Topic/Type: 2.2 Kinetic methods, Particle-In-Cell and Vlasov, Poster

### Forward Semi-Lagrangian method for the Vlasov Poisson equations

N. Crouseilles1, 2, T. Respaud1, 2, E. Sonnendrucker1, 2

1 IRMA STRASBOURG
2 INRIA GRAND EST

Understanding the dynamics of charged particles in a plasma is of great importance
for a large variety of physical phenomena, such as the confinement
of strongly magnetized plasmas, or laser-plasma interaction problems
for example. Thanks to recent
developments in computational science and in numerical methods, meaningful comparisons
between experience and numerics are becoming possible.

An accurate model for the motion of charged particles, is given by the Vlasov equation. It is based on a phase space description so that non-equilibrium dynamics can accurately be investigated. The unknown $\small f(t, x, v)$ depends
on the time $\small t$, the space $\small x$ and the velocity $\small v$. The electromagnetic fields
are computed self-consistently through the Maxwell or Poisson equations, which
leads to the nonlinear Vlasov-Maxwell or Vlasov-Poisson system.

In this work, we consider the numerical resolution
of the two-dimensional Vlasov equation on a mesh of the phase space
using a forward semi-Lagrangian numerical scheme.
In the present method, the characteristics curves are advanced in time and
a deposition procedure on the phase space grid, similar to the procedure used in PIC methods for the configuration space only, enables to update the distribution function.

Forward Semi-Lagrangian method offers more simplicity to compute high orders algorithms, which lead to good numerical results. Moreover, it will enable to preserve charge, which is one of the most important thing dealing with Maxwell\'s equations.