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Japan Journal of Industrial and Applied Mathematics

, Volume 35, Issue 3, pp 1245–1281 | Cite as

A stable and structure-preserving scheme for a non-local Allen–Cahn equation

  • Makoto OkumuraEmail author
Original Paper Area 2
  • 114 Downloads

Abstract

We propose a stable and structure-preserving finite difference scheme for a non-local Allen–Cahn equation which describes a process of phase separation in a binary mixture. The proposed scheme inherits characteristic properties, the conservation of mass and the decrease of the global energy from the equation. We show the stability and unique existence of the solution of the scheme. We also prove the error estimate for the scheme. Numerical experiments demonstrate the effectiveness of the proposed scheme.

Keywords

Non-local Allen–Cahn equation Discrete variational derivative method 

Mathematics Subject Classification

65M06 

Notes

Acknowledgements

I thank the reviewer for helpful and attentive comments that helped me to improve this manuscript. I also thank Prof. D. Furihata of Osaka University, Prof. N. Yamazaki of Kanagawa University, Prof. T. Fukao of Kyoto University of Education and Assoc. Prof. K. Takasao of Kyoto University for helpful advice and discussions.

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Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Pure and Applied Mathematics, Graduate School of Information Science and TechnologyOsaka UniversitySuitaJapan

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