Convergence of the kinetic hydrostatic reconstruction scheme for the Saint Venant system with topography

Bouchut, François; Lhébrard, Xavier

doi:10.1090/mcom/3600

Convergence of the kinetic hydrostatic reconstruction scheme for the Saint Venant system with topography
HTML articles powered by AMS MathViewer

by François Bouchut and Xavier Lhébrard HTML | PDF

Math. Comp. 90 (2021), 1119-1153 Request permission

Abstract:

We prove the convergence of the hydrostatic reconstruction scheme with kinetic numerical flux for the Saint Venant system with continuous topography with locally integrable derivative. We use a recently derived fully discrete sharp entropy inequality with dissipation, that enables us to establish an estimate in the inverse of the square root of the space increment $\Delta x$ of the $L^2$ norm of the gradient of approximate solutions. By DiPerna’s method we conclude the strong convergence towards bounded weak entropy solutions.

References

Debora Amadori and Laurent Gosse, Transient $L^1$ error estimates for well-balanced schemes on non-resonant scalar balance laws, J. Differential Equations 255 (2013), no. 3, 469–502. MR 3053474, DOI 10.1016/j.jde.2013.04.016
Debora Amadori and Laurent Gosse, Stringent error estimates for one-dimensional, space-dependent $2\times 2$ relaxation systems, Ann. Inst. H. Poincaré C Anal. Non Linéaire 33 (2016), no. 3, 621–654. MR 3489631, DOI 10.1016/j.anihpc.2015.01.001
Nikolai Andrianov, Performance of numerical methods on the non-unique solution to the Riemann problem for the shallow water equations, Internat. J. Numer. Methods Fluids 47 (2005), no. 8-9, 825–831. MR 2123795, DOI 10.1002/fld.846
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, François Bouchut, Marie-Odile Bristeau, and Jacques Sainte-Marie, Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system, Math. Comp. 85 (2016), no. 302, 2815–2837. MR 3522971, DOI 10.1090/mcom/3099
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
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, Kinetic invariant domains and relaxation limit from a BGK model to isentropic gas dynamics, Asymptot. Anal. 31 (2002), no. 2, 153–176. MR 1938602
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
Christophe Berthon and Christophe Chalons, A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations, Math. Comp. 85 (2016), no. 299, 1281–1307. MR 3454365, DOI 10.1090/mcom3045
Christophe Berthon and Fabien Marche, A positive preserving high order VFRoe scheme for shallow water equations: a class of relaxation schemes, SIAM J. Sci. Comput. 30 (2008), no. 5, 2587–2612. MR 2429480, DOI 10.1137/070686147
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 and Hermano Frid, Finite difference schemes with cross derivatives correctors for multidimensional parabolic systems, J. Hyperbolic Differ. Equ. 3 (2006), no. 1, 27–52. MR 2214161, DOI 10.1142/S0219891606000719
François Bouchut and Xavier Lhébrard, A multi well-balanced scheme for the shallow water MHD system with topography, Numer. Math. 136 (2017), no. 4, 875–905. MR 3671591, DOI 10.1007/s00211-017-0865-y
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
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
Vivien Desveaux, Markus Zenk, Christophe Berthon, and Christian Klingenberg, Well-balanced schemes to capture non-explicit steady states: Ripa model, Math. Comp. 85 (2016), no. 300, 1571–1602. MR 3471101, DOI 10.1090/mcom/3069
R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), no. 1, 27–70. MR 684413, DOI 10.1007/BF00251724
Laurent Gosse and Athanasios E. Tzavaras, Convergence of relaxation schemes to the equations of elastodynamics, Math. Comp. 70 (2001), no. 234, 555–577. MR 1813140, DOI 10.1090/S0025-5718-00-01256-4
Pierre-Louis Lions, Benoît Perthame, and Panagiotis E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math. 49 (1996), no. 6, 599–638. MR 1383202, DOI 10.1002/(SICI)1097-0312(199606)49:6<599::AID-CPA2>3.0.CO;2-5
Carlos Parés, Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J. Numer. Anal. 44 (2006), no. 1, 300–321. MR 2217384, DOI 10.1137/050628052
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
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. MR 2053160
L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
Alexis Vasseur, Well-posedness of scalar conservation laws with singular sources, Methods Appl. Anal. 9 (2002), no. 2, 291–312. MR 1957491, DOI 10.4310/MAA.2002.v9.n2.a6
Nikolai Andrianov and Gerald Warnecke, On the solution to the Riemann problem for the compressible duct flow, SIAM J. Appl. Math. 64 (2004), no. 3, 878–901. MR 2068446, DOI 10.1137/S0036139903424230