A stable and structure-preserving scheme for a non-local Allen–Cahn equation
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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.
KeywordsNon-local Allen–Cahn equation Discrete variational derivative method
Mathematics Subject Classification65M06
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.
- 25.Takasao, K.: Existence of weak solution for volume preserving mean curvature flow via phase field method, pp. 1–16 (2015). arXiv:1511.01687 [math.AP]