The generalized regularized long wave (GRLW) equation is an important nonlinear equation for describing a large number of physical phenomena, for examples, the shallow water waves and plasma waves. In this study, numerical approximation of the GRLW using the element-free improved moving least-squares Ritz (IMLS-Ritz) method is performed. In the solution procedure, the IMLS approximation is employed to reduce the number of unknown coefficients in the trial functions. The Ritz minimization procedure is then used to derive the final algebraic equation system through discretizing the constructed energy formulation of the nonlinear GRLW equation. Time difference technique and Newton-Raphson method are adopted to solve the nonlinear equation system. Numerical experiments are conducted on the final form of the governing equation system to demonstrate the accuracy and efficiency of the element-free IMLS-Ritz method by comparing the computed IMLS-Ritz results with the existing available analytical solutions.

1. Introduction

The damped GRLW equation is established as a model for small-amplitude long waves on the surface of water [1, 2]. For some special cases, such as the regularized long wave (RLW) or the Benjamin-Bona-Mahony equation [3] which is used to describe a large number of physical phenomena with weak nonlinearity and dispersion waves.

The GRLW equation has been extensively studied for generating its solutions by analytical and approximate methods. Unlike the RLW and the Benjamin-Bona-Mahony equations, the stability of solutions to the GRLW equation depends on the solitary wave velocity [4]. Due to its potentially high nonlinearity, many efforts have been made to generate its solutions accurately and efficiently by means of, for examples, the finite difference method [5], the Adomian decomposition method [6, 7], the finite element method [814], and the element-free method [1518]. Moreover, the separation of temporal and spatial derivatives was also used to study the wave interactions [19, 20].

The element-free or meshless method has become a popular numerical tool in recent years. It has been developed and successfully applied to obtain accurate solutions for PDEs deriving from the physical and engineering fields [2126]. These include the element-free Galerkin method [27, 28], smooth particle hydrodynamics method [29], radial basis function method [30], element-free kp-Ritz method [3136], and meshless local Petrov-Galerkin method [37]. The major advantage of the element-free method for solving partial differential equations (PDEs) is that it does not require domain or boundary discretization. With this advantage together with its flexibility and simplicity in implementation [3840], element-free methods have also been employed for solving many mathematical models of wave equation [1720, 41, 42], such as the kp-Ritz method [17], the radial basis functions method [41], and the element-free Galerkin method [20, 42].

In this paper, we present an element-free computational framework to predict numerical solutions for the nonlinear GRLW equation using an improved moving least square Ritz (IMLS-Ritz) method. This novel IMLS-Ritz method consists of two essential parts: (i) the improved moving least-squares (IMLS) approximation and (ii) the Ritz procedure. The IMLS technique is employed for construction of the shape functions. An energy formulation for the nonlinear GRLW equation is formulated and discretized by the Ritz minimization procedure to obtain its final algebraic equation system. In the solution procedure, the penalty method is adopted to impose the essential boundary conditions. Time difference technique and Newton-Raphson method are employed to solve the nonlinear system equations. Computational simulations for several numerical examples are presented to examine the affectivity and efficiency of the IMLS-Ritz method on the nonlinear GRLW equation.

2. Theoretical Formulation

2.1. Equivalent Functional of GRLW Equation

The general form of the GRLW equation can be written as where , , is a known positive integer, and are two known positive parameters. The subscripts and denote space and time derivatives, respectively. The function will be determined when functions , , and are given. is the computational domain with boundary .

The corresponding initial condition for the problem is and the boundary conditions are

The functional is constructed from the weak form of (1), that is,

2.2. IMLS Shape Functions

The IMLS approximation was proposed for construction of the shape functions [21] in the element-free method. In one-dimensional IMLS approximation, for , , we define where is an inner product, and is the Hilbert space.

In , for the set of points and weight functions , if functions satisfy the conditions we furnish the function set as a weighted orthogonal function set with a weight function about points . If are polynomials, the function set is called a weighted orthogonal polynomials set with a weight function about points .

Consider an equation system from MLS approximation as follows: where is the moment matrix. Equation (7) can be expressed as

If the basis function set , , is a weighted orthogonal function set about points , that is, if then (8) becomes

Subsequently, coefficients can be determined accordingly: that is, where

From (7) and (11), the expression of approximation function is where is the shape function and

The abovementioned formulation details an IMLS approximation in which coefficients are obtained directly. It is, therefore, avoiding forming an ill-conditioned or singular equation system.

From (15), we have which represents the shape function of the IMLS approximation corresponding to node . From (16), the partial derivatives of lead to

The weighted orthogonal basis function set is formed by using the Schmidt method as

Moreover, using the Schmidt method, the weighted orthogonal basis function set can be formed from the monomial basis function. For example, for the monomial basis function the weighted orthogonal basis function set can be generated by

Using the weighted orthogonal basis functions described in (19) and (20), fewer coefficients existed in the trial function.

3. Ritz Minimization Procedure for the GRLW Equation

In the present element-free IMLS-Ritz method, the shape functions do not possess the Kronecker delta property, yielding to special techniques to impose the Dirichlet boundary conditions to the method. Lagrange’s multiplier approach, the penalty method, and modified variational principles are those techniques which are often adopted for imposition of boundary conditions. In the present work, we employ the penalty method to modify the constructed functional in implementing the specified Dirichlet boundary conditions. The variational form of the penalty function is described as follows: where is the specified function on the Dirichlet boundary and is the penalty parameter; normally it is chosen as 103~107 which is case-dependent.

The total functional involving the Dirichlet boundary conditions can be expressed as

Substituting (4) into (22), we have

The approximation of the field function can be obtained from (14) as follows: where

Substituting (24) into (23) and applying the Ritz minimization procedure to the maximum energy function , one has the following:

That yields the following matrix form: where

To solve the above system, time discretization of (27) is forming with the center difference method as follows: where is the time of the step and

Iteration with Newton-Raphson method is implemented to solve the above equation and the numerical solution of the GRLW equation will be obtained.

4. Numerical Examples and Discussion

Numerical analysis for three selected example problems is performed in order to demonstrate the applicability and examine the accuracy of the IMLS-Ritz method for the GRLW equation. The problems are solved using regular node arrangements.

The convergence study is carried out for the results of the GRLW equation for (i) a single solitary wave, (ii) an interaction of two solitary waves, and (iii) an interaction of three solitary waves. Accuracy of the numerical solutions by the IMLS-Ritz method is measured by using the following equations: where and denote the exact solution and numerical approximation, respectively.

4.1. Single Solitary Wave

The analytical solution of (1) is given in the general form of [1, 35] as follows:

When , (32) can be simplified as where , , , and for all examples. The initial and boundary conditions are extracted from the exact solution. Equation (29) is solved numerically with , , and .

We examine the convergence of the element-free IMLS-Ritz method on this example by varying the number of nodes (). The penalty factor is set as and . The -norm and errors of are computed with the number of nodes varied from 11 to 201. The results are tabulated in Table 1. It is apparent that both -norm and errors decrease as increases, indicating convergent results are obtained by the IMLS-Ritz method. Subsequently, we investigated the influence of time steps () on the accuracy of the IMLS-Ritz method by keeping and and varying from 0.01 to 2. As illustrated in Table 2, it is obvious that a smaller time step leads to a more precise result for this example. Moreover, as shown in Table 3, by varying from 2 to 4, accurate results can be furnished when .

Furthermore, the predicted results are compared with the analytical solutions at . As shown in Figure 1, these results and the absolute error are obtained when . A close agreement is obtained from the illustrated results. The computed results of for a time history is also predicted between  s and  s () (Figure 2(a)). The corresponding absolute errors are plotted in Figure 2(b). To illustrate clearly the influence of number of nodes, we display -norm errors in a time period from 0 s to 0.01 s in Figure 3(a). Here we set because when is smaller than 0.001, the -norm errors will increase slightly, presumably due to the increase in round-off error. To further investigate the influence of different time steps, we examine the variation trend of the -norm errors as time step varies in different time period. As exhibited in Figure 3(b), generally, the -norm errors tend to decline linearly as time steps decreased. From the presented results in the tables and figures, we can conclude that the approximate solutions generated by the IMLS-Ritz method are in close agreement with the analytical results.

4.2. Interaction of Two Solitary Waves

Consider an interaction of two solitary waves; we have the following exact solution [1, 35]: where and are arbitrary constants.

In this analysis, parameters are chosen to be , , , , , and . The problem is analyzed with 101 nodes. The numerical solutions are predicted and compared with the analytical solutions at . As presented in Figure 4, the comparison study shows that the IMLS-Ritz method provides a very similar solution as the exact result. In Figure 5, the computed results and corresponding absolute errors of for a time period from 0 s to 0.002 are displayed at . Again, further examination of the influence of number of nodes and in predicting the interaction between two solitary waves, we display the time history of -norm errors by varying time from 0 s to 0.01 s. As exhibited in Figure 6(a), the -norm errors decrease substantially as the arranged nodes increased while keeping the other variations as constants. Moreover, in this case, the results of numerical analysis suggested that satisfied accuracy can be achieved when .

4.3. Interaction of Three Solitary Waves

A third example considered the interaction of three waves of various amplitudes and traveling in the same direction. The analytical solutions has the same form as in Section 4.2, when choosing the following parameters: , , , , , , and .

Firstly, a regular 401 node is used in the IMLS-Ritz analysis with , the penalty factor , and . In Figure 7, the comparison results of IMLS-Ritz solutions and the analytical results are illustrated at  s. It is observed that the results obtained by implementing the IMLS-Ritz method are very close to the exact solutions. It is worth mentioning that the maximum error occurs near the peak position of the solitary wave, showing a good agreement with the results in [35]. Solutions at initial and different time levels are plotted in Figure 8(a), while Figure 8(b) gives the corresponding absolute error at . To examine the influence of number of nodes and in predicting the interaction between three solitary waves, we display the time history of -norm errors by varying time from 0 s to 0.002 s. As observed from Figure 9, convergent results are obtained as increases up to 450 while keeping , and smaller -norm errors are produced as increases.

5. Conclusion

An accurate numerical solution of the GRLW equation is important in investigating the creation of secondary solitary waves corresponding to particle physics. In this paper, the element-free IMLS-Ritz method is applied to provide an alternative solution for the GRLW equation. In this numerical solving process, IMLS approximation is employed to estimate the one-dimensional field function. The total functional is established by enforcement of Dirichlet boundary conditions using the penalty approach. The system of nonlinear discrete equations is furnished through Ritz minimization procedure. Time difference technique and Newton-Raphson method are used to solve the nonlinear equation system. The accuracy and efficiency of the IMS-Ritz method are examined through carefully selected numerical examples. From the computational results, it is concluded that the presented element-free method with satisfied performance can be extended to other PDEs in engineering problems.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


The work described in this paper was fully supported by the National Natural Science Foundation of China (Grant no. 61272098 and Grant no. 11402142).