methods for hyperbolic and kinetic equations with multiple scales. The main ingredients in the schemes are a suitable merging of particle methods in non stiff regimes with high resolution shock capturing techniques in stiff ones. The key aspect in the development of the algorithms is the choice of a suitable hybrid representation of the solution. First, after a brief review on sampling methods and Monte Carlo techniques, we introduce the hybrid schemes for hyperbolic systems with relaxation and present numerical applications to the simple Jin-Xin relaxation model both in one and two space dimensions. Next, we show how to extend the above methodology to the case of kinetic models of BGK type and discuss the challenging case of the full Boltzmann equation. Some numerical results are also presented