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Analysis of a conservative fourth-order compact finite difference scheme for the Klein–Gordon–Dirac equation

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Abstract

In this paper, we propose and analyze a conservative fourth-order compact finite difference scheme for the Klein–Gordon–Dirac equation with periodic boundary conditions. Based on matrix knowledge, we convert the point-wise form of the proposed compact scheme into equivalent vector form and analyze its conservative and convergence properties. We prove that the new scheme conserves the total mass and energy in the discrete level and the convergence rate of the scheme, without any restrictions on the grid ratio, is at the order of \(O(h^4 +\tau ^2)\) in \(l^\infty \)-norm, where h and \(\tau \) are spatial and temporal steps, respectively. The techniques for error analysis include the energy method and the mathematical induction. The numerical experiments are carried out to confirm our theoretical analysis.

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Acknowledgements

The authors are grateful to the anonymous reviewers for their valuable suggestions, which help improve this paper significantly.

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Authors and Affiliations

  1. College of Mathematics Science, Hebei Normal University, Shijiazhuang, 050024, People’s Republic of China

    Jiyong Li

  2. Hebei International Joint Research Center for Mathematics and Interdisciplinary Science, Shijiazhuang, 050024, People’s Republic of China

    Jiyong Li

  3. Hebei Key Laboratory of Computational Mathematics and Applications, Shijiazhuang, 050024, People’s Republic of China

    Jiyong Li

  4. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, People’s Republic of China

    Tingchun Wang

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  1. Jiyong Li
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  2. Tingchun Wang
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Correspondence to Jiyong Li.

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Communicated by Jose Alberto Cuminato.

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The research was supported in part by Hebei Natural Science Foundation of China (no. A2014205136), the National Natural Science Foundation of China (no. 11571181) and the Natural Science Foundation of Jiangsu Province (No. BK20171454).

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Li, J., Wang, T. Analysis of a conservative fourth-order compact finite difference scheme for the Klein–Gordon–Dirac equation. Comp. Appl. Math. 40, 114 (2021). https://doi.org/10.1007/s40314-021-01508-4

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  • DOI: https://doi.org/10.1007/s40314-021-01508-4

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