  We have simulated the thermodynamics of vortices in a thin film of a type-II
superconductor. We make the lowest Landau level approximation, and use
quasi-periodic boundary conditions. Our work is consistent with the results of
previous simulations where evidence was found for an apparent first order
transition between the vortex liquid state and the vortex crystal state. We
show, however, that these results are just an artifact of studying systems
which are too small. There are substantial difficulties in simulating larger
systems using traditional approaches. By means of the optimal energy diffusion
algorithm we have been able to study systems containing up to about one
thousand vortices, and for these larger systems the evidence for a first order
transition disappears. By studying both crystalline and hexatic order, we show
that the KTHNY scenario seems to apply, where melting from the crystal is first
to the hexatic liquid state and next to the normal vortex liquid, in both cases
via a continuous transition.
