Comparison of Numerical Solutions of the Vlasov Equation with Particle Simulations of Collisionless Plasmas

Numerical solutions of the Vlasov and Poisson equations for several physically significant problems are presented and compared with the results of particle simulations. The numerical solutions of the Vlasov equation are based on the Fourier‐Fourier transform method. The spatial representation includes up to 85 modes and is capable of representing strong nonlinear effects. The particle simulations are based on a multipole expansion of finite‐size particles about their nearest grid point location. Special techniques are used to suppress the noise and to accurately control the initial conditions of the plasmas so that quantitative comparisons with Vlasov solutions can be made. Close quantitative agreement between the results of the two simulation techniques is observed. The problems considered are one‐dimensional, with periodic boundary conditions, and involve (1) two‐stream instabilities, with equal and unequal electron beams, and (2) large‐amplitude electron oscillations, with sideband instabilities.