Journal of Scientific Computing

, Volume 80, Issue 1, pp 538–554 | Cite as

A New Approach for Designing Moving-Water Equilibria Preserving Schemes for the Shallow Water Equations

  • Yuanzhen Cheng
  • Alina Chertock
  • Michael Herty
  • Alexander KurganovEmail author
  • Tong Wu


We construct a new second-order moving-water equilibria preserving central-upwind scheme for the one-dimensional Saint-Venant system of shallow water equations. The idea is based on a reformulation of the source terms as integral in the flux function. Reconstruction of the flux variable yields then a third order equation that can be solved exactly. This procedure does not require any further modification of existing schemes. Several numerical tests are performed to verify the ability of the proposed scheme to accurately capture small perturbations of steady states.


Shallow water equations Central-upwind scheme Well-balanced method Steady-state solutions (equilibria) Moving-water and still-water equilibria 

Mathematics Subject Classification

76M12 65M08 35L65 86-08 86A05 



The work of A. Chertock was supported in part by NSF Grants DMS-1521051 and DMS-1818684. The work of Y. Cheng was supported in part by NSF Grant DMS-1521009. The work of A. Kurganov was supported in part by NSFC Grant 11771201 and NSF Grant DMS-1521009. The work of M. Herty was supported in part by DFG HE5386/13-15, the cluster of excellence DFG EXC128 “Integrative Production Technology for High-Wage Countries” and the BMBF project KinOpt.


  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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berthon, C., Marche, F., Turpault, R.: An efficient scheme on wet/dry transitions for shallow water equations with friction. Comput. Fluids 48, 192–201 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bollermann, A., Chen, G., Kurganov, A., Noelle, S.: A well-balanced reconstruction for wet/dry fronts for the shallow water equations. J. Sci. Comput. 56, 267–290 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bouchut, F., Morales, T.: A subsonic-well-balanced reconstruction scheme for shallow water flows. SIAM J. Numer. Anal. 48, 1733–1758 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bryson, S., Epshteyn, Y., Kurganov, A., Petrova, G.: Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system. M2AN Math. Model. Numer. Anal. 45, 423–446 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Castro, M.J., López-García, J.A., Parés, C.: High order exactly well-balanced numerical methods for shallow water systems. J. Comput. Phys. 246, 242–264 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Castro Díaz, M. J., Kurganov, A., Morales de Luna, T.: Path-conservative central-upwind schemes for nonconservative hyperbolic systems. ESAIM Math. Model. Numer. Anal. To appearGoogle Scholar
  9. 9.
    Cea, L., Vázquez-Cendón, M.E.: Unstructured finite volume discretisation of bed friction and convective flux in solute transport models linked to the shallow water equations. J. Comput. Phys. 231, 3317–3339 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cheng, Y., Kurganov, A.: Moving-water equilibria preserving central-upwind schemes for the shallow water equations. Commun. Math. Sci. 14, 1643–1663 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chertock, A., Cui, S., Kurganov, A., Özcan, Ş.N., Tadmor, E.: Well-balanced schemes for the Euler equations with gravitation: conservative formulation using global fluxes. J. Comput. Phys. 358, 36–52 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chertock, A., Cui, S., Kurganov, A., Tong, W.: Steady state and sign preserving semi-implicit Runge–Kutta methods for ODEs with stiff damping term. SIAM J. Numer. Anal. 53, 1987–2008 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chertock, A., Cui, S., Kurganov, A., Wu, T.: Well-balanced positivity preserving central-upwind scheme for the shallow water system with friction terms. Int. J. Numer. Meth. Fluids 78, 355–383 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chertock, A., Dudzinski, M., Kurganov, A., Lukáčová-Medvid’ová, M.: Well-balanced schemes for the shallow water equations with Coriolis forces. Numer. Math. 138, 939–973 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chertock, A., Herty, M., Özcan, Ş. N.: Well-balanced central-upwind schemes for \(2\times 2\) systems of balance laws. In: Klingenberg, C., Westdickenberg, M. (eds.) Theory, Numerics and Applications of Hyperbolic Problems I, vol. 236 of Springer Proceedings in Mathematics and Statistics, Springer International Publishing, pp. 345–361 (2018)Google Scholar
  16. 16.
    de Saint-Venant, A.: Thèorie du mouvement non-permanent des eaux, avec application aux crues des rivière at à l’introduction des marèes dans leur lit., C.R. Acad. Sci. Paris, vol. 73, pp. 147–154 (1871)Google Scholar
  17. 17.
    Gallardo, J., 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. 227, 574–601 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gosse, L.: A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comput. Math. Appl. 39, 135–159 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Goutal, N., Maurel, F.: Proceedings of the second workshop on dam-break wave simulation, tech. report, Technical Report HE-43/97/016/A, Electricité de France, Département Laboratoire National d’Hydraulique, Groupe Hydraulique Fluviale, (1997)Google Scholar
  20. 20.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Khan, A., Lai, W.: Modeling Shallow Water Flows Using the Discontinuous Galerkin Method. CRC Press, Boca Raton (2014)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kurganov, A.: Finite-volume schemes for shallow-water equations. Acta Numer. 27, 289–351 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kurganov, A., Levy, D.: Central-upwind schemes for the Saint-Venant system. M2AN Math. Model. Numer. Anal. 36, 397–425 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kurganov, A., Lin, C.-T.: On the reduction of numerical dissipation in central-upwind schemes. Commun. Comput. Phys. 2, 141–163 (2007)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Kurganov, A., Noelle, S., Petrova, G.: Semi-discrete central-upwind scheme for hyperbolic conservation laws and Hamilton–Jacobi equations. SIAM J. Sci. Comput. 23, 707–740 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kurganov, A., Petrova, G.: A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun. Math. Sci. 5, 133–160 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kurganov, A., Polizzi, A.: Non-oscillatory central schemes for a traffic flow model with Arrhenius look-ahead dynamics. Netw. Heterog. Media 4, 416–451 (2009)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Kurganov, A., Prugger, M., Wu, T.: Second-order fully discrete central-upwind scheme for two-dimensional hyperbolic systems of conservation laws. SIAM J. Sci. Comput. 39, A947–A965 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kurganov, A., Tadmor, E.: New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160, 241–282 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kurganov, A., Tadmor, E.: Solution of two-dimensional riemann problems for gas dynamics without riemann problem solvers. Numer. Methods Partial Differ. Equ. 18, 584–608 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    LeVeque, R.: Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146, 346–365 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Nessyahu, H., Tadmor, E.: Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Noelle, S., Xing, Y., Shu, C.-W.: High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys. 226, 29–58 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Perthame, B., Simeoni, C.: A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38, 201–231 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Ricchiuto, M., Bollermann, A.: Stabilized residual distribution for shallow water simulations. J. Comput. Phys. 228, 1071–1115 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Russo, G.: Central schemes for conservation laws with application to shallow water equations. In: Trends and Applications of Mathematics to Mechanics. Springer Milan, pp. 225–246 (2005)Google Scholar
  37. 37.
    Russo, G., Khe, A.: High order well balanced schemes for systems of balance laws. In: Hyperbolic Problems: Theory, Numerics and Applications, vol. 67 of Proceedings of Symposium. Applied Mathematics, American Mathematical Society, Providence, RI, pp. 919–928 (2009)Google Scholar
  38. 38.
    Sweby, P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21, 995–1011 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    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)CrossRefzbMATHGoogle Scholar
  40. 40.
    Vazquez-Cendon, M.: Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregualr geometry. J. Comput. Phys. 148, 497–526 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Xing, Y.: Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium. J. Comput. Phys. 257, 536–553 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Xing, Y., Shu, C.-W.: A survey of high order schemes for the shallow water equations. J. Math. Study 47, 221–249 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Xing, Y., Shu, C.-W., Noelle, S.: On the advantage of well-balanced schemes for moving-water equilibria of the shallow water equations. J. Sci. Comput. 48, 339–349 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Yuanzhen Cheng
    • 1
  • Alina Chertock
    • 2
  • Michael Herty
    • 3
  • Alexander Kurganov
    • 1
    • 4
    Email author
  • Tong Wu
    • 1
  1. 1.Mathematics DepartmentTulane UniversityNew OrleansUSA
  2. 2.Department of MathematicsNorth Carolina State UniversityRaleighUSA
  3. 3.Department of MathematicsRWTH Aachen UniversityAachenGermany
  4. 4.Department of MathematicsSouthern University of Science and TechnologyShenzhenChina

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