Section: New Results
Two-Scale Asymptotic-Preserving schemes
Participants : Nicolas Crouseilles, Emmanuel Frénod, Michaël Gutnic, Sever Hirstoaga.
In paper  , we build a Two-Scale Macro-Micro decomposition of the Vlasov equation with a strong magnetic field. This consists in writing the solution of this equation as a sum of two oscillating functions with circumscribed oscillations. The first of these functions has a shape which is close to the shape of the Two-Scale limit of the solution and the second one is a correction built to offset this imposed shape. The aim of such a decomposition is to be the starting point for the construction of Two-Scale Asymptotic-Preserving schemes.
During CEMRACS 2011, we have started the project to test on a simplified model the Two-Scale Asymptotic-Preserving Schemes. The model, a two dimensional in phase space Vlasov-Poisson equation with small parameter, is used for a long time simulation of a beam in a focusing channel. This work was already done in  in the case where the solution is approximated by the two scale limit. The first goal is to improve this approximation, by going further, to the first order one; this was done in  . The second goal is to replace this approximation by an exact decomposition, using the macro-micro framework. This last approach will permit to treat the case of a not necessary small parameter. In order to accomplish the first task we have writen a Particle-In-Cell code in SeLaLib.