TY - JOUR
T1 - An efficient and stable compact fourth-order finite difference scheme for the phase field crystal equation
AU - Li, Yibao
AU - Kim, Junseok
N1 - 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.
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
KW - Fourth-order compact scheme
KW - Phase-field crystal equation
UR - http://www.scopus.com/inward/record.url?scp=85015394091&partnerID=8YFLogxK
U2 - 10.1016/j.cma.2017.02.022
DO - 10.1016/j.cma.2017.02.022
M3 - Article
AN - SCOPUS:85015394091
SN - 0045-7825
VL - 319
SP - 194
EP - 216
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
ER -