Congested shallow water model: on floating body
The SMAI Journal of computational mathematics, Volume 6 (2020), pp. 227-251.

We consider the floating body problem in the vertical plane on a large space scale. More precisely, we are interested in the numerical modeling of a body floating freely on the water such as icebergs or wave energy converters. The fluid-solid interaction is formulated using a congested shallow water model for the fluid and Newton’s second law of motion for the solid. We make a particular focus on the energy transfer between the solid and the water since it is of major interest for energy production. A numerical approximation based on the coupling of a finite volume scheme for the fluid and a Newmark scheme for the solid is presented. An entropy correction based on an adapted choice of discretization for the coupling terms is made in order to ensure a dissipation law at the discrete level. Simulations are presented to verify the method and to show the feasibility of extending it to more complex cases.

Published online:
DOI: 10.5802/smai-jcm.67
Classification: 35Q35, 70E15, 74F10, 76B07, 76M12
Keywords: shallow water equations, wave-body interaction, congested model, coupling, entropy satisfying scheme
Edwige Godlewski 1; Martin Parisot 1; Jacques Sainte-Marie 1; Fabien Wahl 1

1 Sorbonne Université, université Paris-Diderot SPC, CNRS, Inria, laboratoire Jacques-Louis Lions, LJLL, ANGE team, F-75005 Paris, France
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Edwige Godlewski and Martin Parisot and Jacques Sainte-Marie and Fabien Wahl},
     title = {Congested shallow water model: on floating body},
     journal = {The SMAI Journal of computational mathematics},
     pages = {227--251},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {6},
     year = {2020},
     doi = {10.5802/smai-jcm.67},
     zbl = {1417.35121},
     language = {en},
     url = {}
AU  - Edwige Godlewski
AU  - Martin Parisot
AU  - Jacques Sainte-Marie
AU  - Fabien Wahl
TI  - Congested shallow water model: on floating body
JO  - The SMAI Journal of computational mathematics
PY  - 2020
SP  - 227
EP  - 251
VL  - 6
PB  - Société de Mathématiques Appliquées et Industrielles
UR  -
DO  - 10.5802/smai-jcm.67
LA  - en
ID  - SMAI-JCM_2020__6__227_0
ER  - 
%0 Journal Article
%A Edwige Godlewski
%A Martin Parisot
%A Jacques Sainte-Marie
%A Fabien Wahl
%T Congested shallow water model: on floating body
%J The SMAI Journal of computational mathematics
%D 2020
%P 227-251
%V 6
%I Société de Mathématiques Appliquées et Industrielles
%R 10.5802/smai-jcm.67
%G en
%F SMAI-JCM_2020__6__227_0
Edwige Godlewski; Martin Parisot; Jacques Sainte-Marie; Fabien Wahl. Congested shallow water model: on floating body. The SMAI Journal of computational mathematics, Volume 6 (2020), pp. 227-251. doi : 10.5802/smai-jcm.67.

[1] E. B. Agamloh; A. K. Wallace; A. von Jouanne Application of Fluid-Structure Interaction Simulation of an Ocean Wave Energy Extraction Device, Renewable Energy, Volume 33 (2008) no. 4, pp. 748-757 | DOI

[2] N. Aïssiouene; M.-O. Bristeau; E. Godlewski; J. Sainte-Marie A combined finite volume - finite element scheme for a dispersive shallow water system, Networks and Heterogeneous Media (NHM) (2016) | Zbl

[3] E. Audusse; M.-O. Bristeau; B. Perthame; J. Sainte-Marie A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation, ESAIM, Math. Model. Numer. Anal., Volume 45 (2011) no. 1, pp. 169-200 | DOI | Numdam | MR | Zbl

[4] K. Benyo Wave-structure interaction for long wave models with a freely moving bottom (2017) (

[5] K. Benyo Numerical analysis of the weakly nonlinear Boussinesq system with a freely moving body on the bottom (2018) (

[6] M. Bergmann; A. Iollo Bioinspired swimming simulations, J. Comput. Phys., Volume 323 (2016), pp. 310-321 | DOI | MR | Zbl

[7] E. Bocchi Floating structures in shallow water: local well-posedness in the axisymmetric case (2018) (

[8] E. Bocchi On the return to equilibrium problem for axisymmetric floating structures in shallow water, 2019 ( | arXiv

[9] P. Bonneton; F. Chazel; D. Lannes; F. Marche; M. Tissier A splitting approach for the fully nonlinear and weakly dispersive Green–Naghdi model, J. Comput. Phys., Volume 230 (2011) no. 4, pp. 1479-1498 | DOI | MR | Zbl

[10] U. Bosi; A. P. Engsig-Karup; C. Eskilsson; M. Ricchiuto A spectral/hp element depth-integrated model for nonlinear wave–body interaction, Comput. Methods Appl. Mech. Eng., Volume 348 (2019), pp. 222-249 | DOI | MR | Zbl

[11] D. Bresch; D. Lannes; G. Metivier Waves Interacting With A Partially Immersed Obstacle In The Boussinesq Regime (2019) ( | arXiv

[12] M.-O. Bristeau; A. Mangeney; J. Sainte-Marie; N. Seguin An energy-consistent depth-averaged Euler system: derivation and properties, Discrete and Continuous Dynamical Systems - Series B, Volume 20 (2015) no. 4, pp. 961-988 | DOI | MR | Zbl

[13] C. Cancès; C. Guichard Numerical analysis of a robust free energy diminishing finite volume scheme for parabolic equations with gradient structure, Found. Comput. Math., Volume 17 (2017) no. 6, pp. 1525-1584 | DOI | MR | Zbl

[14] V. Chabannes; G. Pena; C. Prud’homme High-order fluid–structure interaction in 2D and 3D application to blood flow in arteries, J. Comput. Appl. Math., Volume 246 (2013), pp. 1-9 Fifth International Conference on Advanced COmputational Methods in ENgineering (ACOMEN 2011) | DOI | MR | Zbl

[15] B. Ducassou; J. Nuñez; M. Cruchaga; S. Abadie A fictitious domain approach based on a viscosity penalty method to simulate wave/structure interaction, Journal of Hydraulic Research, Volume 55 (2017) no. 6, pp. 847-862 | arXiv | DOI

[16] N. Favrie; S. Gavrilyuk A rapid numerical method for solving Serre–Green–Naghdi equations describing long free surface gravity waves, Nonlinearity, Volume 30 (2017) no. 7, p. 2718 | DOI | MR | Zbl

[17] E. D. Fernández-Nieto; M. Parisot; Y. Penel; J. Sainte-Marie A hierarchy of dispersive layer-averaged approximations of Euler equations for free surface flows, Commun. Math. Sci., Volume 16 (2018) no. 5, pp. 1169-1202 | DOI | MR | Zbl

[18] M. Folley; A. Babarit; B. Child; D. Forehand; L. O’Boyle; K. Silverthorne; J. Spinneken; P. Troch, ASME 2012 International Conference on Ocean, Offshore and Artic Engineering (2012) | DOI | HAL

[19] J. Fritz On the motion of floating bodies. I, Commun. Pure Appl. Math., Volume 2 (1949) no. 1, pp. 13-57 | DOI | MR | Zbl

[20] E. Godlewski; M. Parisot; J. Sainte-Marie; F. Wahl Congested shallow water model: roof modeling in free surface flow, ESAIM, Math. Model. Numer. Anal., Volume 52 (2018) no. 5, pp. 1679-1707 | DOI | MR | Zbl

[21] E. Godlewski; P.-A. Raviart Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, 118, Springer, 1996, viii+509 pages | MR | Zbl

[22] E. Guerber; B. Benoit; S. T. Grilli; C. Buvat A fully nonlinear implicit model for wave interactions with submerged structures in forced or free motion, Eng. Anal. Bound. Elem., Volume 36 (2012) no. 7, pp. 1151-1163 | DOI | MR | Zbl

[23] J. Harris; K. Kuznetsov; C. Peyrard; S. Saviot; A. Mivehchi; S. T. Grilli; M. Benoit, 27th Offshore and Polar Engineering Conference (2017)

[24] T. Iguchi; D. Lannes Hyperbolic free boundary problems and applications to wave-structure interactions (2018) (

[25] M Kashiwagi Non-linear simulations of wave-induced motions of a floating body by means of the mixed Eulerian-Lagrangian method, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Volume 214 (2000) no. 6, pp. 841-855 | DOI

[26] J. M. Knudsen; P. G. Hjorth Elements of Newtonian Mechanics, Springer, 2012 | Zbl

[27] D. Lannes On the Dynamics of Floating Structures, Ann. PDE, Volume 3 (2017) no. 1, p. 11 | DOI | MR | Zbl

[28] D. Lannes; P. Bonneton Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation, Physics of Fluids, Volume 21 (2009) no. 1, 016601 | DOI | Zbl

[29] D. Lannes; F. Marche A new class of fully nonlinear and weakly dispersive Green–Naghdi models for efficient 2D simulations, J. Comput. Phys., Volume 282 (2015), pp. 238-268 | DOI | MR | Zbl

[30] R. J. LeVeque Finite volume methods for hyperbolic problems, Cambridge University Press, 2002 | Zbl

[31] J. Palm; C. Eskilsson; G. Moura Paredes; L. Bergdahl, Proceedings of the 10th European Wave and Tidal Energy Conference (2013)

[32] M. Parisot Entropy-satisfying scheme for a hierarchy of dispersive reduced models of free surface flow, International Journal for Numerical Methods in Fluids, Volume 91 (2019) no. 10, pp. 509-531 | DOI | MR

[33] M. Parisot; J.-P. Vila Centered-Potential Regularization for the advection upstream splitting method, SIAM J. Numer. Anal., Volume 54 (2016) no. 5, pp. 3083-3104 | DOI | MR | Zbl

[34] N. Parolini; A. Quarteroni Mathematical models and numerical simulations for the America’s Cup, Comput. Methods Appl. Mech. Eng., Volume 194 (2005) no. 9, pp. 1001-1026 | DOI | MR | Zbl

[35] M. Penalba; I Touzón; J. Lopez-Mendia; V. Nava A numerical study on the hydrodynamic impact of device slenderness and array size in wave energy farms in realistic wave climates, Ocean Engineering, Volume 142 (2017), pp. 224-232 | DOI

[36] A. Quarteroni; M. Tuveri; A. Veneziani Computational vascular fluid dynamics: problems, models and methods, Computing and Visualization in Science, Volume 2 (2000) no. 4, pp. 163-197 | DOI | Zbl

[37] K. Steen Energy conservation in Newmark based time integration algorithms, Comput. Methods Appl. Mech. Eng., Volume 195 (2006) no. 44, pp. 6110-6124 | MR | Zbl

[38] G. X. Wu; R. Eatock Taylor The coupled finite element and boundary element analysis of nonlinear interactions between waves and bodies, Ocean Engineering, Volume 30 (2003) no. 3, pp. 387-400 | DOI

[39] Y.-H. Yu; Y. Li Reynolds-Averaged Navier–Stokes simulation of the heave performance of a two-body floating-point absorber wave energy system, Computers & Fluids, Volume 73 (2013), pp. 104-114 | DOI | Zbl

Cited by Sources: