This paper presents a convolution tensor decomposition based model reduction method for solving the Allen-Cahn equation. The Allen-Cahn equation is usually used to characterize phase separation or the motion of anti-phase boundaries in materials. Its solution is time-consuming when high-resolution meshes and large time scale integration are involved. To resolve these issues, the convolution tensor decomposition method is developed, in conjunction with a stabilized semi-implicit scheme for time integration. The development enables a powerful computational framework for high-resolution solutions of Allen-Cahn problems, and allows the use of relatively large time increments for time integration without violating the discrete energy law. To further improve the efficiency and robustness of the method, an adaptive algorithm is also proposed. Numerical examples have confirmed the efficiency of the method in both 2D and 3D problems. Orders-of-magnitude speedups were obtained with the method for high-resolution problems, compared to the finite element method. The proposed computational framework opens numerous opportunities for simulating complex microstructure formation in materials on large-volume high-resolution meshes at a deeply reduced computational cost.

翻译：本文提出了一种基于卷积张量分解的模型降阶方法，用于求解Allen-Cahn方程。Allen-Cahn方程通常用于描述材料中的相分离或反相边界运动。当涉及高分辨率网格和大时间尺度积分时，其求解过程非常耗时。为解决这些问题，本文发展了卷积张量分解方法，并结合用于时间积分的稳定化半隐式格式。该进展为Allen-Cahn问题的高分辨率求解构建了一个强大的计算框架，并允许在时间积分中使用相对较大的时间步长而不违反离散能量定律。为进一步提升方法的效率和鲁棒性，本文还提出了一种自适应算法。数值算例在二维和三维问题上均验证了该方法的有效性。与有限元方法相比，该方法在高分辨率问题上实现了数量级的速度提升。所提出的计算框架为在大体积高分辨率网格上以显著降低的计算成本模拟材料中复杂微观结构的形成开辟了广阔前景。