The Compact Scheme Of Time-Fractional Allen-Cahn Equation And Its Fast Algorithms

In recent years,the research on fractional phase field model has received extensive attention.Due to the nonlocality of fractional operator,it can more accurately describe the phase transition mechanism,such as continuous time random walk particle motion,long-distance interaction between particles,and the evolution process of structural damage and fatigue.In this work,we introduce a compact finite difference scheme for the time-fractional Allen-Cahn equation under nonuniform time grids.And its convergence and stability are analyzed.Moreover,we propose three kinds of acceleration methods for large time simulations.The model is as follows:(?)where Ω=(0,L)2 with closure Ω,the nonlinear bulk force f(u)=u3-u,the constant 0<ε(?)1 is the interaction length describing the thickness of the transition boundary between materials,and ?tα:=0CDtα represents the Caputo derivative of the order α:(?) The solution of the time-fractional Allen-Cahn equation has an initial singularity,and its numerical solution admits multiple time scales,i.e.,the initial dynamics evolve on a fast time scale,while the subsequent coarsening phase develops at a very slow time scale.Therefore,it is reasonable and necessary to use non-uniform time steps such as adaptive time grids.Due to the historical dependence of the time fractional derivative,when we consider using the finite difference method to solve the numerical solution of the equation at a certain time step,the function values of all previous time steps need to be used,resulting in a huge computational workload.Since fractional differential equations can be obtained by using a power-law memory kernel with non-local relations in time or space,we consider approximating the kernel function in a certain time period,that is,using the Sum-of-Exponentials(SOE)technique to solve the time-fractional Allen-Cahn equation.When we solve the numerical solution of a certain time step,we only need to use the function values of the previous part of the time steps,thereby reducing the amount of calculation to save calculation time.In addition,an adaptive timestepping strategy and a method of reducing degrees of freedom can also be considered to speed up the calculation process of the equation.In this paper,the convergence and stability of the compact finite-difference scheme for the time-fractional Allen-Cahn equation are mainly studied,and the acceleration method of the model is discussed.The full text is divided into five chapters:Chapter 1:The background of the time-fractional Allen-Cahn equation,the model described as above and domestic and foreign research literature review are introduced.Chapter 2:The Alikhanov-type compact finite-difference scheme under non-uniform time grids is proposed,and the existence and uniqueness of the solution is proved.Chapter 3:The convergence and stability of the compact finite difference scheme for the time-fractional Allen-Cahn equation are analyzed using the convolution structure.Chapter 4:Combining the difference scheme,three types of acceleration methods are derived,and the rationality of the theoretical results is verified by numerical experiments.Chapter 5:A summary and outlook of this paper are given.