Efficient numerical schemes with unconditional energy stabilities for the modified phase field crystal equation
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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.
KeywordsModified phase field crystal equation Unconditionally energy stable Pseudo energy Invariant energy quadratization
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The first author would like to thank Zhen Xu at Beijing Normal University and Yali Gao at Northwestern Polytechnical University for the valuable discussions. The authors also would like to thank the reviewers for their helpful comments and suggestions.
The work of Q. Li is supported by the China Scholarship Council (CSC No. 201806280137). The work of L. Mei is partially supported by the NSFC under grant no. 11371289. The work of X. Yang is partially supported by the NSF Grant DMS-1720212 and USC ASPIRE I Track-III/IV Fund. The work of Y. Li is partially supported by the NSFC under grant no. 11601416.
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