High-Order Embedded WENO Schemes

  • Bart S. van LithEmail author
  • Jan H. M. ten Thije Boonkkamp
  • Wilbert L. IJzerman
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)


Embedded WENO schemes are a new family of weighted essentially nonoscillatory schemes that always utilise all adjacent smooth substencils. This results in increased control over the convex combination of lower-order interpolations. We show that more conventional WENO schemes, such as WENO-JS and WENO-Z (Borges et al., J. Comput. Phys., 2008; Jiang and Shu, J. Comput. Phys., 1996), do not exhibit this feature and as such do not always provide a desirable linear combination of smooth substencils. In a previous work, we have already developed the theory and machinery needed to construct embedded WENO methods and shown some five-point schemes (van Lith et al., J. Comput. Phys., 2016). Here, we construct a seven-point scheme and show that it too performs well using some numerical examples from the one-dimensional Euler equations.



This work was generously supported by Philips Lighting and the Intelligent Lighting Institute. The authors would especially like to thank Rafael Borges and Wai-Sun Don for the fruitful discussions and the generous sharing of their Matlab code.


  1. 1.
    R. Borges, M. Carmona, B. Costa, W.-S. Don, An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227, 3191–3211 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations (Wiley, Chichester, 1987)zbMATHGoogle Scholar
  3. 3.
    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
  4. 4.
    L. Fu, X. Y. Hu, N.A. Adams, A family of high-order targeted ENO schemes for compressible-fluid simulations. J. Comput. Phys. 305, 333–359 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    G.-S. Jiang, C.-W. Shu, Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    C.-W. Shu, High order weighted essentially non-oscillatory schemes for convection dominated problems. SIAM Rev. 51(1), 82–126 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    C.-W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J. Comput. Phys. 83(1), 32–78 (1989)CrossRefzbMATHGoogle Scholar
  8. 8.
    R.J. Spiteri, S.J. Ruuth, A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM J. Numer. Anal. 40(2), 469–491 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics (Springer, Berlin, 1997)CrossRefzbMATHGoogle Scholar
  10. 10.
    B.S. van Lith, J.H.M. ten Thije Boonkkamp, W.L. IJzerman, Embedded WENO: a design strategy to improve existing WENO schemes. J. Comput. Phys. 330, 529–549 (2016)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Bart S. van Lith
    • 1
    Email author
  • Jan H. M. ten Thije Boonkkamp
    • 1
  • Wilbert L. IJzerman
    • 1
    • 2
  1. 1.CASAEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Philips LightingEindhovenThe Netherlands

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