Mathematical Problems in Engineering

Volume 2015, Article ID 274579, 9 pages

http://dx.doi.org/10.1155/2015/274579

## Runge-Kutta Integration of the Equal Width Wave Equation Using the Method of Lines

Department of Mathematics, Science Faculty for Girls, King Abdulaziz University, P.O. Box 80200, Jeddah 21589, Saudi Arabia

Received 31 August 2014; Revised 5 January 2015; Accepted 8 January 2015

Academic Editor: K. M. Liew

Copyright © 2015 M. A. Banaja and H. O. Bakodah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The equal width (EW) equation governs nonlinear wave phenomena like waves in shallow water. Numerical solution of the (EW) equation is obtained by using the method of lines (MOL) based on Runge-Kutta integration. Using von Neumann stability analysis, the scheme is found to be unconditionally stable. Solitary wave motion and interaction of two solitary waves are studied using the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Accuracy of the proposed method is discussed by computing the and error norms. The results are found in good agreement with exact solution.

#### 1. Introduction

Nonlinear equations are widely used to describe complex phenomena in various fields of science, such as fluid mechanics, plasma physics, solid-state physics, elastodynamic problems, nonlinear Schrodinger equations, elasticity problems, free vibration analysis, and optical fibers [1–16]. Benjamin et al. [17] advocated that the partial differential equation (PDE) modeled the same physical phenomena equally well as the (KdV) equation given the same assumptions and approximations that were originally used by Korteweg and de Vries [18]. This PDE of Benjamin et al. [17] is now often called the BBM equation, although it is also known as the regularized long wave (RLW) equation. Morrison et al. [19] proposed the one-dimensional PDE as an equally valid and accurate model for the same wave phenomena simulated by the KdV and RLW equations. This PDE is called the equal width (EW) equation because the solutions for solitary waves with a permanent form and speed, for a given value of the parameter , are waves with an equal width or wavelength for all wave amplitudes. The equal width (EW) equation, which was introduced by Morrison et al. [19], is an important special kind of a nonlinear dispersive wave equation. It is defined as

For a smooth function on a domain with . Except for a single travelling solitary wave solution, no analytic solutions are known, and therefore numerical methods have to be used. The equation has solution of the form

That represents a solution for a solitary wave traveling at constant speed and vanishing at . When the EW equation is used to model waves generated in shallow water channel, the variables are normalized so that the distance and water elevation are scaled to the water depth , and time is scaled to , where is the acceleration due to gravity. There is experimental evidence to suggest that both descriptions break down if the amplitude of any wave exceeds 0.28 [20]. Different numerical solution methods were applied to solve EW equation such as L. R. Gardner and G. A. Gardner [21] using Galerkin’s method-based cubic B-spline finite elements, Ramos [22] using finite difference methods, Raslan [23] using quartic B-spline method, Saka [24] using finite element method, Ali et al. [25] using He’s Exp-function method, and Cheng and Liew [26] using the improved element-free Galerkin’s method.

In this paper, the method of lines (MOL) solution of the EW equation is presented. This method consists of converting the EW equation with auxiliary conditions into a system of ordinary differential equations with corresponding auxiliary conditions. Then the fourth-order Runge-Kutta method is used to solve the system of first-order ordinary differential equations instead of finite difference methods [22] that are accurate and efficient as shown in [27].

#### 2. The Method of Lines Solution of the EW Equation

The method of lines is a general technique for solving partial differential equations (PDEs) by typically using finite difference relationships for the spatial derivatives and ordinary differential equations for the time derivative. For time-dependent partial differential equations, meshless kernel-based methods were similarly based on a fixed spatial interpolation, but now the coefficients are time-dependent, and one obtains a system of ordinary differential equations for these. This is the well-known method of lines, and it turned to be experimentally useful in various cases.

To apply the method of lines for solving the EW equation, firstly we subdivide the solution domain into uniform rectangular meshes by the linesIn numerical calculation, is usually replaced by and with zero boundary condition at both ends. Hence, the solution domain of the EW equation is the rectangle defined as , . Then, the partial derivatives depending on spatial variables, and , in (1) are replaced by the well-known finite difference approximation at point :By substituting (4) into (1) and introducing the boundary conditions , this yields a system of ordinary differential equations which depend on in the following form: whereThus, we have the system of differential equations of one independent variable . This system can be solved using Runge-Kutta method. The resulting system of ordinary differential equations is integrated with respect to time.

Many existing numerical schemes for evolutionary problems in partial differential equations (PDEs) can be viewed as method of lines (MOL) schemes. In 1984, Verwer and Sanz-Serna [28] treated the convergence of one-step MOL schemes. Their main purpose was to set up a general framework for a convergence analysis applicable to nonlinear problems.

#### 3. Stability Analysis

By solving EW equation using the method of lines, firstly the spatial variables are discretized to obtain an ordinary differential equations system in the time variable that can be discretized by an ordinary differential equation (ODE) solver. The simplest ODE solver is the forward Euler method and it is used widely for analysis of the stability properties of the spatial discretization. However, while forward Euler is ideal for analysis of the stability properties of a given spatial discretization, it is only first-order accurate. In practice, high order time discretization that preserves all the stability properties of forward Euler is needed. In [29], high order strong stability preserving time discretization method for the semidiscrete method of lines approximations of PDE is developed. These methods are derived by assuming that the first-order forward Euler time discretization of the method of lines ODE is strongly stable under a certain norm when the time step is suitably restricted, and then try to find a higher order time discretization (Runge-Kutta or multistep) that maintains strong stability for the same norm perhaps under a different time-step restriction.

The stability analysis constitutes the essential study of the numerical solution of PDEs. In general, this is because such study provides the means by which the step size and the numerical integration scheme for the given differential equation could be selected so as to secure manageable numerical solution.

The stability analysis is based on the Neumann theory in which the growth factor of the error in a typical mode of amplitude iswhere is a Fourier number and , the finite difference size, is determined from a linearization of the numerical scheme. Assuming in the nonlinear term as a constant ; this enables one to discuss the stability in the linearized sense. The numerical method of lines of EW equation gives the system of ordinary differential equations A trial solution is assumed and substituted into (9). The trial solution must take into account the variation of with both and or and . So In accordance with a method proposed by von Neumann, the function can be of the following form: Substituting (10) and (11) into (9) gives Equation (12) shows the growth factor for the error of the following form: In [30], Evans and Raslan proved that and the scheme is virtually unconditionally stable, at least for any practical problem, where is a small quantity and represents the single speed and will usually be around unity.

Kreiss and Scherer in [31] derived the conditions of local stability of Runge-Kutta methods when applied to hyperbolic partial differential equations, as when the time in (12) is discretized by using a locally stable Runge-Kutta method, the resulting completely discretized method is stable provided that with locally stable Runge-Kutta methods whose stability region contains a half circle. Consider where denotes the time step and for computational purposes the Runge-Kutta method is only useful if it is stable for sufficient small .

#### 4. Test Problems

In this section, the results of the numerical solution of the EW equation based on the MOL using Runge-Kutta time discretization are presented. The numerical simulation includes the propagation of a solitary wave, the interaction of two solitary waves, the development of an undular bore, and the temporal evaluation of a Maxwellian initial pulse.

In order to confirm the accuracy and efficiency of the method, and error norms are used and defined by where denotes the exact solution and denotes the numerical solution. According to [32], in many cases, there are the following time invariants:By sufficiently fine spatial resolution, there is no problem to maintain these invariants to reasonable accuracy. Wherever possible, this statement is supported by providing numerical results. The constants of the motion, for a solitary wave of amplitude and depending on as given by (2), may be evaluated analytically to give

##### 4.1. Single Solitary Wave

We first study the motion of a single solitary wave. This is derived from the initial condition. Considerwhere and is a constant. This follows from the analytic solution (2) of the EW equation. We choose , , , , , and through the interval . In this case the problem reduces toTo apply the method, letBy substituting (4) into (19) and introducing the boundary conditions , this yields a system of ordinary differential equations depending on in the following form: where is the inverse of the matrix and This system can be solved using Runge-Kutta method. The solution for times from and 40.0 is shown in Figure 1.