Allen-Cahn方程是一个应用广泛的相场方程. 本文主要应用算子分裂法数值求解AllenCahn方程. 首先文章介绍了Allen-Cahn方程的当下的研究进展以及Allen-Cahn方程的算子分裂格式. 然后针对算子分裂后出现热方程, 我们分别用向前欧拉差分离散, Crank-Nicolson差分离散和谱方法数值求解, 给出了完整的一阶算子裂格式. 借助数值结果, 分析得到了一阶算子分裂法在时间上具有一阶精度, 二阶算子分裂法在时间上具有二阶精度. 向前欧拉差分离散和Crank-Nicolson差分离散在空间上具有二阶精度, 谱方法在空间上具有谱精度. 这三种不的一阶算子分裂法在数值上都观察到了能量递减的性质. 向前欧拉差分离散和Crank-Nicolson差分离散保持最大模上界原理, 但是谱方法不保持最大模上界原理. 最后我们利用一阶算子分裂法数值模拟了二维圆形反相界面的收缩以及亚稳态分解现象.

Allen-Cahn equation is a wide-used phase field equation. In this article, the operator splitting scheme will be applied to solve Allen-Cahn equation numerically. First, we will review the current research progress of Allen-Cahn equation and introduce the first order operator splitting scheme of Allen-Cahn equation. Then we will use forward Euler scheme, Crank-Nicolson scheme and spectrum method to solve the heat equation numerically which is one of subequations in operator splitting scheme. Using numerical results, we found that the first order operator splitting scheme has first order accuracy in time and the second order operator splitting scheme has second order accuracy in time. The spatial accuracy of forward Euler scheme and Crank-Nicolson scheme is second order, and the spectrum scheme has better saptial accuracy, which is known as ’spectral accuracy’. These three different first order operator splitting schemes all have the properties of decreasing energy. Besides, forward Euler scheme and CrankNicolson scheme satisfy the maximum bound principle. But the spectrum method does not satisfy the maximum bound principle. Finally, we simulate the motion of a circle by its mean curvature and spinodal decomposition in 2dimension.