## Section: New Results

### High order schemes for Vlasov-Poisson system

Participant : Nicolas Crouseilles.

In [44] , we derive the order conditions for fourth order time splitting schemes in the case of the $1D$ Vlasov-Poisson system. Computations to obtain such conditions are motivated by the specific Poisson structure of the Vlasov-Poisson system : this structure is similar to Runge-Kutta-Nyström systems. The obtained conditions are proved to be the same as RKN conditions derived for ODE up to the fourth order. Numerical results are performed and show the benefit of using high order splitting schemes in that context.

In [19] , we present a discontinuous Galerkin scheme for the numerical approximation of the one- dimensional periodic Vlasov-Poisson equation. The scheme is based on a Galerkin-characteristics method in which the distribution function is projected onto a space of discontinuous functions. We present comparisons with a semi-Lagrangian method to emphasize the good behavior of this scheme when applied to Vlasov-Poisson test cases.

The CEMRACS is an annual summer research session promoted by the SMAI. The 15th edition of 2010 has been organized by N. Crouseilles, H. Guillard, B. Nkonga and E. Sonnendrücker around "Numerical modeling of fusion plasmas". The volume [38] gathers artless resulting from research projects initiated during the CEMRACS 2010.