Energy stability of variable-step L1-type schemes for time-fractional Cahn-Hilliard model

TL;DR

This paper develops energy-stable variable-step L1-type schemes for the time-fractional Cahn-Hilliard model, ensuring numerical stability and energy dissipation, with proven theoretical properties and validated by numerical experiments.

Contribution

It introduces a novel technique to estimate eigenvalues of discrete kernels and establishes energy dissipation laws for variable-step L1 schemes in fractional phase-field models.

Findings

Discrete kernels are positive definite, ensuring stability.

Energy dissipation laws are established and verified.

Numerical examples demonstrate effectiveness and adaptability.

Abstract

The positive definiteness of discrete time-fractional derivatives is fundamental to the numerical stability (in the energy sense) for time-fractional phase-field models. A novel technique is proposed to estimate the minimum eigenvalue of discrete convolution kernels generated by the nonuniform L1, half-grid based L1 and time-averaged L1 formulas of the fractional Caputo's derivative. The main discrete tools are the discrete orthogonal convolution kernels and discrete complementary convolution kernels. Certain variational energy dissipation laws at discrete levels of the variable-step L1-type methods are then established for time-fractional Cahn-Hilliard model.They are shown to be asymptotically compatible, in the fractional order limit $\alpha\rightarrow1$, with the associated energy dissipation law for the classical Cahn-Hilliard equation. Numerical examples together with an adaptive time-stepping procedure are provided to demonstrate the effectiveness of the proposed methods.