An efficient pure meshless method for phase separation dominated by time-fractional Cahn-Hilliard equations

In this paper, an efficient Twice Finite Point-set Method (TFPM) coupled with difference scheme is proposed to solve the time-fractional Cahn-Hilliard (TF-CH) equation, and then it is extended to predict the phase separation process under nonlocal memory dominated by two-component TF-CH equations for the first time. The proposed meshless schemes are motivated by the following: (a) a high-order accuracy difference scheme is employed to approximate the time Caputo fractional derivative; (b) the fourth-order spatial derivative is divided into two second-order derivatives, and it is discretized by the FPM scheme continuously twice based on Taylor expansion and weighted least squares; (c) the Neumann boundary can be accurately imposed on the FPM scheme. In the numerical experiments, the error and numerical convergence of the proposed meshless method are first tested, which has near second-order convergent rate. Subsequently, the evolution of phase separation under memory dominated by single CH equation versus time is numerically investigated by the proposed method, and compared with the results in other literatures. The influence of fractional parameter on the separation phenomena is also discussed. Finally, the proposed method is used to predict the phase separation process under different parameters dominated by coupled TF-CH equations. All the numerical results show that the proposed coupled method is accurate in solving the TF-CH and efficient in predicting the phase separation evolution.