A variable time-step, second-order, and MBP-preserving linear stabilized scheme for the time-fractional Allen-Cahn equation

Bingyin Zhang; Ao Zhang; Hongfei Fu

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arXiv · 2606.12799

A variable time-step, second-order, and MBP-preserving linear stabilized scheme for the time-fractional Allen-Cahn equation

Bingyin Zhang·Ao Zhang·Hongfei Fu

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Abstract

In this paper, we present a second-order linear scheme based on the variable-step Alikhanov formula and central difference discretization for the time-fractional Allen-Cahn equation. The nonlinear potential is treated explicitly via a second-order extrapolation with preprocessing, which enables the discrete maximum-bound principle (MBP) to be preserved through an appropriate stabilization technique. Moreover, by developing a discrete fractional Grönwall inequality together with the uniform boundedness of numerical solutions guaranteed by the MBP, we establish an $α$-robust and optimal second-order maximum-norm error estimate under initial weak singularity assumption. In addition, energy stability is proved in the sense that the discrete original energy is uniformly bounded by the initial energy plus a high-order spatiotemporal correction term. Finally, extensive numerical experiments are presented to demonstrate the effectiveness of the proposed scheme.

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Bingyin Zhang, Ao Zhang, Hongfei Fu. 2026-06-11. A variable time-step, second-order, and MBP-preserving linear stabilized scheme for the time-fractional Allen-Cahn equation. https://arxiv.org/abs/2606.12799

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