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Error Estimates of Energy Stable Numerical Schemes for Allen–Cahn Equations with Nonlocal Constraints

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Abstract

We present error estimates for four unconditionally energy stable numerical schemes developed for solving Allen–Cahn equations with nonlocal constraints. The schemes are linear and second order in time and space, designed based on the energy quadratization (EQ) or the scalar auxiliary variable (SAV) method, respectively. In addition to the rigorous error estimates for each scheme, we also show that the linear systems resulting from the energy stable numerical schemes are all uniquely solvable. Then, we present some numerical experiments to show the accuracy of the schemes, their volume-preserving as well as energy dissipation properties in a drop merging simulation.

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Acknowledgements

Qi Wang’s research is partially supported by NSF awards DMS-1517347, DMS-1815921 and OIA-1655740, and NSFC awards #11571032, #91630207 and NSAF-U1530401.

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Authors and Affiliations

  1. Beijing Computational Science Research Center, Beijing, 100193, People’s Republic of China

    Shouwen Sun, Xiaobo Jing & Qi Wang

  2. Department of Mathematics, University of South Carolina, Columbia, SC, 29028, USA

    Qi Wang

  3. School of Materials Science and Engineering, Nankai University, Tianjin, 300350, China

    Qi Wang

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  1. Shouwen Sun
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  2. Xiaobo Jing
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Correspondence to Qi Wang.

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Sun, S., Jing, X. & Wang, Q. Error Estimates of Energy Stable Numerical Schemes for Allen–Cahn Equations with Nonlocal Constraints. J Sci Comput 79, 593–623 (2019). https://doi.org/10.1007/s10915-018-0867-7

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  • DOI: https://doi.org/10.1007/s10915-018-0867-7

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