  An implicit finite difference method with non-uniform timesteps for solving
the fractional diffusion equation in the Caputo form is proposed. The method
allows one to build adaptive methods where the size of the timesteps is
adjusted to the behaviour of the solution in order to keep the numerical errors
small without the penalty of a huge computational cost. The method is
unconditionally stable and convergent. In fact, it is shown that consistency
and stability implies convergence for a rather general class of fractional
finite difference methods to which the present method belongs. The huge
computational advantage of adaptive methods against fixed step methods for
fractional diffusion equations is illustrated by solving the problem of the
dispersion of a flux of subdiffusive particles stemming from a point source.
