| The temporal fractional Cahn-Hilliard(TF-CH)equation is often used to describe the phase separation phenomenon with non-local memory effect.It is difficult to find TF-CH equation’s exact solution due to its complexity,such as the multiple spatial and temporal scales as well as the spatial fourth-order derivative term.Therefore,it is of great practical significance to find an effective numerical method to solve the time-fractional Cahn-Hilliard equation.Up to now,many numerical algorithms for the TF-CH equation,such as finite difference method and mixed finite element method,etc.,have been presented,but most of them are mesh-based.Compared with the traditional grid method,the pure meshless method can distribute particles arbitrarily in the support domain,is not limited by the grid,and has strong adaptability.Among grid-free particle methods,the finite pointset method(FPM)based on Taylor expansion and moving least squares is very popular,and has attracted widespread attention due to its mesh-free and flexibility to be applied to the problems in complex high-dimensional domains.FPM performs well for free surface flow problems and those with various computational domains.Moreover,it has also been applied to the numerical solution of partial differential equations.However,it is rare for FPM to be applied to solving the time-fractional Cahn-Hilliard equation.Based on the aforementioned analysis,this paper proposes an efficient method by combining the double finite point set method(TFPM)with the Caputo fractional time difference scheme to numerically study the fractional time Cahn-Hilliard equation.Moreover,this method is extended to the phase separation process under the non-local memory dominated by the two-component TF-CH equation for the first time.This work is organized as follows.(1)For the fractional time Cahn-Hilliard(TF-CH)equation,a pure meshless TFPM method that can accurately solve the TF-CH equation is presented for the first time.the fourth-order spatial derivative is discretized twice by FPM based on Taylor expansion and weighted least squares method,and the Caputo fractional time derivative is approached by high-precision different scheme.Thus,,and the Neumann boundary conditions can be precisely imposed in this discretization process.The error and numerical convergence of the proposed TFPM method are tested by numerically solving the two-dimensional TF-CH equation with analytical solutions.The numerical results show that the proposed TFPM method has a second-order convergence rate.(2)The proposed TFPM method is applied to simulate the single component TF-CH equation.The influence of the parameters on the phase separation phenomenon is discussed in the simulation.In the meantime,the simulation in the irregular region is also investigated.The numerical results indicate that the proposed TFPM method is flexible,and can effectively predict the evolution process of phase separation.(3)The proposed method is further applied to predict the phase separation process governed by the two-component TF-CH equation,and the phase separation evolution process under different time fractional orders is also discussed.All numerical results show that the proposed method can accurately predict the phase separation evolution. |
