The efficiency of a wind turbine depends largely on the wake of the upstream turbine. Seeking to contribute towards the development of a wind farm solver using a Lagrangian scheme to analyze the wake, this thesis analyses and validates (using MATLAB) a Vortex Particle Method (VPM) algorithm by simulating the behavior of vortex rings. Due to the computationally expensive nature of VPM schemes to solve n-body problems, such as the simulation of vortex rings and wind turbine wake, a Fast Multipole Method (FMM) library called BBFMM3D has been studied and validated (using C++) for the acceleration of computation of matrix-vector products which are essential in solving n-body problems. The execution of the VPM algorithm has been shown to be possible to be accelerated using the validated FMM library by implementing the library in the VPM scheme to simulate the case of a single vortex ring. Using the VPM solver developed to simulate vortex rings, the simulation of a wake of a wind turbine, that was modeled using actuator lines, was performed (without FMM) and the results have been analyzed and attempted to be validated. It has been found that the VPM scheme generates less than accurate results of the velocity profile of the wake with respect to other CFD simulations and the model could not be validated due to the results not being accurate up to the mark. However, the accuracy of the results has been found to rely significantly on the formulation of the strengths of the vortices shed into the wake. Two formulations for this purpose have been presented with the results showing signs of improvement from one formulation to another.