  The nonlocal Allen-Cahn (NAC) equation is a generalization of the classic
Allen-Cahn equation by replacing the Laplacian with a parameterized nonlocal
diffusion operator, and satisfies the maximum principle as its local
counterpart. In this paper, we develop and analyze first and second order
exponential time differencing (ETD) schemes for solving the NAC equation, which
unconditionally preserve the discrete maximum principle. The fully discrete
numerical schemes are obtained by applying the stabilized ETD approximations
for time integration with the quadrature-based finite difference discretization
in space. We derive their respective optimal maximum-norm error estimates and
further show that the proposed schemes are asymptotically compatible, i.e., the
approximate solutions always converge to the classic Allen-Cahn solution when
the horizon, the spatial mesh size and the time step size go to zero. We also
prove that the schemes are energy stable in the discrete sense. Various
experiments are performed to verify these theoretical results and to
investigate numerically the relation between the discontinuities and the
nonlocal parameters.
