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The stabilized-trigonometric scalar auxiliary variable approach for gradient flows and its efficient schemes

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Abstract

We develop a trigonometric scalar auxiliary variable (TSAV) approach for constructing linear, totally decoupled, and energy-stable numerical methods for gradient flows. An auxiliary variable r based on the trigonometric form of the nonlinear potential functional removes the bounded-from-below restriction. By adding a positive constant greater than 1, the positivity preserving property of r can be satisfied. Furthermore, the phase-field variables and auxiliary variable r can be treated in a totally decoupled manner, which simplifies the algorithm. A practical stabilization method is employed to suppress the effect of an explicit nonlinear term. Using our proposed approach, temporally first-order and second-order methods are easily constructed. We prove analytically the discrete energy dissipation laws of the first- and second-order schemes. Furthermore, we propose a multiple TSAV approach for complex systems with multiple components. A comparison of stabilized-SAV (S-SAV) and stabilized-TSAV (S-TSAV) approaches is performed to show their efficiency. Two-dimensional numerical experiments demonstrated the desired accuracy and energy stability.

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Acknowledgements

J. Yang was 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 are grateful to the reviewers for their contributions to improve 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. The stabilized-trigonometric scalar auxiliary variable approach for gradient flows and its efficient schemes. J Eng Math 129, 18 (2021). https://doi.org/10.1007/s10665-021-10155-x

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