I will present some numerical methods that are able to efficiently capture the multiscale aspects of some problems described by kinetic equations. These schemes are based on the micro-macro decomposition, in which the distribution function is split into equilibrium and non-equilibrium parts, and on a projection technique that allows to reformulate the kinetic equation into a coupled system of an evolution equation for the macroscopic variables and a kinetic equation for the non-equilibrium part. By using a suitable time semi-implicit discretization, these schemes are able to accurately approximate the solution in both kinetic and fluid regimes. They can also be used to localize the non-equilibrium part wherever it is necessary, which leads to a kind of extended fluid model with a localized kinetic upscaling. This method is rather general, since it can be applied both to linear transport problems with diffusive asymptotic regimes, and to nonlinear Boltzmann equation of rarefied gas dynamics with Euler and compressible Navier-Stokes regimes.
Various parts of this work have been obtained in collaboration with M. Lemou (Universite de Rennes 1, France), P. Degond (Universite de Toulouse, France), J.-G. Liu (University of Maryland, College-Park, USA), and G. Dimarco (Universita di Ferrara, Italy).