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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Stability and convergence analysis for the implicit-explicit method to the Cahn-Hilliard equation
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by Dong Li, Chaoyu Quan and Tao Tang HTML | PDF
Math. Comp. 91 (2022), 785-809 Request permission

Abstract:

Implicit-explicit methods have been successfully used for the efficient numerical simulation of phase field problems such as the Cahn-Hilliard equation or thin film type equations. Due to the lack of maximum principle and stiffness caused by the effect of small dissipation coefficient, most existing theoretical analysis relies on adding additional stabilization terms, mollifying the nonlinearity or introducing auxiliary variables which implicitly either changes the structure of the problem or trades accuracy for stability in a subtle way. In this work, we introduce a robust theoretical framework to analyze directly the stability and accuracy of the standard implicit-explicit approach without stabilization or any other modification. We take the Cahn-Hilliard equation as a model case and provide a rigorous stability and convergence analysis for the original semi-discrete scheme under certain time step constraints. These settle several questions which have been open since the work of Chen and Shen [Comput. Phys. Comm. 108 (1998), pp. 147–158].
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Additional Information
  • Dong Li
  • Affiliation: SUSTech International Center for Mathematics; and Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, People’s Republic of China
  • MR Author ID: 723577
  • ORCID: 0000-0003-2367-4764
  • Email: lid@sustech.edu.cn
  • Chaoyu Quan
  • Affiliation: SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen 518055, People’s Republic of China
  • MR Author ID: 1178215
  • ORCID: 0000-0003-3246-8989
  • Email: quancy@sustech.edu.cn
  • Tao Tang
  • Affiliation: SUSTech International Center for Mathematics, Shenzhen, China; and Division of Science and Technology, BNU-HKBU United International College, Zhuhai, Guangdong Province, People’s Republic of China
  • MR Author ID: 248423
  • Email: tangt@sustech.edu.cn
  • Received by editor(s): June 16, 2020
  • Received by editor(s) in revised form: June 20, 2021, and July 6, 2021
  • Published electronically: November 9, 2021
  • Additional Notes: The second author’s work was supported by the NSFC Grant 11901281, the Guangdong Basic and Applied Basic Research Foundation (2020A1515010336), and the Stable Support Plan Program of Shenzhen Natural Science Fund (Program Contract No. 20200925160747003). The third author’s work was partially supported by the NSFC grants 11731006 and K20911001, and the science challenge project (No. TZ2018001).
  • © Copyright 2021 American Mathematical Society
  • Journal: Math. Comp. 91 (2022), 785-809
  • MSC (2020): Primary 35K55, 65M12
  • DOI: https://doi.org/10.1090/mcom/3704
  • MathSciNet review: 4379976