Two Finite Difference Schemes For Time-fractional Cahn-Hilliard Equations

Classical phase field models are widely used in materials,biology and other fields,but more detailed and global models are needed in real life.With the help of global dependence embodied by fractional calculus,non-local models can be constructed to simulate complex physical phenomena,so the study of fractional phase field models becomes more and more important.In view of the representative position of Cahn-Hilliard equation in the phase field model,this paper chooses to study its time fractional form.Firstly,for the time-fractional Cahn-Hilliard equation with homogeneous Neumann boundary conditions,an implicit difference scheme with 2-α and 2 order accuracy in temporal and spatial is proposed via the L1 formula to approximate the Caputo fractional derivative in the equation.The stability and convergence of the implicit scheme are discussed through the discrete energy method.In order to compare and observe the evolution of solutions,a generalized difference scheme based on time grade grid is presented.The numerical results show that for α∈(0,1),the solutions of the time-fractional Cahn-Hilliard equation always tend to an equilibrium state with time.In order to preserve some of the original properties of the equation,we also consider the preserving structure algorithm of the time-fractional Cahn-Hilliard equation with periodic boundary conditions.A compact difference scheme with high-order accuracy is proposed by combining exponential scalar auxiliary variable method and BDF method based on the relationship between Caputo fractional derivative and Riemann-Liouville fractional derivative,and the application of the L1 time discrete formula and the fourth order compact difference method in space.The compact scheme is proved to be able to maintain energy dissipation similar to the original equation.The results of numerical examples also demonstrate the efficiency and energy dissipation of the high order compact scheme.