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A Second-order Time-Accurate Unconditionally Stable Method for a Gradient Flow for the Modica–Mortola Functional

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Abstract

In this study, we present a second-order time-accurate unconditionally stable numerical method for a gradient flow for the Modica–Mortola functional with an equispaced multiple well potential. The proposed second-order time-accurate unconditionally stable numerical method is based on the operator splitting method. The nonlinear and linear terms in the gradient flow are solved analytically and using the Fourier spectral method, respectively. The numerical solutions in each step are bounded for any time step size and the overall scheme is temporally second-order accurate. We prove theoretically the unconditional stability and boundedness of the numerical solutions. In addition, several numerical tests are conducted to demonstrate the performance of the proposed method.

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Acknowledgements

The first author (S. Ham) was supported by the National Research Foundation (NRF), Korea, under project BK21 FOUR. The corresponding author (J.S. Kim) expresses thanks for the support from the BK21 FOUR program. The authors would like to thank the reviewers for their useful comments and suggestions that helped to improve the paper.

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  1. Department of Mathematics, Korea University, Seoul, 02841, Republic of Korea

    Seokjun Ham, Soobin Kwak, Chaeyoung Lee, Gyeonggyu Lee & Junseok Kim

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  1. Seokjun Ham
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  2. Soobin Kwak
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  3. Chaeyoung Lee
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  4. Gyeonggyu Lee
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  5. Junseok Kim
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Correspondence to Junseok Kim.

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Ham, S., Kwak, S., Lee, C. et al. A Second-order Time-Accurate Unconditionally Stable Method for a Gradient Flow for the Modica–Mortola Functional. J Sci Comput 95, 63 (2023). https://doi.org/10.1007/s10915-023-02198-2

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  • DOI: https://doi.org/10.1007/s10915-023-02198-2

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