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Energy stability and convergence of SAV block-centered finite difference method for gradient flows


Authors: Xiaoli Li, Jie Shen and Hongxing Rui
Journal: Math. Comp. 88 (2019), 2047-2068
MSC (2010): Primary 65M06, 65M12, 65M15, 35K20, 35K35, 65Z05
DOI: https://doi.org/10.1090/mcom/3428
Published electronically: April 1, 2019
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Abstract: We present in this paper construction and analysis of a block-centered finite difference method for the spatial discretization of the scalar auxiliary variable Crank-Nicolson scheme (SAV/CN-BCFD) for gradient flows, and show rigorously that the scheme is second-order in both time and space in various discrete norms. When equipped with an adaptive time strategy, the SAV/CN-BCFD scheme is accurate and extremely efficient. Numerical experiments on typical Allen-Cahn and Cahn-Hilliard equations are presented to verify our theoretical results and to show the robustness and accuracy of the SAV/CN-BCFD scheme.


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Additional Information

Xiaoli Li
Affiliation: School of Mathematics, Shandong University, Jinan 250100, People’s Republic of China
Address at time of publication: Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing and School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, 361005, People’s Republic of China
Email: xiaolisdu@163.com

Jie Shen
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: shen7@purdue.edu

Hongxing Rui
Affiliation: School of Mathematics, Shandong University, Jinan 250100, People’s Republic of China
Email: hxrui@sdu.edu.cn

DOI: https://doi.org/10.1090/mcom/3428
Keywords: Scalar auxiliary variable (SAV), gradient flows, energy stability, block-centered finite difference, error estimates, adaptive time stepping
Received by editor(s): June 26, 2018
Received by editor(s) in revised form: November 18, 2018
Published electronically: April 1, 2019
Additional Notes: The first author thanks the China Scholarship Council for financial support.
The work of the second author was supported in part by NSF grants DMS-1620262, DMS-1720442, and AFOSR grant FA9550-16-1-0102.
The second author is the corresponding author.
The work of the third author was supported by the National Natural Science Foundation of China grant 11671233.
Article copyright: © Copyright 2019 American Mathematical Society