# Simple maximum-principle preserving time-stepping methods for time-fractional Allen-Cahn equation

## TL;DR

This paper introduces fast, adaptive time-stepping methods for the time-fractional Allen-Cahn equation that preserve the maximum principle and are supported by rigorous error analysis and numerical validation.

## Contribution

The paper develops and analyzes fast L1 time-stepping schemes with adaptive meshes for the fractional Allen-Cahn equation, ensuring maximum principle preservation and providing sharp error estimates.

## Key findings

- The proposed schemes preserve the discrete maximum principle.
- Error estimates reflect the solution's time regularity.
- Numerical experiments confirm effectiveness and theoretical analysis.

## Abstract

Two fast L1 time-stepping methods, including the backward Euler and stabilized semi-implicit schemes, are suggested for the time-fractional Allen-Cahn equation with Caputo's derivative. The time mesh is refined near the initial time to resolve the intrinsically initial singularity of solution, and unequal time-steps are always incorporated into our approaches so that an adaptive time-stepping strategy can be used in long-time simulations. It is shown that the proposed schemes using the fast L1 formula preserve the discrete maximum principle. Sharp error estimates reflecting the time regularity of solution are established by applying the discrete fractional Gr\"{o}nwall inequality and global consistency analysis. Numerical experiments are presented to show the effectiveness of our methods and to confirm our analysis.