Enhancing numerical performance by enforcing discrete maximum principle with stabilized term for the Allen–Cahn equation with high-order polynomial free energy

In this paper, we investigate the numerical solution of the Allen–Cahn equation with high-order polynomial free energy using the finite difference method. Due to the inherent integral nonlinearity, numerical schemes typically become nonlinear, resulting in increased computational costs. To overcome this challenge, we employ the Crank–Nicolson/Adams–Bashforth technique to develop a linear scheme. We introduce an effective second-order difference scheme for solving both 1D and 2D problems. Our proposed scheme maintains the well-known maximum-principle property of the equation in a discrete sense through the use of a stabilized term. We provide discrete maximum error estimate demonstrating second-order accuracy in both time and space. To validate our approach, we conduct numerical experiments to empirically support its theoretical performance. Furthermore, we utilize our scheme to explore the impact of high-order polynomial free energy compared to the classical Allen–Cahn equation.