# High-order nonuniform time-stepping and MBP-preserving linear schemes for the time-fractional Allen-Cahn equation

## TL;DR

This paper introduces high-order, nonuniform time-stepping linear schemes for the time-fractional Allen-Cahn equation that preserve energy stability and the maximum-bound principle, with improved MBP preservation for large time steps.

## Contribution

The paper develops a new prediction strategy and auxiliary functional to design unconditionally energy-stable, MBP-preserving linear schemes with enhanced large-time-step MBP preservation.

## Key findings

- The L1 scheme unconditionally preserves the discrete MBP.
- The L2-1σ scheme requires a mild time-step restriction.
- An improved L2-1σ scheme enhances MBP preservation for large time steps.

## Abstract

In this paper, we present a class of nonuniform time-stepping, high-order linear stabilized schemes that can preserve both the discrete energy stability and maximum-bound principle (MBP) for the time-fractional Allen-Cahn equation. To this end, we develop a new prediction strategy to obtain a second-order and MBP-preserving predicted solution, which is then used to handle the nonlinear potential explicitly. Additionally, we introduce an essential nonnegative auxiliary functional that enables the design of an appropriate stabilization term to dominate the predicted nonlinear potential, and thus to preserve the discrete MBP. Combining the newly developed prediction strategy and auxiliary functional, we propose two unconditionally energy-stable linear stabilized schemes, L1 and L2-$1_\sigma$ schemes. We show that the L1 scheme unconditionally preserves the discrete MBP, whereas the L2-$1_\sigma$ scheme requires a mild time-step restriction. Furthermore, we develop an improved L2-$1_\sigma$ scheme with enhanced MBP preservation for large time steps, achieved through a novel unbalanced stabilization term that leverages the boundedness and monotonicity of the auxiliary functional. Representative numerical examples validate the accuracy, effectiveness, and physics-preserving of the proposed methods.