Communications in Mathematical Sciences

Volume 21 (2023)

Number 7

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

Pages: 1767 – 1789

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n7.a2

Authors

Bingquan Ji (School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, China; and Institute of Applied Physics and Computational Mathematics, Beijing, China)

Xiaohan Zhu (School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, China)

Hong-Lin Liao (School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, China; and Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing, China)

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 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.

Keywords

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

Received 10 April 2022

Received revised 1 January 2023

Accepted 5 January 2023

Published 9 October 2023