Journal of Scientific Computing

, Volume 74, Issue 2, pp 640–666 | Cite as

High Order Positivity- and Bound-Preserving Hybrid Compact-WENO Finite Difference Scheme for the Compressible Euler Equations



Based on the same hybridization framework of Don et al. (SIAM J Sci Comput 38:A691–A711 2016), an improved hybrid scheme employing the nonlinear 5th-order characteristic-wise WENO-Z5 finite difference scheme for capturing high gradients and discontinuities in an essentially non-oscillatory manner and the linear 5th-order conservative compact upwind (CUW5) scheme for resolving the fine scale structures in the smooth regions of the solution in an efficient and accurate manner is developed. By replacing the 6th-order non-dissipative compact central scheme (CCD6) with the CUW5 scheme, which has a build-in dissipation, there is no need to employ an extra high order smoothing procedure to mitigate any numerical oscillations that might appear in an hybrid scheme. The high order multi-resolution algorithm of Harten is employed to detect the smoothness of the solution. To handle the problems with extreme conditions, such as high pressure and density ratios and near vacuum states, and detonation diffraction problems, we design a positivity- and bound-preserving limiter by extending the one developed in Hu et al. (J Comput Phys 242, 2013) for solving the high Mach number jet flows, detonation diffraction problems and detonation passing multiple obstacles problems. Extensive one- and two-dimensional shocked flow problems demonstrate that the new hybrid scheme is less dispersive and less dissipative, and allows a potential speedup up to a factor of more than one and half times faster than the WENO-Z5 scheme.


Weighted essentially non-oscillatory Compact upwind Hybrid Positivity-preserving 

Mathematics Subject Classfication

65P30 77Axx 



The authors would like to acknowledge the funding support of this research by National Natural Science Foundation of China (11201441, 11325209, 11521062), Science Challenge Project (TZ2016001), Natural Science Foundation of Shandong Province (ZR2012AQ003) and Fundamental Research Funds for the Central Universities (201562012). The author (Don) also likes to thank the Ocean University of China for providing the startup fund (201712011) that is used in supporting this work.


  1. 1.
    Adams, N., Shariff, K.: High-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems. J. Comput. Phys. 127, 27–51 (1996)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227, 3101–3211 (2008)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Costa, B., Don, W.S.: High order hybrid central-WENO finite difference scheme for conservation laws. J. Comput. Appl. Math. 204(2), 209–218 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Don, W.S., Gao, Z., Li, P., Wen, X.: Hybrid compact-WENO finite difference scheme with conjugate Fourier shock detection algorithm for hyperbolic conservation laws. SIAM J. Sci. Comput. 38(2), A691–A711 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Don, W.S., Borges, R.: Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes. J. Comput. Phys. 250, 347–372 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Guo, Y., Xiong, T., Shi, Y.: A positivity-preserving high order finite volume compact-WENO scheme for compressible Euler equations. J. Comput. Phys. 274, 505–523 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357–393 (1983)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hu, X.Y., Adams, N.A., Shu, C.-W.: Positivity-preserving method for high-order conservative schemes solving compressible Euler equations. J. Comput. Phys. 242, 169–180 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hu, C., Shu, C.-W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 50, 97–127 (1999)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Jia, F., Gao, Z., Don, W.S.: A spectral study on the dissipation and dispersion of the WENO schemes. J. Sci. Comput. 63, 69–77 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Jiang, G.S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lele, S.A.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103(1), 16–42 (1992)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Niu, Y., Gao, Z., Don, W.S., Xie, S.S., Li, P.: Hybrid compact-WENO finite difference scheme for detonation waves simulations, spectral and high order methods for partial differential equations ICOSAHOM 2014. In: Kirby, R.M., Berzins, M., Hesthaven, J.S. (eds.) Lecture Notes on Computer Science Engineering, vol. 106. Springer, New York (2015)Google Scholar
  15. 15.
    Pirozzoli, S.: Conservative hybrid compact-WENO schemes for shock-turbulence interaction. J. Comput. Phys. 178(1), 81–117 (2002)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ren, Y.X., Liu, M., Zhang, H.: A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 192(2), 365–386 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Shu, C.-W.: High order weighted essentially non-oscillatory schemes for convection dominated problems. SIAM Rev. 51, 82–126 (2009)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Tolstykh, A.I., Lipavskii, M.V.: On performance of methods with third- and fifth-order compact upwind differencing. J. Comput. Phys. 140(2), 205–232 (1998)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Vasilyev, O., Lund, T., Moin, P.: A general class of commutative filters for LES in complex geometries. J. Comput. Phys. 146(1), 82–104 (1998)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Wang, C., Zhang, X.X., Shu, C.-W., Ning, J.: Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations. J. Comput. Phys. 231, 653–665 (2012)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54(1), 115–173 (1984)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Zhang, X.X., Shu, C.-W.: On positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229, 8918–8934 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Zhang, X.X., Shu, C.-W.: Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments. Proc. R. Soc. A Math. Phys. Eng. Sci. 467, 2752–2776 (2011)MATHGoogle Scholar
  24. 24.
    Zhang, X.X., Shu, C.-W.: Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms. J. Comput. Phys. 230, 1238–1248 (2011)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Zhang, X.X., Shu, C.-W.: Positivity-preserving high order finite difference WENO schemes for compressible Euler equations. J. Comput. Phys. 231, 2245–2258 (2012)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Zhuang, M., Cheng, R.F.: Optimized upwind dispersion-relation-preserving finite difference schemes for computational aeroacoustics. AIAA J. 36(11), 2146–2148 (1998)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.State Key Laboratory of Explosion Science and TechnologyBeijing Institute of TechnologyBeijingChina
  2. 2.School of Mathematical SciencesOcean University of ChinaQingdaoChina

Personalised recommendations