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.

翻译：非局部Allen-Cahn方程是非局部扩散算子对经典Allen-Cahn方程的推广。该方程满足能量耗散律和最大界原理，对模拟涉及空间长程相互作用的物理与生物现象具有重要意义。本文基于稳定化的指数标量辅助变量（sESAV）方法，为非局部Allen-Cahn型模型构建了时间方向一阶和二阶精度的无条件能量稳定且保持最大界原理的数值格式。一方面，我们在全离散层面上严格证明了最大界原理与无条件能量稳定性；另一方面，采用基于快速傅里叶变换的高效求解器处理空间离散产生的近乎满系数矩阵，从而提高了计算效率。最后，通过典型数值实验验证了所提出格式的性能。