The Barotropic Vorticity Equation describes the balance of vorticity and the Coriolis force for a fluid moving on a rotating sphere, and is important in geophysical fluid dynamics. In this talk, I will discuss some numerical considerations, first presenting some basics of geophysical fluid dynamics, before presenting a short derivation of the barotropic vorticity equation, and then moving on to discuss various numerical aspects of the problem. I will focus on a Lagrangian particle discretization, as well as a new fast summation technique based on a spherical tree code. I will also touch on the role of adaptive mesh refinement.