Communications in Mathematical Sciences
Volume 21 (2023)
Energy stability of variable-step L1-type schemes for time-fractional Cahn–Hilliard model
Pages: 1767 – 1789
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 non-uniform 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 $\alpha \to 1$, with the associated energy dissipation law for the classical Cah–Hilliard equation. Numerical examples together with an adaptive time-stepping procedure are provided to demonstrate the effectiveness of the proposed methods.
time-fractional Cahn–Hilliard model, variable-step L1-type formulas, discrete convolution tools, positive definiteness, variational energy dissipation law
2010 Mathematics Subject Classification
35Q99, 65M06, 65M12, 74A50
Bingquan Ji is supported by grants 2022TQ0046 and 2022M720019 from Postdoctoral Science Foundation of China.
Hong-Lin Liao is supported by a grant 12071216 from National Natural Science Foundation of China.
Received 10 April 2022
Received revised 1 January 2023
Accepted 5 January 2023
Published 9 October 2023