INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2006; 52:213–235 Published online 10 February 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/d.1175 Solving a fully nonlinear highly dispersive Boussinesq model with mesh-less least square-based nite dierence method Benlong Wang ‡ and Hua Liu ∗; † School of Naval Architecture; Ocean and Civil Engineering; Shanghai Jiao Tong University; 200030; Shanghai; China SUMMARY Combining mesh-less nite dierence method and least square approximation, a new numerical model is developed for water wave propagation model in two horizontal dimensions. In the numerical formulation of the method, the approximation of the unknown functions and their derivatives are constructed on a set of nodes in a local circular-shaped region. The Boussinesq equations studied in this paper is a fully nonlinear and highly dispersive model, which is composed of the exact boundary conditions and the truncated series expansion solution of the Laplace equation. The resultant system involves a sparse, unsymmetrical matrix to be solved at each time step of the simulation. Matrix solutions are studied to reduce the computing resource requirements and improve the eciency and accuracy. The convergence properties of the present numerical method are investigated. Preliminary verications are given for nonlinear wave shoaling problems; the numerical results agree well with experimental data available in the literature. Copyright ? 2006 John Wiley & Sons, Ltd. KEY WORDS: nite dierence method; least square approximation; Boussinesq equations; dispersive wave 1. INTRODUCTION In the last decades, mesh-less or mesh-free methods have attracted great attention in the eld of computational mechanics, e.g. References [1, 2] and references therein. Apart from the studies in this eld, applications of mesh-free methods in the area of geophysics and coastal engineering are some of the promising extensions. For example, Reference [3] introduced the mesh-less Galerkin method in hydraulics, where the stationary, shallow water ows in rivers ∗ Correspondence to: Hua Liu, Department of Engineering Mechanics, Shanghai Jiao Tong University, Hua Shan Road 1954#, 200030, Shanghai, China. † E-mail: hliu@sjtu.edu.cn ‡ E-mail: wblsjtu@hotmail.com Contract=grant sponsor: National Science Foundation of China and Doctoral Program Foundation for Higher Education from the Ministry of Education of China; contract=grant numbers: No. 10172058 and No. 2000024817 Received 10 June 2005 Revised 27 November 2005 Copyright ? 2006 John Wiley & Sons, Ltd. Accepted 28 November 2005