  The nonlocal Allen-Cahn equation with nonlocal diffusion operator is a
generalization of the classical Allen-Cahn equation. It satisfies the energy
dissipation law and maximum bound principle (MBP), and is important for
simulating a series of physical and biological phenomena involving
long-distance interactions in space. In this paper, we construct first- and
second-order (in time) accurate, unconditionally energy stable and
MBP-preserving schemes for the nonlocal Allen-Cahn type model based on the
stabilized exponential scalar auxiliary variable (sESAV) approach. On the one
hand, we have proved the MBP and unconditional energy stability carefully and
rigorously in the fully discrete levels. On the other hand, we adopt an
efficient FFT-based fast solver to compute the nearly full coefficient matrix
generated from the spatial discretization, which improves the computational
efficiency. Finally, typical numerical experiments are presented to demonstrate
the performance of our proposed schemes.
