  Superconductivity occurs in the proximity of other competing orders in a wide
variety of materials. Such competing phases may reveal themselves when
superconductivity is locally suppressed by a magnetic field in the core of a
vortex. We explore the competition between superconductivity and charge density
wave order in the attractive Hubbard model on a square lattice. Using
Bogoliubov-deGennes mean field theory, we study how vortex structures form and
evolve as the magnetic flux is tuned. Each vortex seeds a CDW region whose size
is determined by the energy cost of the competing phase. The vortices form a
lattice whose lattice parameter shrinks with increasing flux. Eventually, their
charge-ordered vortex cores overlap, leading to a field-driven coexistence
phase exhibiting both macroscopic charge order and superconductivity -- a
`supersolid'. Ultimately, superconductivity disappears via a first-order phase
transition into a purely charge ordered state. We construct a phase diagram
containing these multiple ordered states, using $t'$, the next-nearest
neighbour hopping, to tune the competition between phases.
