Journal of Scientific Computing

, Volume 39, Issue 1, pp 67–114 | Cite as

High Order Extensions of Roe Schemes for Two-Dimensional Nonconservative Hyperbolic Systems

  • M. J. Castro
  • E. D. Fernández-Nieto
  • A. M. Ferreiro
  • J. A. García-Rodríguez
  • C. Parés


This paper is concerned with the development of well-balanced high order Roe methods for two-dimensional nonconservative hyperbolic systems. In particular, we are interested in extending the methods introduced in (Castro et al., Math. Comput. 75:1103–1134, 2006) to the two-dimensional case. We also investigate the well-balance properties and the consistency of the resulting schemes. We focus in applications to one and two layer shallow water systems.


Generalized Roe schemes 2d Nonconservative hyperbolic systems Nonconservative products Finite volume schemes Conservation laws Source terms Shallow water systems Two-layer problems Geophysical flows 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Armi, L., Farmer, D.: Maximal two-layer exchange through a contraction with barotropic net flow. J. Fluid Mech. 164, 27–51 (1986) MATHCrossRefGoogle Scholar
  2. 2.
    Bermúdez, A., Vázquez, M.E.: Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23(8), 1049–1071 (1994) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Castro, M.J., Ferreiro, A., García, J.A., González-Vida, J., Macías, J., Parés, C., Vázquez-Cendón, M.E.: On the numerical treatment of wet/dry fronts in shallow flows: application to one-layer and two-layers systems. Math. Comput. Model. 42(3–4), 419–439 (2005) MATHCrossRefGoogle Scholar
  4. 4.
    Castro, M.J., Gallardo, J.M., Parés, C.: High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow water systems. Math. Comput. 75, 1103–1134 (2006) MATHCrossRefGoogle Scholar
  5. 5.
    Castro, M.J., LeFloch, P.G., Muñoz, M.L., Parés, C.: Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes. J. Comput. Phys. 227(17), 8107–8129 (2008) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Castro, M.J., Pardo, A., Parés, C., Toro, E.F.: Coefficient-splitting numerical schemes for nonconservative hyperbolic systems and high order extensions (2008, submitted) Google Scholar
  7. 7.
    Dal Maso, G., LeFloch, P.G., Murat, F.: Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74, 483–548 (1995) MATHMathSciNetGoogle Scholar
  8. 8.
    Godlewski, E., Raviart, P.A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Applied Mathematical Sciences, vol. 118. Springer, New York (1996) MATHGoogle Scholar
  9. 9.
    Gosse, L.: A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comput. Math. with Appl. 39, 135–159 (2000) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gosse, L.: A well-balanced scheme using non-conservative products designed for hyperbolic system of conservation laws with source terms. Math. Model. Meth. Appl. Sci. 11, 339–365 (2001) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gottlieb, S., Shu, C.W.: Total variation diminishing Runge–Kutta schemes. Mat. Comput. 67, 73–85 (1998) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Greenberg, J.M., LeRoux, A.Y.: A well balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33, 1–16 (1996) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Greenberg, J.M., LeRoux, A.Y., Baraille, R., Noussair, A.: Analysis and approximation of conservation laws with source terms. SIAM J. Numer. Anal. 34, 1980–2007 (1997) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Harten, A., Hyman, J.M.: Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comput. Phys. 50, 235–269 (1983) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hou, T.Y., LeFloch, P.G.: Why nonconservative schemes converge to wrong solutions: error analysis. Math. Comput. 62, 497–530 (1994) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Krüner, D., Rokyta, M., Wierse, M.: A Lax-Wendroff type theorem for upwind finite volume schemes in 2-D. East–West J. Numer. Math. 4(4), 279–292 (1996) MathSciNetGoogle Scholar
  17. 17.
    LeFloch, P.G.: Shock waves for nonlinear hyperbolic systems in nonconservative form. Preprint 593, Institute for Math. and Its Appl., Minneapolis (1989) Google Scholar
  18. 18.
    LeFloch, P.G., Liu, T.-P.: Existence theory for nonlinear hyperbolic systems in nonconservative form. Forum Math. 5, 261–280 (1993) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Marquina, A.: Local piecewise hyperbolic reconstruction of numerical fluxes for non linear scalar conservation laws. SIAM J. Sci. Comput. 15(4), 892–915 (1994) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Muñoz, M.L., Parés, C.: Godunov’s method for nonconservative hyperbolic systems. ESAIM Math. Model. Numer. Anal. 41(1), 169–185 (2007) MATHCrossRefGoogle Scholar
  21. 21.
    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) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Parés, C., Castro, M.J.: On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems. ESAIM Math. Model. Numer. Anal. 38(5), 821–852 (2004) MATHCrossRefGoogle Scholar
  23. 23.
    Schroll, H.J., Svensson, F.: A bihyperbolic finite volume method for quadrilateral meshes. SIAM J. Sci. Comput. 26(2), 237–260 (2006) MathSciNetGoogle Scholar
  24. 24.
    Serna, S.: A class of extended limiters applied to piecewise hyperbolic methods. SIAM J. Sci. Comput. 28(1), 123–140 (2006) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Shu, C.-W.: Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9(6), 1073–1084 (1988) MATHCrossRefGoogle Scholar
  26. 26.
    Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schems. J. Comput. Phys. 77, 439–471 (1988) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Toro, E.F., Titarev, V.A.: MUSTA fluxes for systems of conservation laws. J. Comput. Phys. 216(2), 403–429 (2006) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Thacker, W.C.: Some exact solutions to the nonlinear shallow-water wave equations. J. Fluid Mech. 107, 499–508 (1981) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Toumi, I.: A weak formulation of Roe’s approximate Riemann solver. J. Comput. Phys. 102(2), 360–373 (1992) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Volpert, A.I.: Spaces BV and quasilinear equations. Math. USSR Sbornik 73, 255–302 (1967) MathSciNetGoogle Scholar
  31. 31.
    Walz, G.: Romberg type cubature over arbitrary triangles. Mannheimer Mathem. Manuskripte Nr. 225, Mannhein (1997) Google Scholar
  32. 32.
    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) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • M. J. Castro
    • 1
  • E. D. Fernández-Nieto
    • 2
  • A. M. Ferreiro
    • 3
  • J. A. García-Rodríguez
    • 3
  • C. Parés
    • 1
  1. 1.Dpto. de Análisis MatemáticoUniversidad de MálagaMálagaSpain
  2. 2.Dpto. de Matemática Aplicada IUniversidad de SevillaSevillaSpain
  3. 3.Dpto. de MatemáticasUniversidad de A CoruñaA CoruñaSpain

Personalised recommendations