  In this paper we describe a version of London Langevin molecular dynamics
simulations that allows for investigations of the vortex lattice melting
transition in the highly anisotropic high-temperature superconductor material
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$. We include the full electromagnetic
interaction as well as the Josephson interaction among pancake vortices. We
also implement periodic boundary conditions in all directions, including the
z-axis along which the magnetic field is applied. We show how to implement flux
cutting and reconnection as an analog to permutations in the multilevel Monte
Carlo scheme and demonstrate that this process leads to flux entanglement that
proliferates in the vortex liquid phase. The first-order melting transition of
the vortex lattice is observed to be in excellent agreement with previous
multilevel Monte Carlo simulations.
