An efficient and stable compact fourth-order finite difference scheme for the phase field crystal equation

Yibao Li; Junseok Kim

doi:10.1016/j.cma.2017.02.022

An efficient and stable compact fourth-order finite difference scheme for the phase field crystal equation

Yibao Li, Junseok Kim

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

59 Citations (Scopus)

Abstract

In this paper, we present a high-order accurate compact scheme for the phase field crystal model in two- and three-dimensional spaces. The proposed scheme is derived by combining a fourth-order compact finite difference formula in space and a backward differentiation for the time derivative term, which is second-order accurate in time. Furthermore, a nonlinearly stabilized splitting scheme is used and thus a larger time step can be allowed. Since the equations at the implicit time level are nonlinear, we introduce a Newton-type iterative method and employ a fast and efficient nonlinear multigrid solver to solve the resulting discrete system. In particular, we implement the compact scheme in the adaptive mesh refinement framework. An adaptive time step method for the phase field crystal model is also proposed. Various numerical experiments are presented and confirm the accuracy, stability, and efficiency of our proposed method.

Original language	English
Pages (from-to)	194-216
Number of pages	23
Journal	Computer Methods in Applied Mechanics and Engineering
Volume	319
DOIs	https://doi.org/10.1016/j.cma.2017.02.022
Publication status	Published - 2017 Jun 1

Bibliographical note

Funding Information:
Y.B. Li is supported by Natural Science Basic Research Plan in Shaanxi Province of China (2016JQ1024), by National Natural Science Foundation of China (Nos. 11601416, 11631012). The corresponding author (J.S. Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03933243). The authors are grateful to the reviewers whose valuable suggestions and comments significantly improved the quality of this paper.

Publisher Copyright:
© 2017 Elsevier B.V.

Keywords

Adaptive mesh refinement
Adaptive time-stepping
Fourth-order compact scheme
Phase-field crystal equation

ASJC Scopus subject areas

Computational Mechanics
Mechanics of Materials
Mechanical Engineering
General Physics and Astronomy
Computer Science Applications

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10.1016/j.cma.2017.02.022

Cite this

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title = "An efficient and stable compact fourth-order finite difference scheme for the phase field crystal equation",

abstract = "In this paper, we present a high-order accurate compact scheme for the phase field crystal model in two- and three-dimensional spaces. The proposed scheme is derived by combining a fourth-order compact finite difference formula in space and a backward differentiation for the time derivative term, which is second-order accurate in time. Furthermore, a nonlinearly stabilized splitting scheme is used and thus a larger time step can be allowed. Since the equations at the implicit time level are nonlinear, we introduce a Newton-type iterative method and employ a fast and efficient nonlinear multigrid solver to solve the resulting discrete system. In particular, we implement the compact scheme in the adaptive mesh refinement framework. An adaptive time step method for the phase field crystal model is also proposed. Various numerical experiments are presented and confirm the accuracy, stability, and efficiency of our proposed method.",

keywords = "Adaptive mesh refinement, Adaptive time-stepping, Fourth-order compact scheme, Phase-field crystal equation",

author = "Yibao Li and Junseok Kim",

note = "Funding Information: Y.B. Li is supported by Natural Science Basic Research Plan in Shaanxi Province of China (2016JQ1024), by National Natural Science Foundation of China (Nos. 11601416, 11631012). The corresponding author (J.S. Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03933243). The authors are grateful to the reviewers whose valuable suggestions and comments significantly improved the quality of this paper. Publisher Copyright: {\textcopyright} 2017 Elsevier B.V.",

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PY - 2017/6/1

Y1 - 2017/6/1

N2 - In this paper, we present a high-order accurate compact scheme for the phase field crystal model in two- and three-dimensional spaces. The proposed scheme is derived by combining a fourth-order compact finite difference formula in space and a backward differentiation for the time derivative term, which is second-order accurate in time. Furthermore, a nonlinearly stabilized splitting scheme is used and thus a larger time step can be allowed. Since the equations at the implicit time level are nonlinear, we introduce a Newton-type iterative method and employ a fast and efficient nonlinear multigrid solver to solve the resulting discrete system. In particular, we implement the compact scheme in the adaptive mesh refinement framework. An adaptive time step method for the phase field crystal model is also proposed. Various numerical experiments are presented and confirm the accuracy, stability, and efficiency of our proposed method.

AB - In this paper, we present a high-order accurate compact scheme for the phase field crystal model in two- and three-dimensional spaces. The proposed scheme is derived by combining a fourth-order compact finite difference formula in space and a backward differentiation for the time derivative term, which is second-order accurate in time. Furthermore, a nonlinearly stabilized splitting scheme is used and thus a larger time step can be allowed. Since the equations at the implicit time level are nonlinear, we introduce a Newton-type iterative method and employ a fast and efficient nonlinear multigrid solver to solve the resulting discrete system. In particular, we implement the compact scheme in the adaptive mesh refinement framework. An adaptive time step method for the phase field crystal model is also proposed. Various numerical experiments are presented and confirm the accuracy, stability, and efficiency of our proposed method.

KW - Adaptive mesh refinement

KW - Adaptive time-stepping

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KW - Phase-field crystal equation

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An efficient and stable compact fourth-order finite difference scheme for the phase field crystal equation

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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