Error analysis of the implicit variable-step BDF2 method for the molecular beam epitaxial model with slope selection

TL;DR

This paper develops and analyzes a stable variable-step BDF2 numerical scheme for the molecular beam epitaxial model with slope selection, proving convergence, energy dissipation, and demonstrating effectiveness through numerical examples.

Contribution

It introduces a new convergence and stability analysis for the variable-step BDF2 method applied to the MBE model with slope selection, including energy law verification.

Findings

Proved second-order convergence in time and space.

Established energy dissipation law with a modified energy functional.

Validated the scheme's effectiveness through numerical experiments.

Abstract

We derive unconditionally stable and convergent variable-step BDF2 scheme for solving the MBE model with slope selection. The discrete orthogonal convolution kernels of the variable-step BDF2 method is commonly utilized recently for solving the phase field models. In this paper, we further prove some new inequalities, concerning the vector forms, for the kernels especially dealing with the nonlinear terms in the slope selection model. The convergence rate of the fully discrete scheme is proved to be two both in time and space in $L^2$ norm under the setting of the variable time steps. Energy dissipation law is proved rigorously with a modified energy by adding a small term to the discrete version of the original free energy functional. Two numerical examples including an adaptive time-stepping strategy are given to verify the convergence rate and the energy dissipation law.