A charge preserving scheme for the numerical resolution of the Vlasov-Ampère equations

In this report, a charge preserving numerical resolution of the 1D Vlasov-Ampere equation is achieved, with a forward Semi-Lagrangian method introduced in \cite{4respaud}. The Vlasov equation belongs to the kinetic way of simulating plasmas evolution, and is coupled with the Poisson's equation, or equivalently under charge conservation, the Ampere's one, which self-consistently rules the electric field evolution. In order to ensure having proper physical solutions, it is necessary that the scheme preserves charge numerically. B-Spline deposition will be used for the interpolation step. The solving of the characteristics will be made with a Runge-Kutta 2 method and with a Cauchy-Kovalevsky procedure.

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