A Meshless Method Using the Radial Basis Functions for Numerical Solution of the Regularized Long...

A Meshless Method Using the Radial BasisFunctions for Numerical Solution of theRegularized Long Wave EquationAli Shokri, Mehdi DehghanDepartment of Applied Mathematics, Faculty of Mathematics and Computer Science,Amirkabir University of Technology, Tehran, Iran

Received 2 November 2008; accepted 26 January 2009Published online 9 April 2009 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/num.20457

This article discusses on the solution of the regularized long wave (RLW) equation, which is introduced todescribe the development of the undular bore, has been used for modeling in many branches of science andengineering. A numerical method is presented to solve the RLW equation. The main idea behind this numer-ical simulation is to use the collocation and approximating the solution by radial basis functions (RBFs).To avoid solving the nonlinear system, a predictor-corrector scheme is proposed. Several test problems aregiven to validate the new technique. The numerical simulation, includes the propagation of a solitary wave,interaction of two positive solitary waves, interaction of a positive and a negative solitary wave, the eval-uation of Maxwellian pulse into stable solitary waves and the development of an undular bore. The threeinvariants of the motion are calculated to determine the conservation properties of the algorithm. The resultsof numerical experiments are compared with analytical solution and with those of other recently publishedmethods to confirm the accuracy and efficiency of the presented scheme. © 2009 Wiley Periodicals, Inc. NumerMethods Partial Differential Eq 26: 807–825, 2010

Keywords: collocation; predictor-corrector; predictor-corrector thin plate splines radial basisfunctions(PC-TPS-RBFs); radial basis functions (RBFs); regularized long wave (RLW) equation; thin platesplines (TPS)

I. INTRODUCTION

Finite difference methods are known as the first techniques for solving partial differential equa-tions. Even though these methods are very effective for solving various kinds of partial differentialequations, conditional stability of explicit finite difference procedures and the need to use largeamount of CPU time in implicit finite difference schemes limit the applicability of these methods[1]. Furthermore, these methods provide the solution of the problem on mesh points only andaccuracy of the techniques is reduced in non-smooth and non-regular domains [1].

Correspondence to: Mehdi Dehghan, Department of Applied Mathematics, Faculty of Mathematics and Computer Science,Amirkabir University of Technology, Tehran, Iran (e-mail: [email protected])

© 2009 Wiley Periodicals, Inc.

808 SHOKRI AND DEHGHAN

To avoid the mesh generation, in recent years meshless techniques have attracted attention ofresearchers. In a meshless (meshfree) method, a set of scattered nodes are used instead of meshingthe domain of the problem [2–7].

In the last 20 years, the radial basis functions method is known as a powerful tool for scattereddata interpolation problem. In the last decade, there has been some advance developments inapplying the radial basis functions (RBFs) for the numerical solution of various types of partialdifferential equations (PDEs). The initial development was due to the pioneering work of Kansa[8,9] who directly collocated the RBFs for the approximated solution of the equations. The mainadvantage of numerical procedures which use radial basis functions over traditional techniques ismeshless property of these methods. In general, the Kansas method has [10] several advantagesover the widely used finite element method (FEM), in that

1. it is a truly meshless method in which the collocation points can be chosen freely (no con-nectivity between points is required as FEM). Hence, the complicated meshing problemhas been avoided;

2. it is an independent spatial dimension which can easily be extended to solve highdimensional problems.

Radial basis functions are used actively for solving partial differential equations. For example see[11] and the references therein. The interested readers can see [12–14] for applications of radialbasis functions to some problems in applied sciences.

To the best of our knowledge, the most popular methods might be the collocation method, thefundamental solutions method, the boundary knots method, the Dual Reciprocity Method (DRM),and the Galerkin method. For more descriptions see [10] and references therein. More recently,Hon and Wu [15] gave a theoretical justification in combining the RBFs with those advancedtechniques of domain decomposition, multilevel/multigrid, Schwarz iterative schemes, and pre-conditioning in the FEM discipline. Though it is very popular to use the Galerkin method in theliterature of FEM, the Galerkin method is seldom used in the meshless RBFs society. One reasonis that the usual Galerkin method is not so efficient as it is in FEM, because the computation of theintegrals would be complicated [10]. The interested reader can see [16–22] for more investigationsin RBFs.

A problem of real interest for applications is the generalized Benjamin-Bona-Mahony (BBM)equations in the form

Ut + βUx − µUxxt + g(U)x = 0, (1.1)

with the initial data

U(x, 0) = f (x) → 0, x → ±∞, (1.2)

where U(x, t) represents the fluid velocity in the horizontal direction x, and µ is a positive con-stant, β ∈ R, and g(U) is a C2-smooth nonlinear function. When g(U)x = UUx with β = 1, Eq.(1.1) is the alternative regularized long-wave (RLW) equation.

II. A BRIEF REVIEW OF THE EXISTING METHODS AND THE APPLICATIONS

The RLW equation is an alternative description of nonlinear dispersive waves to the more usualKorteweg-de Vries (KdV) equation. It has been shown to have solitary wave solutions and to

Numerical Methods for Partial Differential Equations DOI 10.1002/num

MESHLESS METHOD FOR RLW EQUATION 809

govern a large number of important physical phenomena such as the nonlinear transverse wavesin shallow water, ion-acoustic, and magnetohydrodynamic waves in plasma and phonon packetsin nonlinear crystals, the anharmonic lattice longtitudinal dispersive waves in elastic rods, pres-sure waves in liquid–gas bubble mixtures, rotating flow down a tube, the lossless propagationof shallow water waves, thermally exited phonon packets in low-temperature nonlinear crystals[23, 24].

As is said in [24] the RLW equation was originally introduced to describe the behavior of theundular bore by Peregrine [25], has been used for modeling in many branches of science andengineering. An analytical solution for the RLW equation was found under the restricted initialand boundary conditions in [26]. Because the analytical solution of the RLW equation is notvery useful, the availability of accurate and efficient numerical methods is essential. Numericalsolutions of the RLW equation have been undertaken by employing various form of the finitedifference, pseudo-spectral, and finite element methods (see [23, 27] and references therein).

As is said in [23] Peregrine [25] used the finite difference scheme, which is only first-order accu-rate in time and second-order in space. Eilbeck and McGuire [28] gave a second-order differencemethod, accurate both in space and time, but it is a three-level scheme and requires another approx-imation initially. Eilbeck and McGuire [29] and Lewis and Tjon [30] have studied the interaction ofsolitary waves for solving RLW equation by using the above three-level finite difference method.

Bhardwaj and Shankar [31] developed a new finite difference scheme based on operator split-ting and quintic spline interpolation functions technique for the solution of the RLW equation.Recently, finite element methods based on both quadratic and cubic B-spline or linear space finiteelements within Galerkin’s method have been used for getting the solutions of the RLW equation(see [23] and references therein). Most recently, Soliman and Hussien [32] have studied RLWequation by collocation method with septic splines. It is worth to point out that various numericaltechniques based on finite difference [33], finite element [34], spectral methods [35] and Galerkinmethod [36] have been extensively studied [23, 24]. Also several finite elements methods basedon B-splines have been applied to RLW equation [37]. These approaches involve both Galerkinand collocation methods over finite elements with various degree B-splines [38]. More researchworks on the RLW equation can be found in [24,39–60]. Solitary waves of the RLW equation arediscussed in [61]. For further details, the reader refers to Ref. [62] which is an overview of thetheory of solitons and the applications.

This articles presents a new numerical scheme to solve the one-dimensional RLW equationusing the collocation method and approximating directly the solution using radial basis functions.To avoid solving the nonlinear system, a predictor-corrector scheme is proposed.

The rest of this article is organized as follows: Section III shows that how we use the radialbasis functions to approximate the solution. An implementation of the numerical method to theproblem is presented in Sections IV. In Section V we show that how we use the predictor-correctorapproach to avoid solving the nonlinear system. To support our findings, we present results ofnumerical experiments in Section VI. Section VII ends this article with a brief conclusion. Finallysome references are introduced at the end.

It is worth to point out that the new method developed in the current paper can be applied tosolve the boundary value problems investigated in [63–66].

III. BASIC KNOWLEDGE ABOUT RADIAL BASIS FUNCTIONS APPROXIMATION

In the interpolation of the scattered data using radial basis functions, the approximation of a dis-tribution u(x), may be written as a linear combination of N radial functions; usually it takes thefollowing form:



TABLE I. Some well-known functions that generate RBFs.

Name of function Definition

Multiquadrics (MQ) ϕ(r) = √r2 + c2

Inverse Multiquadrics (IMQ) ϕ(r) = (√

r2 + c2)−1

Thin Plate Splines (TPS) ϕ(r) = (−1)k+1r2k log(r)

Gaussian (GA) ϕ(r) = exp(−cr2)

u(x) �N∑

j=1

λjϕ(rj ) + ψ(x) for x ∈ � ⊂ Rd , (3.1)

where N is the number of data points, x = (x1, x2, . . . , xd), d is the dimension of the problem, λ’sare coefficients to be determined, ϕ is the radial basis function and rj = ‖x −xj‖ is the Euclideannorm. Some well-known radial basis functions (RBFs) are listed in Table I.

Equation (3.1) can be written without the additional polynomial ψ . In that case ϕ must beunconditionally positive definite to guarantee the solvability of the resulting system (e.g. Gauss-ian or inverse multiquadrics). However, ψ is usually required when ϕ is conditionally positivedefinite, i.e., when ϕ has a polynomial growth toward infinity. Examples are thin plate splines andmultiquadrics.

For the numerical scheme introduced in Section IV, we will use thin plate splines (TPS). Sinceϕ in TPS is C2k−1 continuous, so higher order thin plate splines must be used for higher orderpartial differential operators. For the RLW equation, k = 2 is used for thin plate splines (i.e. thesecond order thin plate splines).

If Pdq denotes the space of d-variate polynomials of order not exceeding q, and letting the poly-

nomials P1, . . . , Pm be the basis of Pdq in R

d , then the polynomial ψ(x), in Eq. (3.1), is usuallywritten in the following form:

ψ(x) =m∑

i=1

ζiPi(x), (3.2)

where m = (q − 1 + d)!/(d!(q − 1)!).To determinate the coefficients (λ1, . . . , λN) and (ζ1, . . . , ζm), the collocation method is used.

However, in addition to the N equations resulting from collocating (3.1) at the N points, an extram equations are required. This is insured by the m conditions for (3.1),

N∑j=1

λjPi(xj ) = 0, i = 1, . . . , m. (3.3)

In a similar representation as (3.1), for any linear partial differential operator L, Lu can beapproximated by

Lu(x) �N∑

j=1

λjLϕ(rj ) + Lψ(x). (3.4)



IV. IMPLEMENTATION OF THE NUMERICAL METHOD

The RLW equation in the following form:

Ut + Ux + εUUx − µUxxt = 0, (4.1)

where ε and µ are positive parameters and subscripts x and t denote differentiation, is consideredwith the boundary conditions U → 0 as x → ±∞. In the equation U shows dimensionlesssurface elevation, x distance and t time. To implement the numerical method, the solution domainis restricted over an interval [a, b]. Boundary conditions

U(a, t) = U(b, t) = 0, (4.2)

and the initial condition

U(x, 0) = f (x),

are chosen over the interval [a, b]. f (x) is a localized disturbance inside the interval.First, let us discretize (4.1) according to the following scheme

U(x, t + δt) − U(x, t)

δt+ ∇U(x, t) + εU(x, t).∇U(x, t) − µ

∇2U(x, t + δt) − ∇2U(x, t)

δt= 0,

(4.3)

where ∇ is the gradient differential operator, and δt is the time step size. Rearranging (4.3), usingthe notation un = u(x, tn) where tn = tn−1 + δt , we obtain

un+1 − µ∇2un+1 = un − µ∇2un − δt(∇un + εun∇un). (4.4)

Assuming that there are a total of (N − 2) interpolation points, un(x) can be approximated by

un(x) �N−2∑j=1

λnjϕ(rj ) + λn

N−1x + λnN . (4.5)

To determine the interpolation coefficients (λ1, λ2, . . . , λN−1, λN), the collocation method is usedby applying (4.5) at every point xi , i = 1, 2, . . . , N − 2. Thus we have

un(xi) �N−2∑j=1

λnjϕ(rij ) + λn

N−1xi + λnN , i = 1, 2, . . . , N − 2, (4.6)

where rij = √(xi − xj )2. The additional conditions due to (3.3) are written as

N−2∑j=1

λnj =

N−2∑j=1

λnjxj = 0. (4.7)

Writing (4.6) together with (4.7) in a matrix form we have

[u]n = A[λ]n, (4.8)



where [u]n = [un1 un

2 . . . unN−2 0 0]T , [λ]n = [λn

1 λn2 . . . λn

N ]T and A = [aij , 1 ≤ i, j ≤ N ] isgiven by

A =

ϕ11 · · · ϕ1(N−2) x1 1...

. . ....

......

ϕ(N−2)1 · · · ϕ(N−2)(N−2) xN−2 1x1 · · · xN−2 0 01 · · · 1 0 0

. (4.9)

There are (N − 2) internal (domain) points and two boundary points. Therefore, the (N × N)

matrix A can be split into: A = Ad + Ab + Ae, where

Ad = [aij for (2 ≤ i ≤ N − 3, 1 ≤ j ≤ N) and 0 elsewhere],Ab = [aij for (i = 1, N − 2, 1 ≤ j ≤ N) and 0 elsewhere], (4.10)

Ae = [ aij for (N − 1 ≤ i ≤ N , 1 ≤ j ≤ N) and 0 elsewhere].Using the notation LA to designate the matrix of the same dimension as A and containing theelements aij , where aij = Laij , 1 ≤ i, j ≤ N , then Eq. (4.4) together with (4.2) can be written,in the matrix form, as

B[λ]n+1 = C[λ]n − δt{∇Ad[λ]n + ε(Ad[λ]n) ∗ (∇Ad[λ]n)}, (4.11)

where

B = Ad − µ∇2Ad + Ab + Ae, C = Ad − µ∇2Ad .

In Eq. (4.11), the accent ∗ means component by component multiplication of two vectors. Equa-tion (4.11) is obtained by combining (4.4), which applies to the domain points, and (4.2) whichapplies to the boundary points.

V. THE PREDICTOR-CORRECTOR SCHEME

To avoid solving the nonlinear system, the following predictor-corrector scheme is proposed. If inEq. (4.1), we move the nonlinear term εUUx to the right-hand side, and eliminate the nonlinearityand applying the RBF scheme (as we described in previous section), then we have the Eq. (4.11),that constitutes an N × N linear algebraic system in each time level. For simplicity, we write Eq.(4.11) as the following form

B[λ]n+1 = C[λ]n − δt([λ]n), (5.1)

where ([λ]n) = ∇Ad[λ]n + ε(Ad[λ]n) ∗ (∇Ad[λ]n).At the first time level, i.e. when n = 0, [λ]0 is determined using the initial condition. For

dealing with the non-linearity, in each time level (for example n + 1), we propose the following(iterative) algorithm:

• Step 1: Calculate an ‘intermediate’ value [λ]n+1(0) (called the predictor) from (5.1) as follows:

B[λ]n+1(0) = C[λ]n − δt([λ]n). (5.2)



Please note that [λ]n is known from the previous time level. By using the predicted value weget the approximation

B[λ]n+1(1) = C[λ]n − δt

2

{([λ]n) +

([λ]n+10

)}. (5.3)

• Step 2: The general iteration is given by

B[λ]n+1(k) = C[λ]n − δt

2

{([λ]n) +

([λ]n+1k−1

)}, k = 1, 2, . . . (5.4)

• Step 3: We compute the left-hand side of (5.4) until∥∥Un+1

(k) − Un+1(k−1)

∥∥∞∥∥Un+1

(k)

∥∥∞

≤ ε for prescribed tolerance ε. (5.5)

Once the prescribed convergence is achieved, we can move on to the following time level. Thisprocess is iterated, until reaching to the desirable time t .

VI. NUMERICAL RESULTS

In this section, we give some computational results of the thin plate spline radial basis functionmethod with predictor-corrector (PC-TPS-RBF) on the RLW equation to support our discussionin the previous sections. Accuracy of the estimated solutions can be worked out by measuringboth the L2 and L∞ error norms which are defined by

L2 = ∥∥U exact − U numerical∥∥

2=

√√√√δx

N∑i=1

(U exact

i − U numericali

)2, (6.1)

L∞ = ∥∥U exact − U numerical∥∥

∞ = maxi

∣∣U exacti − U numerical

i

∣∣. (6.2)

Note that in this section we follow a procedure which is already employed in the literature[23, 24] to interpret the numerical results. With our best of knowledge, this is the first time thatthe method of radial basis functions is proposed to solve the regularized long wave equation.

Conservation property of the RLW equation will be validated by computing quantitiescorresponding to mass, momentum and energy [67]

C1 =∫ b

a

Udx � δx

N∑i=1

Ui ,

C2 =∫ b

a

(U 2 + µ(Ux)

2)dx � δx

N∑i=1

((Ui)

2 + µ((Ux)i)2)

, (6.3)

C3 =∫ b

a

(U 3 + 3U 2

)dx � δx

N∑i=1

((Ui)

3 + 3(Ui)2)

,

respectively. Then the algorithm is said to be significant and useful. It can be seen from Eq. (6.3)that integrals are approximated by employing the trapezium rule.



TABLE II. Error norms and invariants for propagation of a solitary wave: amplitude 3c = 0.3, δt = 0.1,and δx = 0.125.

Method Time L2 × 105 L∞ × 105 C1 C2 C3

PC-TPS-RBF 0 0 0 3.9799497 0.8104624 2.57900745 6.0324 2.4618 3.9799497 0.8104624 2.5790074

10 11.852 4.8501 3.9799497 0.8104624 2.579007415 17.343 6.9527 3.9799497 0.8104625 2.579007420 22.483 8.8006 3.9799497 0.8104625 2.5790074

PC-TPS-RBF (δt = 0.01) 20 0.2526 0.1010 3.9799497 0.8104624 2.5790074PC-TPS-RBF (δt = 0.001) 20 0.0351 0.0139 3.9799498 0.8104624 2.5790074High-order compact 20 0.0395 0.0155 3.9799067 0.8104625 2.5790074Galerkin linear 20 51.1 19.8 3.98206 0.81164 2.58133Galerkin quadratic 20 22.0 8.6 3.97989 0.810467 2.57902Least square cubic 20 1.84 156.64 3.961597 0.804185 2.558292Petrov-Galerkin quadratic 20 22.7 8.1 3.91986 0.810399 2.57880Pseudo-spectral 20 18.2 6.66 3.97992 0.810462 2.57901Cubic B-spline collocation 20 30 11.6 3.979883 0.8102762 2.5783926

We would like to mention that we employ five test problems to demonstrate the efficiencyof the new method developed in the current paper. These test problems include: motion of thesolitary wave, interaction of two positive solitary waves, interaction of a positive and a negativesolitary wave, the evaluation of Maxwellian pulse into stable solitary waves, the development ofan undular bore.

A. Motion of the Solitary Wave

Choosing examples with known solutions allows for a more complete error analysis. In this exam-ple, we consider the types of solitary wave solutions to the RLW equation were extensively studiedby many authors (for example, see [23, 27, 32]).

The theoretical solution [27]

U(x, t) = 3c sech2(k[x − x0 − (1 + εc)t]), (6.4)

represents a single solitary wave of amplitude 3c, υ = 1 + εc is the wave velocity andk = √

εc/4µ(1 + εc).The initial condition

U(x, 0) = 3c sech2(k[x − x0]), (6.5)

and the boundary conditions

U(a, t) = U(b, t) = 0, (6.6)

are used with parameters c = 0.1 and c = 0.03, x0 = 0 and ε = µ = 1. This initial solitary wavepropagates to the right across the interval −80 ≤ x ≤ 100 in the time period 0 ≤ t ≤ 20 [23].

To compare the new technique with the recent methods collected in [23], the spatial grid spac-ing δx = 0.125 and time step δt = 0.1 are taken. Errors in L2 and L∞ norms and the threeinvariants C1, C2, C3 for the two cases taken at several times are tabulated in Tables II and III,respectively. The plot of estimated solution in different times and distributions of the errors att = 20 for amplitudes 0.3 and 0.09 are shown in Figs. 1 and 2, respectively.

Also note that the new method developed in this paper is denoted by PC-TPS-RBF.



TABLE III. Error norms and invariants for propagation of a solitary wave: amplitude 3c = 0.09, δt = 0.1,and δx = 0.125.

Method Time L2 × 105 L∞ × 105 C1 C2 C3

PC-TPS-RBF 0 0 0 2.1094049 0.1273017 0.38880595 0.3632 0.1061 2.1094049 0.1273017 0.3888059

10 0.7216 0.2157 2.1094046 0.1273017 0.388805915 1.0788 0.3269 2.1094043 0.1273017 0.388805920 1.4342 0.4382 2.1094022 0.1273017 0.3888059

PC-TPS-RBF (δt = 0.01) 20 0.0632 0.0469 2.1094022 0.1273017 0.3888059PC-TPS-RBF (δt = 0.001) 20 0.0614 0.0469 2.1094025 0.1273017 0.3888061High-order compact 20 0.3457 0.1497 2.1067778 0.1273012 0.3888043Galerkin linear 20 53.5 19.8 2.10906 0.127305 0.388815Galerkin quadratic 20 56.3 43.2 2.10460 0.127302 0.388803Least square cubic 20 2.81 155.06 2.128869 0.1272228 0.388571Petrov-Galerkin quadratic 20 53.7 31.6 2.10908 0.127318 0.388854Pseudo-spectral 20 1.34 0.392 2.109405 0.127302 0.388806Cubic B-spline collocation 20 57 43.2 2.104584 0.1272937 0.388778

The L2 and L∞ error norms for the simulations are computed to be L2 = 2.2483 × 10−5 andL∞ = 8.8006×10−5 (δt = 0.1) and L2 = 0.0351×10−5 and L∞ = 0.0139×10−5 (δt = 0.001)

for amplitude 0.3 at time t = 20. For amplitude 0.09 at time t = 20 we have L2 = 0.4382 × 10−5

and L∞ = 1.4342 × 10−5 (δt = 0.1) and L2 = 0.0469 × 10−5 and L∞ = 0.0614 × 10−5

(δt = 0.001). It can be observed from Tables II and III that L2 and L∞ error norms calculatedwith the present PC-TPS-RBF method are less than those of other schemes considered hereinfrom [23].

On the other hand, we can see from Tables II and III that there are only small changes in thethree invariants; quantities C1, C2, and C3 for amplitude 0.3 change from their initial values byless than 1 × 10−7, 2 × 10−7, and 1 × 10−7, respectively, during the simulation, and quantities C1,C2, and C3 for amplitude 0.09 change from their initial values by less than 3 × 10−6, 1 × 10−7,and 1 × 10−7, respectively. Hence, it can be mentioned that the conservation quantities remainsconstant during the simulation [27].

FIG. 1. (Left) plot of estimated solution in different times, (right) error=exact-numerical solution at timet = 20, for solitary wave with amplitude 3c = 0.3 and δt = 0.01. [Color figure can be viewed in the onlineissue, which is available at www.interscience.wiley.com.]



FIG. 2. (Left) plot of estimated solution in different times, (right) error=exact-numerical solution at timet = 20, for solitary wave with amplitude 3c = 0.09 and δt = 0.01. [Color figure can be viewed in the onlineissue, which is available at www.interscience.wiley.com.]

B. Interaction of Two Positive Solitary Waves

In this part, the interaction of two positive solitary waves is also investigated using the initialcondition [23]

U(x, 0) =2∑

j=1

3cj sech2(kj (x − xj )), (6.7)

where cj = 4k2j /(1 − 4k2

j ) and the boundary conditions are

U(a, t) = U(b, t) = 0. (6.8)

The test was run from t = 0 to 25 in the region 0 ≤ x ≤ 120. The parameters used are k1 = 0.4,k2 = 0.3, x1 = 15, x2 = 35, δx = 0.2, δt = 0.01, and ε = µ = 1. The plot of approxi-mated solution in different times is demonstrated in Fig. 3, in which the two solitary waves have

FIG. 3. Plot of the estimated solution in different times for the interaction of two positive solitary waves.[Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]



TABLE IV. Invariants for the interaction of two positive solitary waves.

Time C1 C2 C3

0 37.916515 120.52323 744.081205 37.916622 120.52325 744.08131

10 37.916632 120.52389 744.0844215 37.916647 120.52533 744.0903720 37.916655 120.52310 744.0796025 37.916663 120.52297 744.07896

apparently passed through one another and emerged unchanged by the encounter. In Table IV,we show the values of the three invariants for various times throughout the numerical simula-tion. One can observe that quantities C1, C2, and C3 change from their initial values by less than2 × 10−4 (0.0004%), 3 × 10−3 (0.002%), and 0.01(0.002%), respectively, during the simulation[23]. It is worth to mention that the solution is conformity with the previous results obtained byseveral authors [23, 24, 27, 32].

C. Interaction of a Positive and a Negative Solitary Wave

Author of [68], simulated the interaction of a positive and a negative solitary wave and found thecollision to produce additional solitary waves an observation observed by Lewis and Tjon [30].Also Gardner and Gardner [69], Soliman and Hussien [32] and Lin et al. [23] have investigatedthis problem.

The initial condition is given as Eq. (6.7), U = 0 is used at the both region ends and theexperiment was run from t = 0 to 20 in the region −10 ≤ x ≤ 80. The concerning parametersare k1 = 0.4, k2 = 0.6, x1 = 23, x2 = 38, δx = 0.2, δt = 0.1, and ε = µ = 1. The profileswhich are sometime after the interaction, are demonstrated in Fig. 4, which show details of thestructure of these solitary waves at time t = 2 and 20, respectively. The three invariants of motionfor various times steps are given in Table V.

FIG. 4. Santarelli experiment: (left) wave profile at time t = 2, (right) wave profile at time t = 20.



TABLE V. Invariants for the interaction of a positive and a negative solitary wave.

Time C1 C2 C3

0 −6.0606060 382.85217 −350.877814 −6.0605980 382.31084 −355.910718 −6.0606191 383.18845 −351.20101

12 −6.0606202 383.18121 −351.4950916 −6.0606288 383.16353 −351.5305020 −6.0722598 383.16563 −351.52559

D. The Evaluation of Maxwellian Pulse into Stable Solitary Waves

Again we follow [23] and as an important initial value problem for the RLW equation we inves-tigate the evolution of the Maxwellian pulse into stable solitary waves [23, 24, 32]. We simulatethe test for various values of the parameter µ. Note that the initial condition is

U(x, 0) = exp[−(x − 7)2], (6.9)

and the following boundary conditions are used:

U(a, t) = U(b, t) = 0. (6.10)

In this test, calculations are carried out with ε = 1 and µ = 0.04, 0.01, and 0.001 when finaltime is t = 9, 12, and 25, respectively. The spatial region is 0 ≤ x ≤ 60 for µ = 0.001 and0 ≤ x ≤ 30 for µ = 0.04, 0.01. For µ = 0.04 the Maxwellian pulse develops into a singlesolitary wave, plus a well developed oscillating tail as graphed in Fig. 5. When µ = 0.01 and0.001, the Maxwellian pulse develops into three and nine solitary waves as demonstrated in Figs.6 and 7, respectively. One can observe from Figs. 5–7 that the numerical results calculated withthe PC-TPS-RBF method are consistent with Gardner and Gardner [69], Soliman and Hussien[32] and Lin et al. [23]. It thus seems that the initial Maxwellian pulse breaks up into a trainof solitary waves, the number depending on the value of µ. In Tables VI–VIII, we demonstratethe three invariants at various time steps. It is worth to point out that the conservation laws arechanged in the ranges which are acceptable.

FIG. 5. (Left) plot of the estimated solution in different times and (right) wave profile at time t = 9 forMaxwellian pulse into stable solitary waves for µ = 0.04. [Color figure can be viewed in the online issue,which is available at www.interscience.wiley.com.]





E. The Development of an Undular Bore

As the last experiment in this section we investigate the development of an undular bore whichhas been proposed by [23, 27] and employ the initial condition

U(x, 0) = 0.5U0

[1 − tanh

(x − xc

d

)], (6.11)

TABLE VI. Invariants for evaluation of Maxwellian pulse into stable solitary waves when µ = 0.04.

Time C1 C2 C3

0 1.7724538 1.3034453 4.78326912 1.7724498 1.3034089 4.78337564 1.7724465 1.3034079 4.78337826 1.7724449 1.3034087 4.78337668 1.7724440 1.3034089 4.78337599 1.7724437 1.3034089 4.7833758



TABLE VII. Invariants for evaluation of Maxwellian pulse into stable solitary waves when µ = 0.01.

Time C1 C2 C3

0 1.7724538 1.2658469 4.78326912 1.7724470 1.2658612 4.78655304 1.7724438 1.2659215 4.79051246 1.7724421 1.2659167 4.79096308 1.7724412 1.2658969 4.7908273

10 1.7724407 1.2659222 4.791117512 1.7724396 1.2659295 4.7912048

TABLE VIII. Invariants for evaluation of Maxwellian pulse into stable solitary waves when µ = 0.001.

Time C1 C2 C3

0 1.7724538 1.2545674 4.78326914 1.7724711 1.2748805 4.96396208 1.7728050 1.2682085 4.9053199

12 1.7727235 1.2677298 4.900466016 1.7726554 1.2745380 4.959734120 1.7725723 1.2816287 5.021860125 1.7723998 1.2721862 4.9376577

FIG. 8. Initial and undulation profiles with d = 5 at various times.



FIG. 9. Initial and undulation profiles with d = 2 at various times.

and boundary conditions

U(a, t) = U0, U(b, t) = 0, (6.12)

where magnitude U(x, 0) gives the amount of the change in the water level. The change in waterlevel of magnitude U0 is centered on x = xc, and d represents the slope between the still waterand deeper water. In order to make comparison with earlier investigations, we take the parametersto have the following values: ε = 1.5, µ = 1/6, U0 = 0.1 and xc = 0. We set the experimentalregion of −60 ≤ x ≤ 300 together with δx = 0.5, δt = 0.1 and run the simulation up to finaltime t = 250.

As the simulation proceeds undulations begin to develop and grow, moving back along theprofile as the leading edge moves to the right. Initial and undulation profiles with d = 5 andd = 2 at various times are drawn in Figs. 8 and 9 respectively. The computational results areconsistent with the results produced by [23, 27]. To see the effect of the initial undulation, viewof the formation of the leading undulation for both cases the gentle slope and the steep slope aredemonstrated (see Fig. 10 in which the maximum u is drawn against the time t). It worth to pointout that the results obtained confirm the observations of the earlier researchers.



FIG. 10. The first undulation: (a) gentle slope d = 5 and (b) steep slope d = 2. [Color figure can be viewedin the online issue, which is available at www.interscience.wiley.com.]

VII. CONCLUSION

In this article, we presented a numerical simulation of the RLW equation using collocation andapproximating the solution by radial basis functions (RBFs). To avoid solving the nonlinearsystem, a predictor-corrector scheme is proposed. The method is tested on the problems of prop-agation of a solitary wave, interaction of two positive solitary waves, interaction of a positive anda negative solitary wave, the evaluation of Maxwellian pulse into stable solitary waves and thedevelopment of an undular bore. In the simulation of the propagation of a solitary wave, highaccuracy was achieved by comparing the numerical solutions with the exact ones in terms of theL2 and L∞ error norms, and comparison was made with recently methods collected in [23, 24].Other experiments are modeled well and the results obtained agree well with earlier works. Thethree invariants of motion are satisfactorily constant in all the simulations described here, so thealgorithm can fairly be described as conservation.

The authors are very thankful to the three referees of this paper.

References

1. M. Dehghan, Finite difference procedures for solving a problem arising in modeling and design ofcertain optoelectronic devices, Math Comput Simul 71 (2006), 16–30.

2. M. Dehghan and M. Tatari, Determination of a control parameter in a one-dimensional parabolic equationusing the method of radial basis functions, Math Compu Model 44 (2006), 1160–1168.

3. M. Dehghan and M. Tatari, The radial basis functions method for identifying an unknown parameter in aparabolic equation with overspecified data, Numer Methods Partial Differential Eq 23 (2007), 984–997.

4. M. Tatari and M. Dehghan, On the solution of the non-local parabolic partial differential equations viaradial basis functions, Appl Math Model 33 (2009), 1729–1738.

5. M. Dehghan and A. Shokri, A numerical method for solution of the two-dimensional sine-Gordonequation using the radial basis functions, Math Comput Simul 79 (2008), 700–715.



6. M. Dehghan and D. Mirzaei, Numerical solution to the unsteady two-dimensional Schrodinger equationusing meshless local boundary integral equation method, Int J Numer Methods Eng 76 (2008), 501–520.

7. M. Dehghan and D. Mirzaei, Meshless Local Petrov-Galerkin (MLPG) method for the unsteady mag-netohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity, App Numer Math 59(2009), 1043–1058.

8. E. J. Kansa, Multiquadrics—A scattered data approximation scheme with applications to computationalfluid dynamics. I., Comput Math Appl 19 (1990), 127–145.

9. E. J. Kansa, Multiquadrics—A scattered data approximation scheme with applications to computationalfluid dynamics. II., Comput Math Appl 19 (1990), 147–161.

10. Y. Zhang, Solving partial differential equations by meshless methods using radial basis functions, ApplMath Comput 185 (2007), 614–627.

11. M. Dehghan and A. Shokri, A numerical method for KdV equation using collocation and radial basisfunctions, Nonlinear Dyn 50 (2007), 111–120.

12. Y. C. Hon and X. Z. Mao, An efficient numerical scheme for Burgers’ equation, Appl Math Comput 95(1998), 37–50.

13. J. Li, Y. C. Hon, and C. S. Chen, Numerical comparisons of two meshless methods using radial basisfunctions, Eng Anal Bound Elem 26 (2002), 205–225.

14. J. Li, Y. Chen, and D. Pepper, Radial basis function method for 1-D and 2-D groundwater contaminanttransport modelling, Comp Mech 32 (2003), 10–15.

15. Y. C. Hon and Z. Wu, Additive Schwarz domain decomposition with radial basis approximation, Int JAppl Math 4 (2000), 599–606.

16. S. N. Atluri and T. Zhu, New concepts in meshless methods, Int J Numer Methods Eng 47 (2000),537–556.

17. C. Franke and R. Schaback, Solving partial differential equations by collocation using radial basisfunctions, Appl Math Comput 93 (1998), 73–82.

18. E. J. Kansa and Y. C. Hon, Circumventing the ill-conditioning problem with multiquadric radialbasis functions: application to elliptic partial differential equations, Comput Math Appl 39 (2000),123–137.

19. M. Zerroukat, H. Power, and C. S. Chen, A numerical method for heat transfer problem using collocationand radial basis functions, Int J Numer Methods Eng 42 (1998), 1263–1278.

20. W. K. Liu, Y. Chen, S. Jun, J. S. Chen, T. Belytschko, C. Pan, R. A. Uras, and C. T. Chang, Overview andapplications of the reproducing kernel particle methods, Arch Comput Methods Eng 3 (1996), 3–80.

21. Y. Zhang and Y. Tan, Convergence of meshless Petrov–Galerkin method using radial basis functions,Appl Math Comput 183 (2006), 307–321.

22. Y. Zhang and Y. Tan, Solve partial differential equations by meshless subdomains method combinedwith RBFs, Appl Math Comput 174 (2006), 700–709.

23. J. Lin, Z. Xie, and J. Zhou, High-order compact difference scheme for the regularized long wave equation,Commun Numer Meth Eng 23 (2007), 135–156.

24. B. Saka and I. Dag, A numerical solution of the RLW equation by Galerkin method using quarticB-splines, Commun Numer Methods Eng 24 (2008), 1339–1361.

25. D. H. Peregrine, Calculations of the development of an undular bore, J Fluid Mech 25 (1966), 321–330.

26. T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equations for long waves in non-linear dispersivesystems, Phil Trans R Soc London, Series A 272 (1972), 47–78.

27. I. Dag, B. Saka, and D. Irk, Application of cubic B-splines for numerical solution of the RLW equation,Appl Math Comput 159 (2004), 373–389.



28. J. C. Eilbeck and G. R. McGuire, Numerical study of RLW equation I: numerical methods, J ComputPhys 19 (1975), 43–57.

29. J. C. Eilbeck and G. R. McGuire, Numerical study of the regularized long-wave equation II: Interactionof solitary waves, J Comput Phys 23 (1977), 63–73.

30. J. C. Lewis and J. A. Tjon, Resonant production of solution in the RLW equation, Phys Lett 73A (1979),275–279.

31. D. Bhardwaj and R. Shankar, A computational method for regularized long wave equation, ComputMath Appl 40 (2000), 1397–1404.

32. A. A. Soliman and M. H. Hussien, Collocation solution for RLW equation with septic spline, Appl MathComput 161 (2005), 623–636.

33. S. Kutluay and A. Esen, A finite difference solution of the regularized long wave equation, Math ProblemsEng 2006 (2006), 1–14.

34. I. Dag, Least squares quadratic B-spline finite element method for the regularized long wave equation,Comput Methods Appl Mech Eng 182 (2000), 205–215.

35. B. Guo and W. Cao, The Fourier pseudospectral method with a restrain operator for the RLW equation,J Comput Phys 74 (1988), 110–126.

36. I. Dag, B. Saka, and D. Irk, Galerkin method for the numerical solution of the RLW equation usingquintic B-splines, J Comput Appl Math 190 (2006), 532–547.

37. I. Dag and M. N. Ozer, Approximation of the RLW equation by the least square cubic B-spline finiteelement method, Appl Math Model 25 (2001), 221–231.

38. L. R. T. Gardner, G. A. Gardner, F. A. Ayoub, and N. K. Amein, Modelling an undular bore withB-splines, Comput Methods Appl Mech Eng 147 (1997), 147–152.

39. L. R. T. Gardner, G. A. Gardner, and I. Dag, A B-spline finite element method for the regularized longwave equation, Commn Numer Methods Eng 11 (1995), 59–68.

40. I. Dag, A. Dogan, and B. Saka, B-spline collocation methods for numerical solutions of the RLWequation, Int J Comput Math 80 (2003), 743–757.

41. B. Saka, I. Dag, and A. Dogan, Galerkin method for the numerical solution of the RLW equation usingquadratic B-splines, Int J Comput Math 81 (2004), 727–739.

42. K. Djidjeli, W. G. Prince, E. H. Twizell, and Q. Cao, A linearized implicit pseudo-spectral method forsome model equations: the regularized long wave equation, Commun Numer Methods Eng 19 (2003),847–863.

43. P. Avilez–Valente and F. J. Seabra–Santos, A Petrov–Galerkin finite element scheme for the regularizedlong wave equation, Comput Mech 34 (2004), 256–270.

44. M. Ayadi and K. Omrani, A fully Galerkin method for the damped generalized regularized long-wave(DGRLW) equation, Numer Methods Partial Differential Eq, to appear.

45. H. Gu and N. Chen, Least-squares mixed finite element methods for the RLW equations, Numer MethodsPartial Differential Eq 24 (2008), 749–758.

46. B. Saka and I. Dag, Quartic B-spline collocation algorithms for numerical solution of the RLW equation,Numer Methods Partial Differential Eq 23 (2007), 731–751.

47. P. C. Jain, R. Shankar, and T. V. Singh, Numerical solution of regularized long-wave equation, CommunNumer Methods Eng 9 (1993), 579–586.

48. A. Dogan, Numerical solution of regularized long wave equation using Petrov-Galerkin method,Commun Numer Methods Eng 17 (2001), 485–494.

49. L. R. T. Gardner, G. A. Gardner, and A. Dogan, A least-squares finite element scheme for the RLWequation, Commun Numer Methods Eng 12 (1996), 795–804.



50. J. L. Bona, W. R. McKinney, and J. M. Restrepo, Stable and unstable solitary-wave solutions of thegeneralized regularized long-wave equation, J Nonlinear Sci 10 (2000), 603–638.

51. L. Guo and H. Chen, H 1-Galerkin mixed finite element method for the regularized long wave equation,Computing 77 (2006), 205–221.

52. B. Saka and I. Dag, A collocation method for the numerical solution of the RLW equation using cubicB-spline basis, Arab J Sci Eng 30 (2005), 39–50.

53. K. R. Raslan, A computational method for the regularized long wave (RLW) equation, Appl MathComput 167 (2005), 1101–1118.

54. R. J. Ramos, Solitary waves of the EW and RLW equations, Chaos Solitons Fractals 34 (2007),1498–1518.

55. M. Dehghan and F. Shakeri, Use of He’s homotpy perturbation method for solving a partial differentialequation arising in modeling of flow in porous media, J Porous Media 11 (2008), 765–778.

56. M. Dehghan and D. Mirzaei, A numerical method based on the boundary integral equation and dualreciprocity methods for one-dimensional Cahn–Hilliard equation, Eng Anal Boundary Elem 33 (2009),522–528.

57. D. Kaya, A numerical simulation of solitary-wave solutions of the generalized regularized long-waveequation, Appl Math Comput 149 (2004), 833–841.

58. H. Jafari, A. Borhanifar, and S. A. Karimi, New solitary wave solutions for generalized regularizedlong-wave equation, Int J Comput Math, to appear.

59. A. Rashid, A three-levels finite difference method for nonlinear regularized long-wave equation, MemDiffer Equ Math Phys 34 (2005), 135–146.

60. A. Rashid, Numerical solution of Korteweg–de Vries equation by the Fourier pseudospectral method,Bull Belgian Math Soc 14 (2009), 709–721.

61. SI. Zaki, Solitary waves of the splitted RLW equation, Comput Phys Commun 138 (2001), 80–91.

62. M. A. Helal, Soliton solution of some nonlinear partial differential equations and its application in fluidmechanics, Chaos Solitons Fractals 13 (2002), 1917–1929.

63. M. Dehghan, Identification of a time-dependent coefficient in a partial differential equation subject toan extra measurement, Numer Methods Partial Differential Eq 21 (2005), 611–622.

64. M. Dehghan, Parameter determination in a partial differential equation from the overspecified data,Math Compu Model 41 (2005), 196–213.

65. M. Dehghan, On the solution of an initial-boundary value problem that combines Neumann and integralcondition for the wave equation, Numer Methods Partial Differential Eq 21 (2005), 24–40.

66. M. Dehghan, A computational study of the one-dimensional parabolic equation subject to nonclassicalboundary specifications, Numer Methods Partial Differential Eq 22 (2006), 220–257.

67. P. J. Olver, Euler operators and conservation laws of the BBM equation, Math Proc Cambridge Phil Soc85 (1979), 143–159.

68. A. R. Santarelli, Numerical analysis of the regularized long wave equation, Nuovo Cimento 46B (1978),179–188.

69. L. R. T. Gardner and G. A. Gardner, Solitary wave of the regularised long wave equation, J ComputPhys 91 (1990), 441–459.


A Meshless Method Using the Radial Basis Functions for Numerical Solution of the Regularized Long...

Documents

Transcript of A Meshless Method Using the Radial Basis Functions for Numerical Solution of the Regularized Long...