A Central Numerical Scheme to 1D Green-Naghdi Wave Equations


A numerical scheme based on hybrid central finite-volume and finite-difference method is presented to model Green-Naghdi water wave equations. The governing equations are reformulated into the conservative form, and the convective flux is estimated using a Godunov-type finite volume method while the remaining terms are discretized using finite difference method. To enhance the robustness of the model, a central-upwind flux evaluation and a well-balanced non- negative water depth construction are incorporated. Numerical tests demonstrate that present model has the advantages of stability preserving and numerical efficiency.

Share and Cite:

Fang, K. , Jiao, Z. , Yin, J. and Sun, J. (2015) A Central Numerical Scheme to 1D Green-Naghdi Wave Equations. Journal of Applied Mathematics and Physics, 3, 1032-1037. doi: 10.4236/jamp.2015.38127.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Serre, F. (1953) Contribution à L’étude des écoulements Permanents et Variables Dans Les Canaux. La Houille Blanche, 6, 830-872. http://dx.doi.org/10.1051/lhb/1953058
[2] Green, A.E. and Naghdi, P.M. (1976) A Derivation of Equations for Wave Propagation in Water of Variable Depth. Journal of Fluid Mechanics, 78, 237-246. http://dx.doi.org/10.1017/S0022112076002425
[3] Le Metayer, O., Gavrilyuk, S. and Hank, S. (2010) A Numerical Scheme for the Green-Naghdi Model. Journal of Computational Physics, 229, 2034-2045. http://dx.doi.org/10.1016/j.jcp.2009.11.021
[4] Bonneton, P., Barthelemy, E., Chazel, F., Cienfuegos, R., Lannes, D., Marche, F. and Tissier, M. (2011) Recent Advances in Serre-Green Naghdi Modelling for Wave Transformation, Breaking and Runup Processes. Journal of Computational Physics, 30, 589-597.
[5] Li, M., Guyenne, P., Li, F. and Xu, L. (2014) High Order Well-Balanced CDG-FE Methods for Shallow Water Waves by A Green-Naghdi Model. J. of Computational Physics, 257, 169-192. http://dx.doi.org/10.1016/j.jcp.2013.09.050
[6] Fang, K., Liu, Z. and Zou, Z. (2015) Fully Nonlinear Modeling Wave Transformation over Fringing Reefs Using Shock-Capturing Boussinesq Model. Journal of Coastal Research, in Press. http://dx.doi.org/10.2112/JCOASTRES-D-15-00004.1
[7] Wang, Y., Liang, Q., Kesserwani, G. and Hall, J.W. (2011) A 2D Shallow Flow Model for Practical Dam-Break Simulations. Journal of Hydraulic Research, 49, 307-316. http://dx.doi.org/10.1080/00221686.2011.566248
[8] Kurganov, A. and Petrova, G. (2007) A Second-Order Well-Balanced Positivity Preserving Central Upwind Scheme for the Saint-Vinant System. Commun. Math. Sci., 5, 133-160. http://projecteuclid.org/euclid.cms/1175797625 http://dx.doi.org/10.4310/CMS.2007.v5.n1.a6
[9] Lannes, D. and Marche, F. (2015) A New Class of Fully Nonlinear and Weakly Dispersive Green-Naghdi Models for Efficient 2D Simulations. Journal of Computational Physics, 282, 238-268. http://dx.doi.org/10.1016/j.jcp.2014.11.016
[10] Craig, W., Guyenne, P., Hammack, J., Henderson, D. and Sulem, C. (2006) Solitary Water Wave Interactions. Physics of Fluids (1994-Present), 18, Article ID: 057106. http://dx.doi.org/10.1063/1.2205916
[11] Synolakis, C.E. (1986) The Runup of Long Waves. Doctoral Dissertation, California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-09122007-111121

Copyright © 2021 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.