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Discontinuous Galerkin methods for a dispersive wave hydro-morphodynamic model with bed-load transport. (English) Zbl 07340445
Summary: A dispersive wave hydro-morphodynamic model coupling the Green-Naghdi equations (the hydrodynamic part) with the sediment continuity Exner equation (the morphodynamic part) is presented. Numerical solution algorithms based on discontinuous Galerkin finite element discretizations of the model are proposed. The algorithms include both coupled and decoupled approaches for solving the hydrodynamic and morphodynamic parts simultaneously and separately from each other, respectively. The Strang operator splitting technique is employed to treat the dispersive terms separately, and it provides the ability to ignore the dispersive terms in specified regions, such as surf zones. Algorithms that can handle wetting-drying and detect wave breaking are presented. The numerical solution algorithms are validated with numerical experiments to demonstrate the ability of the algorithms to accurately resolve hydrodynamics of solitary and regular waves, and morphodynamic changes induced by such waves. The results indicate that the model has the potential to be used in studies of coastal morphodynamics driven by dispersive water waves, given that the hydrodynamic part resolves the water motion and dispersive wave effects with sufficient accuracy up to swash zones, and the morphodynamic model can capture the major features of bed erosion and deposition.
76-XX Fluid mechanics
65-XX Numerical analysis
Blaze; PETSc; HLLE; Eigen
Full Text: DOI
[1] Canestrelli, A.; Siviglia, A.; Dumbser, M.; Toro, E. F., Well-balanced high-order centred schemes for non-conservative hyperbolic systems. Applications to shallow water equations with fixed and mobile bed, Adv. Water Resour., 32, 6, 834-844 (2009)
[2] Canestrelli, A.; Dumbser, M.; Siviglia, A.; Toro, E. F., Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed, Adv. Water Resour., 33, 3, 291-303 (2010)
[3] Castro Díaz, M. J.; Fernández-Nieto, E. D.; Ferreiro, A. M., Sediment transport models in shallow water equations and numerical approach by high order finite volume methods, Comput. & Fluids, 37, 3, 299-316 (2008) · Zbl 1237.76082
[4] Castro Díaz, M. J.; Fernández-Nieto, E. D.; Ferreiro, A. M.; Parés, C., Two-dimensional sediment transport models in shallow water equations. A second order finite volume approach on unstructured meshes, Comput. Methods Appl. Mech. Engrg., 198, 33, 2520-2538 (2009) · Zbl 1228.76091
[5] Dal Maso, G.; LeFloch, P. G.; Murat, F., Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9), 74, 483-548 (1995) · Zbl 0853.35068
[6] Kozyrakis, G. V.; Delis, A. I.; Alexandrakis, G.; Kampanis, N. A., Numerical modeling of sediment transport applied to coastal morphodynamics, Appl. Numer. Math., 104, 30-46 (2016) · Zbl 1341.86003
[7] Liang, Q., A coupled morphodynamic model for applications involving wetting and drying, J. Hydrodyn. B, 23, 3, 273-281 (2011)
[8] Barzgaran, M.; Mahdizadeh, H.; Sharifi, S., Numerical simulation of bedload sediment transport with the ability to model wet/dry interfaces using an augmented Riemann solver, J. Hydroinform., 21, 5, 834-850 (2019)
[9] Rehman, K.; Cho, Y.-S., A novel well-balanced scheme for spatial and temporal bed evolution in rapidly varying flow, J. Hydro-environ. Res., 27, 87-101 (2019)
[10] Serrano-Pacheco, A.; Murillo, J.; Garcia-Navarro, P., Finite volumes for 2D shallow-water flow with bed-load transport on unstructured grids, J. Hydraul. Res., 50, 2, 154-163 (2012)
[11] García-Navarro, P.; Murillo, J.; Fernández-Pato, J.; Echeverribar, I.; Morales-Hernández, M., The shallow water equations and their application to realistic cases, Environ. Fluid Mech., 19, 5, 1235-1252 (2019)
[12] Kubatko, E. J.; Westerink, J. J.; Dawson, C., An unstructured grid morphodynamic model with a discontinuous Galerkin method for bed evolution, Ocean Model., 15, 1, 71-89 (2006)
[13] Izem, N.; Seaid, M.; Wakrim, M., A high-order nodal discontinuous Galerkin method for 1D morphodynamic modelling, Int. J. Comput. Appl., 41, 15, 19-27 (2012)
[14] Tassi, P.; Rhebergen, S.; Vionnet, C.; Bokhove, O., A discontinuous Galerkin finite element model for river bed evolution under shallow flows, Comput. Methods Appl. Mech. Engrg., 197, 33, 2930-2947 (2008) · Zbl 1194.76143
[15] Rhebergen, S.; Bokhove, O.; van der Vegt, J. J.W., Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations, J. Comput. Phys., 227, 3, 1887-1922 (2008) · Zbl 1153.65097
[16] Mirabito, C.; Dawson, C.; Kubatko, E. J.; Westerink, J. J.; Bunya, S., Implementation of a discontinuous Galerkin morphological model on two-dimensional unstructured meshes, Comput. Methods Appl. Mech. Engrg., 200, 1, 189-207 (2011) · Zbl 1225.76201
[17] Aizinger, V.; Dawson, C., A discontinuous Galerkin method for two-dimensional flow and transport in shallow water, Adv. Water Resour., 25, 1, 67-84 (2002)
[18] Kubatko, E. J.; Westerink, J. J.; Dawson, C., \( h p\) discontinuous Galerkin methods for advection dominated problems in shallow water flow, Comput. Methods Appl. Mech. Engrg., 196, 1, 437-451 (2006) · Zbl 1120.76348
[19] Dawson, C.; Kubatko, E. J.; Westerink, J. J.; Trahan, C.; Mirabito, C.; Michoski, C.; Panda, N., Discontinuous Galerkin methods for modeling Hurricane storm surge, Adv. Water Resour., 34, 9, 1165-1176 (2011)
[20] Bunya, S.; Kubatko, E. J.; Westerink, J. J.; Dawson, C., A wetting and drying treatment for the Runge-Kutta discontinuous Galerkin solution to the shallow water equations, Comput. Methods Appl. Mech. Engrg., 198, 17, 1548-1562 (2009) · Zbl 1227.76026
[21] Bremer, M.; Kazhyken, K.; Kaiser, H.; Michoski, C.; Dawson, C., Performance comparison of HPX versus traditional parallelization strategies for the discontinuous Galerkin method, J. Sci. Comput., 80, 2, 878-902 (2019) · Zbl 1427.65238
[22] Green, A. E.; Naghdi, P. M., A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78, 2, 237-246 (1976) · Zbl 0351.76014
[23] Bonneton, P.; Chazel, F.; Lannes, D.; Marche, F.; Tissier, M., A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model, J. Comput. Phys., 230, 4, 1479-1498 (2011) · Zbl 1391.76066
[24] Samii, A.; Dawson, C., An explicit hybridized discontinuous Galerkin method for Serre-Green-Naghdi wave model, Comput. Methods Appl. Mech. Engrg., 330, 447-470 (2018) · Zbl 1439.76094
[25] Lannes, D.; Marche, F., A new class of fully nonlinear and weakly dispersive Green-Naghdi models for efficient 2D simulations, J. Comput. Phys., 282, 238-268 (2015) · Zbl 1351.76114
[26] Duran, A.; Marche, F., A discontinuous Galerkin method for a new class of Green-Naghdi equations on simplicial unstructured meshes, Appl. Math. Model., 45, 840-864 (2017) · Zbl 1446.76016
[27] Exner, F. M., Über Die Wechselwirkung Zwischen Wasser Und Geschiebe in Flüssen, Sitzungsberichte. Abt. 2a, Mathematik, Astronomie, Physik Und Meteorologie (1925), Akademie Der Wissenschaften in Wien, Mathematisch- Naturwissenschaftliche Klasse
[28] Cordier, S.; Le, M.; Morales de Luna, T., Bedload transport in shallow water models: Why splitting (may) fail, how hyperbolicity (can) help, Adv. Water Resour., 34, 8, 980-989 (2011)
[29] Grass, A. J., Sediment Transport by Waves and CurrentsReport No. FL29 (1981), SERC London Centre for Marine Technology
[30] E. Meyer-Peter, R. Müller, Formulas for bed-load transport, in: Proceedings of 2nd Meeting of the International Association for Hydraulic Structures Research, 1948, pp. 39-64.
[31] Fernandez Luque, R.; van Beek, R., Erosion and transport of bed-load sediment, J. Hydraul. Res., 14, 2, 127-144 (1976)
[32] Nielsen, P., (Coastal Bottom Boundary Layers and Sediment Transport. Coastal Bottom Boundary Layers and Sediment Transport, Advanced series on ocean engineering (1992), World Scientific)
[33] Ribberink, J. S., Bed-load transport for steady flows and unsteady oscillatory flows, Coast. Eng., 34, 1, 59-82 (1998)
[34] Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5, 3, 506-517 (1968) · Zbl 0184.38503
[35] Samii, A.; Kazhyken, K.; Michoski, C.; Dawson, C., A comparison of the explicit and implicit hybridizable discontinuous Galerkin methods for nonlinear shallow water equations, J. Sci. Comput., 80, 3, 1936-1956 (2019) · Zbl 1428.65093
[36] Harten, A.; Lax, P. D.; v. Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 1, 35-61 (1983) · Zbl 0565.65051
[37] Krivodonova, L.; Xin, J.; Remacle, J.-F.; Chevaugeon, N.; Flaherty, J., Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws, Appl. Numer. Math., 48, 3, 323-338 (2004) · Zbl 1038.65096
[38] Cockburn, B.; Shu, C., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 3, 173-261 (2001) · Zbl 1065.76135
[39] Xu, Z.; Liu, Y.; Shu, C., Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells, J. Comput. Phys., 228, 6, 2194-2212 (2009) · Zbl 1165.65392
[40] Guennebaud, G.; Jacob, B., Eigen v3 (2010), www.eigen.tuxfamily.org
[41] Iglberger, K., Blaze C++ linear algebra library (2012), www.bitbucket.org/blaze-lib
[42] Balay, S.; Abhyankar, S.; Adams, M. F.; Brown, J.; Brune, P.; Buschelman, K.; Dalcin, L.; Dener, A.; Eijkhout, V.; Gropp, W. D.; Karpeyev, D.; Kaushik, D.; Knepley, M. G.; May, D. A.; McInnes, L. C.; Mills, R. T.; Munson, T.; Rupp, K.; Sanan, P.; Smith, B. F.; Zampini, S.; Zhang, H.; Zhang, H., Portable, extensible toolkit for scientific computation (2019), www.mcs.anl.gov/petsc
[43] Kaiser, H.; Lelbach, B. A.; Heller, T.; Simberg, M.; Bergé, A.; Biddiscombe, J.; Bikineev, A.; Mercer, G.; Schäfer, A.; Huck, K.; Lemoine, A. S.; Kwon, T.; Habraken, J.; Anderson, M.; Copik, M.; Brandt, S. R.; Stumpf, M.; Bourgeois, D.; Blank, D.; Jakobovits, S.; Amatya, V.; Viklund, L.; Khatami, Z.; Diehl, P.; Pathak, T.; Bacharwar, D.; Yang, S.; Schnetter, E., STEllAR-GROUP/hpx: HPX V1.4.1: The C++ standards library for parallelism and concurrency (2020), http://dx.doi.org/10.5281/zenodo.3675272
[44] Dubiner, M., Spectral methods on triangles and other domains, J. Sci. Comput., 6, 4, 345-390 (1991) · Zbl 0742.76059
[45] Walkley, M.; Berzins, M., A finite element method for the one-dimensional extended Boussinesq equations, Internat. J. Numer. Methods Fluids, 29, 2, 143-157 (1999) · Zbl 0941.76055
[46] Dodd, N., Numerical model of wave run-up, overtopping, and regeneration, J. Waterw. Port Coast. Ocean Eng., 124, 2, 73-81 (1998)
[47] Dingemans, M. W., Comparison of computations with Boussinesq-like models and laboratory measurements (1994)
[48] Sumer, B. M.; Sen, M. B.; Karagali, I.; Ceren, B.; Fredsøe, J.; Sottile, M.; Zilioli, L.; Fuhrman, D. R., Flow and sediment transport induced by a plunging solitary wave, J. Geophys. Res. Ocean., 116, C1 (2011)
[49] Li, J.; Qi, M.; Fuhrman, D. R., Numerical modeling of flow and morphology induced by a solitary wave on a sloping beach, Appl. Ocean Res., 82, 259-273 (2019)
[50] Pacheco, A.; Carrasco, A. R.; Vila-Concejo, A.; Ferreira, Ó.; Dias, J. A., A coastal management program for channels located in backbarrier systems, Ocean Coast. Manag., 50, 1, 119-143 (2007)
[51] Carrasco, A. R.; Plomaritis, T.; Reyns, J.; Ferreira, Ó.; Roelvink, D., Tide circulation patterns in a coastal lagoon under sea-level rise, Ocean Dyn., 68, 1121-1139 (2018)
[52] Portuguese Hydrographic Institute, Ria Formosa bathymetric model (100 meters spatial resolution), www.hidrografico.pt/recursos/files/download_gratuito/Modelo_batimetrico_Ria_Formosa_100m.rar.
[53] SCORE - Sustainability of using Ria Formosa currents on renewable energy production (2020), Funded by the Portuguese Foundation for Science and Technology (FCT - PTDC/AAG-TEC/1710/2014), www.http://w3.ualg.pt/ ampacheco/Score/index.html, last accessed: 13/05/2020
[54] González-Gorbeña, E.; Pacheco, A.; Plomaritis, T. A.; Óscar. Ferreira, Ó.; Sequeira, C., Estimating the optimum size of a tidal array at a multi-inlet system considering environmental and performance constraints, Appl. Energy, 232, 292-311 (2018)
[55] SCORE project database (2020), http://w3.ualg.pt/ ampacheco/Score/database.html, last accessed: 13/05/2020
[56] EMODnet Bathymetry Consortium, EMODnet Digital Bathymetry (DTM 2018), http://dx.doi.org/10.12770/18ff0d48-b203-4a65-94a9-5fd8b0ec35f6.
[57] Dias, J. M.; Sousa, M. C.; Bertin, X.; Fortunato, A. B.; Oliveira, A., Numerical modeling of the impact of the Ancão inlet relocation (Ria Formosa, Portugal), Environ. Model. Softw., 24, 6, 711-725 (2009)
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