Numerical Algorithm Of Fractional Phase Field Model

Phase field model is an important tool to simulate interface problems.It is widely employed in mechanics,biology,materials,engineering and so on.The fractional phase field model is non-local,which is difficult to be solved accurately.Therefore,its numerical simulation and theoretical analysis are of great significance.In this paper,we will study three kinds of fractional phase field models,namely the time fractional Allen-Chan equation,the space fractional Swift-Hohenberg equation and the fractional Cahn-Hilliard equation,and the numerical schemes for these three models are given.Firstly,we consider the time fractional Allen-Cahn equation.It is transformed into integer order equation by the Laplace transform.A precise and effective numerical scheme can be obtained by using center difference method in space and Crank-Nicolson scheme in time.At the same time,the validity of the numerical scheme and the energy dissipation property are verified by numerical experiments.Secondly,the spatial fractional Swift-Hoenberg equation is considered.Based on the operator splitting method,the finite difference method and the Fourier spectral method,a first-order numerical scheme is obtained,and the stability and convergence analysis of the scheme are given.Then,combined with the spectral deferred correction method,an efficient numerical scheme with high accuracy in both time and space is obtained.Besides,numerical experiments show that the algorithm is efficient and satisfies the energy dissipation property.Finally,the time-space fractional Cahn-Hilliard equation is considered.The time-space fractional Cahn-Hilliard equation is converted into the space fractional Cahn-Hilliard equation by the Laplace transform.Then,using the Fourier spectral method in space and the Crank-Nicolson scheme in time,an efficient numerical scheme with the second-order convergence is obtained.Meanwhile,the validity of the proposed algorithm is verified by numerical experiments,and it satisfies the energy dissipation law and the law of conservation of mass.