期刊
JOURNAL OF COMPUTATIONAL PHYSICS
卷 228, 期 15, 页码 5323-5339出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2009.04.020
关键词
Phase field crystal; Finite difference; Energy stability; Multigrid
资金
- National Science Foundation Division of Mathematical Sciences (DMS)
- Division of Materials Research (DMR)
- Graduate School at the University of California, Irvine
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [915128, 0818030] Funding Source: National Science Foundation
In this paper we present and compare two unconditionally energy stable finite-difference schemes for the phase field crystal equation. The first is a one-step scheme based on a convex splitting of a discrete energy by Wise et al. [S.M. Wise, C. Wang, J.S. Lowengrub, An energy stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., in press]. In this scheme, which is first order in time and second order in space, the discrete energy is non-increasing for any time step. The second scheme we consider is a new, fully second-order two-step algorithm. In the new scheme, the discrete energy is bounded by its initial value for any time step. In both methods, the equations at the implicit time level are nonlinear but represent the gradients of strictly convex functions and are thus uniquely solvable, regardless of time step-size. We solve the nonlinear equations using an efficient nonlinear multigrid method. Numerical. simulations are presented and confirm the stability, efficiency and accuracy of the schemes. (C) 2009 Elsevier Inc. All rights reserved.
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