A Relaxed Step-Ratio Constraint for Time-Fractional Cahn--Hilliard Equations: Analysis and Computation

TL;DR

This paper introduces a relaxed step-ratio constraint for time-fractional Cahn-Hilliard equations, enabling more flexible time-step choices, and develops a high-order numerical scheme with proven stability, convergence, and physical property preservation.

Contribution

It advances the theoretical framework by relaxing the step-size ratio restriction and proposes a high-order, energy-stable numerical scheme with practical mesh construction.

Findings

Relaxed the step-size ratio constraint to over 4.7476.

Established the scheme's stability, convergence, and energy dissipation.

Designed a nonuniform mesh satisfying the new constraint.

Abstract

Numerical solutions of time-fractional differential equations encounter significant challenges arising from solution singularities at the initial time. To address this issue, the construction of nonuniform temporal meshes satisfying $\tau_k/\tau_{k-1} \geq 1$ has emerged as an effective strategy, where $\tau_k$ represents the $k$-th time-step size. For the time-fractional Cahn-Hilliard equation, Liao et al.~[\textit{IMA J. Numer. Anal.}, \textbf{45} (2025), 1425--1454] developed an analytical framework using a variable-step L2 formula with the constraint $0.3960 \leq \tau_k/\tau_{k-1} \leq r^*(\alpha)$, where $r^*(\alpha) \geq 4.660$ for $\alpha \in (0,1)$. The present work makes substantial theoretical progress by introducing innovative splitting techniques that relax the step-size ratio restriction to $\tau_k/\tau_{k-1} \leq \rho^*(\alpha)$, with $\rho^*(\alpha) > \overline{\rho} \approx 4.7476114$. This advancement provides significantly greater flexibility in time-step selection. Building on this theoretical foundation, we propose a refined L2-type temporal approximation coupled with a fourth-order compact difference spatial discretization, yielding an efficient numerical scheme for the time-fractional Cahn-Hilliard problem. Our rigorous analysis establishes the scheme's fundamental properties, including unique solvability, exact discrete volume conservation, proper energy dissipation laws, and optimal convergence rates. For practical implementation, we construct a specialized nonuniform mesh that automatically satisfies the relaxed constraint $\rho^*(\alpha) > 4.7476114$.