  We develop an efficient numerical scheme for the 3D mean-field spherical
dynamo equation. The scheme is based on a semi-implicit discretization in time
and a spectral method in space based on the divergence-free spherical harmonic
functions. A special semi-implicit approach is proposed such that at each time
step one only needs to solve a linear system with constant coefficients. Then,
using expansion in divergence-free spherical harmonic functions in the
transverse directions allows us to reduce the linear system at each time step
to a sequence of one-dimensional equations in the radial direction, which can
then be efficiently solved by using a spectral-element method. We show that the
solution of fully discretized scheme remains bounded independent of the number
of unknowns, and present numerical results to validate our scheme.
