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Superconvergence of the space-time discontinuous Galerkin method for linear nonhomogeneous hyperbolic equations

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Abstract

In this study, we discuss the superconvergence of the space-time discontinuous Galerkin method for the first-order linear nonhomogeneous hyperbolic equation. By using the local differential projection method to construct comparison function, we prove that the numerical solution is \((2n+1)\)-th order superconvergent at the downwind-biased Radau points in the discrete \(L^2\)-norm. As a by-product, we obtain a point-wise superconvergence with order \(2n+\frac{1}{2}\) in vertices. We also find that, in order to obtain these superconvergence results, the source integral term has to be approximated by \((n+1)\)-point Radau-quadrature rule. Numerical results are presented to verify our theoretical findings.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 12071128), the Excellent Youth Foundation of Hunan Province of China (Grant No. 2018JJ1042), the Research Foundation of Education Bureau of Hunan Province of China (Grant Nos. 18B002, 16C0951) and the Construct Program of the Key Discipline in Hunan Province. We would also like to thank the referees for their valuable comments.

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Authors and Affiliations

  1. Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, China

    Hongling Hu & Chuanmiao Chen

  2. Institute of Mathematics and Physics, College of Science, Central South University of Forestry and Technology, Changsha, 410004, China

    Shufang Hu

  3. School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, 410083, China

    Kejia Pan

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  1. Hongling Hu
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Hu, H., Chen, C., Hu, S. et al. Superconvergence of the space-time discontinuous Galerkin method for linear nonhomogeneous hyperbolic equations. Calcolo 58, 16 (2021). https://doi.org/10.1007/s10092-021-00408-7

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