An Adaptive Time-Stepping Algorithm for the Allen–Cahn Equation. (16th July 2022) - Lee, Chaeyoung; Park, Jintae; Kwak, Soobin; Kim, Sangkwon; Choi, Yongho; Ham, Seokjun; Kim, Junseok
Abstract : In this paper, we present a simple and accurate adaptive time-stepping algorithm for the Allen–Cahn (AC) equation. The AC equation is a nonlinear partial differential equation, which was first proposed by Allen and Cahn for antiphase boundary motion and antiphase domain coarsening. The mathematical equation is a building block for modelling many interesting interfacial phenomena such as dendritic crystal growth, multiphase fluid flows, and motion by mean curvature. The proposed adaptive time-stepping algorithm is based on the Runge–Kutta–Fehlberg method, where the local truncation error is estimated by using fourth- and fifth-order numerical schemes. Computational experiments demonstrate that the proposed time-stepping technique is efficient in multiscale computations, i.e., both the fast and slow dynamics.