Advertisement

Journal of Scientific Computing

, Volume 56, Issue 2, pp 267–290 | Cite as

A Well-Balanced Reconstruction of Wet/Dry Fronts for the Shallow Water Equations

  • Andreas Bollermann
  • Guoxian Chen
  • Alexander Kurganov
  • Sebastian NoelleEmail author
Article

Abstract

In this paper, we construct a well-balanced, positivity preserving finite volume scheme for the shallow water equations based on a continuous, piecewise linear discretization of the bottom topography. The main new technique is a special reconstruction of the flow variables in wet–dry cells, which is presented in this paper for the one dimensional case. We realize the new reconstruction in the framework of the second-order semi-discrete central-upwind scheme from (Kurganov and Petrova, Commun. Math. Sci., 5(1):133–160, 2007). The positivity of the computed water height is ensured following (Bollermann et al., Commun. Comput. Phys., 10:371–404, 2011): The outgoing fluxes are limited in case of draining cells.

Keywords

Hyperbolic systems of conservation and balance laws Saint-Venant system of shallow water equations Finite volume methods Well-balanced schemes Positivity preserving schemes Wet/dry fronts 

Notes

Acknowledgments

The first ideas for this work were discussed by the authors at a meeting at the “Mathematisches Forschungsinstitut Oberwolfach”. The authors are grateful for the support and inspiring atmosphere there. The research of A. Kurganov was supported in part by the NSF Grant DMS-1115718 and the ONR Grant N000141210833. The research of A. Bollermann, G. Chen and S. Noelle was supported by DFG Grant NO361/3-1 and No361/3-2. G. Chen is partially supported by the National Natural Science Foundation of China (No. 11001211, 51178359).

References

  1. 1.
    Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R., Perthame, B.: A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25, 2050–2065 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bollermann, A., Noelle, S., Lukáčová-Medvid’ová, M.: Finite volume evolution Galerkin methods for the shallow water equations with dry beds. Commun. Comput. Phys. 10, 371–404 (2011)MathSciNetGoogle Scholar
  3. 3.
    de Saint-Venant, A.J.C.: Théorie du mouvement non-permanent des eaux, avec application aux crues des rivière at à l’introduction des warées dans leur lit. C. R. Acad. Sci. Paris 73, 147–154 (1871)zbMATHGoogle Scholar
  4. 4.
    Gallardo, J.M., Parés, C., Castro, M.: On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas. J. Comput. Phys. 2227, 574–601 (2007)CrossRefGoogle Scholar
  5. 5.
    Gallouët, T., Hérard, J.-M., Seguin, N.: Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32, 479–513 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Godlewski, E., Raviart, P.-A.: Numerical approximation of hyperbolic systems of conservation laws. Springer, New York (1996)zbMATHGoogle Scholar
  7. 7.
    Gottlieb, S., Shu, C.-W., Tadmor, E.: High order time discretization methods with the strong stability property. SIAM Rev. 43, 89–112 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Jin, S.: A steady-state capturing method for hyperbolic system with geometrical source terms. M2AN Math. Model. Numer. Anal. 35, 631–645 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Jin, S., Wen, X.: Two interface-type numerical methods for computing hyperbolic systems with geometrical source terms having concentrations. SIAM J. Sci. Comput. 26, 2079–2101 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kröner, D.: Numerical schemes for conservation laws. Wiley, Chichester (1997)zbMATHGoogle Scholar
  11. 11.
    Kurganov, A., Levy, D.: Central-upwind schemes for the Saint-Venant system. M2AN Math. Model. Numer. Anal. 36, 397–425 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Kurganov, A., Petrova, G.: A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun. Math. Sci. 5(1), 133–160 (2007)MathSciNetzbMATHGoogle Scholar
  13. 13.
    LeVeque, R.: Finite volume methods for hyperbolic problems. Cambridge texts in applied mathematics. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  14. 14.
    LeVeque, R.J.: Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146, 346–365 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Liang, Q., Marche, F.: Numerical resolution of well-balanced shallow water equations with complex source terms. Adv. Water Resour. 32(6), 873–884 (2009)CrossRefGoogle Scholar
  16. 16.
    Lie, K.-A., Noelle, S.: On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws. SIAM J. Sci. Comput. 24, 1157–1174 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Morris, M.: CADAM: concerted action on dam break modelling—final report. HR Wallingford, Wallingford (2000)Google Scholar
  18. 18.
    Nessyahu, H., Tadmor, E.: Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Noelle, S., Pankratz, N., Puppo, G., Natvig, J.: Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213, 474–499 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Perthame, B., Simeoni, C.: A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38, 201–231 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Ricchiuto, M., Bollermann, A.: Stabilized residual distribution for shallow water simulations. J. Comput. Phys. 228, 1071–1115 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Russo, G.: Central schemes for balance laws. In: Hyperbolic problems: theory, numerics, applications, vols. I, II (Magdeburg, 2000), pp. 821–829. International Series of Numerical Mathematics vols. 140, 141, Birkhäuser, Basel (2001)Google Scholar
  23. 23.
    Russo, G.: Central schemes for conservation laws with application to shallow water equations. In: Rionero, S., Romano, G. (eds.) Trends and applications of mathematics to mechanics: STAMM 2002, pp. 225–246. Springer Italia SRL, Berlin (2005)CrossRefGoogle Scholar
  24. 24.
    Singh, J., Altinakar, M.S., Ding, Y.: Two-dimensional numerical modeling of dam-break flows over natural terrain using a central explicit scheme. Adv. Water Resour. 34, 1366–1375 (2011)CrossRefGoogle Scholar
  25. 25.
    Sweby, P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21, 995–1011 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Synolakis, C.E.: The runup of long waves. Ph.D. thesis, California Institute of Technology (1986)Google Scholar
  27. 27.
    Synolakis, C.E.: The runup of solitary waves. J. Fluid Mech. 185, 523–545 (1987)zbMATHCrossRefGoogle Scholar
  28. 28.
    Tai, Y.C., Noelle, S., Gray, J.M.N.T., Hutter, K.: Shock-capturing and front-tracking methods for granular avalanches. J. Comput. Phys. 175, 269–301 (2002)zbMATHCrossRefGoogle Scholar
  29. 29.
    van Leer, B.: Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136 (1979)CrossRefGoogle Scholar
  30. 30.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Xing, Y., Shu, C.-W.: A new approach of high order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. Commun. Comput. Phys. 1, 100–134 (2006)zbMATHGoogle Scholar
  32. 32.
    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(12), 1476–1493 (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Andreas Bollermann
    • 1
  • Guoxian Chen
    • 1
    • 2
  • Alexander Kurganov
    • 3
  • Sebastian Noelle
    • 1
    Email author
  1. 1.IGPM, RWTH Aachen University of TechnologyAachenGermany
  2. 2.School of Mathematics and StatisticsWuhan UniversityWuhanPeople’s Republic of China
  3. 3.Mathematics DepartmentTulane UniversityNew OrleansUSA

Personalised recommendations