The space nonlocal Allen-Cahn equation is a famous example of fractional reaction-diffusion equations. It is also an extension of the classical Allen-Cahn equation, which is widely used in physics to describe the phenomenon of two-phase fluid flows.Due to the nonlocality of the nonlocal operator, numerical solutions to these equations face considerable challenges.It is worth noting that whether we use low-order or high-order numerical differential formulas to approximate the operator, the corresponding matrix is always dense, which implies that the storage space and computational cost required for the former and the latter are the same. However, the higher-order formula can significantly improve the accuracy of the numerical scheme.Therefore, the primary goal of this paper is to construct a high-order numerical formula that approximates the nonlocal operator.To reduce the time step limitation in existing numerical algorithms, we employed a technique combining the compact integration factor method with the Pad\'{e} approximation strategy to discretize the time derivative.A novel high-order numerical scheme, which satisfies both the maximum principle and energy stability for the space nonlocal Allen-Cahn equation, is proposed.Furthermore, we provide a detailed error analysis of the differential scheme, which shows that its convergence order is $\mathcal{O}\left(\tau^2+h^6\right)$.Especially, it is worth mentioning that the fully implicit scheme with sixth-order accuracy in spatial has never been proven to maintain the maximum principle and energy stability before.Finally, some numerical experiments are carried out to demonstrate the efficiency of the proposed method.

翻译：空间非局部Allen-Cahn方程是分数阶反应-扩散方程的典型代表，也是经典Allen-Cahn方程的推广形式，在物理学中广泛用于描述两相流体流动现象。由于非局部算子的非局部特性，这类方程的数值求解面临显著挑战。值得注意的是，无论采用低阶还是高阶数值微分公式逼近该算子，对应的矩阵始终是稠密的，这意味着两者所需的存储空间和计算成本相同。然而，高阶公式能显著提升数值格式的精度。因此，本文的主要目标是构建逼近非局部算子的高阶数值公式。为降低现有数值算法中的时间步长限制，我们采用紧致积分因子法与Padé逼近策略相结合的技术对时间导数进行离散化。针对空间非局部Allen-Cahn方程，本文提出了一种满足最大值原理与能量稳定性的新型高阶数值格式。此外，我们对差分格式进行了详细的误差分析，证明其收敛阶为$\mathcal{O}\left(\tau^2+h^6\right)$。特别需要指出的是，具有六阶空间精度的全隐式格式此前从未被证明能同时保持最大值原理与能量稳定性。最后，通过数值实验验证了所提方法的有效性。