The Semi-Lagrangian Method for the Numerical Resolution of the Vlasov Equation

The numerical resolution of kinetic equations and, in particular, of Vlasov-type equations is performed most of the time using particle in cell methods which consist in describing the time evolution of the equation through a finite number of particles which follow the characteristic curves of the equation, the interaction with the external and self-consistent fields being resolved using a grid. Another approach consists in computing directly the distribution function on a grid by following the characteristics backward in time for one time step and interpolating the value at the feet of the characteristics using the grid point values of the distribution function at the previous time step. In this report we introduce this last method, which couples the Lagrangian and Eulerian points of view and its use for the Vlasov equation and equations derived from it.