Elaborated Modeling of Synchrotron Motion in Vlasov-Fokker-Planck Solvers

Solving the Vlasov-Fokker-Planck equation is a well-tested approach to simulate dynamics of electron bunches self-interacting with their own wake-field. Typical implementations model the dynamics of a charge density in a damped harmonic oscillator, with a small perturbation due to collective effects. This description imposes some limits to the applicability: Because after a certain simulation time coherent synchrotron motion will be damped down, effectively only the incoherent motion is described. Furthermore {even though computed {the tune spread is typically masked by the use of a charge density instead of individual particles. As a consequence, some effects are not reproduced. In this contribution, we present methods that allow to consider single-particle motion, coherent synchrotron oscillations, non-linearities of the accelerating voltage, higher orders of the momentum compaction factor, as well as modulations of the accelerating voltage. We also provide exemplary studies - based on the KIT storage ring KARA (KArlsruhe Research Accelerator) - to show the potentiality of the methods.