Much effort, in particle physics, goes into investigating whether we can observe phase transitions that occurred but nanoseconds past the Big Bang. These phase transitions, if they are first-order, involve extremely high temperatures and can generate gravitational waves through bubble-nucleation. Yet it is hard to study such phase transitions, in quantum field theory, analytically. And robust predictions require numerical techniques. Luckily a lot of information can be obtained through the partition function, which in turn can be calculated numerically by discretizing the fields on a lattice and performing the large number of resulting integrals with Monte-Carlo techniques. In this project I want to perform such simulations for the Abelian-Higgs model; meaning that I want to calculate the free energy for a charged spin-0 particle interacting with a photon.These calculations are valuable for several reasons. First, these models show up (in one form or another) in many new-physics scenarios. Second, there exist only a few simulations of first-order phase transitions in the Abelian-Higgs model, and most of the phase diagram remains unexplored. As such, I want to fully explore the phase-diagram of the abelian Higgs model; determine whether a first-order phase transition takes place (for given parameter values), and if so, calculate the released latent heat. The results from these simulations can not only be used in phenomenological studies, but will also provide an essential test of pertrubative calculations.