On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 5, p. 821-852

This paper is concerned with the numerical approximations of Cauchy problems for one-dimensional nonconservative hyperbolic systems. The first goal is to introduce a general concept of well-balancing for numerical schemes solving this kind of systems. Once this concept stated, we investigate the well-balance properties of numerical schemes based on the generalized Roe linearizations introduced by [Toumi, J. Comp. Phys. 102 (1992) 360-373]. Next, this general theory is applied to obtain well-balanced schemes for solving coupled systems of conservation laws with source terms. Finally, we focus on applications to shallow water systems: the numerical schemes obtained and their properties are compared, in the case of one layer flows, with those introduced by [Bermúdez and Vázquez-Cendón, Comput. Fluids 23 (1994) 1049-1071]; in the case of two layer flows, they are compared with the numerical scheme presented by [Castro, Macías and Parés, ESAIM: M2AN 35 (2001) 107-127].

DOI : https://doi.org/10.1051/m2an:2004041
Classification:  65M99,  76B55,  76B70
Keywords: nonconservative hyperbolic systems, well-balanced schemes, Roe method, source terms, shallow-water systems
@article{M2AN_2004__38_5_821_0,
     author = {Par\'es, Carlos and Castro, Manuel},
     title = {On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {5},
     year = {2004},
     pages = {821-852},
     doi = {10.1051/m2an:2004041},
     zbl = {1130.76325},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2004__38_5_821_0}
}
Parés, Carlos; Castro, Manuel. On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 5, pp. 821-852. doi : 10.1051/m2an:2004041. http://www.numdam.org/item/M2AN_2004__38_5_821_0/

[1] N. Andronov and G. Warnecke, On the solution to the Riemann problem for the compressible duct flow. SIAM J. Appl. Math. 64 (2004) 878-901. | Zbl 1065.35191

[2] F. Bouchut, An introduction to finite volume methods for hyperbolic systems of conservation laws with source, in Free surface geophysical flows. Tutorial Notes. INRIA, Rocquencourt (2002).

[3] A. Bermúdez and M.E. Vázquez, Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 1049-1071. | Zbl 0816.76052

[4] M.J. Castro, J. Macías and C. Parés, A Q-Scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system. ESAIM: M2AN 35 (2001) 107-127. | Numdam | Zbl 1094.76046

[5] M.J. Castro, J.A. García-Rodríguez, J.M. González-Vida, J. Macías, C. Parés and M.E. Vázquez-Cendón, Numerical simulation of two-layer Shallow Water flows through channels with irregular geometry. J. Comp. Phys. 195 (2004) 202-235. | Zbl 1087.76077

[6] T. Chacón, A. Domínguez and E.D. Fernández, A family of stable numerical solvers for Shallow Water equations with source terms. Comp. Meth. Appl. Mech. Eng. 192 (2003) 203-225. | Zbl 1083.76557

[7] T. Chacón, A. Domínguez and E.D. Fernández, An entropy-correction free solver for non-homogeneous shallow water equations. ESAIM: M2AN 37 (2003) 755-772. | Numdam | Zbl 1033.76032

[8] T. Chacón, E.D. Fernández and M. Gómez Mármol, A flux-splitting solver for shallow water equations with source terms. Int. Jour. Num. Meth. Fluids 42 (2003) 23-55. | Zbl 1033.76033

[9] T. Chacón, A. Domínguez and E.D. Fernández, Asymptotically balanced schemes for non-homogeneous hyperbolic systems - application to the Shallow Water equations. C.R. Acad. Sci. Paris, Ser. I 338 (2004) 85-90. | Zbl 1038.65073

[10] J.F. Colombeau, A.Y. Le Roux, A. Noussair and B. Perrot, Microscopic profiles of shock waves and ambiguities in multiplications of distributions. SIAM J. Num. Anal. 26 (1989) 871-883. | Zbl 0674.76049

[11] G. Dal Masso, P.G. Lefloch and F. Murat, Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483-548. | Zbl 0853.35068

[12] E.D. Fernández Nieto, Aproximación Numérica de Leyes de Conservación Hiperbólicas No Homogéneas. Aplicación a las Ecuaciones de Aguas Someras. Ph.D. Thesis, Universidad de Sevilla (2003).

[13] A.C. Fowler, Mathematical Model in the Applied Sciences. Cambridge (1997). | Zbl 0997.00535

[14] P. García-Navarro and M.E. Vázquez-Cendón, On numerical treatment of the source terms in the shallow water equations. Comput. Fluids 29 (2000) 17-45. | Zbl 0986.76051

[15] P. Goatin and P.G. Lefloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, preprint (2003). | Numdam | MR 2097035

[16] E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996). | MR 1410987 | Zbl 0860.65075

[17] L. Gosse, A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comp. Math. Appl. 39 (2000) 135-159. | Zbl 0963.65090

[18] L. Gosse, A well-balanced scheme using non-conservative products designed for hyperbolic system of conservation laws with source terms. Mat. Mod. Meth. Appl. Sc. 11 (2001) 339-365. | Zbl 1018.65108

[19] J.M. Greenberg and A.Y. Leroux, A well balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1-16. | Zbl 0876.65064

[20] J.M. Greenberg, A.Y. Leroux, R. Baraille and A. Noussair, Analysis and approximation of conservation laws with source terms. SIAM J. Numer. Anal. 34 (1997) 1980-2007. | Zbl 0888.65100

[21] A. Harten and J.M. Hyman, Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comp. Phys. 50 (1983) 235-269. | Zbl 0565.65049

[22] P.G. Lefloch, Propagating phase boundaries; formulation of the problem and existence via Glimm scheme. Arch. Rat. Mech. Anal. 123 (1993) 153-197. | Zbl 0784.73010

[23] R. Leveque, Numerical Methods for Conservation Laws. Birkhäuser (1990). | MR 1077828 | Zbl 0723.65067

[24] R. Leveque, Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comp. Phys. 146 (1998) 346-365. | Zbl 0931.76059

[25] R. Leveque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002). | MR 1925043 | Zbl 1010.65040

[26] B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201-231. | Zbl 1008.65066

[27] B. Perthame and C. Simeoni, Convergence of the upwind interface source method for hyperbolic conservation laws, in Proc. of Hyp 2002, Thou and Tadmor Eds., Springer (2003). | MR 2053160 | Zbl 1064.65098

[28] P.A. Raviart and L. Sainsaulieu, A nonconservative hyperbolic system modeling spray dynamics. I. Solution of the Riemann problem. Math. Mod. Meth. Appl. Sci. 5 (1995) 297-333. | Zbl 0837.76089

[29] P.L. Roe, Approximate Riemann solvers, parameter vectors and difference schemes. J. Comp. Phys. 43 (1981) 357-371. | Zbl 0474.65066

[30] P.L. Roe, Upwinding difference schemes for hyperbolic conservation laws with source terms, in Proc. of the Conference on Hyperbolic Problems, Carasso, Raviart and Serre Eds., Springer (1986) 41-51. | Zbl 0626.65086

[31] J.J. Stoker, Water Waves. Interscience, New York (1957). | Zbl 0078.40805

[32] E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction. Springer-Verlag (1997). | MR 1474503 | Zbl 0801.76062

[33] E.F. Toro, Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley (2001). | Zbl 0996.76003

[34] E.F. Toro and M.E. Vázquez-Cendón, Model hyperbolic systems with source terms: exact and numerical solutions, in Proc. of Godunov methods: Theory and Applications (2000). | MR 1963646 | Zbl 0989.65095

[35] I. Toumi, A weak formulation of Roe's approximate Riemann Solver. J. Comp. Phys. 102 (1992) 360-373. | Zbl 0783.65068

[36] M.E. Vázquez-Cendón, Estudio de Esquemas Descentrados para su Aplicación a las Leyes de Conservación Hiperbólicas con Términos Fuente. Ph.D. Thesis, Universidad de Santiago de Compostela (1994).

[37] M.E. Vázquez-Cendón, Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comp. Phys. 148 (1999) 497-526. | Zbl 0931.76055

[38] A.I. Volpert, The space BV and quasilinear equations. Math. USSR Sbornik 73 (1967) 225-267. | Zbl 0168.07402