The recurrence of the initial state in the numerical solution of the Vlasov equation

The approximate recurrence of the initial state, observed recently in the numerical solution of Vlasov's equation by a finite-difference Eulerian model, is shown to be a property of three independent numerical methods. Some of the methods have exponentially growing modes (Dawson's beaming instabilities), and some others do not. The recurrence is in fact a manifestation of the finite velocity resolution of the numerical methods-a property which is independent of the approximation of a plasma by a finite number of electron beams. The recurrence is shown explicitly in the numerical simulation of Landau damping by three different methods: Fourier-Hermite, the recent variational method of Lewis, and the Eulerian finite-difference method. © 1974.