Numerical Analysis of a Convex-Splitting BDF2 Method with Variable Time-Steps for the Cahn–Hilliard Model

In this paper, the convex-splitting BDF2 method with variable time-steps (proposed in Chen et al. SIAM J Numer Anal 57:495–525, 2019) is reconsidered for the Cahn–Hilliard model. We adopt the Fourier pseudo-spectral discretization in space. With the help of the discrete gradient structure of the BDF2 formula and some embedded inequalities, we prove that the scheme preserves a modified energy dissipation law under the updated time-step-ratio restriction 0<rk=τk/τk-1<4.864\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<r_{k}=\tau _{k}/\tau _{k-1}<4.864$$\end{document}. By utilizing the discrete orthogonal convolution kernels, some discrete convolution inequalities and some proof techniques (Lemma 4.6), we tackle the difficulty brought from the Douglas-Dupont regularization stabilized term and prove the robust L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}$$\end{document} norm convergence of the scheme under the same mild time-step-ratio restriction 0<rk<4.864.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<r_{k}<4.864.$$\end{document} Numerical experiments are carried out to support our theoretical analysis and a time adaptive strategy is applied to accelerate the simulation of the multi-scale characteristics of the Cahn–Hilliard model.