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Convergence Analysis for the Invariant Energy Quadratization (IEQ) Schemes for Solving the Cahn–Hilliard and Allen–Cahn Equations with General Nonlinear Potential

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Abstract

In this paper, we carry out stability and error analyses for two first-order, semi-discrete time stepping schemes, which are based on the newly developed invariant energy quadratization approach, for solving the well-known Cahn–Hilliard and Allen–Cahn equations with general nonlinear bulk potentials. Some reasonable sufficient conditions about boundedness and continuity of the nonlinear functional are given in order to obtain optimal error estimates. The well-posedness, unconditional energy stabilities and optimal error estimates of the numerical schemes are proved rigorously. Through the comparisons with some other prevalent schemes for several benchmark numerical examples, we demonstrate the stability and the accuracy of the schemes numerically.

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Authors and Affiliations

  1. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Xiaofeng Yang

  2. School of Mathematics and Information Sciences, Yantai University, Yantai, 264005, Shandong, People’s Republic of China

    Guo-Dong Zhang

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  1. Xiaofeng Yang
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Correspondence to Xiaofeng Yang.

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Xiaofeng Yang: This author’s research is partially supported by the U.S. National Science Foundation under Grant Numbers DMS-1720212 and DMS-1818783.

Guo-Dong Zhang: This author’s research is partially supported by National Science Foundation of China under Grant Numbers 11601468 and 11771375 and Shandong Province Natural Science Foundation (ZR2018MA008).

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Yang, X., Zhang, GD. Convergence Analysis for the Invariant Energy Quadratization (IEQ) Schemes for Solving the Cahn–Hilliard and Allen–Cahn Equations with General Nonlinear Potential. J Sci Comput 82, 55 (2020). https://doi.org/10.1007/s10915-020-01151-x

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  • DOI: https://doi.org/10.1007/s10915-020-01151-x

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