A second-order Strang splitting scheme with exponential integrating factor for the Allen–Cahn equation with logarithmic Flory–Huggins potential

Abstract

In this paper, we mainly consider the numerical approximation for the Allen-Cahn(AC) equation with logarithmic Flory–Huggins potential. It is well-known that the logarithmic Flory–Huggins potential is more physically significant than the generally employed fourth-order polynomial potential in various phase-field models. Nevertheless, owing to the singularity of the logarithmic term and the strong nonlinearity of the energy potential function, it is extremely challenging to develop an effective and easy-to-implement numerical scheme that preserves the maximum principle and energy dissipation law. To overcome these difficulties, we construct a second-order Strang splitting scheme with exponential integrating factor. Firstly, base on the idea of dimensional splitting, a new time dependent function, which is called exponential integrating factor is introduced. Then we propose the Strang splitting approach which is aim to decompose the original equation into linear part and nonlinear part. In particular, the logarithmic term is treated by explicit 2-stage strong stability preserving Runge–Kutta(SSP-RK2) method. Furthermore, we rigorously demonstrate the maximum principle, energy stability and convergence in details for the proposed scheme. A variety of numerical simulations in 2D and 3D are presented to confirm the validity of the proposed method.