Efficient numerical schemes with unconditional energy stabilities for the modified phase field crystal equation

We consider numerical approximations for the modified phase field crystal equation in this paper. The model is a nonlinear damped wave equation that includes both diffusive dynamics and elastic interactions. To develop easy-to-implement time-stepping schemes with unconditional energy stabilities, we...

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Bibliographic Details
Published inAdvances in computational mathematics Vol. 45; no. 3; pp. 1551 - 1580
Main Authors Li, Qi, Mei, Liquan, Yang, Xiaofeng, Li, Yibao
Format Journal Article
LanguageEnglish
Published New York Springer US 01.06.2019
Springer Nature B.V
Subjects
Computational mathematics
Computational Mathematics and Numerical Analysis
Computational Science and Engineering
Mathematical and Computational Biology
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Visualization
Wave equations
Invariant energy quadratization
Modified phase field crystal equation
Pseudo energy
Unconditionally energy stable
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Summary:We consider numerical approximations for the modified phase field crystal equation in this paper. The model is a nonlinear damped wave equation that includes both diffusive dynamics and elastic interactions. To develop easy-to-implement time-stepping schemes with unconditional energy stabilities, we adopt the “Invariant Energy Quadratization” approach. By using the first-order backward Euler, the second-order Crank–Nicolson, and the second-order BDF2 formulas, we obtain three linear and symmetric positive definite schemes. We rigorously prove their unconditional energy stabilities and implement a number of 2D and 3D numerical experiments to demonstrate the accuracy, stability, and efficiency.
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ISSN:1019-7168
1572-9044
DOI:10.1007/s10444-019-09678-w