  Symplectic tracking is important in accelerator beam dynamics simulation. So
far, to the best of our knowledge, there is no self-consistent symplectic
space-charge tracking model available in the accelerator community. In this
paper, we present a two-dimensional and a three-dimensional symplectic
multi-particle spectral model for space-charge tracking simulation. This model
includes both the effect from external fields and the effect of self-consistent
space-charge fields using a split-operator method. Such a model preserves the
phase space structure and shows much less numerical emittance growth than the
particle-in-cell model in the illustrative examples.
