Research Article  Open Access
DongMei Huang, L. W. Zhang, "ElementFree Approximation of Generalized Regularized Long Wave Equation", Mathematical Problems in Engineering, vol. 2014, Article ID 206017, 10 pages, 2014. https://doi.org/10.1155/2014/206017
ElementFree Approximation of Generalized Regularized Long Wave Equation
Abstract
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 elementfree improved moving leastsquares Ritz (IMLSRitz) 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 NewtonRaphson 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 elementfree IMLSRitz method by comparing the computed IMLSRitz results with the existing available analytical solutions.
1. Introduction
The damped GRLW equation is established as a model for smallamplitude long waves on the surface of water [1, 2]. For some special cases, such as the regularized long wave (RLW) or the BenjaminBonaMahony 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 BenjaminBonaMahony 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 [8–14], and the elementfree method [15–18]. Moreover, the separation of temporal and spatial derivatives was also used to study the wave interactions [19, 20].
The elementfree 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 [21–26]. These include the elementfree Galerkin method [27, 28], smooth particle hydrodynamics method [29], radial basis function method [30], elementfree kpRitz method [31–36], and meshless local PetrovGalerkin method [37]. The major advantage of the elementfree 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 [38–40], elementfree methods have also been employed for solving many mathematical models of wave equation [17–20, 41, 42], such as the kpRitz method [17], the radial basis functions method [41], and the elementfree Galerkin method [20, 42].
In this paper, we present an elementfree computational framework to predict numerical solutions for the nonlinear GRLW equation using an improved moving least square Ritz (IMLSRitz) method. This novel IMLSRitz method consists of two essential parts: (i) the improved moving leastsquares (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 NewtonRaphson 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 IMLSRitz 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 elementfree method. In onedimensional 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 illconditioned 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 elementfree IMLSRitz 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 10^{3}~10^{7} which is casedependent.
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 NewtonRaphson 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 IMLSRitz 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 IMLSRitz 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 elementfree IMLSRitz 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 IMLSRitz method. Subsequently, we investigated the influence of time steps () on the accuracy of the IMLSRitz 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 roundoff 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 IMLSRitz method are in close agreement with the analytical results.
(a)
(b)
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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 IMLSRitz 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 .
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(b)
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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 IMLSRitz analysis with , the penalty factor , and . In Figure 7, the comparison results of IMLSRitz solutions and the analytical results are illustrated at s. It is observed that the results obtained by implementing the IMLSRitz 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.
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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 elementfree IMLSRitz method is applied to provide an alternative solution for the GRLW equation. In this numerical solving process, IMLS approximation is employed to estimate the onedimensional 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 NewtonRaphson method are used to solve the nonlinear equation system. The accuracy and efficiency of the IMSRitz method are examined through carefully selected numerical examples. From the computational results, it is concluded that the presented elementfree 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.
Acknowledgments
The work described in this paper was fully supported by the National Natural Science Foundation of China (Grant no. 61272098 and Grant no. 11402142).
References
 D. H. Peregrine, “Calculations of the development of an undular bore,” Journal of Fluid Mechanics, vol. 25, pp. 321–330, 1966. View at: Publisher Site  Google Scholar
 D. H. Peregrine, “Long waves on a beach,” Journal of Fluid Mechanics, vol. 27, no. 4, pp. 815–827, 1967. View at: Publisher Site  Google Scholar
 C. M. GarcíaLópez and J. I. Ramos, “Effects of convection on a modified GRLW equation,” Applied Mathematics and Computation, vol. 219, no. 8, pp. 4118–4132, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 J. L. Bona and A. Soyeur, “On the stability of solitarywaves solutions of model equations for long waves,” Journal of Nonlinear Science, vol. 4, no. 5, pp. 449–470, 1994. View at: Publisher Site  Google Scholar  MathSciNet
 L. M. Zhang, “A finite difference scheme for generalized regularized longwave equation,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 962–972, 2005. View at: Publisher Site  Google Scholar  MathSciNet
 D. Kaya, “A numerical simulation of solitarywave solutions of the generalized regularized longwave equation,” Applied Mathematics and Computation, vol. 149, no. 3, pp. 833–841, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 D. Kaya and S. M. ElSayed, “An application of the decomposition method for the generalized KdV and RLW equations,” Chaos, Solitons and Fractals, vol. 17, no. 5, pp. 869–877, 2003. View at: Publisher Site  Google Scholar  MathSciNet
 I. Daǧ, “Leastsquares quadratic Bspline finite element method for the regularised long wave equation,” Computer Methods in Applied Mechanics and Engineering, vol. 182, no. 12, pp. 205–215, 2000. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 L. R. Gardner, G. A. Gardner, and I. Dag, “A Bspline finite element method for the regularized long wave equation,” Communications in Numerical Methods in Engineering, vol. 11, no. 1, pp. 59–68, 1995. View at: Publisher Site  Google Scholar  MathSciNet
 L. R. T. Gardner, G. A. Gardner, and A. Dogan, “A leastsquares finite element scheme for the RLW equation,” Communications in Numerical Methods in Engineering, vol. 12, no. 11, pp. 795–804, 1996. View at: Publisher Site  Google Scholar  MathSciNet
 İ. Daǧa, B. Saka, and D. Irk, “Application of cubic Bsplines for numerical solution of the RLW equation,” Applied Mathematics and Computation, vol. 159, no. 2, pp. 373–389, 2004. View at: Publisher Site  Google Scholar  MathSciNet
 A. A. Soliman and M. H. Hussien, “Collocation solution for RLW equation with septic spline,” Applied Mathematics and Computation, vol. 161, no. 2, pp. 623–636, 2005. View at: Publisher Site  Google Scholar  MathSciNet
 A. K. Khalifa, K. R. Raslan, and H. M. Alzubaidi, “A collocation method with cubic Bsplines for solving the MRLW equation,” Journal of Computational and Applied Mathematics, vol. 212, no. 2, pp. 406–418, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 L. R. Gardner, G. A. Gardner, F. A. Ayoub, and N. K. Amein, “Approximations of solitary waves of the MRLW equation by $B$spline finite elements,” The Arabian Journal for Science and Engineering Section A: Sciences, vol. 22, no. 2, pp. 183–193, 1997. View at: Google Scholar  MathSciNet
 J.F. Wang, F.N. Bai, and Y.M. Cheng, “A meshless method for the nonlinear generalized regularized long wave equation,” Chinese Physics B, vol. 20, no. 3, Article ID 030206, 2011. View at: Publisher Site  Google Scholar
 T. Roshan, “A PetrovGalerkin method for solving the generalized regularized long wave (GRLW) equation,” Computers & Mathematics with Applications, vol. 63, no. 5, pp. 943–956, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 P. F. Guo, L. W. Zhang, and K. M. Liew, “Numerical analysis of generalized regularized long wave equation using the elementfree kpRitz method,” Applied Mathematics and Computation, vol. 240, pp. 91–101, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 R. J. Cheng and K. M. Liew, “Analyzing modified equal width (MEW) wave equation using the improved elementfree Galerkin method,” Engineering Analysis with Boundary Elements, vol. 36, no. 9, pp. 1322–1330, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 S. Hamdi, W. H. Enright, W. E. Schiesser, and J. J. Gottlieb, “Exact solutions and invariants of motion for general types of regularized long wave equations,” Mathematics and Computers in Simulation, vol. 65, no. 45, pp. 535–545, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. I. Ramos, “Solitary wave interactions of the GRLW equation,” Chaos, Solitons and Fractals, vol. 33, no. 2, pp. 479–491, 2007. View at: Publisher Site  Google Scholar
 L. X. Peng, Y.P. Tao, H.Q. Li, and G.K. Mo, “Geometric nonlinear meshless analysis of ribbed rectangular plates based on the {FSDT} and the moving leastsquares approximation,” Mathematical Problems in Engineering, vol. 2014, Article ID 548708, 13 pages, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 D. Huang, G. Zou, and L. W. Zhang, “Numerical approximation of nonlinear KleinGordon equation using an elementfree approach,” Mathematical Problems in Engineering, vol. 2014, Article ID 548905, 2014. View at: Publisher Site  Google Scholar
 F. X. Sun, C. Liu, and Y. M. Cheng, “An improved interpolating elementfree Galerkin method based on nonsingular weight functions,” Mathematical Problems in Engineering, vol. 2014, Article ID 323945, 13 pages, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 N. Zhao and H. Ren, “The interpolating elementfree Galerkin method for 2D transient heat conduction problems,” Mathematical Problems in Engineering, vol. 2014, Article ID 712834, 9 pages, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 Q. Wei and R. Cheng, “The improved moving leastsquare Ritz method for the onedimensional sineGordon equation,” Mathematical Problems in Engineering, vol. 2014, Article ID 383219, 10 pages, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 F. Li and X. Li, “The interpolating boundary elementfree method for unilateral problems arising in variational inequalities,” Mathematical Problems in Engineering, vol. 2014, Article ID 518727, 11 pages, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 K. M. Liew, Y. Cheng, and S. Kitipornchai, “Boundary elementfree method (BEFM) and its application to twodimensional elasticity problems,” International Journal for Numerical Methods in Engineering, vol. 65, no. 8, pp. 1310–1332, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 T. Belytschko, Y. Y. Lu, and L. Gu, “Elementfree Galerkin methods,” International Journal for Numerical Methods in Engineering, vol. 37, no. 2, pp. 229–256, 1994. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. J. Monaghan, “An introduction to SPH,” Computer Physics Communications, vol. 48, no. 1, pp. 89–96, 1988. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 W. Chen, “New RBF collocation methods and kernel RBF with applications,” in Meshfree Methods for Partial Differential Equations, vol. 26 of Lecture Notes in Computational Science and Engineering, pp. 75–86, Springer, Berlin, Germany, 2000. View at: Publisher Site  Google Scholar  MathSciNet
 K. M. Liew, X. Zhao, and T. Y. Ng, “The elementfree kpRitz method for vibration of laminated rotating cylindrical panels,” International Journal of Structural Stability and Dynamics, vol. 2, pp. 523–558, 2002. View at: Publisher Site  Google Scholar
 X. Zhao, Q. Li, K. M. Liew, and T. Y. Ng, “The elementfree kpRitz method for free vibration analysis of conical shell panels,” Journal of Sound and Vibration, vol. 295, no. 35, pp. 906–922, 2006. View at: Publisher Site  Google Scholar
 X. Zhao, Y. Y. Lee, and K. M. Liew, “Free vibration analysis of functionally graded plates using the elementfree kpRitz method,” Journal of Sound and Vibration, vol. 319, no. 3–5, pp. 918–939, 2009. View at: Publisher Site  Google Scholar
 X. Zhao and K. M. Liew, “Geometrically nonlinear analysis of functionally graded plates using the elementfree kpRitz method,” Computer Methods in Applied Mechanics and Engineering, vol. 198, no. 3336, pp. 2796–2811, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 L. W. Zhang, Z. X. Lei, K. M. Liew, and J. L. Yu, “Large deflection geometrically nonlinear analysis of carbon nanotubereinforced functionally graded cylindrical panels,” Computer Methods in Applied Mechanics and Engineering, vol. 273, pp. 1–18, 2014. View at: Publisher Site  Google Scholar
 L. W. Zhang, Z. X. Lei, K. M. Liew, and J. L. Yu, “Static and dynamic of carbon nanotube reinforced functionally graded cylindrical panels,” Composite Structures, vol. 111, no. 1, pp. 205–212, 2014. View at: Publisher Site  Google Scholar
 S. N. Atluri and T. Zhu, “A new meshless local PetrovGalerkin (MLPG) approach in computational mechanics,” Computational Mechanics, vol. 22, no. 2, pp. 117–127, 1998. View at: Publisher Site  Google Scholar  MathSciNet
 K. M. Liew, T. Y. Ng, X. Zhao, and J. N. Reddy, “Harmonic reproducing kernel particle method for free vibration analysis of rotating cylindrical shells,” Computer Methods in Applied Mechanics and Engineering, vol. 191, no. 3738, pp. 4141–4157, 2002. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 K. M. Liew, X. L. Chen, and J. N. Reddy, “Meshfree radial basis function method for buckling analysis of nonuniformly loaded arbitrarily shaped shear deformable plates,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 3–5, pp. 205–224, 2004. View at: Publisher Site  Google Scholar
 K. M. Liew, Z. X. Lei, J. L. Yu, and L. W. Zhang, “Postbuckling of carbon nanotubereinforced functionally graded cylindrical panels under axial compression using a meshless approach,” Computer Methods in Applied Mechanics and Engineering, vol. 268, pp. 1–17, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 “A meshfree method for the numerical solution of the RLW equation,” Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 997–1012, 2009. View at: Publisher Site  Google Scholar  MathSciNet
 İ. Dağ, B. Saka, and D. Irk, “Galerkin method for the numerical solution of the RLW equation using quintic Bsplines,” Journal of Computational and Applied Mathematics, vol. 190, no. 12, pp. 532–547, 2006. View at: Publisher Site  Google Scholar  MathSciNet
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Copyright © 2014 DongMei Huang and L. W. Zhang. 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.