A linear unconditionally structure-preserving L1 scheme for the time-fractional Allen-Cahn equation

TL;DR

Addressing the numerical challenge of the time-fractional Allen-Cahn equation with memory effects, the authors develop linear, structure-preserving L1-based time-stepping schemes (including a first-order and a min{$1+\alpha$, $2-\alpha$}-order predictor–corrector) with fast SOE acceleration. They prove unconditional MBP preservation and discrete energy dissipation on general time grids, derive sharp error estimates using an extended fractional Grönwall inequality for $\theta=1$, and validate the methods through adaptive-time-stepping simulations and long-time coarsening dynamics. The work delivers an efficient, stable, nonlinear-solve-free framework for TFAC that supports graded/adaptive time grids and high reliability in thermodynamically consistent simulations.

Abstract

As a variational phase-field model, the time-fractional Allen-Cahn (TFAC) equation enjoys the maximum bound principle (MBP) and a variational energy dissipation law. In this work, we develop and analyze linear, structure-preserving time-stepping schemes for TFAC, including first-order and $\min\{1+α, 2-α\}$-order L1 discretizations, together with fast implementations based on the sum-of-exponentials (SOE) technique. A central feature of the proposed linear schemes is their unconditional preservation of both the discrete MBP and the variational energy dissipation law on general temporal meshes, including graded meshes commonly used for these problems. Leveraging the MBP of the numerical solutions, we establish sharp error estimates by employing the time-fractional Gronwall inequality. Finally, numerical experiments validate the theoretical results and demonstrate the effectiveness of the proposed schemes with an adaptive time-stepping strategy.

Figures (4)

• Figure 1: The error behavior with respect to the time step size for the first order scheme \ref{['scheme1']} and the predictor-corrector L1 scheme \ref{['fL1_h']}.
• Figure 2: The evolutions in time of the supremum norm (left) and the variant energy (right) of the simulated solution produced by the proposed fast linear predictor--corrector stabilized L1 scheme \ref{['fL1_h']} with some uniform time steps for the grain coarsening problem.
• Figure 3: Snapshots of the iso-surfaces of the phase function $\phi=0$ (the first and third columns) and density field $\phi$ (the second and last columns) for the 3D model \ref{['TFAC']} with $\alpha=0.3$ (the first two columns) and $\alpha=0.9$ (the last two columns) around the times $t=1$, $10$, $20$, $50$ and $100$.
• Figure 4: The evolutions in time of the supremum norm (left) and the energy (right) of the simulated solution produced by the fast linear predictor--corrector stabilized L1 scheme \ref{['fL1_h']}.

Theorems & Definitions (19)

• Lemma 2.1: LSR19WW88
• Lemma 2.2
• Lemma 2.3: HTY17LTZ20TY16
• Lemma 2.4: HJQ22TY16
• Theorem 2.5
• Proof 1
• Theorem 2.6
• Proof 2
• Lemma 2.7
• Theorem 2.8