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On the well-balance property of Roe's method for nonconservative hyperbolic systems. applications to shallow-water systems

  • Carlos Parés (a1) and Manuel Castro (a1)


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].



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[1] Andronov, N. and Warnecke, G., On the solution to the Riemann problem for the compressible duct flow. SIAM J. Appl. Math. 64 (2004) 878901.
[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] Bermúdez, A. and Vázquez, M.E., Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 10491071.
[4] Castro, M.J., Macías, J. and Parés, C., 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) 107127.
[5] Castro, M.J., García-Rodríguez, J.A., González-Vida, J.M., Macías, J., Parés, C. and Vázquez-Cendón, M.E., Numerical simulation of two-layer Shallow Water flows through channels with irregular geometry. J. Comp. Phys. 195 (2004) 202235.
[6] Chacón, T., Domínguez, A. and Fernández, E.D., A family of stable numerical solvers for Shallow Water equations with source terms. Comp. Meth. Appl. Mech. Eng. 192 (2003) 203225.
[7] Chacón, T., Domínguez, A. and Fernández, E.D., An entropy-correction free solver for non-homogeneous shallow water equations. ESAIM: M2AN 37 (2003) 755772.
[8] Chacón, T., Fernández, E.D. and Gómez Mármol, M., A flux-splitting solver for shallow water equations with source terms. Int. Jour. Num. Meth. Fluids 42 (2003) 2355.
[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.
[10] Colombeau, J.F., Le Roux, A.Y., Noussair, A. and Perrot, B., Microscopic profiles of shock waves and ambiguities in multiplications of distributions. SIAM J. Num. Anal. 26 (1989) 871883.
[11] Dal Masso, G., LeFloch, P.G. and Murat, F., Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483548.
[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).
[14] García-Navarro, P. and Vázquez-Cendón, M.E., On numerical treatment of the source terms in the shallow water equations. Comput. Fluids 29 (2000) 1745.
[15] P. Goatin and P.G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, preprint (2003).
[16] E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996).
[17] Gosse, L., A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comp. Math. Appl. 39 (2000) 135159.
[18] Gosse, L., 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) 339365.
[19] Greenberg, J.M. and LeRoux, A.Y., A well balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 116.
[20] Greenberg, J.M., LeRoux, A.Y., Baraille, R. and Noussair, A., Analysis and approximation of conservation laws with source terms. SIAM J. Numer. Anal. 34 (1997) 19802007.
[21] Harten, A. and Hyman, J.M., Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comp. Phys. 50 (1983) 235269.
[22] LeFloch, P.G., Propagating phase boundaries; formulation of the problem and existence via Glimm scheme. Arch. Rat. Mech. Anal. 123 (1993) 153197.
[23] R. LeVeque, Numerical Methods for Conservation Laws. Birkhäuser (1990).
[24] LeVeque, R., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comp. Phys. 146 (1998) 346365.
[25] R. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002).
[26] Perthame, B. and Simeoni, C., A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201231.
[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).
[28] Raviart, P.A. and Sainsaulieu, L., A nonconservative hyperbolic system modeling spray dynamics. I. Solution of the Riemann problem. Math. Mod. Meth. Appl. Sci. 5 (1995) 297333.
[29] Roe, P.L., Approximate Riemann solvers, parameter vectors and difference schemes. J. Comp. Phys. 43 (1981) 357371.
[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.
[31] J.J. Stoker, Water Waves. Interscience, New York (1957).
[32] E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction. Springer-Verlag (1997).
[33] E.F. Toro, Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley (2001).
[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).
[35] Toumi, I., A weak formulation of Roe's approximate Riemann Solver. J. Comp. Phys. 102 (1992) 360373.
[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] Vázquez-Cendón, M.E., Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comp. Phys. 148 (1999) 497526.
[38] Volpert, A.I., The space BV and quasilinear equations. Math. USSR Sbornik 73 (1967) 225267.


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On the well-balance property of Roe's method for nonconservative hyperbolic systems. applications to shallow-water systems

  • Carlos Parés (a1) and Manuel Castro (a1)


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