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Linearly implicit and second-order energy-preserving schemes for the modified Korteweg-de Vries equation

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Abstract

In this paper, some linearly implicit modified energy-conserving schemes are proposed for the modified Korteweg-de Vries equation (mKdV). The proposed schemes are based on the recently developed invariant energy quadratization (IEQ) approach and the scalar auxiliary variable (SAV) approach. We first introduce an auxiliary variable to transform the original model into an equivalent system, with a modified energy functional law. Then, the Fourier pseudospectral method is employed for the spatial discretization, and Crank-Nicolson, and Leap-Frog methods are used for the temporal discretization. We analyze the conservation properties, existence and uniqueness and the linear stability of the proposed schemes. The optimal order convergence rate of the semi-discrete scheme and the fully discrete schemes were analyzed, respectively. At last, some numerical examples are presented to illustrate the effectiveness of the proposed schemes.

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Acknowledgements

The authors are very grateful to the anonymous reviewers for their insightful comments and suggestions.

Funding

This work is partially supported by the National Natural Science Foundation of China (Grant No. 11861047, 41901323, 11801226), Natural Science Foundation of Fujian Province (Grant No. 2019J01831), PhD Start-up Fund of Wuyi University (Grant No.YJ201702), Teacher and Student Scientific Team Fund of Wuyi University (Grant No. 2020-SSTD-003) and Jiangsu Key Laboratory for NSLSCS (Grant No. 201804).

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Authors and Affiliations

  1. Fujian Key Laboratory of Big Data Application and Intellectualization for Tea Industry, College of Mathematics and Computer, Wuyi University, Wu Yi Shan, 354300, China

    Jinliang Yan & Sihui Zheng

  2. Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Jiangsu, 210023, China

    Jinliang Yan

  3. Department of Mathematics and Physics, Jiangsu University of Science and Technology, Zhenjiang, 212003, China

    Ling Zhu

  4. Changzhou Institute of Technology, Changzhou, 213032, China

    Fuqiang Lu

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  1. Jinliang Yan
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Correspondence to Jinliang Yan.

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Yan, J., Zhu, L., Lu, F. et al. Linearly implicit and second-order energy-preserving schemes for the modified Korteweg-de Vries equation. Numer Algor 91, 1511–1546 (2022). https://doi.org/10.1007/s11075-022-01312-z

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