A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations

Abstract

In this work, we present a second-order nonuniform time-stepping scheme for the time-fractional Allen-Cahn equation. We show that the proposed scheme preserves the discrete maximum principle, and by using the convolution structure of consistency error, we present sharp maximum-norm error estimates which reflect the temporal regularity. As our analysis is built on nonuniform time steps, we may resolve the intrinsic initial singularity by using the graded meshes. Moreover, we propose an adaptive time-stepping strategy for large time simulations. Numerical experiments are presented to show the effectiveness of the proposed scheme. This seems to be the first second-order maximum principle preserving scheme for the time-fractional Allen-Cahn equation.

Numerical simulation for time‐fractional nonlinear reaction–diffusion system on a uniform and nonuniform time stepping

“…For the numerical methods dealing with the non‐integer‐order differential equations, many authors have developed numerical methods such as exponentially fitted methods and 32–35 high‐order difference schemes 36,37 . On the other hand, some numerical methods have been developed for nonuniform mesh such as the L1‐approximation scheme, 38,39 error analysis of time stepping with nonsmooth data, 40 and the finite difference with nonuniform time stepping (NUTS) 41 . Recently, previous studies 42–46 proposed a finite difference method with nonuniform time steps to solve diffusion, advection, and Allen–Cahn equations.…”

Simple Positivity Preserving Nonlinear Finite Volume Scheme for Subdiffusion Equations on General Non-conforming Distorted Meshes

“…Especially, for lumped mass method positivity is preserved if and only if the triangulation is of Delaunay type. Based on energy stable, Liao et al [15,16] considered the second-order and nonuniform adaptive time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations, where the convolution structure of consistency error is used and sharp maximum-norm error estimates with the temporal regularity is proved. Ji et al [17] provided the fast L1 formula preserving the discrete maximum principle for the time-fractional Allen-Cahn equation with Caputo's derivative, then extended to volume constraint problem [18].…”

A Crank–Nicolson Finite Volume Element Method for Time Fractional Sobolev Equations on Triangular Grids

“…Fractional partial differential equations (FPDEs) have received extensive attentions by more and more scholars, and have been widely applied in many fields of science and engineering [1,2]. Many practical problems can be portrayed very well by the some FPDEs, such as fractional (reaction) diffusion equations [3][4][5][6][7][8][9][10][11][12], fractional Allen-Cahn equations [13][14][15], fractional Cable equations [16][17][18], and fractional mobile/immobile transport equations [19][20][21]. In the past few decades, a large number of numerical methods [22][23][24] have been proposed and used to solve the FPDEs, which have achieved excellent theoretical and numerical results.…”