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A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations

Authors: Christophe Berthon and Christophe Chalons
Journal: Math. Comp. 85 (2016), 1281-1307
MSC (2010): Primary 65M60, 65M12, 76M12, 35L65
Published electronically: September 15, 2015
MathSciNet review: 3454365
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Abstract: This work is devoted to the derivation of a fully well-balanced numerical scheme for the well-known shallow-water model. During the last two decades, several well-balanced strategies have been introduced with special attention to the exact capture of the stationary states associated with the so-called lake at rest. By fully well-balanced, we mean here that the proposed Godunov-type method is also able to preserve stationary states with non zero velocity. The numerical procedure is shown to preserve the positiveness of the water height and satisfies a discrete entropy inequality.

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  • [1] Emmanuel Audusse, François Bouchut, Marie-Odile Bristeau, Rupert Klein, and Benoît Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput. 25 (2004), no. 6, 2050-2065. MR 2086830 (2005f:76069),
  • [2] Alfredo Bermudez and Ma. Elena Vazquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. & Fluids 23 (1994), no. 8, 1049-1071. MR 1314237 (95i:76065),
  • [3] Christophe Berthon, Numerical approximations of the 10-moment Gaussian closure, Math. Comp. 75 (2006), no. 256, 1809-1831 (electronic). MR 2240636 (2007e:65107),
  • [4] Christophe Berthon, Robustness of MUSCL schemes for 2D unstructured meshes, J. Comput. Phys. 218 (2006), no. 2, 495-509. MR 2269374 (2008c:65195),
  • [5] Christophe Berthon and Françoise Foucher, Efficient well-balanced hydrostatic upwind schemes for shallow-water equations, J. Comput. Phys. 231 (2012), no. 15, 4993-5015. MR 2929930,
  • [6] 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. MR 2128209 (2005m:65002)
  • [7] François Bouchut, Christian Klingenberg, and Knut Waagan, A multiwave approximate Riemann solver for ideal MHD based on relaxation. I. Theoretical framework, Numer. Math. 108 (2007), no. 1, 7-42. MR 2350183 (2009f:35260),
  • [8] François Bouchut, Christian Klingenberg, and Knut Waagan, A multiwave approximate Riemann solver for ideal MHD based on relaxation II: numerical implementation with 3 and 5 waves, Numer. Math. 115 (2010), no. 4, 647-679. MR 2658158 (2011e:76086),
  • [9] François Bouchut and Tomás Morales de Luna, A subsonic-well-balanced reconstruction scheme for shallow water flows, SIAM J. Numer. Anal. 48 (2010), no. 5, 1733-1758. MR 2733096 (2012b:65119),
  • [10] Steve Bryson, Yekaterina Epshteyn, Alexander Kurganov, and Guergana Petrova, Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system, ESAIM Math. Model. Numer. Anal. 45 (2011), no. 3, 423-446. MR 2804645 (2012f:76085),
  • [11] Thierry Buffard, Thierry Gallouet, and Jean-Marc Hérard, Un schéma simple pour les équations de Saint-Venant, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 3, 385-390 (French, with English and French summaries). MR 1648505 (99g:76089),
  • [12] Manuel Castro, José M. Gallardo, Juan A. López-García, and Carlos Parés, Well-balanced high order extensions of Godunov's method for semilinear balance laws, SIAM J. Numer. Anal. 46 (2008), no. 2, 1012-1039. MR 2383221 (2008m:65210),
  • [13] Manuel J. Castro, Alberto Pardo Milanés, and Carlos Parés, Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique, Math. Models Methods Appl. Sci. 17 (2007), no. 12, 2055-2113. MR 2371563 (2009c:65186),
  • [14] Christophe Chalons, Frédéric Coquel, Edwige Godlewski, Pierre-Arnaud Raviart, and Nicolas Seguin, Godunov-type schemes for hyperbolic systems with parameter-dependent source. The case of Euler system with friction, Math. Models Methods Appl. Sci. 20 (2010), no. 11, 2109-2166. MR 2740716 (2011m:65179),
  • [15] Christophe Chalons and Jean-François Coulombel, Relaxation approximation of the Euler equations, J. Math. Anal. Appl. 348 (2008), no. 2, 872-893. MR 2446042 (2010b:35349),
  • [16] Ashwin Chinnayya, Alain-Yves LeRoux, and Nicolas Seguin, A Well-balanced Numerical Scheme for the Approximation of the Shallow-water Equations with Topography: The Resonance Phenomenon, Int. J. Finite Vol. 1 (2004), no. 1, 33 pp.. MR 2465450 (2009j:65186)
  • [17] Gérard Gallice, Solveurs simples positifs et entropiques pour les systèmes hyperboliques avec terme source, C. R. Math. Acad. Sci. Paris 334 (2002), no. 8, 713-716 (French, with English and French summaries). MR 1903376,
  • [18] Gérard Gallice, Positive and entropy stable Godunov-type schemes for gas dynamics and MHD equations in Lagrangian or Eulerian coordinates, Numer. Math. 94 (2003), no. 4, 673-713. MR 1990589 (2004e:65094)
  • [19] Thierry Gallouët, Jean-Marc Hérard, and Nicolas Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography, Comput. & Fluids 32 (2003), no. 4, 479-513. MR 1966639 (2004a:76095),
  • [20] Edwige Godlewski and Pierre-Arnaud Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, vol. 118, Springer-Verlag, New York, 1996. MR 1410987 (98d:65109)
  • [21] 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. MR 1753567 (2002a:65130),
  • [22] N. Goutal and F. Maurel, A finite volume solver for 1D shallow-water equations applied to an actual river, Int. J. Numer. Meth. Fluids 38 (2002), 1-19.
  • [23] N. Goutaland F. Maurel, Proceedings of the 2nd workshop on dam-break wave simulation, Technical report, EDF-DER, HE-43/97/016/B (1997).
  • [24] 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. MR 1377240 (97c:65144),
  • [25] Amiram Harten, Peter D. Lax, and Bram van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev. 25 (1983), no. 1, 35-61. MR 693713 (85h:65188),
  • [26] Shi Jin, A steady-state capturing method for hyperbolic systems with geometrical source terms, M2AN Math. Model. Numer. Anal. 35 (2001), no. 4, 631-645. MR 1862872 (2002i:65084),
  • [27] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537-566. MR 0093653 (20 #176)
  • [28] Philippe G. LeFloch and Mai Duc Thanh, A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime, J. Comput. Phys. 230 (2011), no. 20, 7631-7660. MR 2823568 (2012g:76028),
  • [29] Philippe G. LeFloch and Mai Duc Thanh, The Riemann problem for the shallow water equations with discontinuous topography, Commun. Math. Sci. 5 (2007), no. 4, 865-885. MR 2375051 (2009k:35179)
  • [30] Q. Liang and F. Marche, Numerical resolution of well-balanced shallow water equations with complex source terms, Advances in Water Resources 32 (6) (2009), 873-884.
  • [31] T. Morales de Luna, M. J. Castro Díaz, C. Parés Madroñal, and E. D. Fernández Nieto, On a shallow water model for the simulation of turbidity currents, Commun. Comput. Phys. 6 (2009), no. 4, 848-882. MR 2672326 (2011j:76051),
  • [32] Sebastian Noelle, Normann Pankratz, Gabriella Puppo, and Jostein R. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comput. Phys. 213 (2006), no. 2, 474-499. MR 2207248 (2006i:76072),
  • [33] Sebastian Noelle, Yulong Xing, and Chi-Wang Shu, High-order well-balanced finite volume WENO schemes for shallow water equation with moving water, J. Comput. Phys. 226 (2007), no. 1, 29-58. MR 2356351 (2008f:76124),
  • [34] 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. MR 2104431 (2006f:65085),
  • [35] Pierre-Arnaud Raviart and Lionel Sainsaulieu, A nonconservative hyperbolic system modeling spray dynamics. I. Solution of the Riemann problem, Math. Models Methods Appl. Sci. 5 (1995), no. 3, 297-333. MR 1330136 (96a:76096),
  • [36] Benoît Perthame and Youchun Qiu, A variant of Van Leer's method for multidimensional systems of conservation laws, J. Comput. Phys. 112 (1994), no. 2, 370-381. MR 1277283 (95a:65140),
  • [37] V. V. Rusanov, The calculation of the interaction of non-stationary shock waves with barriers, Ž. Vyčisl. Mat. i Mat. Fiz. 1 (1961), 267-279 (Russian). MR 0147083 (26 #4601)
  • [38] G. Russo and A. Khe, High order well-balanced schemes based on numerical reconstruction of the equilibrium variables, Proceedings ``WASCOM 2009'' 15th Conference on Waves and Stability in Continuous Media, World Sci. Publ., Hackensack, NJ, 2010, pp. 230-241. MR 2762021,
  • [39] Giovanni Russo and Alexander Khe, High order well balanced schemes for systems of balance laws, Hyperbolic problems: theory, numerics and applications, Proc. Sympos. Appl. Math., vol. 67, Amer. Math. Soc., Providence, RI, 2009, pp. 919-928. MR 2605287 (2011g:65164),
  • [40] Yulong Xing, Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium. part A, J. Comput. Phys. 257 (2014), no. part A, 536-553. MR 3129548,
  • [41] Yulong Xing and Chi-Wang Shu, High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms, J. Comput. Phys. 214 (2006), no. 2, 567-598. MR 2216604 (2006k:65235),
  • [42] Yulong Xing, Chi-Wang Shu, and Sebastian Noelle, On the advantage of well-balanced schemes for moving-water equilibria of the shallow water equations, J. Sci. Comput. 48 (2011), no. 1-3, 339-349. MR 2811708 (2012e:65182),
  • [43] Y. Xing, X. Zhang, and C.-W. Shu, Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations, Adv. Water Resour. 33 (2010), 1476-1493.

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Additional Information

Christophe Berthon
Affiliation: Université de Nantes, Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, 2 rue de la Houssinière, BP 92208, 44322 Nantes, France

Christophe Chalons
Affiliation: Laboratoire de Mathématiques de Versailles, UMR 8100, Université de Versailles Saint-Quentin-en-Yvelines, UFR des Sciences, bâtiment Fermat, 45 avenue des Etats-Unis, 78035 Versailles cedex, France

Keywords: Shallow-water equations, steady states, finite volume schemes, well-balanced property, positive preserving scheme, entropy preserving scheme
Received by editor(s): May 15, 2014
Received by editor(s) in revised form: December 3, 2014
Published electronically: September 15, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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