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

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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:= τk/τk−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-step** strategy to accelerate the computations of the steady state solution and confirm our theoretical findings by numerical examples.

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Acknowledgements

The first and second authors were supported by National Natural Science Foundation of China (Grant No. 12071216). The third author was supported by National Natural Science Foundation of China (Grant No. 11731006) and the NNW2018-ZT4A06 project. The fourth author was supported by National Natural Science Foundation of China (Grant Nos. 11822111, 11688101 and 11731006) and the Science Challenge Project (Grant No. TZ2018001). The authors thank the anonymous referees for their valuable comments that are very helpful in improving the quality of this article (ar**v:2008.03185v1).

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Authors and Affiliations

  1. Department of Mathematics, Nan**g University of Aeronautics and Astronautics, Nan**g, 211101, China

    Hong-Lin Liao & Xuehua Song

  2. Department of Mathematics and International Center for Mathematics, Southern University of Science and Technology, Shenzhen, 518055, China

    Tao Tang

  3. Division of Science and Technology, BNU-HKBU United International College, Zhuhai, 519087, China

    Tao Tang

  4. NCMIS & LSEC, Institute of Computational Mathematics and Scientific Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Bei**g, 100190, China

    Tao Zhou

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  1. Hong-Lin Liao
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  2. Xuehua Song
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  3. Tao Tang
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  4. Tao Zhou
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Correspondence to Tao Tang.

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Liao, HL., Song, X., Tang, T. et al. Analysis of the second-order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection. Sci. China Math. 64, 887–902 (2021). https://doi.org/10.1007/s11425-020-1817-4

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  • DOI: https://doi.org/10.1007/s11425-020-1817-4

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