  Conservative symmetric second-order one-step integrators are derived using
the Discrete Multiplier Method for a family of vortex-blob models approximating
the incompressible Euler's equations on the plane. Conservative properties and
second order convergence are proved. A rational function approximation was used
to approximate the exponential integral that appears in the Hamiltonian.
Numerical experiments are shown to verify the conservative property of these
integrators, their second-order accuracy, and as well as the resulting spatial
and temporal accuracy of the vortex blob method. Moreover, the derived implicit
conservative integrators are shown to be better at preserving conserved
quantities than standard higher-order explicit integrators on comparable
computation times.
