This paper deals with the discrete maximum principle and energy stability of a new difference scheme for solving Allen-Cahn equations. By combining the second-order central difference approximation in space and the Crank-Nicolson method with Newton linearized technique in time, a two-level linearized difference scheme for Allen-Cahn equations is derived, which can yield accuracy of order two both in time and space. Under appropriate conditions, the scheme is proved to be uniquely solvable and able to preserve the maximum principle and energy stability of the equations in the discrete sense. With some numerical experiments, the theoretical results and computational effectiveness of the scheme are further illustrated.

本文讨论了求解Allen-Cahn方程的新差分格式的离散最大原理和能量稳定性。通过将空间的二阶中心差分近似和Crank-Nicolson方法与牛顿线性化技术在时间上相结合，得出了Allen-Cahn方程的两级线性化差分方案，该方案可以在时间和时间上获得二阶精度。空间。在适当的条件下，该方案被证明是唯一可解的，并且能够在离散意义上保留方程的最大原理和能量稳定性。通过一些数值实验，进一步说明了该方案的理论结果和计算有效性。