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We study Landau damping in the 1+1D Vlasov-Poisson system using aFourier-Hermite spectral representation. We describe the propagation of freeenergy in phase space using forwards and backwards propagating Hermite modesrecently developed for gyrokinetics [Schekochihin et al. (2014)]. The change inthe electric field corresponds to the net Hermite flux via a free energyevolution equation. In linear Landau damping, decay in the electric fieldcorresponds to forward propagating Hermite modes; in nonlinear damping, theinitial decay is followed by a growth phase characterised by the generation ofbackwards propagating Hermite modes by the nonlinear term. The free energycontent of the backwards propagating modes increases exponentially untilbalancing that of the forward propagating modes. Thereafter there is nosystematic net Hermite flux, so the electric field cannot decay and thenonlinearity effectively suppresses Landau damping. These simulations areperformed using the fully-spectral 5D gyrokinetics code SpectroGK [Parker etal. 2014], modified to solve the 1+1D Vlasov-Poisson system. This capturesLandau damping via an iterated L\'enard-Bernstein collision operator or viaHou-Li filtering in velocity space. Therefore the code is applicable even inregimes where phase-mixing and filamentation are dominant.

Fourier-Hermite spectral representation for the Vlasov-Poisson system in the weakly collisional limit