Stability and convergence of a variable-step stabilized BDF2 stepping for the MBE model with slope selection

A second order variable-step stabilized convex splitting BDF2 time-stepping is investigated for the molecular beam epitaxy model with slope selection. The common Douglas–Dupont regularization term $A\tau_n\Delta(\phi^n-\phi^{n-1})$ with a properly stabilized parameter $A \gt 0$ is considered such that the numerical scheme preserves the discrete energy dissipation law unconditionally. The present error analysis is essentially different from the traditional energy method [W. Chen, X. Wang, Y. Yan, and Z. Zhang, SIAM J. Numer. Anal., 57:495–525, 2019] by combining the $L^2$ norm with $H^1$ norm analysis, which always require relatively stringent step-ratio restriction. Our main tools are the so-called discrete orthogonal convolution kernels and the associated convolution embedding inequalities. Under the adjacent step ratios constraint $\tau_k/\tau_{k-1} \lt 4.864$, which is stemmed from the positive definiteness of BDF2 convolution kernels, an optimal $L^2$ norm error estimate is achieved for the first time by carefully handling the Douglas–Dupont regularization term. Numerical experiments are presented to support our theoretical results.

Keywords

MBE model with slope selection, adaptive BDF2 method, discrete energy dissipation law, stabilized convex splitting scheme, error estimate

2010 Mathematics Subject Classification

35Q99, 65M06, 65M12, 74A50

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Received 16 March 2022

Received revised 6 March 2023

Accepted 2 October 2023

Published 12 July 2024