# A finite difference solution of the regularized long-wave equation

**ABSTRACT** A linearized implicit finite difference method to obtain numerical solution of the one-dimensional regularized long-wave (RLW) equation is presented. The performance and the accuracy of the method are illustrated by solving three test examples of the problem: a single solitary wave, two positive solitary waves interaction, and an undular bore. The obtained results are presented and compared with earlier work.

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**ABSTRACT:**Geometric integrators are presented for a class of nonlinear dispersive equations which includes the Camassa–Holm equation, the BBM equation and the hyperelastic-rod wave equation. One group of schemes is designed to preserve a global property of the equations: the conservation of energy; while the other one preserves a more local feature of the equations: the multi-symplecticity.Journal of Computational and Applied Mathematics. 01/2011; - [show abstract] [hide abstract]

**ABSTRACT:**The regularized long-wave equation has been solved numerically using the collocation method based on the Adams-Moulton method for the time integration and quintic B-spline functions for the space integration. The method is tested on the problems of propagation of a solitary wave and interaction of two solitary waves. The three conserved quantities of motion are calculated to determine the conservation properties of the proposed algorithm. The L ∞ error norm is used to measure the difference between exact and numerical solutions. A comparison with the previously published numerical methods is performed.Physics of Wave Phenomena 20(3). · 0.19 Impact Factor - SourceAvailable from: link.springer.com[show abstract] [hide abstract]

**ABSTRACT:**In this work, our main purpose is to develop of a sufficiently robust, accurate and efficient numerical scheme for the solution of the regularized long wave (RLW) equation, an important partial differential equation with quadratic nonlinearity, describing a large number of physical phenomena. The crucial idea is based on the discretization of the RLW equation with the aid of a combination of the discontinuous Galerkin method for the space semi-discretization and the backward difference formula for the time discretization. Furthermore, a suitable linearization preserves a linear algebraic problem at each time level. We present error analysis of the proposed scheme for the case of nonsymmetric discretization of the dispersive term. The appended numerical experiments confirm theoretical results and investigate the conservative properties of the RLW equation related to mass, momentum and energy. Both procedures illustrate the potency of the scheme consequently. PACS Codes: 02.70.Dh, 02.60 Cb, 02.60.Lj, 03.65.Pm, 02.30.-f. MSC: 65M60, 65M15, 65M12, 65L06, 35Q53.Boundary Value Problems 2013(1). · 0.92 Impact Factor

Page 1

A FINITE DIFFERENCE SOLUTION OF THE REGULARIZED

LONG-WAVE EQUATION

S. KUTLUAY AND A. ESEN

Received 26 July 2005; Accepted 24 January 2006

A linearized implicit finite difference method to obtain numerical solution of the one-

dimensional regularized long-wave (RLW) equation is presented. The performance and

the accuracy of the method are illustrated by solving three test examples of the problem:

a single solitary wave, two positive solitary waves interaction, and an undular bore. The

obtained results are presented and compared with earlier work.

Copyright © 2006 S. Kutluay and A. Esen.Thisisanopenaccessarticledistributedunder

theCreativeCommonsAttributionLicense,whichpermitsunrestricteduse,distribution,

and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In this study, we will consider the one-dimensional RLW equation

∂U

∂t+∂U

∂x+εU∂U

∂x−μ∂

∂t

?∂2U

∂x2

?

= 0,(1.1)

with the physical boundary conditions U → 0 as x → ±∞, where t is time, x is the space

coordinate, U(x,t) is the wave amplitude, and ε and μ are positive parameters. The RLW

equation(1.1)wasfirstintroducedbyPeregrine[1]todescribethedevelopmentofanun-

dular bore. This equation is one of the most important nonlinear wave equations which

can be used to model a large number of problems arising in various areas of applied sci-

ences [2, 3]. The RLW equation has been solved analytically for a restricted set of bound-

ary and initial conditions. Therefore, the numerical solution of the RLW equation has

been the subject of many papers. Various numerical techniques particularly including fi-

nitedifference[4–8],finiteelement[9–19],andspectral[20–23]methodshavebeenused

for the solution of the RLW equation.

In this paper, we have used a linearized implicit finite difference method to investigate

the motion of a single solitary wave, development of two positive solitary waves interac-

tion, and an undular bore for the RLW equation (1.1).

Hindawi Publishing Corporation

Mathematical Problems in Engineering

Volume 2006, Article ID 85743, Pages 1–14

DOI 10.1155/MPE/2006/85743

Page 2

2Regularized long-wave equation

2. Method ofsolution

For the numerical treatment, the spatial variable x of the problem is restricted over an in-

tervala ≤x ≤b.Inthisstudy,weconsidertheRLWequation(1.1)withthehomogeneous

boundary conditions

U(a,t) =0,

t >0,

U(b,t) =0,

t >0,(2.1)

and the initial condition

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

a ≤x ≤b,(2.2)

where f (x) is a prescribed function.

The solution domain a ≤ x ≤ b, t > 0 is divided into subintervals Δx in the direction

of the spatial variable x and Δt in the direction of time t such that xi= iΔx, i = 0(1)N

(NΔx = b −a); tj= jΔt, j = 0(1)J, and the numerical solution of U at the grid point

(iΔx, jΔt) is denoted by Ui,j.

In the finite difference method, the dependent variable and its derivatives are approx-

imated by the finite difference approximation. This approximation will lead to either a

single explicit equation or a system of difference equations. Applying the classical im-

plicit finite difference method to nonlinear problems normally gives nonlinear system of

equations which cannot be solved directly.

Equation (1.1) can be written as

∂U

∂t+∂U

∂x+ε

2

∂U2

∂x

−μ∂

∂t

?∂2U

∂x2

?

= 0.

(2.3)

Usingtheforwarddifferenceapproximationfor∂U/∂t,theCrank-Nicolsondifferenceap-

proximationfor∂U/∂x and∂U2/∂x,andthecentraldifferenceapproximationfor∂2U/∂x2

at the point (i, j +1),

∂U

∂t

∼=Ui,j+1−Ui,j

Δt

,

∂U

∂x

∂U2

∂x

∼=1

∼=1

2

?

?

1

2Δx

1

2Δx

∂2U

∂x2

?Ui+1,j+1−Ui−1,j+1

?U2

∼=

(Δx)2

?+

?+

1

2Δx

1

2Δx

?Ui+1,j−Ui−1,j

?U2

?,

??

??

,

2

i+1,j+1−U2

1

i−1,j+1

i+1,j−U2

i−1,j

,

?Ui+1,j−2Ui,j+Ui−1,j

(2.4)

respectively, (2.3) yields the system of algebraic equations

Ui,j+1−Ui,j

Δt

+

+

?U2

μ

1

4Δx

i+1,j+1−U2

?Ui+1,j+1−2Ui,j+1+Ui−1,j+1−Ui+1,j+2Ui,j−Ui−1,j

?Ui+1,j+1−Ui−1,j+1+Ui+1,j−Ui−1,j

i−1,j+1+U2

?

ε

8Δx

i+1,j−U2

i−1,j

?

−

Δt(Δx)2

?= 0

(2.5)

Page 3

S. Kutluay and A. Esen3

for i = 1(1)N −1 and j = 0(1)J with a truncation error of O(Δt)+O(Δx)2. The scheme

is a nonlinear system of equations in Ui,j+1and it needs to use an iteration technique to

evaluate the solution.

Using the central difference operator δ defined by δxUi,j= Ui+1,j−Ui−1,j, (2.5) can be

written as

Ui,j+1−Ui,j

Δt

+

+

?δx

μ

1

4Δx

?U2

?Ui+1,j+1−2Ui,j+1+Ui−1,j+1−Ui+1,j+2Ui,j−Ui−1,j

?Ui+1,j+1−Ui−1,j+1+Ui+1,j−Ui−1,j

i,j+1

i,j

?

ε

8Δx

?+δx

?U2

??

−

Δt(Δx)2

?= 0.

(2.6)

By Taylor expansion of U2

i,j+1about the point (i, j) we obtain

U2

i,j+1= U2

i,j+Δt∂U2

i,j

∂t

+··· =U2

i,j+Δt∂U2

i,j

∂Ui,j

∂Ui,j

∂t

+··· .

(2.7)

Hence in terms of order Δt, U2

i,j+1∼= U2

i,j+2Ui,j(Ui,j+1−Ui,j), and taking

Wi=Ui,j+1−Ui,j,(2.8)

(2.6), with some manipulations, leads to

?

ε

4ΔxUi−1,j+

μ

Δt(Δx)2+

μ

Δt(Δx)2−

?Ui+1,j−Ui−1,j

1

4Δx

1

4ΔxUi+1,j−

?+

?

Wi−1−

?1

?

i+1,j−U2

Δt+

2μ

Δt(Δx)2

?

Wi

+

?

1

1

4Δx

?U2

Wi+1

=

2Δx

ε

4Δx

i−1,j

?,

(2.9)

(i = 1(1)N −1) a system of linear equations for Wi. This approximation is second order

in both space and time as regards truncation error. Obviously, the solution at the (j +

1)th time level is obtained from (2.8) as Ui,j+1= Ui,j+Wi. Since the stability parameter

Δt/(Δx)2depends not only on the form of the finite difference scheme (2.9) but also

generallyuponthesolutionU(x,t)beingobtained,thecomplicationsanddifficultiesmay

arise in the analysis of stability. In order to show how good the numerical solutions are in

comparison with the exact ones, we will use the L2and L∞error norms defined by

L2=??Uexact−Unum??2=

L∞=??Uexact−Unum??∞=max

?

Δx

N?

i=1

??Uexact

??Uexact

i

−Unum

i

??2

?1/2

??.

,

i

i

−Unum

i

(2.10)

Page 4

4 Regularized long-wave equation

3. Numericalexamples and results

All computations were executed on a Pentium 4 PC in the Fortran code using double

precision arithmetic. The RLW equation (1.1) satisfies only three conservation laws given

as

I1=

?+∞

?2?

−∞Udx ? Δx

N?

??Ui,j

N?

i=1

Ui,j,

I2=

?+∞

−∞

?

U2+μ?Ux

?+∞

dx ?Δx

N?

i=1

?2+μ

?3+3?Ui,j

??Ux

?

?2?

i,j

?2?

,

I3=

−∞

?U3+3U2?dx ?Δx

i=1

??Ui,j

(3.1)

which respectively correspond to mass, momentum, and energy [24]. In the simulations

the invariants I1, I2, and I3are monitored to check the conservation of the numerical

scheme. For the computation of Uxin (3.1), we used a central finite difference approxi-

mation.

To implement the performance of the method, three test problems will be considered:

the motion of a single solitary wave, the interaction of two positive solitary waves, and

the undular bore.

3.1. The motion of a single solitary wave. We first consider (1.1) with the boundary

conditions U →0 as x → ±∞ and the initial condition

U(x,0) =3csech2?k?x−x0

The exact solution of this problem is

U(x,t) =3csech2?k?x−vt −x0

This solution corresponds to the motion of a single solitary wave with amplitude 3c

and width k, initially centered at x0, where v = 1+εc is the wave velocity and k = (1/

2)(εc/μv)1/2. This solution will also be used over an interval a ≤ x ≤ b. For this problem

the theoretical values of the invariants are [14]

??.

(3.2)

??.

(3.3)

I1=6c

k,

I2=12c2

k

+48kc2μ

5

,

I3=36c2

k

+144c3

5k

(3.4)

which are recorded throughout the simulations. For the purpose of comparing with the

earlier work, all computations are done for the parameters ε =1, μ = 1, and x0=0.

Table 3.1 displays a comparison of the values of the invariants and error norms ob-

tainedbythepresentmethodwiththoseobtainedusingthecubicfinitedifferencemethod

developedbyJainetal.[6]andimplementedbyGardneretal.[10]forc = 0.1.Asitisseen

from the table, the numerical values of invariants obtained from (3.1) are in very good

agreement with their analytical values obtained from (3.4). The quantities in the invari-

ants remain almost constant during the computer run. For the proposed finite difference

Page 5

S. Kutluay and A. Esen5

Table 3.1. Invariants and error norms for the single soliton with c =0.1, Δx =0.1, Δt =0.1, and over

the region −40 ≤x ≤60.

tI1

I2

I3

L2×103

L∞×103

Present method

0

4

8

12 3.97999

16 3.97999

20 3.97997

3.97992

3.97995

3.97997

0.810459

0.810459

0.810459

0.810459

0.810459

0.810459

2.57901

2.57901

2.57901

2.57901

2.57901

2.57901

0.00

0.12

0.23

0.34

0.45

0.55

0.00

0.05

0.09

0.14

0.18

0.21

Finite difference cubic method [6, 10]

0

4

8

12 4.41623

16 4.41423

20 4.41219

3.97992

4.42017

4.41822

0.810459

0.899873

0.899236

0.898601

0.897967

0.897342

2.57901

2.86339

2.86106

2.85863

2.85613

2.85361

0.00

39.82

79.46

118.8

157.7

196.1

0.00

13.74

27.66

41.35

54.60

67.35

method at times t = 0 and t = 20, change in I1is 0.5×10−4, and I2and I3are exact up to

the last recorded digit, whereas for the cubic finite difference method, they are 0.43227,

0.086883, and 0.2746, respectively. The error norms at each time obtained by the present

methodaresmallerthanthosegivenin[6,10].Forthepresentmethodatt = 20,theerror

normsareL2=0.55×10−3andL∞= 0.21×10−3,whereastheyareL2= 196.1×10−3and

L∞= 67.35×10−3for the cubic finite difference method. In Table 3.2 the time evolution

of the invariants I1, I2, and I3, and of the error norms L2and L∞for c =0.03, is compared

with the cubic finite difference method [6, 10]. Again the present method produces good

results.

The rates of convergence for the proposed numerical method in space sizes Δxmand

time steps Δtmcan be calculated by

Order =log10

???Uexact−Unum

???Uexact−Unum

Δxm

?Δxm/Δxm+1

Δtm

?Δtm/Δtm+1

??/??Uexact−Unum

??/??Uexact−Unum

Δxm+1

???

???

log10

?

,

Order =log10

Δtm+1

log10

?

,

(3.5)

respectively [18].

The convergence rates computed by the present method for values of space size Δxm

and a fixed value of the time step Δt are recorded in Table 3.3. It is clearly seen that the

scheme provides remarkable reductions in convergence rates for the smaller space sizes.

Page 6

6 Regularized long-wave equation

Table3.2. Invariantsanderrornormsforthesinglesolitonwithc =0.03,Δx =0.1,Δt =0.1,andover

the region −40 ≤x ≤60.

tI1

I2

I3

L2×103

L∞×103

Present method

0

4

8

12

16

20

2.107

2.108

2.109

2.110

2.110

2.109

0.127301

0.127302

0.127302

0.127302

0.127302

0.127302

0.388804

0.388806

0.388807

0.388807

0.388808

0.388807

0.000

0.150

0.321

0.467

0.567

0.638

0.000

0.123

0.166

0.179

0.185

0.233

Finite difference cubic method [6, 10]

0

4

8

12

16

20

2.107

2.340

2.339

2.337

2.336

2.333

0.127301

0.141322

0.141195

0.141067

0.140940

0.140815

0.388804

0.431621

0.431231

0.430834

0.430440

0.430052

0.000

2.928

5.816

8.698

11.58

14.45

0.000

0.786

1.582

2.384

3.190

3.996

Table 3.3. The order of convergence at t = 20, Δt = 0.1, c = 0.1 (−40 ≤ x ≤ 60), and c = 0.03 (−80 ≤

x ≤120).

L2×103

0.1133.66668

0.5 8.7678861.941021

0.25 2.358203 1.894541

0.1250.7446911.662974

0.0250.229367 0.731713

0.01250.213601 0.102739

c

Δxj

Order

L∞×103

12.74833

3.381133

0.910513

0.286720

0.086429

0.080163

Order

——

1.914730

1.892755

1.667037

0.745094

0.108579

0.031 2.620662

0.667923

0.177379

0.054606

0.015359

0.014146

—0.794513

0.202298

0.053656

0.016471

0.004569

0.004198

—

0.5

0.25

0.125

0.025

0.0125

1.972178

1.912847

1.699704

0.788127

0.118690

1.973589

1.914671

1.703811

0.796742

0.122176

Table 3.4 displays the computed convergence rates for various values of time step Δtjand

a fixed value of the space size Δx. Again a noticeable decrease in convergence rates is

observed when the time step decreases.

Page 7

S. Kutluay and A. Esen7

Table 3.4. The order of convergence at t = 20, Δx = 0.1, c = 0.1 (−40 ≤ x ≤ 60), and c = 0.03 (−80 ≤

x ≤120).

L2×103

0.1120.292460

0.5 5.4615491.893562

0.25 1.6314321.743171

0.1250.6667241.290977

0.0250.358400 0.385679

0.01250.348799 0.039175

c

Δtj

Order

L∞×103

7.618319

2.062621

6.197000

0.255439

0.138568

0.134914

Order

——

1.884994

1.734837

1.278591

0.380022

0.038554

0.031 1.380075

0.367151

0.111587

0.047558

0.027068

0.026428

—0.411315

0.109469

0.033363

0.014296

0.008194

0.008003

—

0.5

0.25

0.125

0.025

0.0125

1.910301

1.718205

1.230409

0.350183

0.034521

1.909721

1.714201

1.222637

0.345821

0.034027

Theprofilesofthesolitarywavesattimest =0andt =20andtheerrordistributionsof

the analytical and numerical solutions at t = 20 for c = 0.1 with the range −40 ≤ x ≤ 60

and for c = 0.03 with the range −80 ≤ x ≤ 120, Δx = 0.125 and Δt = 0.1 are shown in

Figure 3.1.Forc = 0.1,theamplitudeis0.3attimet =0whileitis0.299919attimet = 20

(Figure 3.1(a)) and so the relative change in the amplitude is about 0.027%. It is seen that

the maximum error is about between −4 ×10−3and 4 ×10−3(Figure 3.1(b)). For c =

0.03, the amplitude is 0.09 at time t =0 while it is 0.089997 at time t = 20 (Figure 3.1(c))

and so the relative change in the amplitude is about 0.0033%. It is observed that the

maximum error is about between −6×10−4and 6×10−4(Figure 3.1(d)).

3.2.Theinteractionoftwopositivesolitarywaves. We secondly consider (1.1) with the

boundary conditions U → 0 as x → ±∞ and the initial condition [17]

U(x,0) =

2?

j=1

3Ajsech2?kj

?x−xj

??, (3.6)

where Aj=4k2

For the simulation, all computations are done for the parameters k1= 0.4, x1= 15,

k2= 0.3, x2= 35, ε = 1, μ = 1, Δx = 0.3, and Δt = 0.1 over the region 0 ≤ x ≤ 120. The

experiment was run from t = 0 to t = 25 to allow the interaction to take place. Figure 3.2

shows the interaction of two positive solitary waves. As it is seen from the figure, at t = 0

a solitary wave with larger amplitude is on the left of the other solitary wave with smaller

amplitude.Thelargerwavecatchesupwiththesmalleroneasthetimeincreases.Att =0,

j/(1−4k2

j) (j = 1,2).

Page 8

8Regularized long-wave equation

−40

−20 02040 60

x

0

0.1

0.2

0.3

U(x,t)

t = 0

t = 20

(a)

−40

−20020 4060

x

−6

−4

−2

0

2

4

6

×10−3

Erorr

(b)

−80

−400 4080 120

x

0

0.3

0.6

0.9

×10−1

U(x,t)

t = 0

t = 20

(c)

−80

−400 4080 120

x

−8

−6

−4

−2

0

2

4

6

8

×10−4

Erorr

(d)

Figure 3.1. Solitary wave profiles at t = 0,20 and error (error = exact-numerical) distributions at

t =20.

theamplitudeofthelargersolitarywaveis5.33338whiletheamplitudeofthesmallerone

is 1.68598, whereas at t = 25, the amplitude of the larger solitary wave is 5.30235 at the

point x = 86.7 while the amplitude of the smaller one is 1.67157 at the point x =69.9. An

oscillation of small amplitude trailing behind the solitary waves was observed. In order

to see this oscillation occurring behind the waves in Figure 3.2 at time t =25, the scale of

the figure is magnified as in Figure 3.3. It is clearly seen that an oscillation of amplitude

∼2.2×10−2is trailing behind the solitary waves.

Table 3.5 displays a comparison of the values of the invariants obtained by the present

method with those obtained in [17]. It is observed that the obtained values of the in-

variants remain almost constant during the computer run. At times t = 0 and t = 25, the

relative changes in the invariants I1, I2, and I3for the present method are respectively

2.558 ×10−3%, 6.647 ×10−3%, and 9.797 ×10−3% whereas they are 0.352%, 0.570%,

and 2.237% for the cubic B-spline collocation finite element method given in [17]. It is

clearly seen that each of the conserved quantities obtained by the present method is very

well preserved.

Page 9

S. Kutluay and A. Esen9

020406080 100 120

x

0

1

2

3

4

5

6

U(x,t)

t = 0

(a)

0 20406080 100 120

x

−1

0

1

2

3

4

5

6

U(x,t)

t = 5

(b)

020406080 100120

x

−1

0

1

2

3

4

5

6

U(x,t)

t = 10

(c)

020 406080 100 120

x

−1

0

1

2

3

4

5

6

U(x,t)

t = 15

(d)

0 204060 80100120

x

−1

0

1

2

3

4

5

6

U(x,t)

t = 20

(e)

020 406080 100 120

x

−1

0

1

2

3

4

5

6

U(x,t)

t = 25

(f)

Figure 3.2. The interaction of two positive solitary waves at different times.

3.3. The undular bore. As our last test problem, we consider (1.1) with the physical

boundary conditions U → 0 as x → ∞ and U →U0as x → −∞, and the initial condition

?

U(x,0) =U0

2

1−tanh

?x−x0

d

??

,(3.7)

Page 10

10Regularized long-wave equation

0204060 80100 120

x

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

×10−1

U(x,t)

Figure 3.3. The interaction of two solitary waves at t =25 in Figure 3.2 (magnified).

where U(x,0), denotes the elevation of the water surface above the equilibrium level at

time t = 0, U0represents the magnitude of the change in water level which is centered on

x = x0, and d measures the steepness of the change. Under the above physical boundary

conditions, the invariants I1, I2, I3are not constant but increase linearly throughout the

simulation at the following rates [14]:

M1=d

dtI1=d

?+∞

?+∞

dt

?+∞

−∞Udx =U0+1

U2+μ?Ux

?U3+3U2?dx = 3U2

2U2

0,

M2=d

dtI2=d

dt

−∞

?

?2?

dx = U2

0+2

3U3

0,

M3=d

dtI3=d

dt

−∞

0+3U3

0+3

4U4

0,

(3.8)

respectively.

For the simulation, all computations are done for the parameters ε = 1.5, μ = 1/6,

U0= 0.1, x0= 0, Δx = 0.24, Δt = 0.1, and d = 2,5 in the region −36 ≤ x ≤ 300. The

simulation is run until time t = 250, and the values of the quantities I1, I2, I3with the

position and amplitude of the leading undulation for the steep slope d = 2 and the gen-

tle slope d = 5 are recorded in Table 3.6. The numerical values of variations in quanti-

ties I1, I2, I3are obtained as M1= 0.107500, M2= 0.010992, M3= 0.034096 for d = 2

and M1= 0.107500, M2= 0.010992, M3= 0.034101 for d = 5 which are in good agree-

ment with the theoretical values M1= 0.105000, M2= 0.010667, M3= 0.033075 ob-

tained from (3.8). The values of I1, I2, and I3increase linearly according to the values

of M1, M2, and M3, respectively. The amplitudes of the leading undulation for d = 5 and

d =2 are 0.17710 and 0.18158, respectively.

Page 11

S. Kutluay and A. Esen11

Table 3.5. Invariants for the interaction of two positive solitary waves.

tI1

I2

I3

I1[17]

I2[17]

I3[17]

037.91648120.35150744.0814037.91652120.52280744.08150

237.91682120.35710744.0387037.91596119.17830725.54580

437.91697120.35840744.0110037.91170121.16020736.94430

537.91704120.35860743.99850———

637.91709120.35830743.9796037.89662 118.12660714.05840

837.91719120.35700743.8679037.85975119.73170 728.51730

1037.91727 120.36380743.42020 37.79221119.73430726.68790

1237.91733 120.39150742.33870 37.69667 119.63340725.72360

1437.91736120.41560741.5781037.59553 119.23590724.70020

1537.91738 120.40600741.89150———

1637.91740120.38860 742.4889037.52916 119.41850725.83990

1837.91741 120.36530743.4752037.54027119.82760727.08860

2037.91744 120.35990743.8638037.64730 119.80410727.19480

2237.91745120.35940743.9750037.82237 119.79820727.25420

24 37.91746 120.35950744.00370 37.99313119.89230727.49210

25 37.91745120.35950 744.0085038.05010 119.83550727.43920

Table 3.6. Invariants, position, and amplitude of the leading undulation for d =2,5.

dtI1

I2

I3

x

Amplitude

20 3.588000.350811.08078——

50 8.96300 0.899052.7858448.96000 0.13940

10014.33799 1.44901 4.49069 102.48000 0.15831

15019.71301 1.998966.19543156.72000 0.17013

20025.08799 2.548927.90013211.200000.17713

25030.46299 3.09887 9.60482265.68000 0.18158

50 3.588000.33565 1.03353——

50 8.96300 0.88391 2.7390248.24000 0.11067

10014.33801 1.43389 4.44424102.240000.13683

15019.71300 1.983856.14918 156.240000.15714

20025.08802 2.53381 7.85395 210.480000.16990

25030.46305 3.08376 9.55868 264.960000.17710

Page 12

12 Regularized long-wave equation

−36 050 100150 200250 300

x

0

0.5

1

1.5

2

U(x,t)

×10−1

d = 5,t = 100

(a)

−36 050100 150 200 250300

x

0

0.5

1

1.5

2

U(x,t)

×10−1

d = 5,t = 250

(b)

−36 050 100150 200 250300

x

0

0.5

1

1.5

2

U(x,t)

×10−1

d = 2,t = 100

(c)

−36 050 100150 200250 300

x

0

0.5

1

1.5

2

U(x,t)

×10−1

d = 2,t = 250

(d)

Figure 3.4. Undulation profiles for the gentle slope d = 5 and steep slope d =2 at t =100 and t =250.

Figure 3.4 illustrates the undular bore profiles at t = 100 and t = 250 for the gentle

slope d = 5 and the steep slope d = 2. As it can be seen that from the figure, the number

of undulations formed increases with the decrease of d from d = 5 to d = 2. The number

of undulations also increases with the increase of t, as expected.

4. Conclusion

A linearized implicit finite difference method was presented to obtain numerical solu-

tions of the RLW equation. The efficiency of the method was tested on three numerical

experiments of wave propagation: the motion of a single solitary wave, the development

of two positive solitary waves interaction, and an undular bore, and its accuracy was ex-

amined by the error norms L2and L∞. The obtained results show that the error norms

are reasonably small and the conservation properties are all very good. The results also

suggest that the present method whose application is easier than many other numerical

techniques such as finite element and spectral methods can be applied to a large number

of physically important nonlinear wave problems with success.

Page 13

S. Kutluay and A. Esen13

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S. Kutluay: Department of Mathematics, Faculty of Arts and Science, Inonu University,

44280 Malatya, Turkey

E-mail address: skutluay@inonu.edu.tr

A. Esen: Department of Mathematics, Faculty of Arts and Science, Inonu University,

44280 Malatya, Turkey

E-mail address: aesen@inonu.edu.tr

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