  A study of the diffusion of a passive Brownian particle on the surface of a
sphere and subject to the effects of an external potential, coupled linearly to
the probability density of the particle's position, is presented through a
numerical algorithm devised to simulate the trajectories of an ensemble of
Brownian particles. The algorithm is based on elementary geometry and
practically only algebraic operations are used, this makes the algorithm
efficient and simple, and converges, in the \textit{weak sense}, to the
solutions of the Smoluchowski equation on the sphere. Our findings show that
the global effects of curvature are taken into account in both, the time
dependent and stationary processes.
