Effective time step analysis for the Allen–Cahn equation with a high‐order polynomial free energy. (17th June 2022)

• Record Type: Journal Article
• Authors: Lee, Seunggyu; Yoon, Sungha; Lee, Chaeyoung; Kim, Sangkwon; Kim, Hyundong; Yang, Junxiang; Kwak, Soobin; Hwang, Youngjin; Kim, Junseok
• Is Part Of: International journal for numerical methods in engineering. Volume 123:Number 19(2022)
• Page Start: 4726
• Page End: 4743
• Publication Date: 2022-06-17
• Subjects: Allen–Cahn equation -- effective time step -- high‐order polynomial free energy -- linear convex splitting; Numerical analysis -- Periodicals; Engineering mathematics -- Periodicals; 620.001518
• DOI: 10.1002/nme.7053
• Languages: English
• ISSNs: 0029-5981

Abstract

Abstract: An effective time step analysis to the linear convex splitting scheme for the Allen–Cahn equation with a high‐order polynomial free energy is presented in this article. Although the convex splitting scheme is unconditionally stable, using a large time step causes a time step rescaling effect, leading to delayed dynamics of the governing equation. We verify this problem by comparing it with a reformulated semi‐implicit scheme using the effective time step. Theoretical results show that the discrete energy stability and maximum‐principle hold, and the numerical results demonstrate that the time step rescaling issue can be resolved using the effective time step. We confirm that slow dynamics due to high‐order potential is alleviated by the time step modification through the results of motion by mean curvature.