Efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model

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• 作者：Xiaofeng Yang^a ; ^b ; ^{xfyang@math.sc.edu} ; Lili Ju^a ; ^c ; ^{ju@math.sc.edu}
• 关键词：Phase field ; Allen&ndash ; Cahn ; Linear ; Elastic bending energy ; Invariant energy quadratization ; Energy stability
• 刊名：Computer Methods in Applied Mechanics and Engineering
• 出版年：2017
• 出版时间：1 March 2017
• 年：2017
• 卷：315
• 期：Complete
• 页码：691-712
• 全文大小：2213 K
• 卷排序：315

文摘

In this paper, we study efficient numerical schemes of the classical phase field elastic bending energy model that has been widely used to describe the shape deformation of biological lipid vesicles, in which the free energy of the system consists of an elastic bending energy, a surface area constraint and a volume constraint. One major challenge in solving such model numerically is how to design appropriate temporal discretizations in order to preserve energy stability with large time step sizes at the semi-discrete level. We develop a first order and a second order time stepping scheme for this highly nonlinear and stiff parabolic PDE system based on the “Invariant Energy Quadratization” approach. In particular, the resulted semi-discretizations lead to linear systems in space with symmetric positive definite operators at each time step, thus can be efficiently solved. In addition, the proposed schemes are rigorously proved to be unconditionally energy stable. Various numerical experiments in 2D and 3D are presented to demonstrate the stability and accuracy of the proposed schemes.