  Allen--Cahn equation with constant and degenerate mobility, and with
polynomial and logarithmic energy functionals is discretized using symmetric
interior penalty discontinuous Galerkin (SIPG) finite elements in space. We
show that the energy stable average vector field (AVF) method as the time
integrator for gradient systems like the Allen-Cahn equation satisfies the
energy decreasing property for the fully discrete scheme. The numerical results
for one and two dimensional Allen-Cahn equation with periodic boundary
condition, using adaptive time stepping, reveal that the discrete energy
decreases monotonically, the phase separation and metastability phenomena can
be observed and the ripening time is detected correctly.
