Recent Results on the Improved WENO-Z+ Scheme

  • Rafael Brandão de Rezende BorgesEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)


The WENO-Z scheme is known to achieve less dissipative results than the classical WENO scheme, especially in problems involving both shocks and smooth structures. In Acker et al. (J Comput Phys 313:726–753, 2016), the cause of the improved results of WENO-Z was shown to be its comparatively higher weights on less-smooth substencils. This knowledge was exploited to develop the fifth-order WENO-Z+ scheme, which generalizes WENO-Z by including an extra term for increasing the weights of less-smooth substencils even further. The new scheme WENO-Z+ was shown to achieve even better results than WENO-Z, while keeping the same numerical robustness. In this study, the third- and seventh-order versions of the WENO-Z+ scheme are presented and discussed. The preliminary numerical results make evident that the approach used by WENO-Z+ is also sound for orders other than 5.



The author would like to thank Profs. Wai-Sun Don and Guus Jacobs for organizing together the minisymposium of High-order Methods at ICOSAHOM 2016; Prof. Chi-Wang Shu and the Division of Applied Mathematics of Brown University, where part of this research was done; and the organizers of ICOSAHOM 2016, especially Prof. Marco Bittencourt, for all the support.


  1. 1.
    F. Acker, R.B. de R. Borges, B. Costa, An improved WENO-Z scheme. J. Comput. Phys. 313, 726–753 (2016)Google Scholar
  2. 2.
    B.S. Balsara, C.-W. Shu, Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160(2), 405–452 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    R. Borges, M. Carmona, B. Costa, W.S. Don, An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227(6), 3191–3211 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    M. Castro, B. Costa, W.S. Don, High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230(5), 1766–1792 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    W.-S. Don, R. Borges, Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes. J. Comput. Phys. 250, 347–372 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Y. Ha, C.H. Kim, Y.J. Lee, J. Yoon, An improved weighted essentially non-oscillatory scheme with a new smoothness indicator. J. Comput. Phys. 232(1), 68–86 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    A. Harten, B. Engquist, S. Osher, S.R. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71(2), 231–303 (1987)CrossRefzbMATHGoogle Scholar
  8. 8.
    A.K. Henrick, T.D. Aslam, J.M. Powers, Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207(2), 542–567 (2005)CrossRefzbMATHGoogle Scholar
  9. 9.
    G.-S. Jiang, C.-W. Shu, Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    X.-D. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    S. Pirozzoli, On the spectral properties of shock-capturing schemes. J. Comput. Phys. 219(2), 489–497 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    J. Shi, Y.-T. Zhang, C.-W. Shu, Resolution of high order WENO schemes for complicated flow structures. J. Comput. Phys. 186(2), 690–696 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. NASA/CR-97-206253 ICASE Report 97–65, (1997)Google Scholar
  14. 14.
    C.-W. Shu, High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51(1), 82–126 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    N.K. Yamaleev, M.H. Carpenter, A systematic methodology for constructing high-order energy stable WENO schemes. J. Comput. Phys. 228(11), 4248–4272 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    S. Zhao, N. Lardjane, I. Fedioun, Comparison of improved finite-difference WENO schemes for the implicit large eddy simulation of turbulent non-reacting and reacting high-speed shear flows. Comput. Fluids 95, 74–87 (2014)CrossRefMathSciNetGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Departamento de Análise MatemáticaUniversidade do Estado do Rio de JaneiroRio de Janeiro / RJBrazil

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