A simple and effective numerical method is proposed for simulating the
temporal diffusive mass transport process through a microstructure
with arbitrary complexity described by a phase-field approach.  The
mass diffusion through a given microstructure is modeled by a
diffusion equation with a variable diffusion coefficient, which is
solved by an efficient and accurate semi-implicit spectral method.  It
is shown that it is possible to extract the effective diffusion
coefficient for any given microstructure from the temporal
concentration profiles. The method is used to simulate the grain
boundary diffusion in a single-phase polycrystalline grain structure
and the heterogeneous diffusion in a two-phase microstructure with
different diffusion coefficient in each phase.  Results are compared
with existing analytical theories and computer simulations.