A Relaxed Step-Ratio Constraint for Time-Fractional Cahn--Hilliard Equations: Analysis and Computation
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| Format: | Preprint |
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2025
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| _version_ | 1866915460327407616 |
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| author | Li, Shipeng Ding, Hengfei |
| author_facet | Li, Shipeng Ding, Hengfei |
| contents | 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 $ρ^*(α) > 4.7476114$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_17178 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Relaxed Step-Ratio Constraint for Time-Fractional Cahn--Hilliard Equations: Analysis and Computation Li, Shipeng Ding, Hengfei Numerical Analysis 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 $ρ^*(α) > 4.7476114$. |
| title | A Relaxed Step-Ratio Constraint for Time-Fractional Cahn--Hilliard Equations: Analysis and Computation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2508.17178 |