JAMP  Vol.3 No.8 , August 2015
A Central Numerical Scheme to 1D Green-Naghdi Wave Equations
Abstract: 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.
Cite this paper: 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.

[1]   Serre, F. (1953) Contribution à L’étude des écoulements Permanents et Variables Dans Les Canaux. La Houille Blanche, 6, 830-872.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]   Synolakis, C.E. (1986) The Runup of Long Waves. Doctoral Dissertation, California Institute of Technology.