Numerical Studies of the Nonlinear Vlasov Equation

The nonlinear one-dimensional Vlasov equation is solved numerically as an initial-value problem. The problem is the same as that considered by Knorr, and is related to, but not the same as, various one-dimensional model calculations using charged sheets. The electron distribution is doubly expanded, the spatial part being expanded as a Fourier series and the velocity part as a Gram-Charlier series. In this representation, the Vlasov equation appears as a matrix ordinary differential equation for the expansion coefficients and is first order in the time. Problems considered are nonlinear Landau damping and the development of strongly unstable initial conditions (two-stream instability). In the latter situation, the limiting amplitudes are computed.