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

Chang Yeol Jung, Thien Binh Nguyen

Research output: Contribution to journalArticleResearchpeer-review

8 Citations (Scopus)


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-6, 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-6 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-6 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.

Original languageEnglish
Pages (from-to)147–174
Number of pages28
JournalAdvances in Computational Mathematics
Issue number1
Publication statusPublished - 2018


  • 2D Riemann problem
  • Euler equations
  • Shock-capturing methods
  • Weighted essentially non-oscillatory (WENO) schemes
  • WENO

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