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A new ADER discontinuous Galerkin method based on differential transformation procedure for hyperbolic conservation laws

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Abstract

This article develops a new discontinuous Galerkin (DG) method with the one-stage arbitrary derivatives in time and space approach to solve one-dimensional hyperbolic conservation laws. This method employs the differential transformation procedure instead of the Cauchy–Kowalewski procedure to recursively express the spatiotemporal expansion coefficients of the solution through the low-order spatial expansion coefficients. The proposed method is free of solving generalized Riemann problems at inter-cells. Compared with the Runge–Kutta DG methods, the current method needs less computer memory due to no intermediate stages. In summary, this method is one step, one stage, fully discrete, and easily achieves arbitrary high-order accuracy in time and space. Extensive numerical results illustrate the good performances of the present method: high-order accuracy for smooth solutions, good resolution for discontinuous solutions and high efficiency.

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Acknowledgements

This research is supported by the National Natural Science Foundation of China (11771228).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Qingdao University, Qingdao, 266071, Shandong, People’s Republic of China

    Yingjuan Zhang, Gang Li, Shouguo Qian & Jinmei Gao

Authors
  1. Yingjuan Zhang
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  2. Gang Li
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  3. Shouguo Qian
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  4. Jinmei Gao
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Corresponding author

Correspondence to Shouguo Qian.

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Communicated by Abdellah Hadjad.

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Appendices

Appendix A: The algorithm of the differential transformation procedure for Burgers’ equation

figure a

Appendix B: The algorithm of the differential transformation procedure for Euler equations

figure b

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Zhang, Y., Li, G., Qian, S. et al. A new ADER discontinuous Galerkin method based on differential transformation procedure for hyperbolic conservation laws . Comp. Appl. Math. 40, 139 (2021). https://doi.org/10.1007/s40314-021-01525-3

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  • DOI: https://doi.org/10.1007/s40314-021-01525-3

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