  The venerable 2D point-vortex model plays an important role as a simplified
version of many disparate physical systems, including superfluids,
Bose-Einstein condensates, certain plasma configurations, and inviscid
turbulence. This system is also a veritable mathematical playground, touching
upon many different disciplines from topology to dynamic systems theory.
Point-vortex dynamics are described by a relatively simple system of nonlinear
ODEs which can easily be integrated numerically using an appropriate adaptive
time stepping method. As the separation between a pair of vortices relative to
all other inter-vortex length scales decreases, however, the computational time
required diverges. Accuracy is usually the most discouraging casualty when
trying to account for such vortex motion, though the varying energy of this
ostensibly Hamiltonian system is a potentially more serious problem. We solve
these problems by a series of coordinate transformations: We first transform to
action-angle coordinates, which, to lowest order, treat the close pair as a
single vortex amongst all others with an internal degree of freedom. We next,
and most importantly, apply Lie transform perturbation theory to remove the
higher-order correction terms in succession. The overall transformation
drastically increases the numerical efficiency and ensures that the total
energy remains constant to high accuracy.
