  In this paper, we propose and analyze a time-stepping method for the time
fractional Allen-Cahn equation. The key property of the proposed method is its
unconditional stability for general meshes, including the graded mesh commonly
used for this type of equations. The unconditional stability is proved through
establishing a discrete nonlocal free energy dispassion law, which is also true
for the continuous problem. The main idea used in the analysis is to split the
time fractional derivative into two parts: a local part and a history part,
which are discretized by the well known L1, L1-CN, and $L1^{+}$-CN schemes.
Then an extended auxiliary variable approach is used to deal with the nonlinear
and history term. The main contributions of the paper are: First, it is found
that the time fractional Allen-Chan equation is a dissipative system related to
a nonlocal free energy. Second, we construct efficient time stepping schemes
satisfying the same dissipation law at the discrete level. In particular, we
prove that the proposed schemes are unconditionally stable for quite general
meshes. Finally, the efficiency of the proposed method is verified by a series
of numerical experiments.
