Journal of Scientific Computing

, Volume 51, Issue 3, pp 527–559 | Cite as

Hybrid Well-balanced WENO Schemes with Different Indicators for Shallow Water Equations

  • Gang Li
  • Changna Lu
  • Jianxian QiuEmail author


In (J. Comput. Phys. 229: 8105–8129, 2010), Li and Qiu investigated the hybrid weighted essentially non-oscillatory (WENO) schemes with different indicators for Euler equations of gas dynamics. In this continuation paper, we extend the method to solve the one- and two-dimensional shallow water equations with source term due to the non-flat bottom topography, with a goal of obtaining the same advantages of the schemes for the Euler equations, such as the saving computational cost, essentially non-oscillatory property for general solution with discontinuities, and the sharp shock transition. Extensive simulations in one- and two-dimensions are provided to illustrate the behavior of this procedure.


WENO approximation Up-wind linear approximation Troubled-cell indicator Shallow water equations Hybrid schemes Source term 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alcrudo, F., Benkhaldoun, F.: Exact solution to the Riemann problem of the shallow water euqaionts with a bottom step. Comput. Fluids 30, 643–671 (2001) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bermudez, A., Vazquez, M.E.: Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23, 1049–1071 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework. Math. Comput. 52, 411–435 (1989) MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cosat, B., Don, W.S.: High order hybrid central-WENO finite difference scheme for conservation laws. J. Comput. Appl. Math. 204, 209–218 (2007) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Crnjaric, N., Vukovic, S., Sopta, L.: Extension of ENO and WENO schemes to one-dimensional sediment transport equations. Comput. Fluids 33, 31–56 (2004) CrossRefzbMATHGoogle Scholar
  6. 6.
    Fennema, R.J., Chaudhry, M.H.: Explicit methods for 2D transient free surface flows. J. Hydraul. Eng. 116, 1013–1034 (1990) CrossRefGoogle Scholar
  7. 7.
    Harten, A.: Adaptive multiresolution schemes for shock computations. J. Comput. Phys. 115, 319–338 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jiang, G., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Krivodonova, L., Xin, J., Remacle, J.-F., Chevaugeon, N., Flaherty, J.: Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48, 323–338 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    LeVeque, R.J.: Balancing source terms and flux gradient on high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146, 346–365 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Li, G., Qiu, J.: Hybrid weighted essentially non-oscillatory schemes with different indicators. J. Comput. Phys. 229, 8105–8129 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pirozzoli, S.: Conservative hybrid compact-WENO schemes for shock-turbulence interaction. J. Comput. Phys. 178, 81–117 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Qiu, J., Shu, C.-W.: A comparison of troubled-cell indicators for Runge-Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters. SIAM J. Sci. Comput. 27, 995–1013 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Rogers, B.D., Borthwick, A.G.L., Taylor, P.H.: Mathematical balancing of flux gradient and source terms prior to using Roe’s approximate Riemann solver. J. Comput. Phys. 192, 422–451 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shu, C.-W., Osher, S.: Efficient implementiation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. NASA/CR-97-206253, ICASE Report NO.97-65 (1997) Google Scholar
  17. 17.
    Shu, C.-W.: High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51, 82–126 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Toro, E.F.: Shock-capturing Methods for Free-surface Shallow Flows. Wiley, New York (2001) zbMATHGoogle Scholar
  19. 19.
    Vukovic, S., Sopta, L.: ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations. J. Comput. Phys. 179, 593–621 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Xing, Y., Shu, C.-W.: High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208, 206–227 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Xing, Y., Shu, C.-W.: High-order well-balanced finite difference WENO schemes for a class of hyperbolic systems with source terms. J. Sci. Comput. 27, 477–494 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Xing, Y., Shu, C.-W.: High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. J. Comput. Phys. 214, 567–598 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Xing, Y., Zhang, X., Shu, C.-W.: Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Adv. Water Resour. 33, 1476–1493 (2010) CrossRefGoogle Scholar
  24. 24.
    Zhou, J.G., Causon, D.M., Mingham, C.G., Ingram, D.M.: The surface gradient method for the treatment of source terms in the shallow-water equations. J. Comput. Phys. 168, 1–25 (2001) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zhu, H., Qiu, J.: Adaptive Runge-Kutta discontinuous Galerkin methods using different indicators: one-dimensional case. J. Comput. Phys. 228, 6957–6976 (2009) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingP.R. China
  2. 2.School of Mathematical ScienceQingdao UniversityQingdaoP.R. China
  3. 3.College of Mathematics & PhysicsNanjing University of Information Science & TechnologyNanjingP.R. China
  4. 4.School of Mathematical SciencesXiamen UniversityXiamenP.R. China

Personalised recommendations