Volume 34, Issue 5
Implicit-Explicit Scheme for the Allen-Cahn Equation Preserves the Maximum Principle

Tao Tang & Jiang Yang

J. Comp. Math., 34 (2016), pp. 451-461.

Published online: 2016-10

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  • Abstract

It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes? To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the maximum principle.

  • Keywords

Allen-Cahn Equations Implicit-explicit scheme Maximum principle Nonlinear energy stability

  • AMS Subject Headings

65GXX 65MXX.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

tangt@sustc.edu.cn (Tao Tang)

jyanghkbu@gmail.com (Jiang Yang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-34-451, author = {Tang , Tao and Yang , Jiang }, title = {Implicit-Explicit Scheme for the Allen-Cahn Equation Preserves the Maximum Principle}, journal = {Journal of Computational Mathematics}, year = {2016}, volume = {34}, number = {5}, pages = {451--461}, abstract = { It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes? To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the maximum principle.}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1603-m2014-0017}, url = {http://global-sci.org/intro/article_detail/jcm/9806.html} }
TY - JOUR T1 - Implicit-Explicit Scheme for the Allen-Cahn Equation Preserves the Maximum Principle AU - Tang , Tao AU - Yang , Jiang JO - Journal of Computational Mathematics VL - 5 SP - 451 EP - 461 PY - 2016 DA - 2016/10 SN - 34 DO - http://dor.org/10.4208/jcm.1603-m2014-0017 UR - https://global-sci.org/intro/article_detail/jcm/9806.html KW - Allen-Cahn Equations KW - Implicit-explicit scheme KW - Maximum principle KW - Nonlinear energy stability AB - It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes? To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the maximum principle.
Tao Tang & Jiang Yang . (2020). Implicit-Explicit Scheme for the Allen-Cahn Equation Preserves the Maximum Principle. Journal of Computational Mathematics. 34 (5). 451-461. doi:10.4208/jcm.1603-m2014-0017
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