An energy stable, hexagonal finite difference scheme for the 2D phase field crystal amplitude equations

JOURNAL OF COMPUTATIONAL PHYSICS

卷:321

Article

Guan, Zhen¹  Heinonen, Vili²  Lowengrub, John¹  Wang, Cheng³  Wise, Steven M.⁴ 

[1] Univ Calif Irvine, Dept Math, Irvine, CA 92717 USA

[2] Aalto Univ, Sch Sci, COMP Ctr Excellence, Dept Appl Phys, Aalto, Finland

[3] Univ Massachusetts, Dept Math, Dartmouth, MA USA

[4] Univ Tennessee, Dept Math, Knoxville, TN 37996 USA

关键词: Phase field crystal;    Amplitude equations;    Energy stable scheme;    Hexagonal finite differences;    Multigrid;   

DOI  :  10.1016/j.jcp.2016.06.007

来源: Elsevier

【 摘 要 】

In this paper we construct an energy stable finite difference scheme for the amplitude expansion equations for the two-dimensional phase field crystal (PFC) model. The equations are formulated in a periodic hexagonal domain with respect to the reciprocal lattice vectors to achieve a provably unconditionally energy stable and solvable scheme. To our knowledge, this is the first such energy stable scheme for the PFC amplitude equations. The convexity of each part in the amplitude equations is analyzed, in both the semi-discrete and fully-discrete cases. Energy stability is based on a careful convexity analysis for the energy (in both the spatially continuous and discrete cases). As a result, unique solvability and unconditional energy stability are available for the resulting scheme. Moreover, we show that the scheme is point-wise stable for any time and space step sizes. An efficient multigrid solver is devised to solve the scheme, and a few numerical experiments are presented, including grain rotation and shrinkage and grain growth studies, as examples of the strength and robustness of the proposed scheme and solver. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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