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Fine structures for the solutions of the two-dimensional Riemann problems by high-order WENO schemes

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Abstract

The two-dimensional Riemann problem with polytropic gas is considered. By a restriction on the constant states of each quadrant of the computational domain such that there is only one planar centered wave connecting two adjacent quadrants, there are nineteen genuinely different initial configurations of the problem. The configurations are numerically simulated on a fine grid and compared by the 5th-order WENO-Z5, 6th-order WENO-𝜃6, and 7th-order WENO-Z7 schemes. The solutions are very well approximated with high resolution of waves interactions phenomena and different types of Mach shock reflections. Kelvin-Helmholtz instability-like secondary-scaled vortices along contact continuities are well resolved and visualized. Numerical solutions show that WENO-𝜃6 outperforms the comparing WENO-Z5 and WENO-Z7 in terms of shock capturing and small-scaled vortices resolution. A catalog of the numerical solutions of all nineteen configurations obtained from the WENO-𝜃6 scheme is listed. Thanks to their excellent resolution and sharp shock capturing, the numerical solutions presented in this work can be served as reference solutions for both future numerical and theoretical analyses of the 2D Riemann problem.

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Acknowledgments

This work was supported by the National Research Foundation of Korea grant funded by the Ministry of Education (2015R1D1A1A01059837). The authors would like to thank the anonymous referees for their valuable comments to improve the manuscript.

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Authors and Affiliations

  1. Department of Mathematical Sciences, School of Natural Science, Ulsan National Institute of Science and Technology, UNIST-gil 50, Ulsan, 689-798, Republic of Korea

    Chang-Yeol Jung & Thien Binh Nguyen

  2. School of Mathematical Sciences, Monash University, 9 Rainforest Walk, Clayton, Melbourne, VIC 3800, Australia

    Thien Binh Nguyen

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  1. Chang-Yeol Jung
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  2. Thien Binh Nguyen
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Correspondence to Chang-Yeol Jung.

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Communicated by: Jan Hesthaven

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Jung, CY., Nguyen, T.B. Fine structures for the solutions of the two-dimensional Riemann problems by high-order WENO schemes. Adv Comput Math 44, 147–174 (2018). https://doi.org/10.1007/s10444-017-9538-8

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  • DOI: https://doi.org/10.1007/s10444-017-9538-8

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