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A Linearly Implicit and Local Energy-Preserving Scheme for the Sine-Gordon Equation Based on the Invariant Energy Quadratization Approach

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Abstract

In this paper, we develop a novel, linearly implicit and local energy-preserving scheme for the sine-Gordon equation. The basic idea is from the invariant energy quadratization approach to construct energy stable schemes for gradient systems, which are energy dispassion. We here take the sine-Gordon equation as an example to show that the invariant energy quadratization approach is also an efficient way to construct linearly implicit and local energy-conserving schemes for energy-conserving systems. Utilizing the invariant energy quadratization approach, the sine-Gordon equation is first reformulated into an equivalent system, which inherits a modified local energy conservation law. The new system are then discretized by the conventional finite difference method and a semi-discretized system is obtained, which can conserve the semi-discretized local energy conservation law. Subsequently, the linearly implicit structure-preserving method is applied for the resulting semi-discrete system to arrive at a fully discretized scheme. We prove that the resulting scheme can exactly preserve the discrete local energy conservation law. Moveover, with the aid of the classical energy method, an unconditional and optimal error estimate for the scheme is established in discrete \(H_h^1\)-norm. Finally, various numerical examples are addressed to confirm our theoretical analysis and demonstrate the advantage of the new scheme over some existing local structure-preserving schemes.

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Acknowledgements

The authors would like to express sincere gratitude to the referees for their insightful comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11771213, 61872422, 11871418), the National Key Research and Development Project of China (Grant Nos. 2016YFC0600310, 2018YFC0603500, 2018YFC1504205), the Major Projects of Natural Sciences of University in Jiangsu Province of China (Grant Nos. 15KJA110002, 18KJA110003), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20171480), the Foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems (201905) and the Yunnan Provincial Department of Education Science Research Fund Project (2019J0956).

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Authors and Affiliations

  1. School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, 650221, China

    Chaolong Jiang

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

    Wenjun Cai & Yushun Wang

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  1. Chaolong Jiang
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Correspondence to Yushun Wang.

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Jiang, C., Cai, W. & Wang, Y. A Linearly Implicit and Local Energy-Preserving Scheme for the Sine-Gordon Equation Based on the Invariant Energy Quadratization Approach. J Sci Comput 80, 1629–1655 (2019). https://doi.org/10.1007/s10915-019-01001-5

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