Vlasov model using kinetic phase point trajectories for the study of BGK modes

Based on the method of characteristics, the system of Vlasov-Poisson equations is simulated. Two observed effects, Landau damping and development of BGK modes, are in good agreement with the reported theoretical and simulation results. Since we study high-frequency electron waves, ions are immobile and to simulate the electron motion, the following algorithm is designed. The simulation method contains a fixed grid (denoted by subscript g) in the phase space. Phase points (denoted by subscript p) are mobile and carry constant distribution function (Liouville theorem). The distribution function (DF) on the mesh, fg, is obtained by an interpolation scheme from the DF in phase space, fp. Then, the electron density is obtained by ∫ fgdv. The electric field on the mesh, Eg, is calculated by the use of Poisson solver. Extrapolating of Eg gives the electric field at the position of phase-space points. Finally, the trajectories of the mobile phase points are obtained following the characteristic equations [1],