High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. (English) Zbl 1096.65082

This paper is concerned with the construction of high order schemes for the numerical solution of one-dimensional nonconservative hyperbolic systems. Based on the use of a first order Roe scheme and weighted essentially nonoscillatory reconstruction of states, the authors give the general expression of a well-balanced high order scheme. For such a general expression, once obtained, particular schemes can be deduced for any system of the form discussed in the paper, where the numerical treatment of source and coupling terms is automatically derived. The authors also study the well-balanced properties of the resulting schemes and apply the schemes to shallow-water systems to verify the well-balanced property numerically.


65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76M12 Finite volume methods applied to problems in fluid mechanics
35L65 Hyperbolic conservation laws
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
Full Text: DOI


[1] N. Andrianov, CONSTRUCT: a collection of MATLAB routines for constructing the exact solution to the Riemann problem for the shallow water equations, avalaible at http://www-ian.math.unimagdeburg.de/home/andriano/CONSTRUCT.
[2] Alfredo Bermudez and Ma. Elena Vazquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. & Fluids 23 (1994), no. 8, 1049 – 1071. · Zbl 0816.76052
[3] François Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2004. · Zbl 1086.65091
[4] V. Caselles, R. Donat and G. Haro, Flux-gradient and source term balancing for certain high resolution shock-capturing schemes, submitted. · Zbl 1237.76100
[5] Manuel Castro, Jorge Macías, and Carlos Parés, A \?-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system, M2AN Math. Model. Numer. Anal. 35 (2001), no. 1, 107 – 127. · Zbl 1094.76046
[6] Tomás Chacón Rebollo, Antonio Domínguez Delgado, and Enrique D. Fernández Nieto, A family of stable numerical solvers for the shallow water equations with source terms, Comput. Methods Appl. Mech. Engrg. 192 (2003), no. 1-2, 203 – 225. · Zbl 1083.76557
[7] Tomás Chacón Rebollo, Antonio Domínguez Delgado, and Enrique D. Fernández Nieto, An entropy-correction free solver for non-homogeneous shallow water equations, M2AN Math. Model. Numer. Anal. 37 (2003), no. 5, 755 – 772. · Zbl 1033.76032
[8] T. Chacón, E.D. Fernández, M.J. Castro and C. Parés, On well-balanced finite volume methods for non-homogeneous non-conservative hyperbolic systems. Preprint, 2005.
[9] A. Chinnaya and A.Y. LeRoux, A new general Riemann solver for the shallow-water equations with friction and topography, available at http://www.math.ntnu.no/ conservation/1999/021.html.
[10] A. Chinnaya and A.Y. LeRoux, A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: the resonance phenomenon. To appear in Int. J. Finite Volume, 2004.
[11] G. Dal Maso, Ph. 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 models in the applied sciences, Cambridge Texts in Applied Mathematics, Cambridge University Press, 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. Comp. & Fluids 29(8) (2000), 17-45.
[15] G. Godinaud, A.Y. LeRoux and M.N. LeRoux, Generation of new solvers involving the source term for a class of nonhomogeneous hyperbolic systems, available at http://www.math.ntnu.no/conservation/2000/029.html.
[16] Edwige Godlewski and Pierre-Arnaud Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, vol. 118, Springer-Verlag, New York, 1996. · Zbl 0860.65075
[17] L. Gosse, A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms, Comput. Math. Appl. 39 (2000), no. 9-10, 135 – 159. · Zbl 0963.65090
[18] Laurent Gosse, A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms, Math. Models Methods Appl. Sci. 11 (2001), no. 2, 339 – 365. · Zbl 1018.65108
[19] Sigal Gottlieb and Chi-Wang Shu, Total variation diminishing Runge-Kutta schemes, Math. Comp. 67 (1998), no. 221, 73 – 85. · Zbl 0897.65058
[20] N. Goutal and F. Maurel, Proceedings of the Second Workshop on Dam-Break Wave Simulation, Technical Report HE-43/97/016/A, Electrité de France, Département Laboratoire National d’Hydraulique, Groupe Hydraulique Fluviale, 1997.
[21] 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), no. 1, 1 – 16. · Zbl 0876.65064
[22] 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), no. 5, 1980 – 2007. · Zbl 0888.65100
[23] Guang-Shan Jiang and Chi-Wang Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996), no. 1, 202 – 228. · Zbl 0877.65065
[24] Ami Harten and James M. Hyman, Self-adjusting grid methods for one-dimensional hyperbolic conservation laws, J. Comput. Phys. 50 (1983), no. 2, 235 – 269. · Zbl 0565.65049
[25] Randall J. LeVeque, Numerical methods for conservation laws, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1990. · Zbl 0723.65067
[26] Randall J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm, J. Comput. Phys. 146 (1998), no. 1, 346 – 365. · Zbl 0931.76059
[27] Xu-Dong Liu, Stanley Osher, and Tony Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys. 115 (1994), no. 1, 200 – 212. · Zbl 0811.65076
[28] Carlos Parés and Manuel Castro, On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems, M2AN Math. Model. Numer. Anal. 38 (2004), no. 5, 821 – 852. · Zbl 1130.76325
[29] B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, Calcolo 38 (2001), no. 4, 201 – 231. · Zbl 1008.65066
[30] Benoît Perthame and Chiara Simeoni, Convergence of the upwind interface source method for hyperbolic conservation laws, Hyperbolic problems: theory, numerics, applications, Springer, Berlin, 2003, pp. 61 – 78. · Zbl 1064.65098
[31] P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys. 43 (1981), no. 2, 357 – 372. · Zbl 0474.65066
[32] J. Shi, C. Hu and C.-W. Shu, A technique of treating negative weights in WENO schemes. J. Comput. Phys. 175 (2002), 108-127. · Zbl 0992.65094
[33] Chi-Wang Shu and Stanley Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988), no. 2, 439 – 471. · Zbl 0653.65072
[34] C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, ICASE Report 97-65, 1997.
[35] E. F. Toro and M. E. Vazquez-Cendon, Model hyperbolic systems with source terms: exact and numerical solutions, Godunov methods (Oxford, 1999) Kluwer/Plenum, New York, 2001, pp. 941 – 948. · Zbl 0989.65095
[36] I. Toumi, A weak formulation of Roe’s approximate Riemann solver, J. Comput. Phys. 102 (1992), no. 2, 360 – 373. · Zbl 0783.65068
[37] María Elena Vázquez-Cendón, Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry, J. Comput. Phys. 148 (1999), no. 2, 497 – 526. · Zbl 0931.76055
[38] Senka Vukovic and Luka Sopta, ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations, J. Comput. Phys. 179 (2002), no. 2, 593 – 621. · Zbl 1130.76389
[39] Yulong Xing and Chi-Wang Shu, High order finite difference WENO schemes with the exact conservation property for the shallow water equations, J. Comput. Phys. 208 (2005), no. 1, 206 – 227. · Zbl 1114.76340
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