Abstract

This paper will introduce the Fourier spectral method combined with the strongly stable exponential time difference method as an attractive and easy-to-implement alternative for the integration of multi-dimensional Allen-Cahn equations with no-flux boundary conditions. The main advantages of the proposed method are that it utilizes the discrete fast Fourier transform which ensures efficiency, allows an extension to two and three spatial dimensions in a similar fashion as one-dimensional problem, and deal with various boundary conditions. Several numerical experiments are carried out on multi-dimensional Allen-Cahn equations to demonstrate the fourth-order temporal accuracy and stability of the method. The numerical results showed that the proposed method is fourth-order accurate in the time direction and is able to satisfy the discrete energy law.