Journal of Scientific Computing

, Volume 48, Issue 1–3, pp 274–295 | Cite as

On the Convergence and Well-Balanced Property of Path-Conservative Numerical Schemes for Systems of Balance Laws

  • María Luz Muñoz-RuizEmail author
  • Carlos Parés


This paper deals with the numerical approximation of one-dimensional hyperbolic systems of balance laws. We consider these systems as a particular case of hyperbolic systems in nonconservative form, for which we use the theory introduced by Dal Maso, LeFloch and Murat (J. Math. Pures Appl. 74:483, 1995) in order to define the concept of weak solutions. This theory is based on the prescription of a family of paths in the phases space. We also consider path-conservative schemes, that were introduced in Parés (SIAM J. Numer. Anal. 44:300, 2006). The first goal is to prove a Lax-Wendroff type convergence theorem. In Castro et al. (J. Comput. Phys. 227:8107, 2008) it was shown that, for general nonconservative systems a rather strong convergence assumption is needed to prove such a result. Here, we prove that the same hypotheses used in the classical Lax-Wendroff theorem are enough to ensure the convergence in the particular case of systems of balance laws, as the numerical results shown in Castro et al. (J. Comput. Phys. 227:8107, 2008) seemed to suggest. Next, we study the relationship between the well-balanced properties of path-conservative schemes applied to systems of balance laws and the family of paths.


Hyperbolic systems of balance laws Hyperbolic nonconservative systems Path-conservative schemes Convergence Well-balanced schemes 


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  1. 1.
    Abgrall, R., Karni, S.: Two-layer shallow water system: a relaxation approach. SIAM J. Sci. Comput. 31, 1603–1627 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Abgrall, R., Karni, S.: A comment on the computation of non-conservative products. J. Comput. Phys. 229, 2759–2763 (2010) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alcrudo, F., Benkhaldoun, F.: Exact solutions to the Riemann problem of the shallow water equations with a bottom step. Comput. Fluids 30, 643–671 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Andrianov, N., Warnecke, G.: On the solution to the Riemann problem for the compressible duct flow. SIAM J. Appl. Math. 64, 878–901 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    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) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bermúdez, A., Vázquez, M.E.: Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23, 1049–1071 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bernetti, R., Titarev, V.A., Toro, E.F.: Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry. J. Comput. Phys. 227, 3212–3243 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bouchut, F.: Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Birkhäuser, Basel (2004) zbMATHGoogle Scholar
  9. 9.
    Castro, M.J., Pardo, A., Parés, C.: Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique. Math. Mod. Meth. Appl. Sci. 17, 2055–2113 (2007) zbMATHCrossRefGoogle Scholar
  10. 10.
    Castro, M.J., LeFloch, P.G., Muñoz-Ruiz, M.L., Parés, C.: Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes. J. Comput. Phys. 227, 8107–8129 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Castro, M.J., Pardo, A., Parés, C., Toro, E.F.: On some fast well-balanced first order solvers for nonconservative systems. Math. Comput. 79, 1427–1472 (2010) zbMATHGoogle Scholar
  12. 12.
    Dal Maso, G., LeFloch, P.G., Murat, F.: Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74, 483–548 (1995) zbMATHMathSciNetGoogle Scholar
  13. 13.
    De Vuyst, F.: Schémas non-conservatifs et schémas cinétiques pour la simulation numérique d’écoulements hypersoniques non visqueux en déséquilibre thermochimique. Thése de Doctorat de l’Université Paris VI (1994) Google Scholar
  14. 14.
    Goatin, P., LeFloch, P.G.: The Riemann problem for a class of resonant hyperbolic systems of balance laws. Ann. Inst. H. Poincaré Anal. Non Lin. 21, 881–902 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Godunov, S.K.: A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47, 357–393 (1959) MathSciNetGoogle Scholar
  16. 16.
    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) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Karni, S.: Computations of shallow water flows in channels. In: The Proceedings of Numhyp 2009, Conference on Numerical Approximations of Hyperbolic Systems with Source Terms and Applications (2009) (to appear) Google Scholar
  18. 18.
    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) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Hou, T., LeFloch, P.G.: Why nonconservative schemes converge to wrong solutions: error analysis. Math. Comput. 62, 497–530 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Lax, P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math. 7, 159–193 (1954) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Lax, P.D., Wendroff, B.: Systems of conservation laws. Commun. Pure Appl. Math. 13, 217–237 (1960) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    LeFloch, P.G.: Shock Waves for Nonlinear Hyperbolic Systems in Nonconservative Form. Institute for Mathematics and its Applications, Minneapolis (1989). Preprint 593 Google Scholar
  23. 23.
    Muñoz-Ruiz, M.L., Parés, C.: Godunov method for nonconservative hyperbolic systems. ESAIM: M2AN 41, 169–185 (2007) zbMATHCrossRefGoogle Scholar
  24. 24.
    Parés, C., Castro, M.J.: On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow water systems. ESAIM: M2AN 38, 821–852 (2004) zbMATHCrossRefGoogle Scholar
  25. 25.
    Parés, C.: Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal. 44, 300–321 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Perthame, B., Simeoni, C.: Convergence of the upwind interface source method for hyperbolic conservation laws. In: Thou, Tadmor (eds.) Hyperbolic Problems: Theory, Numerics, Applications. Proceedings of the Ninth International Conference on Hyperbolic Problems, pp. 61–78. Springer, Berlin (2003) CrossRefGoogle Scholar
  27. 27.
    Roe, P.L.: Approximate Riemann solvers, paremeter vectors, and difference schemes. J. Comput. Phys. 43, 357–372 (1981) zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Roe, P.L.: Upwinding difference schemes for hyperbolic conservation laws with source terms. In: Carasso, Raviart, Serre (eds.) Nonlinear Hyperbolic Problems, Lect. Notes Math., pp. 41–51. Springer, Berlin (1986) Google Scholar
  29. 29.
    Rosatti, G., Begnudelli, L.: The Riemann problem for the one-dimensional free-surface shallow water equations with a bed step: theoretical analysis and numerical simulations. J. Comput. Phys. 229, 760–787 (2010) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Toumi, I.: A weak formulation of Roe’s approximate Riemann solver. J. Comput. Phys. 102, 360–373 (1992) zbMATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Dept. Matemática AplicadaUniversidad de MálagaMálagaSpain
  2. 2.Dept. Análisis MatemáticoUniversidad de MálagaMálagaSpain

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