Cosine expansion-based differential quadrature method for numerical solution of the KdV equation Bu ¨ lent Saka * Mathematics Department, Eskis ßehir Osmangazi University, 26480 Eskis ßehir, Turkiye Accepted 3 October 2007 Communicated by Prof. L. Marek-Crnjac Abstract Numerical solution of the Korteweg–de Vries equation is obtained using space-splitting technique and the differen- tial quadrature method based on cosine expansion (CDQM). The details of the CDQM and its implementation to the KdV equation are given. Three test problems are studied to demonstrate the accuracy and efficiency of the proposed method. Accuracy and efficiency are discussed by computing the numerical conserved laws and L 2 , L 1 error norms. Ó 2007 Elsevier Ltd. All rights reserved. 1. Introduction Many physical phenomena such as propagation of long waves in shallow water and waves in plasmas can be described by the Korteweg–de Vries (KdV) equation which was first derived by the two scientists Korteweg and de Vries [1]. The KdV equation is a completely integrable Hamiltonian system which can be solved explicitly. Thus some analytical solu- tions of the KdV equation are found and their existence and uniqueness have been studied for a certain class of initial functions [6]. For example, the KdV equation is solved by Adomian decomposition method which provides series solu- tions [12]. In general, usefulness of these solutions is limited. Therefore, numerical solutions of KdV equation are nec- essary for various boundary and initial conditions to model many physical events. Various numerical methods have been proposed for numerical treatment of the KdV equation [8,11,14–17]. Ideally, a numerical method should be free of phase errors and the conservation properties of the equation must be satisfied. Since the KdV equation is an integrable Ham- iltonian system, there exist infinitely many independent conserved quantities. We will observe the well-known three con- served quantities of the KdV equation for our numerical solutions in section of numerical examples. Recently, differential quadrature method (DQM) has been used in obtaining the numerical solutions of the time- dependent partial differential equations [9,13]. DQM is an approximation to derivatives of a function using the weighted sum of the functional values at certain discrete points. Shu presented the approach of generalized differential quadrature, which computes the weighting coefficients of the first order derivative by a simple algebraic formulation, and the weighting coefficients of the second and higher order derivatives by a recurrence relationship [7]. Recent 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.10.004 * Tel.: +90 222 2393750. E-mail address: bsaka@ogu.edu.tr Available online at www.sciencedirect.com Chaos, Solitons and Fractals 40 (2009) 2181–2190 www.elsevier.com/locate/chaos