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Energy Stability of BDF Methods up to Fifth-Order for the Molecular Beam Epitaxial Model Without Slope Selection

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Abstract

The backward differential formulas of order \(\mathrm {k}\) (BDF-\(\mathrm {k}\)) for \(3\le \mathrm {k}\le 5\) are analyzed for the molecular beam epitaxial (MBE) model without slope selection. We show that the fully implicit uniform BDF-\(\mathrm {k}\) schemes are convex and uniquely solvable under a weak time-step constraint. Then the BDF methods are proved to preserve the modified discrete energy dissipation laws by using the discrete gradient structures of BDF-\(\mathrm {k}\) \((3\le \mathrm {k}\le 5)\) formulas. Furthermore, with the help of discrete orthogonal convolution kernels and the corresponding convolution Young inequalities, the \(L^2\) norm stability and convergence analysis of the BDF-\(\mathrm {k}\) \((3\le \mathrm {k}\le 5)\) schemes for the MBE model are established by the recent discrete energy technique. To the best of our knowledge, it is the first time to present a unified approach to establish the discrete energy dissipation laws and \(L^2\) norm convergence of non-A-stable BDF-\(\mathrm {k}\) methods for the MBE model. Numercial simulations are presented to demonstrate the accuracy and efficiency of the proposed schemes.

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Funding

This work is supported by NSF of China under Grant Number 12071216.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, People’s Republic of China

    Yuanyuan Kang

  2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, People’s Republic of China

    Hong-lin Liao

  3. Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing, 211106, People’s Republic of China

    Hong-lin Liao

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  1. Yuanyuan Kang
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  2. Hong-lin Liao
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Correspondence to Hong-lin Liao.

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Kang, Y., Liao, Hl. Energy Stability of BDF Methods up to Fifth-Order for the Molecular Beam Epitaxial Model Without Slope Selection. J Sci Comput 91, 47 (2022). https://doi.org/10.1007/s10915-022-01830-x

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  • DOI: https://doi.org/10.1007/s10915-022-01830-x

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