The main objective of this thesis is to describe the codes that we have developed in order to simulate the ground state of a spin one and spin two Bose-Einstein condensate as well as codes that we have developed to solve the Bogoliubov equations used to examine the elementary excitations of these ground states. Motivated by viewing non-abelian vortices within a spin two condensate as anyons that can be used for quantum computation, we also look at introducing vortices into our codes by means of suitable initial conditions. To do this, we have developed methods for setting up and looking at vortices in our ground state systems. Finally, we will put all our code together by numerically solving the Bogoliubov equations for a ground state system with a vortex present and provide analysis of these results.