## Section: New Results

### Multiple time scales solvers and Magneto HydroDynamics

Participants : Emmanuel Frénod, Michael Gutnic, Philippe Helluy, Hélène Mathis, Alexandre Mouton, Eric Sonnendrücker.

#### Multiple time scales solvers

Problems in which two very different time scales co-exist put a strong constraint on the time step which needs to be adapted to the smallest time scale in order to avoid instabilities. We have developed a new technique based on two-scale convergence tools for handling such problems. The idea is to find the two-scale limit of the original equation and to solve numerically this two-scale limit, in which the fast scale is handled via an integral term and where the small time step constraint is removed. We have used this technique in order to develop a two-scale PIC method for a problem in accelerator physics [25] and also for the gyrokinetic limit in the thesis of Alexandre Mouton.

#### Magneto HydroDynamics

The CALVI project is mainly devoted to the construction of numerical schemes for kinetic plasma physics simulations. We have started a new research effort directed towards fluid models such as the ideal magneto-hydro-dynamics (MHD) model. A summer research session has been organized in Luminy in July and august 2008 during the CEMRACS'08 http://smai.emath.fr/cemracs/cemracs08/fr_projects.html

The project, entitled "GADMHD: approximation GAlerkin Discontinu pour la MagnétoHydroDynamique" allowed to achieve several objectives: to develop a new high order discontinuous galerkin code for solving the MHD equations, to investigate several practical aspects of this kind of
problems (validation, analytical solution, Riemann problem, multiplicity of the entropy solutions,
*etc.* ), and finally to test and analyze a new general multi-time step algorithm based on the Adams-Bashforth time integration. This last algorithm permits to reduce significantly the computational cost when large and big cells are mixed in the mesh. The results obtained during the
research session are available in the final report
[41] .