Congested shallow water model: roof modeling in free surface flow
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1679-1707.

We are interested in the modeling and the numerical approximation of flows in the presence of a roof, for example flows in sewers or under an ice floe. A shallow water model with a supplementary congestion constraint describing the roof is derived from the Navier-Stokes equations. The congestion constraint is a challenging problem for the numerical resolution of hyperbolic equations. To overcome this difficulty, we follow a pseudo-compressibility relaxation approach. Eventually, a numerical scheme based on a finite volume method is proposed. The well-balanced property and the dissipation of the mechanical energy, acting as a mathematical entropy, are ensured under a non-restrictive condition on the time step in spite of the large celerity of the potential waves in the congested areas. Simulations in one dimension for transcritical steady flow are carried out and numerical solutions are compared to several analytical (stationary and non-stationary) solutions for validation.

DOI : 10.1051/m2an/2018032
Classification : 35L65, 76B07, 76M12, 35Q35, 74F10
Mots clés : Shallow water equations, congested hyperbolic model, unilateral constraint, well-balanced scheme, entropic scheme
Godlewski, Edwige 1 ; Parisot, Martin 1 ; Sainte-Marie, Jacques 1 ; Wahl, Fabien 1

1
@article{M2AN_2018__52_5_1679_0,
     author = {Godlewski, Edwige and Parisot, Martin and Sainte-Marie, Jacques and Wahl, Fabien},
     title = {Congested shallow water model: roof modeling in free surface flow},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1679--1707},
     publisher = {EDP-Sciences},
     volume = {52},
     number = {5},
     year = {2018},
     doi = {10.1051/m2an/2018032},
     zbl = {1417.35121},
     mrnumber = {3878609},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2018032/}
}
TY  - JOUR
AU  - Godlewski, Edwige
AU  - Parisot, Martin
AU  - Sainte-Marie, Jacques
AU  - Wahl, Fabien
TI  - Congested shallow water model: roof modeling in free surface flow
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2018
SP  - 1679
EP  - 1707
VL  - 52
IS  - 5
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2018032/
DO  - 10.1051/m2an/2018032
LA  - en
ID  - M2AN_2018__52_5_1679_0
ER  - 
%0 Journal Article
%A Godlewski, Edwige
%A Parisot, Martin
%A Sainte-Marie, Jacques
%A Wahl, Fabien
%T Congested shallow water model: roof modeling in free surface flow
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2018
%P 1679-1707
%V 52
%N 5
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2018032/
%R 10.1051/m2an/2018032
%G en
%F M2AN_2018__52_5_1679_0
Godlewski, Edwige; Parisot, Martin; Sainte-Marie, Jacques; Wahl, Fabien. Congested shallow water model: roof modeling in free surface flow. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1679-1707. doi : 10.1051/m2an/2018032. http://archive.numdam.org/articles/10.1051/m2an/2018032/

[1] E. Audusse, A multilayer Saint-Venant model: derivation and numerical validation. Discrete Contin. Dyn. Syst. Ser. B 5 (2005) 189–214. | DOI | MR | Zbl

[2] E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 (2004) 2050–2065. | DOI | MR | Zbl

[3] E. Audusse, M.-O. Bristeau, B. Perthame and J. Sainte-Marie, A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation. ESAIM: M2AN 45 (2011) 169–200. | DOI | Numdam | MR | Zbl

[4] A. Bermudez and M.E Vazquez, Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 1049–1071. | DOI | MR | Zbl

[5] F. Berthelin, Existence and weak stability for a pressureless model with unilateral constraint. Math. Model. Methods Appl. Sci. 12 (2002) 249–272. | DOI | MR | Zbl

[6] F. Berthelin and F. Bouchut, Weak solutions for a hyperbolic system with unilateral constraint and mass loss. Ann. Inst. Henri Poincaré (C) Non Linear Anal. 20 (2003) 975–997. | DOI | Numdam | MR | Zbl

[7] F. Berthelin, P. Degond, V. Le Blanc, S. Moutari, M. Rascle and J. Royer, A traffic-flow model with constraints for the modeling of traffic jams. Math. Model. Methods Appl. Sci. 18 (2008) 1269–1298. | DOI | MR | Zbl

[8] F. Bouchut, Y. Brenier, J. Cortes and J.-F. Ripoll, A hierarchy of models for two-phase flows. J. Nonlinear Sci. 10 (2000) 639–660. | DOI | MR | Zbl

[9] C. Bourdarias, M. Ersoy and S. Gerbi, A mathematical model for unsteady mixed flows in closed water pipes. Sci. Chin. Math. 55 (2012) 221–244. | DOI | MR | Zbl

[10] C. Bourdarias and S. Gerbi, A finite volume scheme for a model coupling free surface and pressurised flows in pipes. J. Comput. Appl. Math. 209 (2007) 109–131. | DOI | MR | Zbl

[11] K. Brenner and C. Cancès, Improving Newton’s method performance by parametrization: the case of Richards equation. SIAM J. Numer. Anal. 55 (2017) 1760–1785. | DOI | MR | Zbl

[12] M.-O. Bristeau, A. Mangeney, J. Sainte-Marie and N. Seguin, An energy-consistent depth-averaged Euler system: derivation and properties. Discrete Contin. Dyn. Syst. Ser. B 20 (2015) 961–988. | DOI | MR | Zbl

[13] H. Capart, X. Sillen and Y. Zech, Numerical and experimental water transients in sewer pipes. J. Hydraul. Res. 35 (1997) 659–672. | DOI

[14] M. Castro, J.M. Gallardo, J.A. López-García and C. Parés, Well-balanced high order extensions of Godunov’s method for semilinear balance laws. SIAM J. Numer. Anal. 46 (2008) 1012–1039. | DOI | MR | Zbl

[15] C. Chalons, M. Girardin and S. Kokh, Large time step and asymptotic preserving numerical schemes for the gas dynamics equations with source terms. SIAM J. Sci. Comput. 35 (2013) A2874–A2902. | DOI | MR | Zbl

[16] H. Chanson, Hydraulics of Open Channel Flow. Elsevier Science (2004).

[17] A. J. Chorin, Numerical solution of the Navier-Stokes equations. Math. Comput. 22 (1968) 745–762. | DOI | MR | Zbl

[18] G. Dal Maso, P. G. Lefloch and F. Murat, Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483–548. | MR | Zbl

[19] S. Dellacherie, Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number. J. Comput. Phys. 229 (2010) 978–1016. | DOI | MR | Zbl

[20] B. Després, A geometrical approach to nonconservative shocks and elastoplastic shocks. Arch. Ration. Mech. Anal. 186 (2007) 275–308. | DOI | MR | Zbl

[21] B. Després, F. Lagoutière and N. Seguin, Weak solutions to Friedrichs systems with convex constraints. Nonlinearity 24 (2011) 3055–3081. | DOI | MR | Zbl

[22] F. Dubois and P. G. Lefloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differ. Equ. 71 (1988) 93–122. | DOI | MR | Zbl

[23] E. D. Fernandez-Nieto, M. Parisot, Y. Penel and J. Sainte-Marie, A hierarchy of non-hydrostatic layer-averaged approximation of Euler equations for free surface flows. Working paper or preprint [] (2017). | HAL | MR

[24] M. Fuamba, Contribution on transient flow modelling in storm sewers. J. Hydraul. Res. 40 (2002) 685–693. | DOI

[25] J.-F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B 1 (2001) 89–102. | MR | Zbl

[26] 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) 1–16. | DOI | MR | Zbl

[27] H. Guillard and A. Murrone, On the behavior of upwind schemes in the low Mach number limit: II. Godunov type schemes. Comput. Fluids 33 (2004) 655–675. | DOI | Zbl

[28] R. Herbin, W. Kheriji and J.-C. Latché, On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations. ESAIM: M2AN 48 (2014) 1807–1857. | DOI | MR | Zbl

[29] S. Jin and X. Wen, Two interface-type numerical methods for computing hyperbolic systems with geometrical source terms having concentrations. SIAM J. Sci. Comput. 26 (2005) 2079–2101. | DOI | MR | Zbl

[30] M. Kashiwagi, Non-linear simulations of wave-induced motions of a floating body by means of the mixed Eulerian-Lagrangian method. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 214 (2000) 841–855. | DOI

[31] S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34 (1981) 481–524. | DOI | MR | Zbl

[32] S. Klainerman and A. Majda, Compressible and incompressible fluids. Commun. Pure Appl. Math. 35 (1982) 629–651. | DOI | MR | Zbl

[33] D. Lannes, On the dynamics of floating structures. Ann. PDE 3 (2017) 11. | DOI | MR | Zbl

[34] D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation. Phys. Fluids 21 (2009) 016601. | DOI | Zbl

[35] L. Levi, Obstacle problems for scalar conservation laws. ESAIM: M2AN 35 (2001) 575–593. | DOI | Numdam | MR | Zbl

[36] P.-L. Lions and N. Masmoudi, On a free boundary barotropic model. Ann. Inst. Henri Poincaré (C) Non Linear Anal. 16 (1999) 373–410. | DOI | Numdam | MR | Zbl

[37] B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion modeling. Netw. Heterog. Media 6 (2011) 485–519. | DOI | MR | Zbl

[38] V. Michel-Dansac, C. Berthon, S. Clain and F. Foucher, A well-balanced scheme for the shallow-water equations with topography or manning friction. J. Comput. Phys. 335 (2017) 115–154. | DOI | MR | Zbl

[39] S. Noelle, Y. Xing and C.-W. Shu, High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys. 226 (2007) 29–58. | DOI | MR | Zbl

[40] M. Parisot and J.-P. Vila, Centered-potential regularization for the advection upstream splitting method. SIAM J. Numer. Anal. 54 (2016) 3083–3104. | DOI | MR | Zbl

[41] C. Perrin and E. Zatorska, Free/congested two-phase model from weak solutions to multi-dimensional compressible Navier-Stokes equations. Commun. PDE 40 (2015) 1558–1589. | DOI | MR | Zbl

[42] B. Perthame, PDE models for chemotactic movements: parabolic, hyperbolic and kinetic. Appl. Math. 49 (2004) 539–564. | DOI | MR | Zbl

[43] A. Quarteroni, M. Tuveri and A. Veneziani, Computational vascular fluid dynamics: problems, models and methods. Comput. Vis. Sci. 2 (2000) 163–197. | DOI | Zbl

[44] R. Rannacher, On Chorin’s Projection Method for the Incompressible Navier-Stokes Equations. Springer, Berlin, Heidelberg (1992) 167–183. | MR | Zbl

[45] J. Shen, Pseudo-compressibility methods for the unsteady incompressible Navier-Stokes equations. Proc. of the 1994 Beijing Symposium on Nonlinear Evolution Equations and Infinite Dynamical Systems (1997) 68–78.

[46] R. Témam, Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (I). Arch. Ration. Mech. Anal. 32 (1969) 135–153. | DOI | MR | Zbl

Cité par Sources :