Comparison of Different Methods to Calculate Induced Voltage in Longitudinal Beam Dynamics Codes

Collective effects in longitudinal beam dynamics simulations are essential for many studies since they can perturb the RF potential, giving rise to instabilities. The beam induced voltage can be computed in frequency or time domain using a slicing of the beam profile. This technique is adopted by many codes including CERN BLonD. The slicing acts as a frequency filter and cuts high frequency noise but also physical contributions if the resolution is not sufficient. Moreover, a linear interpolation usually defines the voltage for all the macro-particles, and this can be another source of unphysical effects. The MuSiC code describes interaction between the macro-particles with the wakes generated only by resonator impedances. The complications related to the slices are avoided, but the voltage can contain high frequency noise. In addition, since the computational time scales with the number of resonators and macro-particles, having a large number of them can be cumbersome. In this paper the features of the different approaches are described together with benchmarks between them and analytical formulas, considering both single and multi-turn wakes.