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

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 $τ_k/τ_{k-1} \geq 1$ has emerged as an effective strategy, where $τ_k$ 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 $0.3960 \leq τ_k/τ_{k-1} \leq r^*(α)$, where $r^*(α) \geq 4.660$ for $α\in (0,1)$. The present work makes substantial theoretical progress by introducing innovative splitting techniques that relax the step-size ratio restriction to $τ_k/τ_{k-1} \leq ρ^*(α)$, with $ρ^*(α) > \overlineρ \approx 4.7476114$. 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 $ρ^*(α) > 4.7476114$.