  In this work, we present a second-order nonuniform time-stepping scheme for
the time-fractional Allen-Cahn equation. We show that the proposed scheme
preserves the discrete maximum principle, and by using the convolution
structure of consistency error, we present sharp maximum-norm error estimates
which reflect the temporal regularity. As our analysis is built on nonuniform
time steps, we may resolve the intrinsic initial singularity by using the
graded meshes. Moreover, we propose an adaptive time-stepping strategy for
large time simulations. Numerical experiments are presented to show the
effectiveness of the proposed scheme. This seems to be the first second-order
maximum principle preserving scheme for the time-fractional Allen-Cahn
equation.
