Vlasov solver for longitudinal dynamics in beam delivery systems for x-ray free electron lasers

Direct numerical methods for solving the Vlasov equation offer some advantages over macroparticle simulations, as they do not suffer from the consequences of the statistical fluctuations inherent in using a number of macroparticles smaller than the bunch population. Unfortunately, these methods are more time consuming and generally considered impractical in a full 6D phase space. However, in a lower-dimension phase space they may become attractive if the beam dynamics is sensitive to the presence of small charge-density fluctuations and a high resolution is needed. In this paper we present a 2D Vlasov solver for studying the longitudinal beam dynamics in single-pass systems of interest for x-ray FELs, where characterization of the microbunching instability stemming from self-field amplified noise is of particular relevance.