# A high-order discrete energy decay and maximum-principle preserving scheme for time fractional Allen-Cahn equation

## TL;DR

This paper introduces a novel finite difference scheme using the shifted fractional trapezoidal rule for the time-fractional Allen-Cahn equation, ensuring energy decay and maximum-principle preservation, with demonstrated accuracy and efficiency.

## Contribution

The paper develops a new SFTR-based scheme that preserves key physical properties and effectively handles initial singularities in the time-fractional Allen-Cahn equation.

## Key findings

- The scheme guarantees discrete energy decay.
- It preserves the maximum principle.
- Numerical results outperform classical methods.

## Abstract

The shifted fractional trapezoidal rule (SFTR) with a special shift is adopted to construct a finite difference scheme for the time-fractional Allen-Cahn (tFAC) equation. Some essential key properties of the weights of SFTR are explored for the first time. Based on these properties, we rigorously demonstrate the discrete energy decay property and maximum-principle preservation for the scheme. Numerical investigations show that the scheme can resolve the intrinsic initial singularity of such nonlinear fractional equations as tFAC equation on uniform meshes without any correction. Comparison with the classic fractional BDF2 and L2-1$_\sigma$ method further validates the superiority of SFTR in solving the tFAC equation. Experiments concerning both discrete energy decay and discrete maximum-principle also verify the correctness of the theoretical results.