Computer Physics Communications 183 (2012) 1609–1616 Contents lists available at SciVerse ScienceDirect Computer Physics Communications www.elsevier.com/locate/cpc Numerical solutions of RLW equation using Galerkin method with extrapolation techniques ✩ Liquan Mei a,b,∗ , Yaping Chen a a School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China b Center for Computational Geosciences, Xi’an Jiaotong University, Xi’an, 710049, China article info abstract Article history: Received 7 December 2010 Received in revised form 7 February 2012 Accepted 27 February 2012 Available online 1 March 2012 Keywords: RLW equation Finite element method Galerkin method Extrapolation technique In this paper, we present a new Galerkin method for the regularized long wave (RLW) equation. Based on the Galerkin method using linear finite elements, the extrapolation technique is proposed to increase the order of the time discretization accuracy to O ((t ) 2 ), giving O ((t ) 2 +h 2 ) overall, which is quite efficient to solve the one-dimensional RLW. A stability analysis based on Von Neumann theory is performed. Propagation of solitary waves, interaction of two solitary waves and undular bores are simulated using the proposed method to validate the method which is found to be accurate and efficient.  2012 Elsevier B.V. All rights reserved. 1. Introduction Nonlinear partial differential equations are useful in describing a variety of phenomena across a range of disciplines. Analytical solutions of these equations are usually not available, especially when nonlinear terms are involved. Since only limited classes of the equations are solvable by analytical means, numerical solu- tion of these nonlinear partial differential equations is of practical importance. The regularized long wave (RLW) equation was first proposed by Peregrine [1] to describe nonlinear dispersive waves. It has been shown that the RLW equation can model a large class of physical phenomena such as the nonlinear transverse waves in shallow water, ion-acoustic waves in plasma, magnetohydrody- namics waves in plasma, longitudinal dispersive waves in elastic rods and pressure waves in liquid–gas bubbles. Various numeri- cal techniques have been proposed to solve the equation. These include finite difference methods [2,3] and various finite element methods such as the Galerkin method [4–7], the least squares method [8–10] and collocation method with quadratic B-splines [11], cubic B-splines [12] and recently septic splines [13]. Jain et al. [14] proposed a numerical method based on a splitting technique and cubic spline interpolation functions to solve the RLW equation and produced a difference equation in which, for computational work, they removed the nonlinearity by using a Taylor series ex- pansion. Gardner [15] used the least-squares method using linear ✩ The project is supported by NSF of China (10971164). * Corresponding author at: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China. E-mail address: lqmei@mail.xjtu.edu.cn (L. Mei). space–time finite elements to solve the RLW equation. Later on, Dogan [16] solved the same equation by Galerkin’s method using linear elements. Both of them linearized the nonlinear terms by allowing the nonlinearities to lag one time step behind. However, this decreases the order of the time discretization error. In order to overcome this drawback, we consider the adoption of extrapola- tion techniques [17] in the linearization procedure of the nonlinear terms. In this paper, we solve the RLW equation based on the Galerkin method using linear space finite elements and the nonlinearity will be removed by using extrapolation techniques. We discuss the properties and advantages of this method and compare its accu- racy in modeling a solitary wave with that of previous numerical algorithms. The interaction of two solitary waves and the evolu- tion of an undular bore are modeled too. The layout of the paper is as follows. In Section 2, we describe the Galerkin method with extrapolation techniques for the RLW equation. In Section 3, a sta- bility analysis is shown for this method. The results of numerical examples are presented in Section 4. The last section is a brief con- clusion. 2. The numerical method We consider the RLW equation u t + u x + εuu x − μu xxt = 0, in Ω × J , u = 0, on Γ × J , u(·, 0) = u 0 , in Ω (1) 0010-4655/$ – see front matter  2012 Elsevier B.V. All rights reserved. doi:10.1016/j.cpc.2012.02.029