  The present paper deals with the numerical solution of time-fractional
advection-diffusion equations involving the Caputo derivative with source term
by means of an unconditionally stable implicit finite difference method on
quasi-uniform grids. We use a special quasi-uniform mesh in order to improve
the numerical accuracy of the classical discrete fractional formula for the
Caputo derivative. The stability and the convergence of the method are
discussed. The error estimates established for a quasi-uniform grid and a
uniform one are reported to support the theoretical results. Numerical
experiments are carried out to demonstrate the effectiveness of the method.
