Spectrally Accurate Methods for Computing Kinetic Electron Plasma Wave Dynamics

We present two numerical methods for computing solutions of the Vlasov-Fokker-Planck-Poisson equations that are spectrally accurate in all three variables; time, space and velocity. The first is a Chebyshev collocation method for solving the Volterra equation for the space-time evolution of the plasma density for the linearized, collisionless case. This is then used to construct the velocity distribution function in Case-van Kampen normal modes, building on the work of Li and Spies, for example. The second is an arbitrary-order exponential time differencing scheme that makes use of the Duhamel principle to fold in the effects of collisions and nonlinearity. We investigate the emergence of a continuous spectrum in the collisionless limit and the embedding of Landau's poles in this general setting. We simultaneously resolve the effects of filamentation, phase mixing, and Landau and collisional damping to arbitrary order of accuracy. We will ultimately incorporate echoes and trapping phenomena in three dimensions by generalizing the latter method as presented here in a simpler setting.