# A Meshless Finite-Point Approximation for Solving the RLW Equation

Authors: L. Pérez Pozo; R. Meneses; C. Spa; O. Durán
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

An alternative meshless finite-point method (FPM) technique for the numerical solution of the Regularized long wave (RLW) equation is presented. In this context, we derive the discretized system by combining finite difference (FD) techniques for the time derivative and FPM for the spatial derivatives. The accuracy of this alternative approach is tested withL2, L∞ error norms and the conservation properties of mass, energy, and momentum under the RLW equation.


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## Body

## 1. Introduction

The numerical analysis of nonlinear dispersive waves has significant importance in physical phenomena, such as shallow water waves [1], ion acoustic solitary waves [2], motion for mixtures of liquid and gas bubbles [3], and so on. Importantly, the regularized long wave (RLW) equation is an alternative description to the well-known Korteweg-de Vries (KdV) equation [4]. Like the KdV equation, the RLW equation also describes a large number of physical phenomena; see [5, 6]. The first formulation of the RLW equation was presented by Peregrine [7] to describe the behavior of an undular bore with a numerical solution based on a finite difference (FD) method with first-order accuracy in terms of time. In the recent years, several numerical methods for the solution of the RLW equation have been developed, including FD methods [8–11], Fourier pseudospectral (PS) methods [12], finite element methods (FEMs) based on Galerkin and collocation techniques [13–19]. Furthermore, focusing on a meshless context, Shokri and Dehghan [20] presented an approach for solving the RLW equation with radial basis functions (RBFs) by using FD to yield the time derivative as well as a predictor-corrector procedure to solve the nonlinear system. In addition, Islam et al. [21] also applied RBF and FD to the time derivative, but they addressed nonlinear behavior with the θ-weighted scheme. These recent works and references mentioned therein present helpful reviews on the RLW equation and related aspects.Meshless methods are a family of numerical techniques that do not require a mesh. In these methods, the body or domain is discretized by a collection of points. It is divided into local interpolation subdomains, which are also called clouds; they consist of one central point, or star node, and several neighboring points. Generally, these methods are computationally efficient and easy to implement, and they have been successfully used in several application. The general characteristics, classifications, advantages, and disadvantages of these methods can be found in [22–26]. Meshless finite point method (FPM) approximation around each point is obtained using weighted least square techniques. The discrete system of equation is constructed based on a point collocation procedure. This method was proposed in [27, 28] to solve convective transport and fluid flow problems. Its application has been extended to advection-diffusion transport [29], incompressible flow problems [30], elasticity [31, 32], solid dynamics [33], solidification modeling [34], nonlinear material behavior problems [35, 36], adaptive refinement [37, 38], and large deflection analysis of flexible plates [39]. The lack of dependence on a mesh or integration procedure is an important feature, making the FPM a truly meshless method. This work is structured as follows. In Section 2, the FPM is introduced. The meshless FPM implementation of the RLW equation is developed in Section 3. In Section 4, the accuracy of this approach is tested with respect to L2, L∞ error norms and the conservation properties of mass, energy and momentum [40]. The numerical simulation includes the propagation of a solitary wave, the interaction of two positive solitary waves, the interaction of a positive and a negative solitary wave, the evaluation of Maxwellian pulse into stable solitary waves, and the development of an undular bore. Finally, conclusions to the current investigation are shown in Section 5.


## 2. The Finite Point Method

Although the FPM introduced by Oñate et al. was originally formulated for the numerical solution of convective transport and fluid flow problems, it can be easily adapted to the RLW equation. In this section, we review the basic formulation of the FPM and provide a brief overview of its main features. In order to obtain the final system of discrete equations, the FPM approximates the local solution of a partial differential equation in each point of the discretized domain by means of a weighted least squares technique and a point collocation procedure. Due to the local character of the approximation procedure used by this method, it is necessary to define a subdomainΩk for each node xk. This Ωk contain neighboring nodes selected by a suitable criterion [31, 41, 42]. This collection of nodes is called a cloud, and its referential central point is the star node. For example, a relevant aspect in the definition of clouds is that their superposition must produce the whole domain, Ω:(2.1)Ω⊆⋃k=1NpΩk,
where Np is the total number of nodes. Note that the definition of clouds is the basic, initial step in implementing the FPM approximation using fixed weighted least squares. With the discretized domain defined, let us define a function u(x), which is locally approximated by û(x) only valid in the subdomain Ωk associated with the star nodexk as a linear combination of known functions p(x):(2.2)u(x)≅û(x)=p(x)αk∀x∈Ωk,
where p(x) is the vector that represents the basis of m linearly independent functions and αk is a vector of constant parameters only valid in Ωk. The elements of the interpolation base may belong to any function family. In this paper the m first monomial polynomials are used; that is, p(x)=[1xx2···xm-1]. Since (2.2) is valid for all Nc points of the kth subdomain, the approximations û(X) conform to a Vandermonde system given by the following relation:(2.3)u(Xk)≅û(Xk)=P(Xk)⋅αk,
where(2.4)Xk=[xk,1⋅⋅⋅xk,Nc]T,u(Xk)=[u(xk,1)⋅⋅⋅u(xk,Nc)]T,û(Xk)=[û(xk,1)⋅⋅⋅û(xk,Nc)]T,αk=[αk,1⋅⋅⋅αk,Nc]T,P(Xk)=[p(xk,1)⋮p(xk,Nc)].In general, the number of pointsNc that conform to the cloud is greater than the number of functions m that define the basis; hence, the matrix P(Xk) is usually rectangular. This means that the property of interpolation is lost, and the problem must be addressed with numerical approximation. The coefficients of the vector αk must be determined in such a way that the weighted sums of the squared differences between the exact values u(x) and the approximated values û(x) of each point are minimized according to the following expression:(2.5)min⁡{∑j=1Ncw(xk,j)⋅(u(xk,j)-û(xk,j))2},
where w(x) is a fixed weighting function defined in Ωk. See [27, 28]. The minimization process described by (2.5) leads to the following expression for vector αk:(2.6)αk=A-1(Xk)B(Xk)λ(Xk),
where λ(Xk) is a vector that represents the unknown parameters sought on the cloudΩk defined as follows:(2.7)λ(Xk)=[λ(xk,1)⋅⋅⋅λ(xk,Nc)]T.
Additionally, matrices A(Xk), B(Xk), and W(Xk) are given as follows:(2.8)A(Xk)=P(Xk)W(Xk)PT(Xk),B(Xk)=PT(Xk)W(Xk),
and W(Xk) is an Nc×Nc diagonal matrix defined by(2.9)W(Xk)=[diag⁡(w(xk,1)⋅⋅⋅w(xk,Nc))],
where the weighting functions w(x) are derived in order to have unit values near the star node and zero values outside the Ωk subdomains. Under the FPM, the common selection for the fixed weighting function is given as follows:(2.10)w(x)={exp⁡(-(hj/ζ)2)-exp⁡(-(r/ζ)2)1-exp⁡(-(r/ξ)2),ifhj≤r,0,ifhj>r,
where hj is the distance between the star nodexk and the point x, r=q·hmax (max. of hj) is a reference distance, and ζ=β·r. A detailed description of the effects of the constant parameters q and β on numerical approximation as well as guidelines for setting their values is presented in [43]. Other considerations in selecting the function w(x) can be found in [27, 28, 44, 45]. Finally, replacing (2.6) in (2.2), the next relation is obtained:(2.11)û(x)=NT(x)λ(Xk),
where N(x) is a matrix called shape functiondefined by(2.12)N(x)=pT(x)C(Xk),
with C(Xk)=A-1(Xk)B(Xk). We remark that p(x) denote [1xx2···xm-1]. Note that according to the least square nature of the approximation, u(x)≅û(x)≠λ(x); that is, the local values of the approximating function do not fit the nodal unknown values. Indeed, û(x) is the true approximation, which we be will used to satisfy the differential equation and the boundary conditions; in this context, λ(x) are simply the unknown parameters we aim to determine. According to the concepts described above and (2.11), it is possible to obtain the following expressions:(2.13)ûx(x)=NxT(x)λ(Xk)ûxx(x)=NxxT(x)λ(Xk),
where (·)x and (·)xx denote the first and the second space derivatives, respectively. Note that these derivatives are computed by taking the derivative of the basis functions p(x) in (2.2).


## 3.  The RLW Equation and the FPM Numerical Implementation

In this section, we present RLW model, which is approximated by the FPM explained above. First, the partial differential equation (PDE) is presented by defining the appropriate initial and boundary conditions, and the relevant parameters needed to understand the behavior of the PDE are briefly discussed. Finally, we derive the discretized system by combining FD techniques for the time derivative and FPM for the spatial derivatives.Let us consider the following form of the RLW equation:(3.1)ut(x,t)+ux(x,t)+ε⋅u(x,t)ux(x,t)-μ⋅uxxt(x,t)=0,x∈R,t>0,
with the physical conditions u→0 as |x|→∞. To the numerical implementation, we consider a≤x≤b and the following boundary conditions:(3.2)u(a,t)=0u(b,t)=0.
The initial condition for the problem (3.1) is given as follows:(3.3)u(x,0)=u0(x),a≤x≤b.
Parameters ε and μ in (3.1) are positive constants and are related by the Stokes number S defined as follows:(3.4)S=εμ.
Usually, this number is set to 1 in order to balance the nonlinear effects of the advective and dispersive terms present in this problem.In order to obtain the numerical approximation of (3.1), the time derivative of the RLW equation is calculated by applying a FD formula and the θ-weighted (0≤θ≤1) scheme to the space derivative at two successive time levels n and n+1, where n is an integer that denotes the time step. The approximation is then given as follows:(3.5)ûn+1-ûnδt+θ((ûx)n+1+ε⋅ûn+1(ûx)n+1)+(1-θ)((ûx)n+ε⋅ûn(ûx)n)-μδt((ûxx)n+1-(ûxx)n)=0,
where ûn=û(x,tn), tn=tn-1+δt, and δt is the size of the time step. With the idea to adequately address the nonlinear term in (3.5), a linearized scheme given by the following expression is used [21, 46]:(3.6)ûn+1(ûx)n+1≈ûn+1(ûx)n+ûn(ûx)n+1-ûn(ûx)n;
therefore, from (3.5) and (3.6), the following expression is obtained:(3.7)ûn+1+δt⋅θ((ûx)n+1+ε⋅[ûn+1(ûx)n+ûn(ûx)n+1])-μ⋅(ûxx)n+1=ûn+δt⋅(ε⋅(2θ-1)ûn(ûx)n-(1-θ)(ûx)n)-μ⋅(ûxx)n.
With the purpose to obtain the local approximation ûn+1, this work presents an alternative method based on the FPM, which was introduced in Section 2. Using (2.11) and (2.13), assume that θ=1/2 and replace into (3.7). Then the following discrete relation is obtained for the time level n+1:(3.8)Gk⋅λ(Xk)=fk,
where matrix Gk and the known quantity fk (based on the time level n) are defined as follows:(3.9)Gk=NT+δt2⋅NxT+δt2ε⋅NT⋅ûxn+δt2ε⋅ûn⋅NxT-μ⋅NxxT,fk=ûn-δt2⋅ûxn-μ⋅ûxx;
here, the subscript k denotes the evaluation in the star nodexk.Note that for simplicity, the termsNT(x), NxT(x), and NxxT(x) have been replaced by NT, NxT, and NxxT, respectively. We remark that this shape functions are computed only at the beginning of calculation.Otherwise, in the case of boundary points (see (3.2)), the matrix Gk and quantity fk adopt the following forms, respectively:(3.10)Ga,b=NTfa,b=0.
Finally, using (3.8) and (3.10) and taking into account the point collocation procedure for each of the Np nodes that comprise the domain Ω, the following system of discrete equation is solved:(3.11)G⋅λ=f,
where G is defined as a stiffness matrix, λ represents the vector collecting of the point parameters λn+1, and f is a vector of known values brought from the time level n.


## 4.  Test Problems

In this section, the results of the numerical solution of the RLW equation based on the FPM approach are presented. The numerical simulation includes the propagation of a solitary wave, the interaction of two positive solitary waves, the interaction of a positive and a negative solitary wave, the evaluation ofMaxwellian pulse into stable solitary waves, and the development of an undular bore. According to [40], the RLW equation must comply with three conservation laws related to mass, momentum, and energy, which are given as follows:(4.1)∫-∞∞udx≈C1=∫abudx,(4.2)∫-∞∞(u2+μ(uxx)2)dx≈C2=∫ab(u2+μ(uxx)2)dx,(4.3)∫-∞∞(u3+3u2)dx≈C3=∫ab(u3+3u2)dx.
In the following test problems, the numerical solutions must control these conservation laws during propagation. Therefore, these quantities are used to measure the accuracy of the proposed method. All of the following test problems were developed using the FPM parameters shown in Table 1. It is worth mentioning that the following experiments have been extracted from previous works; see [10, 11, 19–21] for further details on the suitability of our proposal.Table 1
FPM parameters for the numerical simulation.
Total discretization pointsNp300Cloud sizeNc5Interpolation base functionm4Weighting function parametersq and β1:1 and 0:25


### 4.1.  Propagation of a Single Solitary Wave

Solitary waves are wave packets or pulses that propagate nonlinearly in dispersive media. Due to the dynamic balance between the nonlinear and dispersive effects, these waves retain a stable waveform known as a soliton. This is a very special type of solitary wave, which also keeps its waveform after collision with other solitons [21]. In the case of solitons, the following explicit solution to (3.1) is given by [7](4.4)u(x,t)=3csech2[k(x-x0-vt)],
which represents a single solitary wave of amplitude 3c centered at x0 with velocity v=1+εc and width k=(1/2)εc/μv. The initial condition of (3.1) is given as follows:(4.5)u(x,0)=3csech2[k(x-x0)].
The RLW parameters used in the numerical simulation are c=0.01,0.055,0.1, x0=0, ε=μ=1(S=1) and δt=0.1. As expected, the solitary wave moves to the right across the space interval -40≤x≤60 in the time interval 0≤t≤20. As in previous work, such as [20, 21], the boundary conditions are extracted from the exact solution (4.4), and the initial condition is given by (4.5). The FPM parameters are shown in Table 1. Moreover, based on the conservation laws Equations (3.9) and (3.10) and the explicit solution (4.4), the following error norms are calculated for this test problem:(4.6)L2=h∑j=1Np|u(xj)-û(xj)|2,L∞=max⁡j|u(xj)-û(xj)|,
where h is the minimum distance between any two points in the domain a≤x≤b. The error norms L2, L∞ and conservation quantities C1, C2, and C3 are shown in Tables 2, 3, and 4. A perspective and superior view of the propagation of a single solitary wave when c=0.1 is shown in Figure 1. Moreover, Figure 2 presents the evolution of the L2 and L∞ error norms at different values of c.Table 2
Error norms and conservation quantities for the propagation of a single solitary wave forc=0.01.
tL2L∞C1C2C300.0000390.0000411.2054600.0241670.07293850.0000600.0000301.2054790.0241670.072938100.0000770.0000301.2054060.0241670.072938150.0000830.0000301.2053030.0241670.072938200.0000890.0000431.2051270.0241670.072938Table 3
Error norms and conservation quantities for the propagation of a single solitary wave forc=0.055.
tL2L∞C1C2C300.0000280.0000122.8906050.3212900.99589650.0000450.0000192.8906050.3212900.995896100.0000760.0000282.8906050.3212900.995896150.0001060.0000372.8906050.3212900.995896200.0001330.0000472.8906050.3212900.995896Table 4
Error norms and conservation quantities for the propagation of a single solitary wave forc=0.1.
tL2L∞C1C2C300.0001230.0000583.9799500.8105202.57920250.0002490.0001113.9799500.8105202.579202100.0004280.0001723.9799500.8105212.579202150.0005800.0002303.9799500.8105212.579202200.0007020.0002683.9799500.8105212.579202(a) Perspective and (b) superior view of the propagation of a single solitary wave forc=0.1.
(a)(b)(a)L2 and (b) L∞ error norms for the propagation of a single solitary wave.
(a)(b)The results obtained in this case show that the conservation quantities are controlled in different time steps, and moreover, the error normsL2 and L∞ are of the order of 10-4, which demonstrate the remarkable accuracy of the proposed method and, therefore, the suitability of FPM for this kind of physical problem.


### 4.2.  Interaction of Two Positive Solitary Waves

In this case, the numerical example consists on the interaction of two positive solitary waves defined by the following initial condition:(4.7)u(x,0)=∑j=123cjsech2(kj(x-xj)),
with cj=4kj2/(1-4kj2) and the boundary conditions:(4.8)u(a,t)=u(b,t)=0.
The RLW parameters used in the numerical simulation are k1=0.4, k2=0.3, x1=15, x2=35, ε=μ=1(S=1), and δt=0.01. This waves move across the space interval 0≤x≤120 and the time interval 0≤t≤30. Likewise, the FPM parameters used in this simulation are shown in Table 1. The results of the FPM approximation at different times are shown in Figures 3 and 4, where it is possible to observe that the higher amplitude solitary wave passes through the smaller wave with no change in its waveform. Once again, our results are in agreement with our expectations for all discrete time measured. As in the case of a solitary wave, the three quantities C1, C2, and C3 are conserved see Table 5.Table 5
Conservation quantities for the interaction of two positive solitary waves.
tC1C2C3037.916479120.674902745.391797537.916674120.676914745.3966941037.916670120.607313744.7496761537.916673120.409138743.0404802037.916671120.663186745.2654792537.916672120.678190745.4124423037.916662120.678701745.417152(a) Perspective and (b) superior view of the interaction of two positive solitary waves.
(a)(b)Detailed view of the interaction of two positive solitary waves at various times.
(a)(b)(c)(d)(e)(f)


### 4.3.  Interaction of a Positive and Negative Solitary Wave

This example focuses on the interaction of a positive and negative solitary wave, starting with the initial condition given by (4.7) and the boundary conditions defined in (4.8). The RLW parameters used in the numerical simulation are k1=0.4, k2=0.6, x1=23, x2=38, ε=μ=1(S=1), and δt=0.1. The waves interact across the space interval -10≤x≤80 and the time interval 0≤t≤20. As in the other cases, the FPM parameters are shown in Table 1. The results of the FPM approximation at different times are shown in Figures 5 and 6, where it is possible to observe that the collision produces additional solitary waves. Finally, note that the three conservation quantities for various times steps are controlled, as shown in Table 6.Table 6
Conservation quantities for the interaction of a positive and negative solitary wave.
tC1C2C30−6.06072383.42957−354.2331562−6.06072383.48023−353.3059895−6.06072406.31404−379.08477710−6.06072383.69552−354.09381115−6.06072383.64904−353.88387920−6.07021383.63883−353.856241(a) Perspective and (b) superior view of the interaction of a positive and negative solitary wave.
(a)(b)Detailed view of the interaction of a positive and negative solitary wave at various times.
(a)(b)(c)(d)(e)(f)


### 4.4.  Evolution of Maxwellian Pulse into Stable Waves

This problem deals with the evolution of theMaxwellian pulse into stable solitary waves at various values of the parameter μ. The initial condition is given as follows:(4.9)u(x,0)=exp⁡[-(x-7)2].
The boundary conditions are defined as in (4.8). For this test, the RLW parameters for the numerical simulation are ε=1, μ=0.1, 0.001, and 0.01, and δt=0.01. This pulse is studied across the space interval 0≤x≤30 and time interval 0≤t≤10 for the three cases of μ. The FPM parameters are shown in Table 1. The results of the FPM approximation at different values of the parameter μ are shown in Figures 7, 8, and 9, where it is possible to observe that the Maxwellian pulse develops into various solitary waves. As in the other experiments, the three conservation quantities for various times steps and values of parameter μ as given in Tables 7, 8, and 9 are controlled throughout the simulation.Table 7
Conservation quantities for the evolution of the Maxwellian pulse into stable waves forμ=0.1.
tC1C2C301.7724541.3786334.78326911.7724541.3786174.78325721.7724541.3785314.78317931.7724541.3784554.78311041.7724541.3784124.78306951.7724541.3783894.78304361.7724541.3783754.78302671.7724541.3783674.78301581.7724541.3783614.78300691.7724541.3783574.783000101.7724541.3783544.782995Table 8
Conservation quantities for the evolution of the Maxwellian pulse into stable waves forμ=0.01.
tC1C2C301.7724541.2658464.78326911.7724541.2660674.78416921.7724541.2670094.78819331.7724541.2678894.79201441.7724541.2682684.79372551.7724541.2683984.79433361.7724541.2684444.79455171.7724541.2684634.79463981.7724541.2684734.79468091.7724541.2684794.794702101.7724541.2684834.794716Table 9
Conservation quantities for the evolution of the Maxwellian pulse into stable waves forμ=0.001.
tC1C2C301.7724541.2545674.78326911.7724541.2548504.78512121.7724541.2651464.85982731.7724541.2749274.92908141.7724541.2780244.95008351.7724541.2788884.95444761.7724541.2792164.95579171.7724541.2795634.95882781.7724541.2797754.96082391.7724541.2796964.959418101.7724541.2795914.957822(a) Perspective, (b) superior, and (c) final detailed view of the evolution of the Maxwellian pulse into stable waves forμ=0.1.
(a)(b)(c)(a) Perspective, (b) superior, and (c) final detailed view of the evolution of the Maxwellian pulse into stable waves forμ=0.01.
(a)(b)(c)(a) Perspective, (b) superior, and (c) final detailed view of the evolution of the Maxwellian pulse into stable waves forμ=0.001.
(a)(b)(c)


### 4.5.  Development of an Undular Bore

This numerical example allows us to study the development of an undular bore from the following initial condition:(4.10)u(x,0)=0.5u0[1-tanh(x-xcd)],
which represents the elevation of a water surface above equilibrium. The parameter u0 is the change in water level at x=xc, and d is the slope between still and deep water. Beside these conditions, the boundary conditions are defined by (4.8). The RLW parameters used for the numerical simulation are ε=3/2, μ=1/6, and δt=0.1. The undular bore is studied in the space interval -36≤x≤300 and time interval 0≤t≤250. In addition, u0=0.1, xc=0, and d=1,2,5. The FPM parameters are shown in Table 1. An initial detailed view at different values of slope d is presented in Figure 10. The results of the FPM approximation at different values of d are shown in Figures 11, 12, and 13, and the conservation quantities for various times steps are given in Tables 10, 11, and 12. In this particular problem, quantities C1, C2, and C3 are not conserved but increase linearly according to the values of M1, M2, and M3, respectively [47]:(4.11)M1=ddtC1=ddt∫-∞∞udx=u0+12u02,M2=ddtC2=ddt∫-∞∞(u2+μ(ux)2)dx=u02+23u03,M3=ddtC3=ddt∫-∞∞(u3+3u2)dx=3u02+3u03+34u04.
Given the fixed value of u0, the values of M1, M2, and M3 are 0.105000, 0.010667 and 0.033075, respectively. With the values of M1, M2, and M3 defined above and using Δt=50, it is possible to calculate the increment in the quantities C1, C2, and C3 as ΔC1=5.25000, ΔC2=0.53335 and ΔC3=1.65375, respectively. These increments are shown in Tables 10, 11, and 12 at different values of slope d.Table 10
Conservation quantities for the development of an undular bore ford=1.
tC1C2C306.0707500.6025471.8661315011.4385801.1525843.57151810016.8053661.7015405.27328915022.1843952.2527706.98217620027.5584932.8029438.68770725032.9332553.35322010.393560Table 11
Conservation quantities for the development of an undular bore ford=2.
tC1C2C306.0707500.5973741.8505125011.4352771.1465593.55329410016.8095171.6966785.25879715022.1852172.2471106.96520120027.5600922.7973818.67106325032.9350363.34766610.376953Table 12
Conservation quantities for the development of an undular bore ford=5.
tC1C2C306.0707500.5822111.8032615011.4350001.1311423.50546310016.8100001.6812695.21121715022.1850012.2314996.91711420027.5600012.7817758.62304625032.9350003.33206410.328972Figure 10
Initial detailed view of the development of an undular bore at different values of sloped.(a) Perspective, (b) superior, and (c) final detailed view of the development of an undular bore ford=1.
(a)(b)(c)(a) Perspective, (b) superior, and (c) final detailed view of the development of an undular bore ford=2.
(a)(b)(c)(a) Perspective, (b) superior, and (c) final detailed view of the development of an undular bore ford=5.
(a)(b)(c)


## 4.1.  Propagation of a Single Solitary Wave

Solitary waves are wave packets or pulses that propagate nonlinearly in dispersive media. Due to the dynamic balance between the nonlinear and dispersive effects, these waves retain a stable waveform known as a soliton. This is a very special type of solitary wave, which also keeps its waveform after collision with other solitons [21]. In the case of solitons, the following explicit solution to (3.1) is given by [7](4.4)u(x,t)=3csech2[k(x-x0-vt)],
which represents a single solitary wave of amplitude 3c centered at x0 with velocity v=1+εc and width k=(1/2)εc/μv. The initial condition of (3.1) is given as follows:(4.5)u(x,0)=3csech2[k(x-x0)].
The RLW parameters used in the numerical simulation are c=0.01,0.055,0.1, x0=0, ε=μ=1(S=1) and δt=0.1. As expected, the solitary wave moves to the right across the space interval -40≤x≤60 in the time interval 0≤t≤20. As in previous work, such as [20, 21], the boundary conditions are extracted from the exact solution (4.4), and the initial condition is given by (4.5). The FPM parameters are shown in Table 1. Moreover, based on the conservation laws Equations (3.9) and (3.10) and the explicit solution (4.4), the following error norms are calculated for this test problem:(4.6)L2=h∑j=1Np|u(xj)-û(xj)|2,L∞=max⁡j|u(xj)-û(xj)|,
where h is the minimum distance between any two points in the domain a≤x≤b. The error norms L2, L∞ and conservation quantities C1, C2, and C3 are shown in Tables 2, 3, and 4. A perspective and superior view of the propagation of a single solitary wave when c=0.1 is shown in Figure 1. Moreover, Figure 2 presents the evolution of the L2 and L∞ error norms at different values of c.Table 2
Error norms and conservation quantities for the propagation of a single solitary wave forc=0.01.
tL2L∞C1C2C300.0000390.0000411.2054600.0241670.07293850.0000600.0000301.2054790.0241670.072938100.0000770.0000301.2054060.0241670.072938150.0000830.0000301.2053030.0241670.072938200.0000890.0000431.2051270.0241670.072938Table 3
Error norms and conservation quantities for the propagation of a single solitary wave forc=0.055.
tL2L∞C1C2C300.0000280.0000122.8906050.3212900.99589650.0000450.0000192.8906050.3212900.995896100.0000760.0000282.8906050.3212900.995896150.0001060.0000372.8906050.3212900.995896200.0001330.0000472.8906050.3212900.995896Table 4
Error norms and conservation quantities for the propagation of a single solitary wave forc=0.1.
tL2L∞C1C2C300.0001230.0000583.9799500.8105202.57920250.0002490.0001113.9799500.8105202.579202100.0004280.0001723.9799500.8105212.579202150.0005800.0002303.9799500.8105212.579202200.0007020.0002683.9799500.8105212.579202(a) Perspective and (b) superior view of the propagation of a single solitary wave forc=0.1.
(a)(b)(a)L2 and (b) L∞ error norms for the propagation of a single solitary wave.
(a)(b)The results obtained in this case show that the conservation quantities are controlled in different time steps, and moreover, the error normsL2 and L∞ are of the order of 10-4, which demonstrate the remarkable accuracy of the proposed method and, therefore, the suitability of FPM for this kind of physical problem.


## 4.2.  Interaction of Two Positive Solitary Waves

In this case, the numerical example consists on the interaction of two positive solitary waves defined by the following initial condition:(4.7)u(x,0)=∑j=123cjsech2(kj(x-xj)),
with cj=4kj2/(1-4kj2) and the boundary conditions:(4.8)u(a,t)=u(b,t)=0.
The RLW parameters used in the numerical simulation are k1=0.4, k2=0.3, x1=15, x2=35, ε=μ=1(S=1), and δt=0.01. This waves move across the space interval 0≤x≤120 and the time interval 0≤t≤30. Likewise, the FPM parameters used in this simulation are shown in Table 1. The results of the FPM approximation at different times are shown in Figures 3 and 4, where it is possible to observe that the higher amplitude solitary wave passes through the smaller wave with no change in its waveform. Once again, our results are in agreement with our expectations for all discrete time measured. As in the case of a solitary wave, the three quantities C1, C2, and C3 are conserved see Table 5.Table 5
Conservation quantities for the interaction of two positive solitary waves.
tC1C2C3037.916479120.674902745.391797537.916674120.676914745.3966941037.916670120.607313744.7496761537.916673120.409138743.0404802037.916671120.663186745.2654792537.916672120.678190745.4124423037.916662120.678701745.417152(a) Perspective and (b) superior view of the interaction of two positive solitary waves.
(a)(b)Detailed view of the interaction of two positive solitary waves at various times.
(a)(b)(c)(d)(e)(f)


## 4.3.  Interaction of a Positive and Negative Solitary Wave

This example focuses on the interaction of a positive and negative solitary wave, starting with the initial condition given by (4.7) and the boundary conditions defined in (4.8). The RLW parameters used in the numerical simulation are k1=0.4, k2=0.6, x1=23, x2=38, ε=μ=1(S=1), and δt=0.1. The waves interact across the space interval -10≤x≤80 and the time interval 0≤t≤20. As in the other cases, the FPM parameters are shown in Table 1. The results of the FPM approximation at different times are shown in Figures 5 and 6, where it is possible to observe that the collision produces additional solitary waves. Finally, note that the three conservation quantities for various times steps are controlled, as shown in Table 6.Table 6
Conservation quantities for the interaction of a positive and negative solitary wave.
tC1C2C30−6.06072383.42957−354.2331562−6.06072383.48023−353.3059895−6.06072406.31404−379.08477710−6.06072383.69552−354.09381115−6.06072383.64904−353.88387920−6.07021383.63883−353.856241(a) Perspective and (b) superior view of the interaction of a positive and negative solitary wave.
(a)(b)Detailed view of the interaction of a positive and negative solitary wave at various times.
(a)(b)(c)(d)(e)(f)


## 4.4.  Evolution of Maxwellian Pulse into Stable Waves

This problem deals with the evolution of theMaxwellian pulse into stable solitary waves at various values of the parameter μ. The initial condition is given as follows:(4.9)u(x,0)=exp⁡[-(x-7)2].
The boundary conditions are defined as in (4.8). For this test, the RLW parameters for the numerical simulation are ε=1, μ=0.1, 0.001, and 0.01, and δt=0.01. This pulse is studied across the space interval 0≤x≤30 and time interval 0≤t≤10 for the three cases of μ. The FPM parameters are shown in Table 1. The results of the FPM approximation at different values of the parameter μ are shown in Figures 7, 8, and 9, where it is possible to observe that the Maxwellian pulse develops into various solitary waves. As in the other experiments, the three conservation quantities for various times steps and values of parameter μ as given in Tables 7, 8, and 9 are controlled throughout the simulation.Table 7
Conservation quantities for the evolution of the Maxwellian pulse into stable waves forμ=0.1.
tC1C2C301.7724541.3786334.78326911.7724541.3786174.78325721.7724541.3785314.78317931.7724541.3784554.78311041.7724541.3784124.78306951.7724541.3783894.78304361.7724541.3783754.78302671.7724541.3783674.78301581.7724541.3783614.78300691.7724541.3783574.783000101.7724541.3783544.782995Table 8
Conservation quantities for the evolution of the Maxwellian pulse into stable waves forμ=0.01.
tC1C2C301.7724541.2658464.78326911.7724541.2660674.78416921.7724541.2670094.78819331.7724541.2678894.79201441.7724541.2682684.79372551.7724541.2683984.79433361.7724541.2684444.79455171.7724541.2684634.79463981.7724541.2684734.79468091.7724541.2684794.794702101.7724541.2684834.794716Table 9
Conservation quantities for the evolution of the Maxwellian pulse into stable waves forμ=0.001.
tC1C2C301.7724541.2545674.78326911.7724541.2548504.78512121.7724541.2651464.85982731.7724541.2749274.92908141.7724541.2780244.95008351.7724541.2788884.95444761.7724541.2792164.95579171.7724541.2795634.95882781.7724541.2797754.96082391.7724541.2796964.959418101.7724541.2795914.957822(a) Perspective, (b) superior, and (c) final detailed view of the evolution of the Maxwellian pulse into stable waves forμ=0.1.
(a)(b)(c)(a) Perspective, (b) superior, and (c) final detailed view of the evolution of the Maxwellian pulse into stable waves forμ=0.01.
(a)(b)(c)(a) Perspective, (b) superior, and (c) final detailed view of the evolution of the Maxwellian pulse into stable waves forμ=0.001.
(a)(b)(c)


## 4.5.  Development of an Undular Bore

This numerical example allows us to study the development of an undular bore from the following initial condition:(4.10)u(x,0)=0.5u0[1-tanh(x-xcd)],
which represents the elevation of a water surface above equilibrium. The parameter u0 is the change in water level at x=xc, and d is the slope between still and deep water. Beside these conditions, the boundary conditions are defined by (4.8). The RLW parameters used for the numerical simulation are ε=3/2, μ=1/6, and δt=0.1. The undular bore is studied in the space interval -36≤x≤300 and time interval 0≤t≤250. In addition, u0=0.1, xc=0, and d=1,2,5. The FPM parameters are shown in Table 1. An initial detailed view at different values of slope d is presented in Figure 10. The results of the FPM approximation at different values of d are shown in Figures 11, 12, and 13, and the conservation quantities for various times steps are given in Tables 10, 11, and 12. In this particular problem, quantities C1, C2, and C3 are not conserved but increase linearly according to the values of M1, M2, and M3, respectively [47]:(4.11)M1=ddtC1=ddt∫-∞∞udx=u0+12u02,M2=ddtC2=ddt∫-∞∞(u2+μ(ux)2)dx=u02+23u03,M3=ddtC3=ddt∫-∞∞(u3+3u2)dx=3u02+3u03+34u04.
Given the fixed value of u0, the values of M1, M2, and M3 are 0.105000, 0.010667 and 0.033075, respectively. With the values of M1, M2, and M3 defined above and using Δt=50, it is possible to calculate the increment in the quantities C1, C2, and C3 as ΔC1=5.25000, ΔC2=0.53335 and ΔC3=1.65375, respectively. These increments are shown in Tables 10, 11, and 12 at different values of slope d.Table 10
Conservation quantities for the development of an undular bore ford=1.
tC1C2C306.0707500.6025471.8661315011.4385801.1525843.57151810016.8053661.7015405.27328915022.1843952.2527706.98217620027.5584932.8029438.68770725032.9332553.35322010.393560Table 11
Conservation quantities for the development of an undular bore ford=2.
tC1C2C306.0707500.5973741.8505125011.4352771.1465593.55329410016.8095171.6966785.25879715022.1852172.2471106.96520120027.5600922.7973818.67106325032.9350363.34766610.376953Table 12
Conservation quantities for the development of an undular bore ford=5.
tC1C2C306.0707500.5822111.8032615011.4350001.1311423.50546310016.8100001.6812695.21121715022.1850012.2314996.91711420027.5600012.7817758.62304625032.9350003.33206410.328972Figure 10
Initial detailed view of the development of an undular bore at different values of sloped.(a) Perspective, (b) superior, and (c) final detailed view of the development of an undular bore ford=1.
(a)(b)(c)(a) Perspective, (b) superior, and (c) final detailed view of the development of an undular bore ford=2.
(a)(b)(c)(a) Perspective, (b) superior, and (c) final detailed view of the development of an undular bore ford=5.
(a)(b)(c)


## 5.  Conclusions

The RLW equation is a PDE that can be used to solve several nonlinear physical problems. In fact, there exist various numerical approximations, such as FD, PS, and FEM, which are also suitable for this kind of problem.In this paper, we present a meshless FPM as an alternative approach for the numerical solution of the RLW equation. About the FPM, we remark that the linearized algebraical system (3.8) is easy to implement, as shape functions are computed only at the beginning of calculation.The efficiency of the proposed technique has been tested using different numerical experiments. The above is remarked by the comparision between exact and numerical solutions shown in Tables2, 3, and 4 (see L2, L∞ error norms).In cases in which precision (L2, L∞) cannot be evaluated, we verify the accuracy of the simulations by measuring the conservation quantities C1, C2 and C3 at different time steps. Note that the values of C1 are stables for different time step, as such quantity is defined in linear terms of u; see (4.3).We have also presented relevant features of this equation in order to better understand the behavior of the model.The numerical test problems presented in Section4 are in agreement with related literature; see [20, 21, 47].According to the numerical results, we conclude that the present method can be considered a useful scheme for solving the type of nonlinear PDE considered here.



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