High-order nonuniform time-stepping and MBP-preserving linear schemes for the time-fractional Allen-Cahn equation
This work provides provably stable and MBP-preserving numerical methods for solving time-fractional phase-field models, which is important for accurate long-time simulations in materials science.
The authors developed high-order linear stabilized schemes for the time-fractional Allen-Cahn equation that preserve discrete energy stability and the maximum-bound principle (MBP). The L1 scheme unconditionally preserves MBP, while the L2-$1_σ$ scheme requires a mild time-step restriction, and an improved version enhances MBP preservation for large time steps.
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-$1_σ$ schemes. We show that the L1 scheme unconditionally preserves the discrete MBP, whereas the L2-$1_σ$ scheme requires a mild time-step restriction. Furthermore, we develop an improved L2-$1_σ$ 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.