Higher-order Algorithm For Time-fractional Allen-Cahn And Cahn-Hilliard Equations

In this paper,we apply the time-fractional phase field models to describe two immiscible mixed flows,in order to explain some anomalous diffusion phenomena that are common in nature.We mainly study higher-efficiency and higher-order numerical algorithms for time-fractional Allen-Cahn and Cahn-Hilliard phase field models.We adopt fast algorithm that based on an efficient sum-of-exponentials approximation for the kernel t^-1-?to reduce the amount of storage caused by the historical memory of the fractional derivative.We use the stabilization method and the scalar auxiliary variable?SAV?strategy to construct efficient and higher-order numerical methods,which have achieved two results as follows:On the one hand,based on the fast L1 format,we apply stabilization method so that our numerical scheme can overcome the limitation of small step length in the time evolution of the phase field model,which makes it possible to simulate the phase field evolution with a relatively larger step.We constructed first-order,?2-??-order unconditional energy stable and efficient numerical algorithm for time-fractional phase field models.The energy dissipation in continuous and discrete state is proved.On the other hand,apply the SAV strategy with backward differentiation for-mula?BDF?.We base on the fast SAV/BDF1 and fast SAV/BDF2 scheme describes efficient numerical algorithms with first-order and?2-??-order accuracy,respec-tively.We proved the dissipation of discrete energy.Based on the fast SAV/BDF3scheme,higher-order numerical algorithms with?2+??-order??<0.5?and?3-??-order??>0.5?accuracy are described.Finally,some numerical experiments are carried out to further verify the higher-efficiency and higher-order numerical algorithm.