Development of a 2D Vlasov Solver for Longitudinal BeamDynamics in Single-Pass Systems

Direct numerical methods for solving the Vlasov equation offer some advantages over macroparticle simulations, as they do not suffer from the numerical noise 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-FEL's, where characterization of the microbunching instability is of particular relevance. The solver includes a model to account for the smearing effect of a finite horizontal emittance on microbunching. We explore the effect of space charge and coherent synchrotron radiation (CSR). The numerical solutions are compared with results from linear theory and good agreement is found in the regime where linear theory applies