A review of the numerical methods for solving the binary Allen–Cahn equation

Abstract

In this review, we present an overview of numerical methods to solve the binary Allen–Cahn (AC) equation, which is extensively used to model phase separation processes in materials science. It describes the time-dependent evolution of interfaces between two phases and accounts for both local reaction kinetics and diffusion effects. This equation plays a critical role in understanding the behavior of interfaces. The AC equation has various applications across fields such as materials science, physics, and biology, where it helps to analyze and predict phenomena such as phase transitions, grain boundary motion, and pattern formation in complex systems. Its importance lies in its ability to model the dynamics of interfaces and help the study of pattern formation and phase transitions in diverse environments. We discuss various computational methodologies developed for this important mathematical model and describe their strengths, limitations, and applications in diverse scientific domains. © 2025 Elsevier B.V.