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.
Similar content being viewed by others
References
Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227, 3191–3211 (2008)
Balsara, D.S., Shu, C.-W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160, 405–452 (2000)
Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-z schemes for hyperbolic conservation laws. J. Comput. Phys. 230, 1766–1792 (2011)
Gottlieb, S., Shu, C.-W.: Total variation dimishing Runge-Kutta schemes. Math. Comput. 67(221), 73–85 (1998)
Henrick, A.K., Aslam, T.D., Powers, J. M.: Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points. J. Comput. Phys. 207, 542–567 (2005)
Ha, Y., Kim, C.H., Lee, Y., Yoon, J.: An improved weighted essentially non-oscillatory scheme with a new smoothness indicator. J. Comput. Phys. 232, 68–86 (2013)
Han, E., Li, J., Tang, H.: Accuracy of the adaptive GRP scheme and the simulation of 2-D Riemann problems for compressible Euler equations. Commun. Comput. Phys. 10(3), 577–606 (2011)
Harten, A., Osher, S.: Uniformly high-order accurate nonoscillatory schemes, I. SIAM J. Numer. Anal. 24(2), 279–309 (1987)
Hu, X.Y., Wang, Q., Adams, N.A.: An adaptive central-upwind weighted essentially non-oscillatory scheme. J. Comput. Phys. 229, 8952–8965 (2010)
Jung, C.-Y., Nguyen, T.B.: A new adaptive weighted essentially non-oscillatory WENO- 𝜃 scheme for hyperbolic conservation laws. Submitted, preprint at arXiv:1504.00731
Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)
Kurganov, A., Tadmor, E.: Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. Numer. Meth. Part. D. E. 18 (5), 584–608 (2002)
Lax, P., Liu, X.-D.: Solution of two-dimensional Riemann problems of gas dynamics by positive schemes. SIAM J. Sci. Comput. 19(2), 319–340 (1998)
Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)
Li, J., Sheng, W., Zhang, T., Zheng, Y.: Two-dimensional Riemann problems: from scalar conservation laws to compressible Euler equations. Acta Math. Sci. 29B(4), 777–802 (2009)
Liska, R., Wendroff, B.: Comparison of several difference schemes on 1D and 2D test problems for the Euler equations. SIAM J. Sci. Comput. 25(3), 995–1017 (2003)
Seaïd, M.: High-resolution relaxation scheme for the two-dimensional Riemann problems in gas dynamics. Numerical Methods for Partial Differential Equations 22 (2), 397–413 (2006)
Shu, C.-W.: High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51(1), 82–126 (2009)
Schulz-Rinne, C.W.: Classification of the Riemann problem for two-dimensional gas dynamics. SIAM J. Math. Anal. 24(1), 76–88 (1993)
Schulz-Rinne, C.W., Collins, J.P., Glaz, H.M.: Numerical solution of the Riemann problem for two-dimensional gas dynamics. SIAM J. Sci. Comput. 14(6), 1394–1414 (1993)
Shen, Y., Zha, G.: Improved Seventh-Order WENO Scheme, AIAA 2010–1451, 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, 4–7 January 2010, Orlando, Florida (2010)
Tesdall, A.M., Sanders, R., Keyfitz, B.L.: The triple point paradox for the nonlinear wave system. SIAM J. Appl. Math. 67(2), 321–336 (2006)
Taylor, E.M., Wu, M., Martiń, M.P.: Optimization of nonlinear error for weighted essentially non-oscillatory methods in direct numerical simulations of compressible turbulence. J Comput. Phys. 223, 384–397 (2007)
Wagner, D.: The Riemann problem in two space dimensions for a single conservation law. SIAM J. Math. Anal. 14(3), 534–559 (1983)
Yamaleev, N.K., Carpenter, M.H.: A systematic methodology for constructing high-order energy stable WENO schemes. J. Comput. Phys. 228, 4248–4272 (2009)
Zhang, T., Chen, G.: Some fundamental concepts about system of two spatial dimensional conservation laws. Acta Math. Sci. 6, 463–474 (1986)
Zhang, T., Chen, G.-C., Yang, S.: On the 2-D Riemann problm for the compressible Euler equations I. Interaction of shocks and rarefaction waves. Dis. Cont. Dyn. Sys. 1, 555–584 (1995)
Zhang, T., Chen, G.-Q., Yang, S.: On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities. Dis. Cont. Dyn. Sys. 6(2), 419–430 (2000)
Zheng, Y.: Systems of conservation laws: two-dimensional Riemann problems. Birkhäuser, Boston (2001)
Zhang, T., Zheng, Y.: Two-dimensional Riemann problem for a single conservation law. Trans. Amer. Math. Soc. 312(2), 589–619 (1989)
Zhang, T., Zheng, Y.: Conjecture on the structure of solutions of the Riemann problem for the two-dimensional gas dynamics systems. SIAM J. Math. Anal. 21(3), 593–630 (1990)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Jan Hesthaven
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-017-9538-8
Keywords
- 2D Riemann problem
- Euler equations
- Shock-capturing methods
- Weighted essentially non-oscillatory (WENO) schemes
- WENO-𝜃