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On the convergence and well-balanced property of path-conservative numerical schemes for systems of balance laws. (English) Zbl 1230.65102
This article is concerned with the numerical approximation of Cauchy problems for one-dimensional systems of conservation laws with source terms or balance laws. The authors prove that, for systems of balance laws with sufficiently smooth right hand side, a Lax-Wendroff type result can be obtained for path-conservative numerical schemes assuming the same hypothesis of convergence used in the classical Lax-Wendroff theorem for conservative problems. Together with the hypothesis concerning the convergence of the approximations, some assumptions both on the chosen family of paths and on the numerical scheme are also required. Moreover, this paper presents some general results showing that some further assumptions have to be imposed to the family of paths, in order to obtain well-balanced schemes.

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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[1] Abgrall, R., Karni, S.: Two-layer shallow water system: a relaxation approach. SIAM J. Sci. Comput. 31, 1603–1627 (2009) · Zbl 1188.76229
[2] Abgrall, R., Karni, S.: A comment on the computation of non-conservative products. J. Comput. Phys. 229, 2759–2763 (2010) · Zbl 1188.65134
[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) · Zbl 1048.76008
[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) · Zbl 1065.35191
[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) · Zbl 1133.65308
[6] Bermúdez, A., Vázquez, M.E.: Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23, 1049–1071 (1994) · Zbl 0816.76052
[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) · Zbl 1132.76027
[8] Bouchut, F.: Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Birkhäuser, Basel (2004) · Zbl 1086.65091
[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) · Zbl 1137.76038
[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) · Zbl 1176.76084
[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) · Zbl 1369.65107
[12] Dal Maso, G., LeFloch, P.G., Murat, F.: Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74, 483–548 (1995) · Zbl 0853.35068
[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)
[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) · Zbl 1086.35069
[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)
[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) · Zbl 0963.65090
[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) · Zbl 1159.76026
[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) · Zbl 0876.65064
[19] Hou, T., LeFloch, P.G.: Why nonconservative schemes converge to wrong solutions: error analysis. Math. Comput. 62, 497–530 (1994) · Zbl 0809.65102
[20] Lax, P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math. 7, 159–193 (1954) · Zbl 0055.19404
[21] Lax, P.D., Wendroff, B.: Systems of conservation laws. Commun. Pure Appl. Math. 13, 217–237 (1960) · Zbl 0152.44802
[22] LeFloch, P.G.: Shock Waves for Nonlinear Hyperbolic Systems in Nonconservative Form. Institute for Mathematics and its Applications, Minneapolis (1989). Preprint 593
[23] Muñoz-Ruiz, M.L., Parés, C.: Godunov method for nonconservative hyperbolic systems. ESAIM: M2AN 41, 169–185 (2007) · Zbl 1124.65077
[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) · Zbl 1130.76325
[25] Parés, C.: Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal. 44, 300–321 (2006) · Zbl 1130.65089
[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) · Zbl 1064.65098
[27] Roe, P.L.: Approximate Riemann solvers, paremeter vectors, and difference schemes. J. Comput. Phys. 43, 357–372 (1981) · Zbl 0474.65066
[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)
[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) · Zbl 1253.76014
[30] Toumi, I.: A weak formulation of Roe’s approximate Riemann solver. J. Comput. Phys. 102, 360–373 (1992) · Zbl 0783.65068
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