Analysis of the second order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection

TL;DR

This paper analyzes the stability and convergence of a second order BDF scheme with variable steps for a molecular beam epitaxial model, establishing energy stability and error estimates under a specific step-ratio constraint, and validating with numerical experiments.

Contribution

It provides the first rigorous stability and error analysis of the variable-step BDF2 scheme for this model, including novel inequalities and an adaptive time-stepping strategy.

Findings

Energy dissipation law is preserved under step-ratio < 3.561.

Established optimal error estimates for the scheme.

Adaptive time-stepping accelerates steady state computations.

Abstract

In this work, we are concerned with the stability and convergence analysis of the second order BDF (BDF2) scheme with variable steps for the molecular beam epitaxial model without slope selection. We first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step constraint. Then we show that it preserves an energy dissipation law if the adjacent time-step ratios $r_k:=\tau_k/\tau_{k-1}<3.561.$ Moreover, with a novel discrete orthogonal convolution kernels argument and some new discrete convolutional inequalities, the $L^2$ norm stability and rigorous error estimates are established, under the same step-ratios constraint that ensuring the energy stability., i.e., $0<r_k<3.561.$ This is known to be the best result in literature. We finally adopt an adaptive time-stepping strategy to accelerate the computations of the steady state solution and confirm our theoretical findings by numerical examples.