Study of a Fast Convolution Method for Solving the Space Charge Fields of Charged Particle Bunches

The kernel of beam dynamics simulations using the particle-in-cell (PIC) model is the solution of Poisson’s equation for the electric potential. A very common way to solve Poisson’s equation is to use the convolution of charge density and Green’s function, the so-called Green’s function method. Additionally, theintegratedGreen’sfunctionmethodisbeing used in order to achieve a higher accuracy. For both methods, the convolutions are done using fast Fourier transform based on the convolution theorem. However, the construction of the integrated Green’s function and the further convolution is still very time-consuming. The computation can be accelerated without loosing precision if the calculation of Green’s function values is limited to that part of the computational domain with non-zero grid charge density. In this paper we present a general numerical study of these Green’s function methods for computing the potential of different bunches: The results can also be used in other simulation codes to improve efficiency.