Numerische Integration der nichtlinearen Vlasov-Gleichung

Abstract

The VLASOV equation, i. e. the BOLTZMANN equation without collision term, was numerically integrated as an initial value problem together with the second MAXWELL equation for a one-dimensional electron plasma with smeared-out ion background. The initial values were taken as a MAXWELLian distribution in velocity space and a cosine distribution in position, resulting in a sinusform for the electric field. The LANDAU damping of the total energy of the electric field turned out to be valid for times larger than those given by a formal estimation of the validity of the linear solution. For still later times, the decay rate of the electric energy is less than that given by LANDAU damping. The period of validity of LANDAU damping was computed by estimating the influence of the trapped particles. As a non-linear effect the first harmonic of the electric field builds up within this period. The growth of the first harmonic is faster for larger wave numbers k.