  We consider time discretizations of the two-dimensional Euler equation
written in vorticity form. The discretization method uses a Crouch-Grossman
integrator that proceeds in two stages: first freezing the velocity vector
field at the beginning of the time step, and then solve the resulting
elementary transport equation by using a symplectic integrator to discretize in
time the flow of the associated Hamiltonian differential equation. We prove
that these schemes yield order one approximations of the exact solutions, and
provide error estimates in Sobolev norms.
