Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-23T17:37:25.867Z Has data issue: false hasContentIssue false
  • English
  • Français

Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations

Published online by Cambridge University Press:  24 March 2015

Arnaud Duran  and
Fabien Marche
Arnaud Duran
Affiliation:
Institut de Mathématiques et de Modélisation de Montpellier (I3M), Université Montpellier 2, CC 051, 34090 Montpellier, France Inria, Project Team LEMON, Bt 5 – CC05 017 – 34095 Montpellier, France
Fabien Marche*
Affiliation:
Institut de Mathématiques et de Modélisation de Montpellier (I3M), Université Montpellier 2, CC 051, 34090 Montpellier, France Inria, Project Team LEMON, Bt 5 – CC05 017 – 34095 Montpellier, France
*
*Corresponding author. Email addresses:Arnaud.Duran@math.univ-montp2.fr (A. Duran), Fabien.Marche@math.univ-montp2.fr (F. Marche)
Get access

Abstract

We describe in this work a discontinuous-Galerkin Finite-Element method to approximate the solutions of a new family of 1d Green-Naghdi models. These new models are shown to be more computationally efficient, while being asymptotically equivalent to the initial formulation with regard to the shallowness parameter. Using the free surface instead of the water height as a conservative variable, the models are recasted under a pre-balanced formulation and discretized using a nodal expansion basis. Independently from the polynomial degree in the approximation space, the preservation of the motionless steady-states is automatically ensured, and the water height positivity is enforced. A simple numerical procedure devoted to stabilize the computations in the vicinity of broken waves is also described. The validity of the resulting model is assessed through extensive numerical validations.

Keywords

76M1065M10Green-Naghdi equationsdiscontinuous Galerkinshallow water equationswave breakingnonlinear and dispersive water waves
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 3 , March 2015 , pp. 721 - 760
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adjerid, S., Devine, K., Flaherty, J., and Krivodonova, L.. A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Comput. Meth. Appl. Mech. Engrg, 191(11):1097–1112, 2002.Google Scholar
[2]Alvarez-Samaniego, B. and Lannes, D.. A Nash-Moser theorem for singular evolution equations. application to the Serre and Green-Naghdi equations. Indiana Univ. Math. J., 57:97–131, 2008.Google Scholar
[3]Ambati, V.R. and Bokhove, O.. Space–time discontinuous Galerkin finite element method for shallow water flows. J. Comput. App. Math., 204:452–462, 2007.Google Scholar
[4]Arnold, D.N., Brezzi, F., Cockburn, B., and Marini, L.D.. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39(5):1749–1779, 2002.Google Scholar
[5]Bassi, F. and Rebay, S.. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys., 131:267–279, 1997.Google Scholar
[6]Beji, S. and Battjes, J.A.. Experimental investigation of wave propagation over a bar. Coast. Engrg., 19:151–162, 1993.Google Scholar
[7]Berthon, C. and Marche, F.. A positive preserving high order VFRoe scheme for shallow water equations: a class of relaxation schemes. SIAM J. Sci. Comput., 30:2587–2612, 2008.Google Scholar
[8]Bonneton, P.. Modelling of periodic wave transformation in the inner surf zone. Ocean Engineering, 34(10):1459–1471, 2007.Google Scholar
[9]Bonneton, P., Chazel, F., Lannes, D., Marche, F., and Tissier, M.. A splitting approach for the fully nonlinear and weakly dispersive Green – Naghdi model. J. Comput. Phys., 230:1479–1498, 2011.Google Scholar
[10]Burbeau, A., Sagaut, P., and Bruneau, Ch.-H.. A problem-independent limiter for high-order runge–kutta discontinuous Galerkin methods. J. Comput. Phys., 169:111–150, 2001.Google Scholar
[11]Chazel, F., Lannes, D., and Marche, F.. Numerical simulation of strongly nonlinear and dispersive waves using a Green – Naghdi model. J. Sci. Comput., pages 105–116, 2011.Google Scholar
[12]Chen, Q., Kirby, J.T., Dalrymple, R., and Kennedy, A.. Boussinesq modeling of wave transformation, breaking, and runup. II: 2 D. J. Water. Port Coast. and Oc. Engrg., 2000.Google Scholar
[13]Cienfuegos, R., Barthélemy, E., and Bonneton, P.. A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. I: Model development and analysis. Int. J. Num. Meth. Fluids, 51(11):1217–1253, 2006.Google Scholar
[14]Cienfuegos, R., Barthelemy, E., and Bonneton, P.. A wave-breaking model for Boussinesq-type equations including mass-induced effects. J. Water. Port Coast. and Oc. Engrg., 136:10–26, 2010.Google Scholar
[15]Cienfuegos, R., Barthélemy, E., and Bonneton, P.. Wave-breaking model for Boussinesq type equations including roller effects in the mass conservation equation. J. Water. Port. Coast. and Oc. Engrg., 136:10–26, 2010.Google Scholar
[16]Cockburn, B. and Shu, C.-W.. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput., 16(3):173–260, 2001.Google Scholar
[17]Cockburn, B. and Shu, S.-W.. The Local Discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal., 141:2440–2463, 1998.Google Scholar
[18]Cox, D.T.. Experimental and numerical modelling of surf zone hydrodynamics. Univ. of Delaware, Newark, 1995.Google Scholar
[19]Davis, T.A. and Duff, I.S.. An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM J. on Matrix Anal. and Applications, 18:140–158, 1997.Google Scholar
[20]Dingemans, M.W.. Comparison of computations with Boussinesq-like models and laboratory measurements. Delft Hydr., Report H-1684.12, 32, 1994.Google Scholar
[21]Duran, A. and Marche, F.. Recent advances on the discontinuous Galerkin method for shallow water equations with topography source terms. Comput. & Fluids, 101:88–104, 2014.Google Scholar
[22]Duran, A., Marche, F., and Liang, Q.. On the well-balanced numerical discretization of shallow water equations on unstructured meshes. J. Comput. Phys., 235:565–586, 2013.Google Scholar
[23]Dutykh, D., Katsaounis, T., and Mitsotakis, D.. Finite volume schemes for dispersive wave propagation and runup. J. Comput. Phys., 230:3035–3061, 2011.Google Scholar
[24]Engsig-Karup, A.P., Hesthaven, J.S., Bingham, H.B., and Madsen, P.A.. Nodal DG-FEM solution of high-order Boussinesq-type equations. J. Eng. Math., 56:351–370, 2006.Google Scholar
[25]Engsig-Karup, A.P., Hesthaven, J.S., Bingham, H.B., and Warburton, T.. DG-FEM solution for nonlinear wave-structure interaction using Boussinesq-type equations. Coast. Engrg., 55:197–208, 2008.Google Scholar
[26]Erduran, K., Ilic, S., and Kutija, V.. Hybrid finite-volume finite-difference scheme for the solution of Boussinesq equations. Int. J. Num. Meth. Fluids, 49:1213–1232, 2005.Google Scholar
[27]Erduran, K.S.. Further application of hybrid solution to another form of Boussinesq equations and comparisons. Int. J. Num. Meth. Fluids, 53:827–849, 2007.Google Scholar
[28]Ern, A. and Guermond, J.-L.. Discontinuous Galerkin methods for Friedrichs’ systems. i. general theory. SIAM J. Numer. Anal., 44(2):753–778, 2006.Google Scholar
[29]Ern, A. and Guermond, J.-L.. Discontinuous Galerkin methods for Friedrichs’ systems. ii. second-order elliptic pdes. SIAM J. Numer. Anal., 44(6):2363–2388, 2006.Google Scholar
[30]Ern, A., Piperno, S., and Djadel, K.. A well- balanced Runge-Kutta discontinuous Galerkin method for the shallow-water equations with flooding and drying. Int. J. Num. Meth. Fluids, 58:1–25, 2008.Google Scholar
[31]Eskilsson, C. and Sherwin, S. J.. An hp/spectral element model for efficient long-time integration of Boussinesq-type equations. Coast. Engrg., 45:295–320, 2003.Google Scholar
[32]Eskilsson, C. and Sherwin, S.J.. Discontinuous Galerkin spectral/hp element modelling of dispersive shallow water systems. J. Sci. Comput., 22 – 23:269–288, 2005.Google Scholar
[33]Eskilsson, C. and Sherwin, S.J.. Spectral/hp discontinuous Galerkin method for modelling 2D Boussinesq equations. J. Comput. Phys., 212:566–589, 2006.Google Scholar
[34]Eskilsson, C., Sherwin, S.J., and Bergdahl, L.. An unstructured spectral/hp element model for enhanced Boussinesq-type equations. Coast. Engrg., 53(947–963), 2006.Google Scholar
[35]Gottlieb, S., Shu, C.-W., and Tadmor, E.Strong stability preserving high order time discretization methods. SIAM Review, 43:89–112, 2001.Google Scholar
[36]Green, A.E. and Naghdi, P.M.. A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech., 78:237–246, 1976.Google Scholar
[37]Guibourg, S.. Modélisation numérique et expérimentale des houles bidimensionnelles en zone cotière, phd thesis. Université Joseph Fourier-Grenoble I, France, 1994.Google Scholar
[38]Hsiao, S-C. and Lin, T-C.. Tsunami-like solitary waves impinging and overtopping an impermeable seawall: Experiment and rans modeling. Coast. Engrg., 57:1–18, 2010.Google Scholar
[39]Kazolea, M., Delis, A.I., Nikolos, I.K., and Synolakis, C.E.. An unstructured finite volume numerical scheme for extended 2DBoussinesq-type equations. Coast. Engrg., 69:42–66, 2012.Google Scholar
[40]Kazolea, M., Delis, A.I., and Synolakis, C.E.. Numerical treatment of wave breaking on unstructured finite volume approximations for extended Boussinesq-type equations. J. Comput. Phys., 271:281–305, 2014.Google Scholar
[41]Kennedy, A., Chen, Q., Kirby, J., and Dalrymple, R.. Boussinesq modeling of wave transformation, breaking, and runup. I: 1 D. J. Water. Port Coast. and Oc. Engrg., 1999.Google Scholar
[42]Krivodonova, L., Xin, J., Remacle, J.-F., Chevaugeon, N., and Flaherty, J.E.. Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Applied Numerical Mathematics, 48:323–338, 2004.Google Scholar
[43]Lannes, D.. Water waves: mathematical analysis and asymptotics. Amer. Mathematical Society, 2013.Google Scholar
[44]Lannes, D. and Bonneton, P.. Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation. Physics of fluids, 21:016601, 2009.Google Scholar
[45]Lannes, D. and Marche, F.. A new class of fully nonlinear and weakly dispersive Green-Naghdi models for efficient 2d simulations. submitted, 2014.Google Scholar
[46]Le Métayer, O., Gavrilyuk, S., and Hank, S.. A numerical scheme for the Green-Naghdi model. J. Comput. Phys., (229):2034–2045, 2010.Google Scholar
[47]M Li, P. Guyenne, Li, F., and Xu, F.. High order well-balanced CDG-FE methods for shallow water waves by a Green-Naghdi model. J. Comput. Phys., 257:169–192, 2014.Google Scholar
[48]Li, Y.S., Liu, S.X., Yu, Y.X., and Lai, G.Z.. Numerical modeling of Boussinesq equations by finite element method. Coast. Engrg., 37:97–122, 1999.Google Scholar
[49]Liang, Q. and Borthwick, A. G. L.. Adaptive quadtree simulation of shallow flows with wetdry fronts over complex topography. Comput. & Fluids, 38(2):221–234, 2009.Google Scholar
[50]Liang, Q. and Marche, F.. Numerical resolution of well-balanced shallow water equations with complex source terms. Adv. Water Res., 32:873–884, 2009.Google Scholar
[51]Madsen, P.A., Bingham, H.B., and Liu, H.. A new Boussinesq method for fully nonlinear waves from shallow to deep water. J. Fluid Mech., 462:1–30, 2002.Google Scholar
[52]Madsen, P.A., Bingham, H.B., and Schaffer, H.A.. Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: derivation and analysis. Proc. R. Soc. A, 459:1075–1104, 2003.Google Scholar
[53]Madsen, P.A., Murray, R., and Sorensen, O.R.. A new form of the Boussinesq equations with improved linear dispersion characteristics. Coast. Engrg., 15(4):371–388, 1991.Google Scholar
[54]Madsen, P.A. and Sørensen, O.R.. A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2: A slowing varying bathymetry. Coast. Engrg., 18:183–204, 1992.Google Scholar
[55]Mitsotakis, D., Ilan, B., and Dutykh, D.. On the Galerkin/finite-element method for the Serre equations. J. Sci. Comput., 2014.Google Scholar
[56]Nwogu, O.. Alternative form of Boussinesq equations for nearshore wave propagation. J. Water. Port Coast. and Oc. Engrg., 119:618–638, 1993.Google Scholar
[57]Orszaghova, J., Borthwick, A.G.L., and Taylor, P.H.. From the paddle to the beach – A Boussinesq shallow water numerical wave tank based on Madsen and Sørensen’s equations. J. Comput. Phys., 231:328–344, 2012.Google Scholar
[58]Pearce, J.D. and Esler, J.G.. A pseudo-spectral algorithm and test cases for the numerical solution of the two-dimensional rotating Green-Naghdi shallow water equations. J. Comput. Phys., 229:7594–7608, 2010.Google Scholar
[59]Peregrine, D.. Long waves on a beach. J. Fluid Mech., 27:815–827, 1967.Google Scholar
[60]Peregrine, D.H.. Long waves on a beach. J. Fluid Mech., 27:815–827, 1967.Google Scholar
[61]Qiu, J. and Shu, C.-W.. A comparison of troubled-cell indicators for Runge–Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters. SIAM J. Sci. Comput., 27:995–1013, 2005.Google Scholar
[62]Ricchiuto, M. and Filippini, A.G.. Upwind residual discretization of enhanced Boussinesq equations for wave propagation over complex bathymetries. J. Comput. Phys., in press, 2014.Google Scholar
[63]Roeber, V. and Cheung, K.F.. Boussinesq-type model for energetic breaking waves in fringing reef environments. Coast. Engrg., 70(0):1–20, 2012.Google Scholar
[64]Rogers, B., Fujihara, M., and Borthwick, A.. Adaptive Q-tree Godunov-type scheme for shallow water equations. Int. J. Num. Meth. Fluids, 35:247–280, 2001.Google Scholar
[65]Schaeffer, H.. A Boussinesq model for waves breaking in shallow water. Coast. Engrg., 20:185–202, 1993.Google Scholar
[66]Serre, F.. Contribution á l’étude des écoulements permanents et variables dans les canaux. Houille Blanche, 6:830–872, 1953.Google Scholar
[67]Shi, F., Kirby, J.T., Harris, J.C., Geiman, J.D., and Grilli, S.T.. A high-order adaptive time-stepping tvd solver for Boussinesq modeling of breaking waves and coastal inundation. Ocean Modelling, 43–44:36–51, 2012.Google Scholar
[68]Soares-Frazão, S. and Guinot, V.. A second-order semi-implicit hybrid scheme for one-dimensional Boussinesq-type waves in rectangular channels. Int. J. Num. Meth. Fluids, 58:237–261, 2008.Google Scholar
[69]Sørensen, O., Schäffer, H., and Madsen, P.. Surf zone dynamics simulated by a Boussinesq type model. III. Wave-induced horizontal nearshore circulation. Coast. Engrg., 33:155–176, 1998.Google Scholar
[70]Sørensen, O.R., Schaffer, H.A., and Sørensen, L.S.. Boussinesq-type modelling using an unstructured finite element technique. Coast. Engrg., 50:181–198, 2004.Google Scholar
[71]Synolakis, C.E.. The runup of solitary waves. J. Fluid Mech., 185:523–545, 1981.Google Scholar
[72]Tissier, M., Bonneton, P., Marche, F., Chazel, F., and Lannes, D.. A new approach to handle wave breaking in fully non-linear Boussinesq models. Coast. Engrg., 67:54–66, 2012.Google Scholar
[73]Tonelli, M. and Petti, M.. Hybrid finite volume- finite difference scheme for 2dh improved Boussinesq equations. Coast. Engrg., 56(5–6):609–620, 2009.Google Scholar
[74]Tonelli, M. and Petti, M.. Finite volume scheme for the solution of 2d extended Boussinesq equations in the surf zone. Ocean Engineering, 37:567–582, 2010.Google Scholar
[75]Tonelli, M. and Petti, M.. Shock-capturing Boussinesq model for irregular wave propagation. Coastal Engineering, 61:8–19, 2012.Google Scholar
[76]Toro, E.F., Spruce, M., and Speare, W.. Restoration of the contact surface in the HLL Riemann solver. Shock waves, 4:25–34, 1994.Google Scholar
[77]Walkley, M. and Berzins, M.. A finite element method for the two-dimensional extended Boussinesq equations. Int. J. Num. Meth. Fluids, 39(10):865–885, 2002.Google Scholar
[78]Wang, Y., Liang, Q., Kesserwani, G., and Hall, J.W.. A 2D shallow flow model for practical dam-break simulations. J. Hydraulic Res., 49(3):307–316, 2011.Google Scholar
[79]Wei, G. and Kirby, J.. Time-dependent numerical code for extended Boussinesq equations. J. Water. Port Coast. and Oc. Engrg., 121:251–261, 1995.Google Scholar
[80]Wei, G., Kirby, J., Grilli, S., and Subramanya, R.. A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. J. Fluid Mech., 294:71–92, 1995.Google Scholar
[81]Xing, Y. and Shu, C.-W.. A new approach of high order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. Commun. Comput. Phys., 1:100–134, 2006.Google Scholar
[82]Xing, Y. and Zhang, X.. Positivity-preserving well-balanced discontinuous Galerkin methods for the shallow water equations on unstructured triangular meshes. J. Sci. Comput., 57:19–41, 2013.Google Scholar
[83]Xing, Y., Zhang, X., and Shu, C.-W.. Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Advances in Water Resources, 33(12):1476 – 1493, 2010.Google Scholar
[84]Xu, Y. and Shu, C.-W.. Local Discontinuous Galerkin methods for high-order time-dependent partial differential equations. Commun. Comput. Phys., 7(1):1–46, 2010.Google Scholar
[85]Zhang, X. and Shu, C.-W.. On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys., 229(9):3091 – 3120, 2010.Google Scholar