An unconditionally stable scheme for the Allen–Cahn equation with high-order polynomial free energy

Abstract

In this paper, we propose an unconditionally stable numerical scheme for the Allen–Cahn (AC) equation with high-order (higher than fourth) polynomial free energy. The AC equation was proposed by Allen and Cahn to model the anti-phase domain coarsening in a binary mixture. The AC equation has been extensively used as a building block equation for modeling many scientific problems such as image processing, dendritic growth, motion by mean curvature, and multi-phase fluid flows. The AC equation can be derived from a gradient flow of a total energy functional which consists of a double-well form potential and a gradient term. Practically, a quartic polynomial has been used for the double-well potential. High-order (greater than fourth) polynomial free energy potentials can be also used in the total energy functional and can better represent interfacial dynamics of the AC equation. However, the AC equation with the high-order polynomial is getting stiffer as the polynomial order increases. Typically, this type of double-well potential is solved using a convex splitting with a stabilizing parameter and effectively modifies the original governing equation.In the proposed method, we use a second-order operator splitting method and an interpolation method. First, we solve the nonlinear double-well potential term using interpolation from the pre-computed values. Second, we solve the diffusion equation using the Crank–Nicolson method and multigrid method. The overall scheme is unconditionally stable and we theoretically prove the unconditional stability. Computational experiments are performed to demonstrate the robustness and accuracy of the proposed method; and investigate the effect of the order of the double-well potential on the dynamics of the AC equation. Finally, we highlight the different dynamics for the AC equation with polynomial free energy of various orders. The computational results suggest that the proposed method will be useful for modeling various interfacial phenomena.