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Linear and energy stable schemes for the Swift–Hohenberg equation with quadratic-cubic nonlinearity based on a modified scalar auxiliary variable approach

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Abstract

In this study, we develop linear and energy stable numerical schemes for the Swift–Hohenberg equation with quadratic-cubic nonlinearity. A modified scalar auxiliary variable (SAV) approach is used to construct the temporally first- and second-order accurate discretizations. Different from the classical SAV approach, the proposed schemes permit us to solve the governing equations in a step-by-step manner, i.e., the calculation of inner product is not needed. We analytically prove the energy stability. We solve the resulting system of discrete equations using the linear multigrid method. We perform various numerical examples to show the accuracy and energy stability of the proposed method. The pattern formations in two- and three-dimensional spaces are also simulated.

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Acknowledgements

J. Yang is supported by China Scholarship Council (201908260060). The corresponding author (J.S. Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A2C1003053). The authors appreciate the reviewers for their constructive comments, which have improved the quality of this paper.

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

    Junxiang Yang & Junseok Kim

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  1. Junxiang Yang
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Correspondence to Junseok Kim.

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Yang, J., Kim, J. Linear and energy stable schemes for the Swift–Hohenberg equation with quadratic-cubic nonlinearity based on a modified scalar auxiliary variable approach. J Eng Math 128, 21 (2021). https://doi.org/10.1007/s10665-021-10122-6

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