We study the canonical momentum based discretizations of a hybrid model with
kinetic ions and mass-less electrons. Two equivalent formulations of the hybrid
model are presented, in which the vector potentials are in different gauges and
the distribution functions depend on canonical momentum (not velocity).
Particle-in-cell methods are used for the distribution functions, and the
vector potentials are discretized by the finite element methods in the
framework of finite element exterior calculus. Splitting methods are used for
the time discretizations. It is illustrated that the second formulation is
numerically superior and the schemes constructed based on the anti-symmetric
bracket proposed have better conservation properties, although the filters can
be used to improve the schemes of the first formulation.Comment: 24 pages, 8 figure