Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system

Audusse, Emmanuel; Bouchut, François; Bristeau, Marie-Odile; Sainte-Marie, Jacques

doi:10.1090/mcom/3099

Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system
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by Emmanuel Audusse, François Bouchut, Marie-Odile Bristeau and Jacques Sainte-Marie PDF

Math. Comp. 85 (2016), 2815-2837 Request permission

Abstract:

A lot of well-balanced schemes have been proposed for discretizing the classical Saint-Venant system for shallow water flows with nonflat bottom. Among them, the hydrostatic reconstruction scheme is a simple and efficient one. It involves the knowledge of an arbitrary solver for the homogeneous problem (for example, Godunov, Roe, kinetic, etc.). If this solver is entropy satisfying, then the hydrostatic reconstruction scheme satisfies a semi-discrete entropy inequality. In this paper we prove that, when used with the classical kinetic solver, the hydrostatic reconstruction scheme also satisfies a fully discrete entropy inequality, but with an error term. This error term tends to zero strongly when the space step tends to zero, including solutions with shocks. We also prove that the hydrostatic reconstruction scheme does not satisfy the entropy inequality without error term.

References

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, DOI 10.1137/S1064827503431090
Emmanuel Audusse and Marie-Odile Bristeau, A well-balanced positivity preserving “second-order” scheme for shallow water flows on unstructured meshes, J. Comput. Phys. 206 (2005), no. 1, 311–333. MR 2135839, DOI 10.1016/j.jcp.2004.12.016
E. Audusse, M.-O. Bristeau, M. Pelanti, and J. Sainte-Marie, Approximation of the hydrostatic Navier-Stokes system for density stratified flows by a multilayer model: kinetic interpretation and numerical solution, J. Comput. Phys. 230 (2011), no. 9, 3453–3478. MR 2780473, DOI 10.1016/j.jcp.2011.01.042
E. Audusse, M.-O. Bristeau, B. Perthame, Kinetic schemes for Saint Venant equations with source terms on unstructured grids, Technical Report 3989, INRIA, Unité de recherche de Rocquencourt, France, 2000. http://www.inria.fr/rrrt/rr-3989.html.
Emmanuel Audusse, Marie-Odile Bristeau, Benoît Perthame, and Jacques Sainte-Marie, A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation, ESAIM Math. Model. Numer. Anal. 45 (2011), no. 1, 169–200. MR 2781135, DOI 10.1051/m2an/2010036
F. Berthelin, Convergence of flux vector splitting schemes with single entropy inequality for hyperbolic systems of conservation laws, Numer. Math. 99 (2005), no. 4, 585–604. MR 2121070, DOI 10.1007/s00211-004-0567-0
F. Berthelin and F. Bouchut, Relaxation to isentropic gas dynamics for a BGK system with single kinetic entropy, Methods Appl. Anal. 9 (2002), no. 2, 313–327. MR 1957492, DOI 10.4310/MAA.2002.v9.n2.a7
F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws, J. Statist. Phys. 95 (1999), no. 1-2, 113–170. MR 1705583, DOI 10.1023/A:1004525427365
F. Bouchut, Entropy satisfying flux vector splittings and kinetic BGK models, Numer. Math. 94 (2003), no. 4, 623–672. MR 1990588, DOI 10.1007/s00211-002-0426-9
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, DOI 10.1007/b93802
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, DOI 10.1137/090758416
François Bouchut and Vladimir Zeitlin, A robust well-balanced scheme for multi-layer shallow water equations, Discrete Contin. Dyn. Syst. Ser. B 13 (2010), no. 4, 739–758. MR 2601337, DOI 10.3934/dcdsb.2010.13.739
Marie-Odile Bristeau, Nicole Goutal, and Jacques Sainte-Marie, Numerical simulations of a non-hydrostatic shallow water model, Comput. & Fluids 47 (2011), 51–64. MR 2949395, DOI 10.1016/j.compfluid.2011.02.013
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, DOI 10.1142/S021820250700256X
Frédéric Coquel, Khaled Saleh, and Nicolas Seguin, A robust and entropy-satisfying numerical scheme for fluid flows in discontinuous nozzles, Math. Models Methods Appl. Sci. 24 (2014), no. 10, 2043–2083. MR 3211117, DOI 10.1142/S0218202514500158
F. Coron and B. Perthame, Numerical passage from kinetic to fluid equations, SIAM J. Numer. Anal. 28 (1991), no. 1, 26–42 (English, with French summary). MR 1083323, DOI 10.1137/0728002
Laurent Gosse, Computing qualitatively correct approximations of balance laws, SIMAI Springer Series, vol. 2, Springer, Milan, 2013. Exponential-fit, well-balanced and asymptotic-preserving. MR 3053000, DOI 10.1007/978-88-470-2892-0
N. Goutal and J. Sainte-Marie, A kinetic interpretation of the section-averaged Saint-Venant system for natural river hydraulics, Internat. J. Numer. Methods Fluids 67 (2011), no. 7, 914–938. MR 2842218, DOI 10.1002/fld.2401
Shi Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, Riv. Math. Univ. Parma (N.S.) 3 (2012), no. 2, 177–216. MR 2964096
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, DOI 10.1016/j.jcp.2007.03.031
B. Perthame, Boltzmann type schemes for gas dynamics and the entropy property, SIAM J. Numer. Anal. 27 (1990), no. 6, 1405–1421 (English, with French summary). MR 1080328, DOI 10.1137/0727081
B. Perthame, Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions, SIAM J. Numer. Anal. 29 (1992), no. 1, 1–19 (English, with French summary). MR 1149081, DOI 10.1137/0729001
Benoît Perthame, Kinetic formulation of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 21, Oxford University Press, Oxford, 2002. MR 2064166
B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, Calcolo 38 (2001), no. 4, 201–231. MR 1890353, DOI 10.1007/s10092-001-8181-3
Yulong Xing and Chi-Wang Shu, A survey of high order schemes for the shallow water equations, J. Math. Study 47 (2014), no. 3, 221–249. MR 3269836, DOI 10.4208/jms.v47n3.14.01