On the Full and Global Accuracy of a Compact Third Order WENO Scheme: Part II

Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 112)


Recently, we showed in (O. Kolb, SIAM J. Numer. Anal., 52 (2014), pp. 2335–2355) for which parameter range the compact third order WENO reconstruction procedure introduced in (D. Levy, G. Puppo, and G. Russo, SIAM J. Sci. Comput., 22 (2000), pp. 656–672) reaches the optimal order of accuracy (h3 in the smooth case and h2 near discontinuities). This is the case for the parameter choice ɛ = Kh q in the weight design with q ≤ 3 and pq ≥ 2, where p ≥ 1 is the exponent used in the computation of the weights in the WENO scheme. While these theoretical results for the convergence rates of the WENO reconstruction procedure could also be validated in the numerical tests, the application within the semi-discrete central scheme of (A. Kurganov, and D. Levy, SIAM J. Sci. Comput., 22 (2000), pp. 1461–1488) together with a third order TVD-Runge-Kutta scheme for the time integration did not yield a third order accurate scheme in total for q > 2. The aim of this follow-up paper is to explain this observation with further analytical and numerical results.


Weighted Essentially Non-oscillatory (WENO) WENO Scheme Order WENO WENO Reconstruction Reconstruction Procedure 
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  1. 1.
    F. Aràndiga, A. Baeza, A.M. Belda, P. Mulet, Analysis of WENO schemes for full and global accuracy. SIAM J. Numer. Anal. 49 (2), 893–915 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    S. Bryson, D. Levy, Mapped WENO and weighted power ENO reconstructions in semi-discrete central schemes for Hamilton-Jacobi equations. Appl. Numer. Math. 56 (9), 1211–1224 (2006)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    I. Cravero, M. Semplice, On the accuracy of WENO and CWENO reconstructions of third order on nonuniform meshes. J. Sci. Comput. 67 (3), 1219–1246 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    H. Feng, F. Hu, R. Wang, A new mapped weighted essentially non-oscillatory scheme. J. Sci. Comput. 51 (2), 449–473 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    S. Gottlieb, C.-W. Shu, Total variation diminishing Runge-Kutta schemes. Math. Comput. 67, 73–85 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. Harten, B. Engquist, S. Osher, S.R. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71 (1), 231–303 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    G.-S. Jiang, C.-W. Shu, Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126 (1), 202–228 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    O. Kolb, On the full and global accuracy of a compact third order WENO scheme. SIAM J. Numer. Anal. 52 (5), 2335–2355 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A. Kurganov, D. Levy, A third-order semidiscrete central scheme for conservation laws and convection-diffusion equations. SIAM J. Sci. Comput. 22 (4), 1461–1488 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    D. Levy, G. Puppo, G. Russo, Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput. 22 (2), 656–672 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    X.-D. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115 (1), 200–212 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    S. Serna, A. Marquina, Power ENO methods: a fifth-order accurate weighted power ENO method. J. Comput. Phys. 194 (2), 632–658 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    N.K. Yamaleev, M.H. Carpenter, Third-order energy stable WENO scheme. J. Comput. Phys. 228 (8), 3025–3047 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MannheimMannheimGermany

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