Maximum bound principle preserving linear schemes for nonlocal Allen–Cahn equation based on the stabilized exponential-SAV approach

The nonlocal Allen–Cahn equation with nonlocal diffusion operator is important for simulating a series of physical and biological phenomena involving long-distance interactions in space. As a generalization of the classical Allen–Cahn equation, the nonlocal Allen–Cahn equation also satisfies the energy dissipation law and maximum bound principle. In this paper, we construct first- and second-order (in time) accurate linear schemes for the nonlocal Allen–Cahn type model based on the stabilized exponential scalar auxiliary variable approach which preserve discrete maximum bound principle and unconditionally energy stability. On the one hand, we prove the discrete maximum bound principle, unconditional energy stability and error convergence analysis 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 which generated from the spatial discretization, which improves the computational efficiency. Finally, typical numerical experiments are presented to demonstrate the performance of our proposed schemes.