A Meshless Finite-Point Approximation for Solving the RLW Equation

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 with 𝐿₂, 𝐿_∞ error norms and the conservation properties of mass, energy, and momentum under the RLW equation.

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 subdomain Ω_𝑘 for each node 𝑥_𝑘. This Ω_𝑘 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:Ω⊆^𝑁𝑝_𝑘=1Ω_𝑘, (2.1) where 𝑁_𝑝 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 𝑢(𝑥) only valid in the subdomain Ω_𝑘 associated with the star node 𝑥_𝑘 as a linear combination of known functions 𝐩(𝑥):𝑢(𝑥)≅𝑢(𝑥)=𝐩(𝑥)𝜶_𝐤∀𝑥∈Ω_𝑘, (2.2) where 𝐩(𝑥) is the vector that represents the basis of 𝑚 linearly independent functions and 𝜶_𝐤 is a vector of constant parameters only valid in Ω_𝑘. 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⋅⋅⋅𝑥^𝑚−1]. Since (2.2) is valid for all 𝑁_𝑐 points of the 𝑘th subdomain, the approximations 𝑢(𝐗) conform to a Vandermonde system given by the following relation:𝐮𝐗^𝐤≅𝐮𝐗^𝐤=𝐏𝐗^𝐤⋅𝜶_𝐤, (2.3) where𝐗^𝐤=𝑥_𝑘,1⋅⋅⋅𝑥_{𝑘,𝑁𝑐}^𝑇,𝐮𝐗^𝐤=𝑢𝑥_𝑘,1⋅⋅⋅𝑢𝑥_{𝑘,𝑁𝑐}^𝑇,𝐮𝐗^𝐤=𝑢𝑥_𝑘,1⋅⋅⋅𝑢𝑥_{𝑘,𝑁𝑐}^𝑇,𝜶_𝐤=𝛼_𝑘,1⋅⋅⋅𝛼_{𝑘,𝑁𝑐}^𝑇,𝐏𝐗^𝐤=𝐩𝑥_𝑘,1⋮𝐩𝑥_{𝑘,𝑁𝑐}. (2.4)

In general, the number of points 𝑁_𝑐 that conform to the cloud is greater than the number of functions 𝑚 that define the basis; hence, the matrix 𝐏(𝐗^𝐤) 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 𝜶_𝐤 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 𝑢(𝑥) of each point are minimized according to the following expression:min^𝑁𝑐_𝑗=1𝑤𝑥_𝑘,𝑗⋅𝑢(𝑥_𝑘,𝑗)−𝑢(𝑥_𝑘,𝑗)², (2.5) where 𝑤(𝑥) is a fixed weighting function defined in Ω_𝑘. See [27, 28]. The minimization process described by (2.5) leads to the following expression for vector 𝜶_𝐤:𝜶_𝐤=𝐀⁻¹𝐗^𝐤𝐁𝐗^𝐤𝝀𝐗^𝐤, (2.6) where 𝝀(𝐗^𝐤) is a vector that represents the unknown parameters sought on the cloud Ω_𝑘 defined as follows:𝝀𝐗^𝐤=𝜆𝑥_𝑘,1⋅⋅⋅𝜆𝑥_{𝑘,𝑁𝑐}^𝑇. (2.7) Additionally, matrices 𝐀(𝐗^𝐤), 𝐁(𝐗^𝐤), and 𝐖(𝐗^𝐤) are given as follows:𝐀𝐗^𝐤=𝐏𝐗^𝐤𝐖𝐗^𝐤𝐏^𝑇𝐗^𝐤,𝐁𝐗^𝐤=𝐏^𝑇𝐗^𝐤𝐖𝐗^𝐤, (2.8) and 𝐖(𝐗^𝐤) is an 𝑁_𝑐×𝑁_𝑐 diagonal matrix defined by𝐖𝐗^𝐤=diag𝑤𝑥_𝑘,1⋅⋅⋅𝑤𝑥_{𝑘,𝑁𝑐}, (2.9) where the weighting functions 𝑤(𝑥) are derived in order to have unit values near the star node and zero values outside the Ω_𝑘 subdomains. Under the FPM, the common selection for the fixed weighting function is given as follows:𝑤(𝑥)=exp−ℎ_𝑗/𝜁²−exp−(𝑟/𝜁)²1−exp−(𝑟/𝜉)²,ifℎ_𝑗≤𝑟,0,ifℎ_𝑗>𝑟, (2.10) where ℎ_𝑗 is the distance between the star node 𝑥_𝑘 and the point 𝑥, 𝑟=𝑞⋅ℎ_max (max. of ℎ_𝑗) 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:𝑢(𝑥)=𝐍^𝑇(𝑥)𝝀𝐗^𝐤, (2.11) where 𝐍(𝑥) is a matrix called shape function defined by𝐍(𝑥)=𝐩^𝑇(𝑥)𝐂𝐗^𝐤, (2.12) with 𝐂(𝐗^𝐤)=𝐀⁻¹(𝐗^𝐤)𝐁(𝐗^𝐤). We remark that 𝐩(𝑥) denote [1𝑥𝑥²⋅⋅⋅𝑥^𝑚−1]. Note that according to the least square nature of the approximation, 𝑢(𝑥)≅𝑢(𝑥)≠𝜆(𝑥); that is, the local values of the approximating function do not fit the nodal unknown values. Indeed, 𝑢(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:𝑢_𝑥(𝑥)=𝐍^𝑇_𝑥(𝑥)𝝀𝐗^𝐤𝑢_𝑥𝑥(𝑥)=𝐍^𝑇_𝑥𝑥(𝑥)𝝀𝐗^𝐤, (2.13) 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:𝑢_𝑡(𝑥,𝑡)+𝑢_𝑥(𝑥,𝑡)+𝜀⋅𝑢(𝑥,𝑡)𝑢_𝑥(𝑥,𝑡)−𝜇⋅𝑢_𝑥𝑥𝑡(𝑥,𝑡)=0,𝑥∈ℝ,𝑡>0, (3.1) with the physical conditions 𝑢→0 as |𝑥|→∞. To the numerical implementation, we consider 𝑎≤𝑥≤𝑏 and the following boundary conditions:𝑢(𝑎,𝑡)=0𝑢(𝑏,𝑡)=0. (3.2) The initial condition for the problem (3.1) is given as follows:𝑢(𝑥,0)=𝑢₀(𝑥),𝑎≤𝑥≤𝑏. (3.3) Parameters 𝜀 and 𝜇 in (3.1) are positive constants and are related by the Stokes number 𝑆 defined as follows:𝜀𝑆=𝜇. (3.4) 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:𝑢^𝑛+1−𝑢^𝑛𝛿𝑡+𝜃𝑢_𝑥^𝑛+1+𝜀⋅𝑢^𝑛+1𝑢_𝑥^𝑛+1+(1−𝜃)𝑢_𝑥^𝑛+𝜀⋅𝑢^𝑛𝑢_𝑥^𝑛−𝜇𝛿𝑡𝑢_𝑥𝑥^𝑛+1−𝑢_𝑥𝑥^𝑛=0, (3.5) where 𝑢^𝑛=𝑢(𝑥,𝑡^𝑛), 𝑡^𝑛=𝑡^𝑛−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]:𝑢^𝑛+1𝑢_𝑥^𝑛+1≈𝑢^𝑛+1𝑢_𝑥^𝑛+𝑢^𝑛𝑢_𝑥^𝑛+1−𝑢^𝑛𝑢_𝑥^𝑛; (3.6) therefore, from (3.5) and (3.6), the following expression is obtained:𝑢^𝑛+1+𝛿𝑡⋅𝜃𝑢_𝑥^𝑛+1+𝜀⋅𝑢^𝑛+1𝑢_𝑥^𝑛+𝑢^𝑛𝑢_𝑥^𝑛+1−𝜇⋅𝑢_𝑥𝑥^𝑛+1=𝑢^𝑛+𝛿𝑡⋅𝜀⋅(2𝜃−1)𝑢^𝑛𝑢_𝑥^𝑛−(1−𝜃)𝑢_𝑥^𝑛−𝜇⋅𝑢_𝑥𝑥^𝑛. (3.7) With the purpose to obtain the local approximation 𝑢^𝑛+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:𝐆_𝑘⋅𝝀𝐗^𝐤=𝑓_𝑘, (3.8) where matrix 𝐆_𝑘 and the known quantity 𝑓_𝑘 (based on the time level 𝑛) are defined as follows:𝐆_𝑘=𝐍^𝑇+𝛿𝑡2⋅𝐍_𝐱^𝑇+𝛿𝑡2𝜀⋅𝐍^𝑇⋅𝑢^𝑛_𝑥+𝛿𝑡2𝜀⋅𝑢^𝑛⋅𝐍^𝑇_𝑥−𝜇⋅𝐍^𝑇_𝑥𝑥,𝑓_𝑘=𝑢^𝑛−𝛿𝑡2⋅𝑢^𝑛_𝑥−𝜇⋅𝑢_𝑥𝑥; (3.9) here, the subscript 𝑘 denotes the evaluation in the star node 𝑥_𝑘.

Note that for simplicity, the terms 𝐍^𝑇(𝑥), 𝐍^𝑇_𝑥(𝑥), and 𝐍^𝑇_𝑥𝑥(𝑥) have been replaced by 𝐍^𝑇, 𝐍^𝑇_𝑥, and 𝐍^𝑇_𝑥𝑥, 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 𝐆_𝑘 and quantity 𝑓_𝑘 adopt the following forms, respectively:𝐆_𝑎,𝑏=𝐍^𝑇𝑓_𝑎,𝑏=0. (3.10) Finally, using (3.8) and (3.10) and taking into account the point collocation procedure for each of the 𝑁_𝑝 nodes that comprise the domain Ω, the following system of discrete equation is solved:𝐆⋅𝝀=𝐟, (3.11) where 𝐆 is defined as a stiffness matrix, 𝝀 represents the vector collecting of the point parameters 𝜆^𝑛+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 of Maxwellian 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:^∞_−∞𝑢d𝑥≈𝐶₁=^𝑏_𝑎𝑢d𝑥,_∞−∞𝑢²+𝜇𝑢_𝑥𝑥²d𝑥≈𝐶₂=^𝑏_𝑎𝑢²+𝜇𝑢_𝑥𝑥²d𝑥,_∞−∞𝑢³+3𝑢²d𝑥≈𝐶₃=^𝑏_𝑎𝑢³+3𝑢²d𝑥. (4.1)(4.2)(4.3) 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.

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]𝑢(𝑥,𝑡)=3𝑐sech²𝑘𝑥−𝑥₀−𝑣𝑡, (4.4) 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:𝑢(𝑥,0)=3𝑐sech²𝑘𝑥−𝑥₀. (4.5) 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:𝐿₂=ℎ^𝑁𝑝_𝑗=1𝑢(𝑥_𝑗)−𝑢(𝑥_𝑗)²,𝐿_∞=max_𝑗𝑢𝑥_𝑗−𝑢𝑥_𝑗, (4.6) 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 for 𝑐=0.01.

Table 3 

Error norms and conservation quantities for the propagation of a single solitary wave for 𝑐=0.055.

Table 4 

Error norms and conservation quantities for the propagation of a single solitary wave for 𝑐=0.1.

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(b)

Figure 1 

(a) Perspective and (b) superior view of the propagation of a single solitary wave for 𝑐=0.1.

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(b)

Figure 2 

(a) 𝐿2 and (b) 𝐿∞ error norms for the propagation of a single solitary wave.

The results obtained in this case show that the conservation quantities are controlled in different time steps, and moreover, the error norms 𝐿₂ 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:𝑢(𝑥,0)=²_𝑗=13𝑐_𝑗sech²𝑘_𝑗𝑥−𝑥_𝑗, (4.7) with 𝑐_𝑗=4𝑘²_𝑗/(1−4𝑘²_𝑗) and the boundary conditions:𝑢(𝑎,𝑡)=𝑢(𝑏,𝑡)=0. (4.8) 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.

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(b)

Figure 3 

(a) Perspective and (b) superior view of the interaction of two positive solitary waves.

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(b)

(c)

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Figure 4 

Detailed view of the interaction of two positive solitary waves at various times.

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.

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(b)

Figure 5 

(a) Perspective and (b) superior view of the interaction of a positive and negative solitary wave.

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(b)

(c)

(d)

(e)

(f)

Figure 6 

Detailed view of the interaction of a positive and negative solitary wave at various times.

4.4. Evolution of Maxwellian Pulse into Stable Waves

This problem deals with the evolution of the Maxwellian pulse into stable solitary waves at various values of the parameter 𝜇. The initial condition is given as follows:𝑢(𝑥,0)=exp−(𝑥−7)². (4.9) 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 for 𝜇=0.1.

Table 8 

Conservation quantities for the evolution of the Maxwellian pulse into stable waves for 𝜇=0.01.

Table 9 

Conservation quantities for the evolution of the Maxwellian pulse into stable waves for 𝜇=0.001.

(a)

(b)

(c)

Figure 7 

(a) Perspective, (b) superior, and (c) final detailed view of the evolution of the Maxwellian pulse into stable waves for 𝜇=0.1.

(a)

(b)

(c)

Figure 8 

(a) Perspective, (b) superior, and (c) final detailed view of the evolution of the Maxwellian pulse into stable waves for 𝜇=0.01.

(a)

(b)

(c)

Figure 9 

(a) Perspective, (b) superior, and (c) final detailed view of the evolution of the Maxwellian pulse into stable waves for 𝜇=0.001.

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:𝑢(𝑥,0)=0.5𝑢₀1−tanh𝑥−𝑥_𝑐𝑑, (4.10) which represents the elevation of a water surface above equilibrium. The parameter 𝑢₀ is the change in water level at 𝑥=𝑥_𝑐, 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, 𝑥_𝑐=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]:𝑀₁=d𝐶d𝑡¹=dd𝑡^∞_−∞𝑢d𝑥=𝑢₀+12𝑢²₀,𝑀₂=d𝐶d𝑡²=dd𝑡^∞_−∞𝑢²+𝜇𝑢_𝑥²d𝑥=𝑢²₀+23𝑢³₀,𝑀₃=d𝐶d𝑡³=dd𝑡^∞_−∞𝑢³+3𝑢²d𝑥=3𝑢²₀+3𝑢³₀+34𝑢⁴₀. (4.11) 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 for 𝑑=1.

Table 11 

Conservation quantities for the development of an undular bore for 𝑑=2.

Table 12 

Conservation quantities for the development of an undular bore for 𝑑=5.

Figure 10 

Initial detailed view of the development of an undular bore at different values of slope 𝑑.

(a)

(b)

(c)

Figure 11 

(a) Perspective, (b) superior, and (c) final detailed view of the development of an undular bore for 𝑑=1.

(a)

(b)

(c)

Figure 12 

(a) Perspective, (b) superior, and (c) final detailed view of the development of an undular bore for 𝑑=2.

(a)

(b)

(c)

Figure 13 

(a) Perspective, (b) superior, and (c) final detailed view of the development of an undular bore for 𝑑=5.

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 Tables 2, 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 Section 4 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.

Acknowledgments

L. P. Pozo is partially supported by the Chilean Agency CONICYT (FONDECYT Project 11100253). C. Spa is partially supported by the Chilean Agency CONICYT (FONDECYT Project 3110046). The authors would also like to thank two anonymous reviewers for many helpful comments and suggestions on previous drafts of this paper.