Numerical study on Landau damping

We present a numerical study of the so-called Landau damping phenomenon in the Vlasov theory for spatially periodic plasmas in a nonlinear setting. It shows that the electric field does decay exponentially to zero as time goes to infinity with general analytical initial data which are close to a Maxwellian. The time decay depends on the length of the period as well as the closeness between the initial data and the Maxwellian. A similar pattern is observed if the Maxwellian is replaced by other algebraically decaying homogeneous equilibria with a single maximum, or even by some homogeneous equilibria with small double-humps. The numerical method used is a high order accurate hybrid spectral and finite difference scheme which is carefully calibrated with the well-known decay theory for the corresponding linear case, to guarantee a reliable resolution free of numerical artifacts for a long time integration.