Chebyshev Finite-Spectral Method for 1D Boussinesq-Type Equations
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 132, Issue 3
Abstract
In this paper, an accurate Chebyshev finite-spectral method for one-dimensional (1D) Boussinesq-type equations is proposed. The method combines the advantages of both the finite-difference and spectral methods. The spatial derivatives in the governing equations can be calculated accurately in an efficient way, while some flexibility is allowed for treating irregular grids. The efficiency and accuracy of the proposed method are verified by successfully solving the problem of solitary wave propagation over a flat bottom where analytical solutions are available for comparison. A simple formula to calculate the wave celerity of the solitary wave propagation has also been derived. Finally, the applicability of the numerical method to periodic and random waves was validated by the simulation of nonlinear wave propagation over a bar where laboratory data are available for comparison. The method can be easily extended to treat 2D problems.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The work reported in this paper is supported by the National Natural Science Foundation of China (NSFC, No. 10272118), the Research Fund for the Doctoral Program of Ministry of Education of China (No. 20020558013), and The Hong Kong Polytechnic University (Research Grant No. A-PE28).
References
Beji, S., and Battjes, J. A. (1993). “Experimental investigation of wave propagation over a bar.” Coastal Eng., 19, 151–162.
Beji, S., and Battjes, J. A. (1994). “Numerical simulation of nonlinear wave propagation over a bar.” Coastal Eng., 23, 1–16.
Beji, S., and Nadaoka, K. (1996). “A formal derivation and numerical modelling of the improved Boussinesq equations for varying depth.” Ocean Eng., 23(8), 691–704.
Ghaddar, N. K., Karniadakis, G. E., and Patera, A. T. (1986). “A conservative isoparametric spectral element method for forced convection: Application to fully developed flow in periodic geometries.” Numer. Heat Transfer, 9(3), 277–300.
Giraldo, F. X. (2003). “Strong and weak Lagrange-Galerkin spectral element methods for the shallow water equations.” Comput. Math. Appl., 45(1–3), 97–121.
Larsen, J., and Dancy, H. (1983). “Open boundaries in short wave simulation—A new approach.” Coastal Eng., 7, 285–297.
Li, Y. S., Liu, S. X., Yu, Y. X., and Lai, G. Z. (1999). “Numerical modeling of Boussinesq equations by finite element method.” Coastal Eng., 37, 97–122.
Li, Y. S., Liu, S. X., Yu, Y. X., and Lai, G. Z. (2000). “Numerical modelling of multi-directional irregular wave through breakwaters.” Appl. Math. Model., 24(8–9), 551–574.
Li, Y. S., and Zhan, J. M. (2001). “Boussinesq-type model with boundary-fitted coordinate system.” J. Waterw., Port, Coastal, Ocean Eng., 127(3), 152–160.
Madsen, P. A. and Sørensen, O. R. (1992). “A new form of Boussinesq equation with improved linear dispersion characteristics. 2: A slowly-varying bathymetry.” Coastal Eng., 18, 183–204.
Nwogu, O. (1993). “Alternative form of Boussinesq equations for nearshore wave propagation.” J. Waterw., Port, Coastal, Ocean Eng., 119(6), 618–638.
Patera, A. T. (1984). “A spectral element method for fluid dynamics—Laminar flow in a channel expansion.” J. Comput. Phys., 54, 468–488.
Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T. (1989). Numerical recipes, Cambridge Univ. Press, New York, 569–572.
Shi, F., Dalrymple, R. A., Kirby, J. T., Chen, Q., and Kennedy, A. (2001). “Boussinesq model in generalized curvilinear coordinates.” Coastal Eng., 42, 337–358.
Sørensen, O. R., and Sørensen, L. S. (2000). “Boussinesq type modeling using unstructured finite element technique.” Coastal Engrg., B. L. Edge, ed., ASCE, Reston, Va., 190–202.
Wang, J. P. (1998). “Finite spectral method based on non-periodic Fourier transform.” Comput. Fluids, 27(5–6), 639–644.
Wei, G., and Kirby, J. T. (1995). “Time-dependent numerical code for extended Boussinesq equations.” J. Waterw., Port, Coastal, Ocean Eng., 121(5), 251–261.
Woo, S.-B., and Liu, P. L.-F. (2004a). “A Petrov-Galerkin finite element model for one-dimensional fully non-linear and weakly dispersive wave propagation.” Int. J. Numer. Methods Fluids, 37, 541–575.
Woo, S.-B., and Liu, P. L.-F. (2004b). “Finite-element model for modified Boussinesq equations. I: Model development.” J. Waterw., Port, Coastal, Ocean Eng., 130(1), 1–16.
Information & Authors
Information
Published In
Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 132 • Issue 3 • May 2006
Pages: 212 - 223
Copyright
© 2006 ASCE.
History
Received: Sep 9, 2004
Accepted: Sep 14, 2005
Published online: May 1, 2006
Published in print: May 2006
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.