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Maximum-Principle-Preserving Local Discontinuous Galerkin Methods for Allen-Cahn Equations

Communications on Applied Mathematics and Computation volume 4, pages 353–379 (2022)Cite this article

Abstract

In this paper, we study the classical Allen-Cahn equations and investigate the maximum-principle-preserving (MPP) techniques. The Allen-Cahn equation has been widely used in mathematical models for problems in materials science and fluid dynamics. It enjoys the energy stability and the maximum-principle. Moreover, it is well known that the Allen-Cahn equation may yield thin interface layer, and nonuniform meshes might be useful in the numerical solutions. Therefore, we apply the local discontinuous Galerkin (LDG) method due to its flexibility on h-p adaptivity and complex geometry. However, the MPP LDG methods require slope limiters, then the energy stability may not be easy to obtain. In this paper, we only discuss the MPP technique and use numerical experiments to demonstrate the energy decay property. Moreover, due to the stiff source given in the equation, we use the conservative modified exponential Runge-Kutta methods and thus can use relatively large time step sizes. Thanks to the conservative time integration, the bounds of the unknown function will not decay. Numerical experiments will be given to demonstrate the good performance of the MPP LDG scheme.

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Funding

Jie Du is supported by the National Natural Science Foundation of China under Grant Number NSFC 11801302 and Tsinghua University Initiative Scientific Research Program. Eric Chung is supported by Hong Kong RGC General Research Fund (Projects 14304217 and 14302018). The third author is supported by the NSF grant DMS-1818467.

Author information

Authors and Affiliations

  1. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Jie Du

  2. Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, 101408, China

    Jie Du

  3. Department of Mathematics, The Chinese University of Hong Kong, Hong Kong SAR, China

    Eric Chung

  4. Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA

    Yang Yang

Authors
  1. Jie Du
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  2. Eric Chung
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  3. Yang Yang
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Corresponding author

Correspondence to Yang Yang.

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Du, J., Chung, E. & Yang, Y. Maximum-Principle-Preserving Local Discontinuous Galerkin Methods for Allen-Cahn Equations. Commun. Appl. Math. Comput. 4, 353–379 (2022). https://doi.org/10.1007/s42967-020-00118-x

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  • DOI: https://doi.org/10.1007/s42967-020-00118-x

Keywords

  • Maximum-principle-preserving
  • Local discontinuous Galerkin methods
  • Allen-Cahn equation
  • Conservative exponential integrations

Mathematics Subject Classification

  • 65M12
  • 65M60

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