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Volume 16, Issue 1
An Energy Stable Filtered Backward Euler Scheme for the MBE Equation with Slope Selection

Jiexin Wang, Hong-Lin Liao & Ying Zhao

DOI: 10.4208/nmtma.OA-2022-0072

Numer. Math. Theor. Meth. Appl., 16 (2023), pp. 165-181.

Published online: 2023-01

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  • Abstract

As a promising strategy to adjust the order in the variable-order BDF algorithm, a time filtered backward Euler scheme is investigated for the molecular beam epitaxial equation with slope selection. The temporal second-order convergence in the $L^2$ norm is established under a convergence-solvability-stability (CSS)-consistent time-step constraint. The CSS-consistent condition means that the maximum step-size limit required for convergence is of the same order to that for solvability and stability (in certain norms) as the small interface parameter $ε → 0^+.$ Similar to the backward Euler scheme, the time filtered backward Euler scheme preserves some physical properties of the original problem at the discrete levels, including the volume conservation, the energy dissipation law and $L^2$ norm boundedness. Numerical tests are included to support the theoretical results.

  • Keywords

MBE model, time filter, energy dissipation law, error estimate.

  • AMS Subject Headings

65M06, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-16-165, author = {Wang , JiexinLiao , Hong-Lin and Zhao , Ying}, title = {An Energy Stable Filtered Backward Euler Scheme for the MBE Equation with Slope Selection}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2023}, volume = {16}, number = {1}, pages = {165--181}, abstract = {

As a promising strategy to adjust the order in the variable-order BDF algorithm, a time filtered backward Euler scheme is investigated for the molecular beam epitaxial equation with slope selection. The temporal second-order convergence in the $L^2$ norm is established under a convergence-solvability-stability (CSS)-consistent time-step constraint. The CSS-consistent condition means that the maximum step-size limit required for convergence is of the same order to that for solvability and stability (in certain norms) as the small interface parameter $ε → 0^+.$ Similar to the backward Euler scheme, the time filtered backward Euler scheme preserves some physical properties of the original problem at the discrete levels, including the volume conservation, the energy dissipation law and $L^2$ norm boundedness. Numerical tests are included to support the theoretical results.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2022-0072}, url = {http://global-sci.org/intro/article_detail/nmtma/21347.html} }
TY - JOUR T1 - An Energy Stable Filtered Backward Euler Scheme for the MBE Equation with Slope Selection AU - Wang , Jiexin AU - Liao , Hong-Lin AU - Zhao , Ying JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 165 EP - 181 PY - 2023 DA - 2023/01 SN - 16 DO - http://doi.org/10.4208/nmtma.OA-2022-0072 UR - https://global-sci.org/intro/article_detail/nmtma/21347.html KW - MBE model, time filter, energy dissipation law, error estimate. AB -

As a promising strategy to adjust the order in the variable-order BDF algorithm, a time filtered backward Euler scheme is investigated for the molecular beam epitaxial equation with slope selection. The temporal second-order convergence in the $L^2$ norm is established under a convergence-solvability-stability (CSS)-consistent time-step constraint. The CSS-consistent condition means that the maximum step-size limit required for convergence is of the same order to that for solvability and stability (in certain norms) as the small interface parameter $ε → 0^+.$ Similar to the backward Euler scheme, the time filtered backward Euler scheme preserves some physical properties of the original problem at the discrete levels, including the volume conservation, the energy dissipation law and $L^2$ norm boundedness. Numerical tests are included to support the theoretical results.

Wang , JiexinLiao , Hong-Lin and Zhao , Ying. (2023). An Energy Stable Filtered Backward Euler Scheme for the MBE Equation with Slope Selection. Numerical Mathematics: Theory, Methods and Applications. 16 (1). 165-181. doi:10.4208/nmtma.OA-2022-0072
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