Journal of Scientific Computing

, Volume 64, Issue 2, pp 477–507 | Cite as

Fifth Order Multi-moment WENO Schemes for Hyperbolic Conservation Laws

  • Chieh-Sen Huang
  • Feng Xiao
  • Todd Arbogast


A general approach is given to extend WENO reconstructions to a class of numerical schemes that use different types of moments (i.e., multi-moments) simultaneously as the computational variables, such as point values and grid cell averages. The key is to re-map the multi-moment values to single moment values (e.g., cell average or point values), which can then be used to invoke known, standard reconstruction coefficients and smoothness indicators for single moment WENO reconstructions. The WENO reconstructions in turn provide the numerical approximations for the flux functions and other required quantities. One major advantage of using multi-moments for WENO reconstructions is its compactness. We present two new multi-moment WENO (MM-WENO) schemes of fifth order that use reconstructions supported over only three grid cells, as opposed to the usual five. This is similar to the Hermite WENO schemes of Qiu and Shu (J Comput Phys 193:115–135, 2003), which can also be derived using our general approach. Numerical tests demonstrate that the new schemes achieve their designed fifth order accuracy and eliminate spurious oscillations effectively. The numerical solutions to all benchmark tests are of good quality and comparable to the classic, single moment WENO scheme of the same order of accuracy. The basic idea presented in this paper is universal, which makes the WENO reconstruction an easy-to-follow method for developing a wide variety of additional multi-moment numerical schemes.


Multi-moment MM-WENO scheme  Hyperbolic conservation law  Finite volume Finite difference 

Mathematics Subject Classification

65M06 65M08 76M12 76M20 


  1. 1.
    Akoh, R., Ii, S., Xiao, F.: A multi-moment finite volume formulation for shallow water equations on unstructured mesh. J. Comput. Phys. 229, 4567–4590 (2010)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    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)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Chen, C.G., Xiao, F.: Shallow water model on cubed-sphere by multi-moment finite volume method. J. Comput. Phys. 227, 5019–5044 (2008)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high-order accurate essentially nonoscillatory schemes III. J. Comput. Phys. 71(2), 231–303 (1987)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    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)MATHCrossRefGoogle Scholar
  6. 6.
    Huang, C.-S., Arbogast, T.: An Eulerian–Lagrangian WENO method for nonlinear conservation laws (in preparation)Google Scholar
  7. 7.
    Huang, C.-S., Arbogast, T., Hung, Ch-H: A re-averaged WENO reconstruction and a third order CWENO scheme for hyperbolic conservation laws. J. Comput. Phys. 262, 291–312 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Huang, C.-S., Arbogast, T., Qiu, J.: An Eulerian–Lagrangian WENO finite volume scheme for advection problems. J. Comput. Phys. 231(11), 4028–4052 (2012). doi: 10.1016/ MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Ii, S., Xiao, F.: CIP/multi-moment finite volume method for Euler equations, a semi-Lagrangian characteristic formulation. J. Comput. Phys. 222, 849–871 (2007)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Ii, S., Xiao, F.: High order multi-moment constrained finite volume method. Part I: basic formulation. J. Comput. Phys. 228, 3669–3707 (2009)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Jiang, J.-S., Peng, D.: Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21(6), 2126–2143 (2000)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Jiang, J.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Qiu, J., Shu, C.-W.: Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method: one-dimensional case. J. Comput. Phys. 193, 115–135 (2003)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Shu,C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. ICASE-1997-65Google Scholar
  16. 16.
    Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes II. J. Comput. Phys. 83, 32–78 (1989)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Xiao, F.: Unified formulation for compressible and incompressible flows by using multi integrated moments I: one-dimensional inviscid compressible flow. J. Comput. Phys. 195, 629–654 (2004)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Xiao, F., Akoh, R., Ii, S.: Unified formulation for compressible and incompressible flows by using multi integrated moments II: multi-dimensional version for compressible and incompressible flows. J. Comput. Phys. 213, 31–56 (2006)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Xiao, F., Ikebata, A., Hasegawa, T.: Numerical simulations of free-interface fluids by a multi integrated moment method. Comput. Struct. 83, 409–423 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Xiao, F., Yabe, T.: Completely conservative and oscillationless semi-Lagrangian schemes for advection transportation. J. Comput. Phys. 170, 498–522 (2001)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Xiao, F., Yabe, T., Ito, T.: Constructing oscillation preventing scheme for advection equation by rational function. Comput. Phys. Commun. 93, 1–12 (1996)MATHCrossRefGoogle Scholar
  22. 22.
    Xiao, F., Yabe, T., Peng, X., Kobayashi, H.: Conservative and oscillation-less atmospheric transport schemes based on rational functions. J. Geophys. Res. 107(D22), 4609 (2002)CrossRefGoogle Scholar
  23. 23.
    Yabe, T., Xiao, F., Utsumi, T.: The constrained interpolation profile method for multiphase analysis. J. Comput. Phys. 169, 556–593 (2001)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Applied Mathematics and National Center for Theoretical SciencesNational Sun Yat-sen UniversityKaohsiungTaiwan, ROC
  2. 2.Tokyo Institute of TechnologyYokohamaJapan
  3. 3.Institute for Computational Engineering and SciencesUniversity of Texas at AustinAustinUSA

Personalised recommendations