We present natural axisymmetric variants of schemes for curvature flows
introduced earlier by the present authors and analyze them in detail. Although
numerical methods for geometric flows have been used frequently in axisymmetric
settings, numerical analysis results so far are rare. In this paper, we present
stability, equidistribution, existence and uniqueness results for the
introduced approximations. Numerical computations show that these schemes are
very efficient in computing numerical solutions of geometric flows as only a
spatially one-dimensional problem has to be solved. The good mesh properties of
the schemes also allow them to compute in very complex axisymmetric geometries.