Open Access Peer-reviewed

Finite Element Modeling of Regular Waves by 1-D Madsen and Sorensen Extended Boussineq Equations

Parviz Ghadimi1,, Amin Rahimzadeh1, Mohammad H. Jabbari1, Rahim Zamanian2

1Department of Marine Technology, Amirkabir University of Technology, Tehran, Iran

2International Campus – Mechanical Engineering Group, Amirkabir University of Technology, Tehran, Iran

American Journal of Marine Science. 2013, 1(1), 1-6. DOI: 10.12691/marine-1-1-1
Published online: August 25, 2017


In this paper, finite element modeling of one-dimensional extended Boussinesq equations derived by Madsen and Sorensen is presented for simulation of propagating regular waves. In order to spatially discretize the finite element equations, method of weighted residual Galerkin approach is used. Discretization of third-order derivative in momentum equation is performed by introducing of an auxiliary equation, which makes it possible to use linear finite element method. Adams-Bashforth-Moulton predictor-corrector method is used for time integration. Regular wave trains are simulated using the proposed numerical scheme. For validation of the developed code, the model is applied to several examples of wave propagation over the computational domain and the obtained results of the current computations are compared against the experimental measurements. In all cases, the proposed model has proved very suitable for simulating the propagation of wave indicating favorable agreements with experimental data.


Enhanced Boussinesq equations, Galerkin finite element method, Auxiliary equation
[1]  Peregrine D. H., “Long waves on a beach,” J. Fluid Mech, 27. 815-827. 1967.View Article
[2]  Witting, J.M., “A unified model for the evolution of nonlinear water waves,” J. Comput. Phys, 56. 203-236. 1984.View Article
[3]  Madsen P. A., Sch¨affer H. A., Higher-order Boussinesq type equations for surface gravity waves: Derivation and analysis, Proc. R. Soc. London, 3123-318, 1998.
[4]  Madsen P. A., Sørensen O. R., “A new form of the Boussinesq equations with improved linear dispersion characteristics: Part 2. A slowly-varying bathymetry,” Coastal Eng, 18. 183-204. 1992.View Article
[5]  Nwogu O., “Alternative form of Boussinesq equations for nearshore wave propagation,” Coastal Eng, 15. 371-388. 1991.
[6]  Wei G., Kirby J. T., Grilli S. T., Subramanya R., “A fully nonlinear Boussinesq model for surface waves, Part 1: Highly nonlinear unsteady waves,” J. Fluid Mech, 294. 71-92. 1995.View Article
[7]  Lynett P. J., Liu P. L. F., “Linear analysis of the multi-layer model,” Coastal Eng, 35 51. 439-454. 2004.
[8]  Kennedy A. B., Kirby J. T., Chen Q., Dalrymple R. A., “Boussinesq-type equations with improved nonlinear behavior,” Wave Motion, 33. 225-243. 2001.View Article
[9]  Li B., “Equations for regular and irregular water wave propagation,” J. Waterway, Port,Coastal, and Ocean Eng, 134(2).121-142. 2008.
[10]  Gobbi M. F., Kirby J. T., Wei G., “A fully nonlinear Boussinesq model for surface waves, Part 2: Extension to O(kh)4,” J. Fluid Mech, 405. 181-210. 2000.View Article
[11]  Nadaoka K., Beji S., Nakagawa Y., A fully dispersive weakly nonlinear model for water waves, Proc. R. Soc. London, Ser. A, 453, 303-318, 1997.View Article
[12]  Madsen P. A., Murray R., Sørensen O. R., “A new form of the Boussinesq equations with improved linear dispersion characteristics,” Coastal Eng, 15. 371-388. 1991.View Article
[13]  Beji S., Nadaoka K., “A formal derivation and numerical modeling of the improved Boussinesq equations for varying depth,” Coastal Eng, 23(8). 691-704. 1996.
[14]  Antunes do Carmo J. S., Seabra Santos F. J., Barth´elemy E., “Surface waves propagation in shallow water: A finite element model,” Int. J. Numer. Meth. Fluids, 16. 447-459. 1993.View Article
[15]  Ambrosi D., Quartapelle L., “A Taylor–Galerkin method for simulating nonlinear dispersive water waves,” J. Comput. Phys, 146. 546-569. 1998.View Article
[16]  Eskilsson C., Sherwin S. J., “A discontinuous spectral element model for Boussinesq type equations,” J. Sci. Comput, 17. 143-152. 2002.View Article
[17]  Eskilsson C., Sherwin S. J., “Discontinuous Galerkin spectral/hp element modeling of dispersive shallow water systems,” J. Sci. Comput, 22. 269-288. 2005.View Article
[18]  Li Y. S., Liu S. X., Yu Y. X., Lai G. Z., “Numerical modeling of Boussinesq equations by finite element method, ” Coastal Eng, 37. 97-122. 1999.View Article
[19]  Walkley M., Berzins M., “A finite element method for the one-dimensional extended Boussinesq equations,” Int. J. Numer. Meth. Fluids, 29. 143-157. 1999.View Article
[20]  Walkley M., Berzins M., “A finite element method for the two-dimensional extended Boussinesq equations,” Int. J. Numer. Meth. Fluids, 39. 865-885. 2002.View Article
[21]  Ole R. Sørensen, Hemming A. Scha¨ffer, Lars S. Sørensen, “Boussinesq-type modelling using an unstructured finite element technique,” Coastal Eng, 50. 181-198. 2004.View Article
[22]  Bradford S. F., Sanders B. F., “Finite-volume models for unidirectional, nonlinear, dispersive waves,” J. Waterway, Port, Coastal, Ocean Eng, 128. 173-182. 2002.View Article
[23]  Stansby P. K., “Solitary wave run up and overtopping by a semi-implicit finite-volume shallow-water Boussinesq model,” J. Hydraulic Res, 41. 639-647. 2003.View Article
[24]  Erduran K. S., Ilic S., Kutija V., “Hybrid finite-volume 1 finite-difference scheme for the solution of Boussinesq equations,” Int. J. Numer. Meth. Fluids, 49. 1213- 1232, 2005.View Article
[25]  Ning D. Z., Zang J., Liang Q., Taylor P. H., Borthwick A. G. L., “Boussinesq cut-cell model for nonlinear wave interaction with coastal structures,” Int. J. Numer. Meth.Fluids, 57. 1459-1483. 2008.View Article
[26]  Dean R. G., Dalrymple R. A., Water Wave Mechanics for Engineers and Scientists, World Scientific, Singapore, 2000.
[27]  Luth H. R., Klopman G., Kitou N., Projects 13G: Kinematics of waves breaking partially on an offshore bar, LVD measurements for waves without a net onshore current, Technical Rep. No. H1573, Delft Hydraulics, Delft, the Netherlands. 1994