A General Method of Propagation of Phase Space Distribution with Application to the Sawtooth Instability

We propose and illustrate a general numerical method to follow the probability distribution in phase space as a function of time. It applies to any multiparticle system governed by Liouville, Vlasov or Vlasov-Fokker-Planck dynamics. The technique, based on discretization of the local Perron-Frobenius operator, is sim-ple in concept, easy to implement, and numerically stable in examples studied to date. We illustrate by treating longitudinal dynamics in electron storage rings with realistic wake eld. Applied to the SLC damping rings, the method gives the observed current threshold for bunch lengthening, and several aspects of ob-served behavior above threshold, including the presence of a bursting or sawtooth mode. In contrast to previous particle-in-cell simulations, we have very low nu-merical noise and the ability to follow the motion over several damping times. The method has also been applied to the coherent beam-beam interaction. It appears likely that this approach will be of interest for some of the central prob-lems of this workshop, for instance matching of space-charge dominated beams to a focusing channel, and coherent synchrotron radiation with self-consistent charge/current density