  We consider a finite difference approximation of mean curvature flow for
axisymmetric surfaces of genus zero. A careful treatment of the degeneracy at
the axis of rotation for the one dimensional partial differential equation for
a parameterization of the generating curve allows us to prove error bounds with
respect to discrete $L^2$- and $H^1$-norms for a fully discrete approximation.
The theoretical results are confirmed with the help of numerical convergence
experiments. We also present numerical simulations for some genus-0 surfaces,
including for a non-embedded self-shrinker for mean curvature flow.
