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

Hong-Lin Liao, Xuehua Song, Tao Tang*, Tao Zhou

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

35 Citations (Scopus)

Abstract

In this work, we are concerned with the stability and convergence analysis of the second-order backward difference formula (BDF2) 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 satisfy rk:= τkk−1 < 3.561. Moreover, with a novel discrete orthogonal convolution kernels argument and some new estimates on the corresponding positive definite quadratic forms, the L2 norm stability and rigorous error estimates are established, under the same step-ratio constraint that ensures the energy stability, i.e., 0 < rk < 3.561. This is known to be the best result in the 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.

Original languageEnglish
Pages (from-to)887-902
Number of pages16
JournalScience China Mathematics
Volume64
Issue number5
DOIs
Publication statusPublished - May 2021

Scopus Subject Areas

  • Mathematics(all)

User-Defined Keywords

  • 35Q99
  • 65M06
  • 65M12
  • 74A50
  • convergence analysis
  • discrete orthogonal convolution kernels
  • energy stability
  • molecular beam epitaxial growth
  • variable-step BDF2 scheme

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