Research ArticleEfficiency of High-Order Accurate Difference Schemes forthe Korteweg-de Vries Equation

Kanyuta Poochinapan1 Ben Wongsaijai1 and Thongchai Disyadej2

1Department of Mathematics Faculty of Science Chiang Mai University Chiang Mai 50200 Thailand2Electricity Generating Authority of Thailand Phitsanulok 65000 Thailand

Correspondence should be addressed to Kanyuta Poochinapan kanyutahotmailcom

Received 5 August 2014 Accepted 2 November 2014 Published 8 December 2014

Academic Editor Igor Andrianov

Copyright copy 2014 Kanyuta Poochinapan et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Two numerical models to obtain the solution of the KdV equation are proposed Numerical tools compact fourth-order andstandard fourth-order finite difference techniques are applied to the KdV equationThe fundamental conservative properties of theequation are preserved by the finite difference methods Linear stability analysis of two methods is presented by the Von Neumannanalysis The new methods give second- and fourth-order accuracy in time and space respectively The numerical experimentsshow that the proposed methods improve the accuracy of the solution significantly

1 Introduction

Researchers in the past have worked onmathematical modelsexplaining the behavior of a nonlinear wave phenomenonwhich is one of the significant areas of applied researchDerived by Korteweg and de Vries [1] the Korteweg-de Vrieseqaution (KdV equation) is one of the mathematical modelswhich are used to study a nonlinear wave phenomenon TheKdV equation has been used in very wide applications suchas magnetic fluid waves ion sound waves and longitudinalastigmatic waves

The KdV equation has been solved numerically byvarious methods such as the collocation method [2ndash4] thefinite element method [5 6] the Galerkin method [7ndash10] thespectral method [11 12] and the finite differencemethod [13ndash18] To create a numerical tool the finite difference methodfor the KdV equation is developed until now Zhu [13] solvedthe KdV equation using the implicit difference method Thescheme is unconditionally linearly stable and has a truncationerror of order 119874(120591 + ℎ

2) Qu and Wang [14] developed the

alternating segment explicit-implicit (ASE-I) differencescheme consisting of four asymmetric difference schemes aclassical explicit scheme and an implicit scheme which isunconditionally linearly stable by the analysis of linearization

procedure Wang et al [15] have proposed an explicit finitedifference scheme for the KdV equationThe scheme is morestable than the Zabusky-Kruskal (Z-K) scheme [16] when itis used to simulate the collisions of multisolitonThe stabilityof the method in [15] was also discussed by using the frozencoefficient Von Neumann analysis method The time steplimitation of themethod in [15] is twice looser than that of theZ-K method Moreover Kolebaje and Oyewande [17]investigated the behavior of solitons generated from the KdVequation that depends on the nature of the initial conditionby using the Goda method [18] the Z-K method and theAdomian decomposition method

The stability accuracy and efficiency which are in con-flict with each other are the desired properties of the finitedifference scheme Implicit approximation is requested inorder to reach the stability of the finite difference scheme Ahigh-order accuracy in the spatial discretization is desired invarious problemsThe stencil becomes wider with increasingorder of accuracy for a high-order method of a conventionalscheme Furthermore using an implicit method results in thesolution of an algebraic system for equations with extensivebandwidth It is required to improve schemes that have abroad range of stability and high order of accuracy Addi-tionally this leads to the solution of the system for linear

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 862403 8 pageshttpdxdoiorg1011552014862403

2 Mathematical Problems in Engineering

equations with a pentadiagonal matrix that is the systemof linear equations arising from a standard second-orderdiscretization of a boundary value problem A method toconquer the conflict between stability accuracy and com-putational cost is the development of a high-order compactscheme

In recent decades many scientists concentrated upon thedifference method that makes a discrete analogue effective inthe fundamental conservation properties This causes us tocreate finite difference schemes which preserve the mass andenergy of solutions for the KdV equation In this paper twofourth-order difference schemes are constructed for the onedimensional KdV equation

119906119905+ 120572119906119909119909119909

+ 120574 (1199062)119909= 0 119909

119871lt 119909 lt 119909

119877 0 le 119905 le 119879 (1)

with an initial condition

119906 (119909 0) = 1199060(119909) 119909

119871le 119909 le 119909

119877 (2)

and boundary conditions

119906 (119909119871 119905) = 119906 (119909

119877 119905) = 0

119906119909(119909119871 119905) = 119906

119909(119909119877 119905) = 0

119906119909119909

(119909119871 119905) = 119906

119909119909(119909119877 119905) = 0

0 le 119905 le 119879

(3)

where 120572 and 120574 are any real numberWhen minus119909119871≫ 0 and 119909

119877≫

0 the initial-boundary value problem (1)ndash(3) is consistent sothe boundary condition (3) is reasonable By assumptions thesolitary wave solution and its derivatives have the followingasymptotic values 119906 rarr 0 as 119909 rarr plusmninfin and for 119899 ge 1120597119899119906120597119909119899

rarr 0 as 119909 rarr plusmninfin Moreover we obtain the solutionproperties as follows [19]

1198681= int

119909119877

119909119871

119906 (119909 119905) 119889119909

1198682= int

119909119877

119909119871

119906 (119909 119905)2119889119909

1198683= int

119909119877

119909119871

[2120574119906 (119909 119905)3minus 3120572 [119906 (119909 119905)

119909]2

]

(4)

The content of this paper is organized as follows In thenext section we create fourth-order finite difference schemesfor the KdV equation with the initial and boundary condi-tions The stability of finite difference schemes is discussedand the conservative approximations are also given Theresults on validation of finite difference schemes are pre-sented in Section 3 where we make a detailed comparisonwith available data to confirm and illustrate our theoreticalanalysis Finally we finish our paper by conclusions in the lastsection

2 Difference Schemes

We start the discussion of finite difference schemes bydefining a grid of points in the (119909 119905) plane For simplicity we

use a uniform grid for a discrete process with states identifiedby 119909119895= 119909119871+119895ℎwhich the grid size is ℎ = (119909

119877minus119909119871)119872 where

119872 is the number of grid pointsTherefore the grid will be thepoints (119909

119895 119905119899) = (119909

119871+ 119895ℎ 119899120591) for arbitrary integers 119895 and 119899

Here 120591 is a time increment (time step length) We write thenotation 119906

119899

119895for a value of a function 119906 at the grid point

(119909119871+ 119895ℎ 119899120591)In this paper we give a complete description of our finite

difference schemes and an algorithm for the formulationof the problem (1)ndash(3) We use the following notations forsimplicity

119906119899

119895=

119906119899+1

119895+ 119906119899minus1

119895

2 (119906

119899

119895)=

119906119899+1

119895minus 119906119899minus1

119895

2120591

(119906119899

119895)119909=

119906119899

119895minus 119906119899

119895minus1

ℎ (119906

119899

119895)119909=

119906119899

119895+1minus 119906119899

119895

ℎ

(119906119899

119895)119909=

119906119899

119895+1minus 119906119899

119895minus1

2ℎ (119906

119899

119895)119909=

119906119899

119895+2minus 119906119899

119895minus2

4ℎ

(119906119899 V119899) = ℎ

119872minus1

sum

119895=1

119906119899

119895V119899119895

10038171003817100381710038171199061198991003817100381710038171003817

2

= (119906119899 119906119899)

10038171003817100381710038171199061198991003817100381710038171003817infin = max

1le119895le119872minus1

10038161003816100381610038161003816119906119899

119895

10038161003816100381610038161003816

(5)

As introduced in the following subsections the tech-niques for determining the value of numerical solution to (1)are used

21 Compact Fourth-Order Finite Difference Scheme By set-ting119908 = minus120572119906

119909119909119909minus120574(1199062)119909 (1) can be written as119908 = 119906

119905 By the

Taylor expansion we obtain

119908119899

119895= (120597119905119906)119899

119895= (119906119899

119895)+ 119874 (120591

2) (6)

119908119899

119895= minus120572[(119906

119899

119895)119909119909119909

minusℎ2

4(1205975

119909119906)119899

119895]

minus 120574 [[(119906119899

119895)2

]119909

minusℎ2

6(1205973

1199091199062)119899

119895] + 119874 (ℎ

4)

(7)

From (6) we have

120572 (1205975

119909119906)119899

119895= minus120574 (120597

3

1199091199062)119899

119895minus (1205972

119909119908)119899

119895 (8)

Substituting (8) into (7) we get

119908119899

119895= minus120572 (119906

119899

119895)119909119909119909

minusℎ2

4(1205972

119909119908)119899

119895minus 120574 [(119906

119899

119895)2

]119909

minusℎ2

12120574 (1205973

1199091199062)119899

119895+ 119874 (ℎ

4)

(9)

Using second-order accuracy for approximation we obtain

(1205973

1199091199062)119899

119895= [(119906

119899

119895)2

]119909119909119909

+ 119874 (ℎ2)

(1205972

119909119908)119899

119895= (119908119899

119895)119909119909

+ 119874 (ℎ2)

(10)

Mathematical Problems in Engineering 3

The following method is the proposed compact finite differ-ence scheme to solve the problem (1)ndash(3)

(119906119899

119895)+

ℎ2

4(119906119899

119895)119909119909

+ 120572 (119906119899

119895)119909119909119909

+ 120574 [(119906119899

119895) (119906119899

119895)]119909

+120574ℎ2

12[(119906119899

119895) (119906119899

119895)]119909119909119909

= 0

(11)

where

1199060

119895= 1199060(119909119895) 0 le 119895 le 119872 (12)

Since the boundary conditions are homogeneous they give

119906119899

0= 119906119899

119872= 0 (119906

119899

0)119909= (119906119899

119872)119909= 0 1 le 119899 le 119873 (13)

At this time let 119890119899

119895= V119899119895minus 119906119899

119895where V119899

119895and 119906

119899

119895are the

solution of (1)ndash(3) and (11)ndash(13) respectivelyThen we obtainthe following error equation

119903119899

119895= (119890119899

119895)+

ℎ2

4(119890119899

119895)119909119909

+ 120572 (119890119899

119895)119909119909119909

+ 120574 [(V119899119895) (V119899119895)]119909

minus 120574 [(119906119899

119895) (119906119899

119895)]119909+

120574ℎ2

12[(V119899119895) (V119899119895)]119909119909119909

minus120574ℎ2

12[(119906119899

119895) (119906119899

119895)]119909119909119909

(14)

where 119903119899

119895denotes the truncation error By using the Taylor

expansion it is easy to see that 119903119899119895= 119874(120591

2+ℎ4) holds as 120591 ℎ rarr

0The Von Neumann stability analysis of (11) with 119906

119899

119895=

120585119899119890119894119896119895ℎ where 119894

2= minus1 and 119896 is a wave number gives the

following the amplification factor

1205852=

119860 minus 119894120591119861

119860 + 119894120591119861 (15)

where

119860 = 6ℎ3(cos (119896ℎ) + 1)

119861 = 12120572 (sin (2119896ℎ) minus 2 sin (119896ℎ))

+ 120574ℎ2(119906119899

119895) (sin (4119896ℎ) + 10 sin (2119896ℎ))

(16)

The amplification factor which is a complex number has itsmodulus equal to one therefore the compact finite differencescheme is unconditionally stable

Theorem 1 Suppose 119906(119909 119905) is smooth enough then the scheme(11)ndash(13) is conservative in a sense

119868119899

1=

ℎ

2

119872minus1

sum

119895=1

(119906119899+1

119895+ 119906119899

119895) = 119868119899minus1

1= sdot sdot sdot = 119868

0

1 (17)

under assumptions 1199061= 119906119872minus1

= 0

Proof By multiplying (11) by ℎ summing up for 119895 from 1 to119872minus1 and considering the boundary condition and assuming1199061= 119906119872minus1

= 0 we get

ℎ

2120591

119872minus1

sum

119895=1

(119906119899+1

119895minus 119906119899minus1

119895) = 0 (18)

Then this gives (17)

22 Standard Fourth-Order Finite Difference Scheme By thefact (1199062)

119909= (23)[119906119906

119909+(1199062)119909] and using an implicit finite dif-

ference method we propose a standard seven-point implicitdifference scheme for the problem (1)ndash(3)

(119906119899

119895)+ 120572(

3

2(119906119899

119895)119909119909119909

minus1

2(119906119899

119895)119909119909 119909

)

+ 2120574 [4

9((119906119899

119895119906119899

119895)119909+ 119906119899

119895(119906119899

119895)119909) minus

1

9((119906119899

119895119906119899

119895)119909+ 119906119899

119895(119906119899

119895)119909)]

= 0

(19)

where

1199060

119895= 1199060(119909119895) 0 le 119895 le 119872 (20)

Since the boundary conditions are homogeneous we obtain

119906119899

0= 119906119899

119872= 0 (21)

4 (119906119899

0)119909minus (119906119899

0)119909= 4 (119906

119899

119872)119909minus (119906119899

119872)119909= 0 (22)

minus (119906119899

minus1)119909119909

+ 14 (119906119899

0)119909119909

minus (119906119899

1)119909119909

= minus (119906119899

119872minus1)119909119909

14 (119906119899

119872)119909119909

minus (119906119899

119872+1)119909119909

= 0 1 le 119899 le 119873

(23)

119906 119906119909 and 119906

119909119909are required by the standard fourth-order

technique to be zero at the upstream and downstreambound-aries because the method utilizes a seven-point finite differ-ence scheme for the approximation of solution 119906 Throughthe analytical technique of contrasting (11) requires twohomogeneous boundary conditions only

Now let 119890119899119895= V119899119895minus 119906119899

119895where V119899

119895and 119906

119899

119895are the solution of

(1)ndash(3) and (19)ndash(22) respectively Then we obtain the fol-lowing error equation

(119890119899

119895)+ 120572

3

2(119890119899

119895)119909119909119909

minus 1205721

2(119890119899

119895)119909119909 119909

+8120574

9[((V119899119895V119899119895)119909+ V119899119895(V119899119895)119909) minus ((119906

119899

119895119906119899

119895)119909+ 119906119899

119895(119906119899

119895)119909)]

minus2120574

9[((V119899119895V119899119895)119909+ V119899119895(V119899119895)119909) minus ((119906

119899

119895119906119899

119895)119909+ 119906119899

119895(119906119899

119895)119909)] = 0

(24)

where 119903119899

119895denotes the truncation error By using the Taylor

expansion it is easy to see that 119903119899119895= 119874(120591

2+ℎ4) holds as 120591 ℎ rarr

0

4 Mathematical Problems in Engineering

The Von Neumann stability analysis of (19) with 119906119899

119895=

120585119899119890119894119896119895ℎ gives the following amplification factor

1205852=

36ℎ3minus 119894120591119860

36ℎ3 + 119894120591119860 (25)

where

119860 = 4120574ℎ2(119906119899

119895) (minus sin (4119896ℎ) + 7 sin (2119896ℎ) + 8 sin (119896ℎ))

+ 9120572 (minus sin (3119896ℎ) + 8 sin (2119896ℎ) minus 13 sin (119896ℎ))

(26)

The amplification factor which is a complex numberhas its modulus equal to one therefore the finite differencescheme is unconditionally stable

Theorem2 Suppose119906(119909 119905) is smooth enough then the scheme(11)ndash(13) is conservative in a sense

119868119899

1=

ℎ

2

119872minus1

sum

119895=1

(119906119899+1

119895+ 119906119899

119895)

+ 120591ℎ120574

119872minus1

sum

119895=1

[4

9119906119899

119895(119906119899+1

119895)119909minus

1

9119906119899

119895(119906119899+1

119895)119909]

= 119868119899minus1

1= sdot sdot sdot = 119868

0

1

(27)

under assumptions 1199061= 1199062= 119906119872minus2

= 119906119872minus1

= 0 Moreoverthe scheme (19)ndash(22) is conservative in a sense

119868119899

2=

1

2

10038171003817100381710038171199061198991003817100381710038171003817

2

+1

2

10038171003817100381710038171003817119906119899+110038171003817100381710038171003817

2

= 119868119899minus1

2= sdot sdot sdot = 119868

0

2 (28)

Proof By multiplying (11) by ℎ summing up for 119895 from 1 to119872minus1 and considering the boundary condition and assuming1199061= 1199062= 119906119872minus2

= 119906119872minus1

= 0 we have

120591ℎ

119872minus1

sum

119895=1

[8

9(119906119899

119895(119906119899

119895)119909) minus

2

9(119906119899

119895(119906119899+1

119895)119909)]

= 120591ℎ

119872minus1

sum

119895=1

[4

9(119906119899

119895(119906119899+1

119895)119909minus 119906119899minus1

119895(119906119899

119895)119909)

minus1

9(119906119899

119895(119906119899+1

119895)119909minus 119906119899minus1

119895(119906119899

119895)119909)]

(29)

As a result we have

ℎ

2

119872minus1

sum

119895=1

(119906119899+1

119895minus 119906119899minus1

119895)

+ 120574120591ℎ

119872minus1

sum

119895=1

[4

9(119906119899

119895(119906119899+1

119895)119909minus 119906119899minus1

119895(119906119899

119895)119909)

minus1

9(119906119899

119895(119906119899+1

119895)119909minus 119906119899minus1

119895(119906119899

119895)119909)] = 0

(30)

Then this gives (27) We then take an inner product between(19) and 2119906

119899 We obtain

1

2120591(10038171003817100381710038171003817119906119899+110038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899minus110038171003817100381710038171003817

2

) +3120572

2((119906119899)119909119909119909

(119906119899))

minus120572

2((119906119899)119909119909 119909

(119906119899)) + 2120574 (120593

119899(119906119899 119906119899) 119906119899) = 0

(31)

where

120593119899(119906119899

119895 119906119899

119895) =

4

9((119906119899

119895119906119899

119895)119909+ 119906119899

119895(119906119899

119895)119909)

minus1

9((119906119899

119895119906119899

119894)119909+ 119906119899

119895(119906119899

119895)119909)

(32)

by considering the boundary condition (13) According to

(119906119899

119909119909119909 119906119899) = 0

(119906119899

119909119909 119909 119906119899) = 0

(33)

indeed

(120593119899(119906119899 119906119899) 119906119899)

=4ℎ

9

119872minus1

sum

119895=1

[119906119899

119895(119906119899+1

119895)119909+ (119906119899

119895119906119899+1

119895)119909] 119906119899+1

119895

minusℎ

9

119872minus1

sum

119895=1

[119906119899

119895(119906119899+1

119895)119909+ (119906119899

119895119906119899+1

119895)119909] 119906119899+1

119895

=2

9

119872minus1

sum

119895=1

[(119906119899

119895119906119899+1

119895119906119899+1

119895+1minus 119906119899

119895minus1119906119899+1

119895minus1119906119899+1

119895)

+ (119906119899

119895+1119906119899+1

119895119906119899+1

119895+1minus 119906119899

119895119906119899+1

119895minus1119906119899+1

119895)]

minus1

36

119872minus1

sum

119895=1

[(119906119899

119895119906119899+1

119895119906119899+1

119895+2minus 119906119899

119895minus2119906119899+1

119895minus2119906119899+1

119895)

+ (119906119899

119895+2119906119899+1

119895119906119899+1

119895+2minus 119906119899

119895119906119899+1

119895minus2119906119899+1

119895)]

= 0

(34)

Therefore

1

2120591(10038171003817100381710038171003817119906119899+110038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899minus110038171003817100381710038171003817

2

) = 0 (35)

Then this gives (28)

A conservative approximation confirms that the energywould not increase in time which allows making the schemestable

3 Numerical Experiments

In this section we present numerical experiments on theclassical KdV equation when 120572 = 1 and 120574 = 3 with both dif-ference schemes The accuracy of the methods is measured

Mathematical Problems in Engineering 5

Table 1 Error and convergence rate of the compact finite difference scheme (11) at 119905 = 60 ℎ = 05 and 120591 = 025

120591 ℎ 1205914 ℎ2 12059116 ℎ4119890 139538 times 10

minus2715872 times 10

minus4449013 times 10

minus5

Rate mdash 428481 399487

119890infin

764991 times 10minus3

332024 times 10minus4

208869 times 10minus5

Rate mdash 452608 399062

Table 2 Error and convergence rate of the standard fourth-order finite difference scheme (19) at 119905 = 60 ℎ = 05 and 120591 = 025

120591 ℎ 1205914 ℎ2 12059116 ℎ4119890 159924 times 10

minus1979739 times 10

minus3609352 times 10

minus4

Rate mdash 402885 400705

119890infin

863999 times 10minus2

533149 times 10minus3

333067 times 10minus4

Rate mdash 401842 400066

Table 3 Invariants of 1198681 1198682 and 119868

3of the compact fourth-order finite difference scheme (11)

119905 1198681

1198682

1198683

0 20000000000 06666666667 1205883634610 19999449243 06666680888 1205920147320 20001106778 06666680896 1205918697830 19999055324 06666679386 1205915516740 20001880153 06666680804 1205919379150 19999670401 06666680255 1205926253860 19998768932 06666679688 12059162036

by the comparison of numerical solutions with the exactsolutions as well as other numerical solutions from methodsin the literatures by using sdot and sdot

infinnorm The initial

conditions for each problem are chosen in such a way that theexact solutions can be explicitly computed In case 120572 = 1 and120574 = 3 the KdV equation has the analytical solution as

119906 (119909 119905) = 05 sech2 (05 (119909 minus 119905)) (36)

Therefore the initial condition of (1) takes the form

1199060(119909) = 05 sech2 (05 (119909)) (37)

For these particular experiments we set 119909119871

= minus40119909119877

= 100 and 119879 = 60 We make a comparison between thecompact fourth-order finite difference scheme (11) and thestandard fourth-order finite difference scheme (19) So theresults on this experiment in terms of errors at the time 119905 = 60

is reported in Tables 1 and 2 respectively It is clear that theresults obtained by the compact fourth-order differencescheme (11) are more accurate than the ones obtained by thestandard fourth-order difference scheme but the estimationof the rate of convergence for both schemes is close to the the-oretically predicted fourth-order rate of convergence It canbe seen that the computational efficiency of the scheme (11) isbetter than that of the scheme (19) in terms of error

Conservative approximation that is a supplementaryconstraint is essential for a suitable difference equation tomake a discrete analogue effective to the fundamental con-servation properties of the governing equationThen we can

calculate three conservative approximations by using discreteforms as follows

1198681asymp

ℎ

2

119872

sum

119895=1

(119906119899+1

119895+ 119906119899

119895)

1198682asymp

ℎ

2

119872

sum

119895=1

[(119906119899+1

119895)2

+ (119906119899

119895)2

]

1198683asymp ℎ

119872

sum

119895=1

[

[

2120574(

(119906119899+1

119895)3

+ (119906119899

119895)3

2)

minus3120572(

(119906119899+1

119895)2

119909+ (119906119899

119895)2

119909

2)]

]

(38)

Here we take ℎ = 025 and 120591 = ℎ2 at 119905 isin [0 60] for the com-

pact fourth-order finite difference scheme (11) and the stan-dard fourth-order finite difference scheme (19) and resultsare presented in Tables 3 and 4 respectively The numericalresults show that both two schemes can preserve the discreteconservation properties

The second-order explicit scheme (Z-K scheme) and thesecond-order implicit scheme (Goda scheme) are used fortesting the numerical performance of the new schemes InFigure 1 we see that the Z-K scheme computes reasonablesolutions using ℎ = 01 and 120591 = 001 except that the approx-imate solution at 119905 = 01 does not maintain the shape ofthe exact solution Similar calculations at 119905 = 01 and 119905 = 011

6 Mathematical Problems in Engineering

Table 4 Invariants of 1198681 1198682 and 119868

3of the standard fourth-order finite difference scheme (19)

119905 1198681

1198682

1198683

0 20000000000 06666666667 1205883634610 20000527573 06666666667 1205911524120 20000219448 06666666667 1205912578330 19999931738 06666666667 1205910591540 20001264687 06666666667 1205909947750 19999456225 06666666667 1205911628160 19998875333 06666666667 12059106816

0002

004006

00801

020

0010203040506

t

x

u(xt)

minus20

Figure 1 Explicit solutions using the Z-K scheme at 119905 isin [0 01]119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

0

01

02

03

04

05

06

07

minus01minus15 minus10 minus5 0 5 10 15

Figure 2 Explicit solution using the Z-K scheme at 10 time steps119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

are demonstrated in Figures 2 and 3 respectively The figuresshow that numerical waveforms begin to oscillate at 119905 = 01

and show a blowup when 119905 = 011 According to the resultsthe Z-K scheme is numerically unstable regardless of howsmall time increment is

As shown in Figure 2 the results of the Z-K scheme aregreatly fluctuating at 10 time steps Therefore It can not beused to predict the behavior of the solution at long timeFigures 4 and 5 present the numerical solutions by using the

10

10 15

5

5

0

0

minus5

minus5

minus10

minus10minus15

Figure 3 Explicit solution using the Z-K scheme at 11 time steps119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

02

46

810

020

0

01

02

03

04

05

06

tx

u(xt)

minus20

Figure 4 Implicit solutions using the Goda scheme at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

Goda schemeWe see that the Goda scheme can run very wellat ℎ = 05 and 120591 = 025 However the result is still slightlyoscillate at the left side of the solution

Using the same parameters as the Goda scheme Figures 6and 7 present waveforms with 119905 isin [0 10] The result obtainedby the fourth-order difference schemes is greatly improvedcompared to that obtained by the second-order schemes

Figure 8 shows the numerical solution at 119905 = 200 Theresult from the compact fourth-order difference scheme (11)is almost perfectly sharp From the point of view for the long

Mathematical Problems in Engineering 7

0

005

01

015

02

025

03

035

04

045

minus40minus005

minus30 minus20 minus10 0 2010 30 40 50

Figure 5 Implicit solution using the Goda scheme at 119905 = 10 119909119871=

minus40 119909119877= 100 ℎ = 05 and 120591 = 025

02

46

810

0

20

0

01

02

03

04

05

06

tx minus20

u(xt)

Figure 6 Numerical solutions using the scheme (11) at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

time behavior of the resolution the compact fourth-orderdifference scheme (11) can be seen to be much better than thestandard implicit fourth-order scheme (19)

The results of this section suffice to claim that bothnumerical implementations offer a valid approach toward thenumerical investigation of a solution of the KdV equationespecially for the compact finite difference method

4 Conclusion

Two conservative finite difference schemes for the KdV equa-tion are introduced and analyzed The construction of thecompact finite difference scheme (11) requires only a regularfive-point stencil at higher time level which is similar to thestandard second-order Crank-Nicolson scheme the explicitscheme [16] and the implicit scheme [18] However the con-struction of the standard fourth-order scheme (19) requires aseven-point stencil at higher time levelThe accuracy and sta-bility of the numerical schemes for the solutions of the KdV

02

46

810

0

20

0

01

02

03

04

05

06

tx minus20

u(xt)

Figure 7 Numerical solutions using the scheme (19) at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

190 195 200 205 210

0

01

02

03

04

05

06

Scheme (11)Scheme (19)Exact solution

minus01

Figure 8 Numerical solutions at 119905 = 200 119909119871

= minus40 119909119877

= 300ℎ = 05 and 120591 = 025

equation can be tested by using the exact solution In thepaper the numerical experiments show that the presentmethods support the analysis of convergence rate The per-formance of the fourth-order schemes is well efficient at longtime by comparing with the second-order schemes [16 18]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was supported by Chiang Mai University

References

[1] D J Korteweg and G de Vries ldquoOn the change of form of longwaves advancing in a rectangular canal and on a new type of

8 Mathematical Problems in Engineering

long stationary waverdquo Philosophical Magazine vol 39 pp 422ndash449 1895

[2] D Pavoni ldquoSingle and multidomain Chebyshev collocationmethods for the Korteweg-de Vries equationrdquo Calcolo vol 25no 4 pp 311ndash346 1988

[3] A A Soliman ldquoCollocation solution of the Korteweg-de Vriesequation using septic splinesrdquo International Journal of ComputerMathematics vol 81 no 3 pp 325ndash331 2004

[4] H Kalisch and X Raynaud ldquoOn the rate of convergence ofa collocation projection of the KdV equationrdquo MathematicalModelling andNumerical Analysis vol 41 no 1 pp 95ndash110 2007

[5] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[6] L R T Gardner G A Gardner and A H A Ali ldquoSimulationsof solitons using quadratic spline finite elementsrdquo ComputerMethods in Applied Mechanics and Engineering vol 92 no 2pp 231ndash243 1991

[7] M E Alexander and J L Morris ldquoGalerkin methods applied tosomemodel equations for non-linear dispersive wavesrdquo Journalof Computational Physics vol 30 no 3 pp 428ndash451 1979

[8] S R Barros and J W Cardenas ldquoA nonlinear Galerkin methodfor the shallow-water equations on periodic domainsrdquo Journalof Computational Physics vol 172 no 2 pp 592ndash608 2001

[9] H Ma and W Sun ldquoA Legendre-Petrov-Galerkin and Cheby-shev collocation method for third-order differential equationsrdquoSIAM Journal on Numerical Analysis vol 38 no 5 pp 1425ndash1438 2000

[10] J Shen ldquoA new dual-Petrov-Galerkin method for third andhigher odd-order differential equations application to the KDVequationrdquo SIAM Journal onNumerical Analysis vol 41 no 5 pp1595ndash1619 2003

[11] W Heinrichs ldquoSpectral approximation of third-order prob-lemsrdquo Journal of Scientific Computing vol 14 no 3 pp 275ndash2891999

[12] YMaday andAQuarteroni ldquoError analysis for spectral approx-imation of the Korteweg-de Vries equationrdquo ModelisationMathematique et Analyse numerique vol 22 no 3 pp 499ndash5291988

[13] S Zhu ldquoA scheme with a higher-order discrete invariant for theKdV equationrdquo Applied Mathematics Letters vol 14 no 1 pp17ndash20 2001

[14] F-l Qu andW-q Wang ldquoAlternating segment explicit-implicitscheme for nonlinear third-orderKdV equationrdquoAppliedMath-ematics and Mechanics English Edition vol 28 no 7 pp 973ndash980 2007

[15] H-P Wang Y-S Wang and Y-Y Hu ldquoAn explicit scheme forthe KdV equationrdquo Chinese Physics Letters vol 25 no 7 pp2335ndash2338 2008

[16] N J Zabusky and M D Kruskal ldquoInteraction of ldquosolitonsrdquo in acollisionless plasma and the recurrence of initial statesrdquoPhysicalReview Letters vol 15 no 6 pp 240ndash243 1965

[17] O Kolebaje and O Oyewande ldquoNumerical solution of theKorteweg-de Vries equation by finite differenece an d adomaindecomposition methodrdquo International Journal of Basic andApplied Sciences vol 1 no 3 pp 321ndash335 2012

[18] K Goda ldquoOn stability of some finite difference schemes for theKorteweg-de Vries equationrdquo Journal of the Physical Society ofJapan vol 39 no 1 pp 229ndash236 1975

[19] S Hamdi W H Enright W E Schiesser and J J GottliebldquoExact solutions and conservation laws for coupled generalized

Korteweg-de Vries and quintic regularized long wave equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 63 no 5ndash7 pp e1425ndashe1434 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

equations with a pentadiagonal matrix that is the systemof linear equations arising from a standard second-orderdiscretization of a boundary value problem A method toconquer the conflict between stability accuracy and com-putational cost is the development of a high-order compactscheme

In recent decades many scientists concentrated upon thedifference method that makes a discrete analogue effective inthe fundamental conservation properties This causes us tocreate finite difference schemes which preserve the mass andenergy of solutions for the KdV equation In this paper twofourth-order difference schemes are constructed for the onedimensional KdV equation

119906119905+ 120572119906119909119909119909

+ 120574 (1199062)119909= 0 119909

119871lt 119909 lt 119909

119877 0 le 119905 le 119879 (1)

with an initial condition

119906 (119909 0) = 1199060(119909) 119909

119871le 119909 le 119909

119877 (2)

and boundary conditions

119906 (119909119871 119905) = 119906 (119909

119877 119905) = 0

119906119909(119909119871 119905) = 119906

119909(119909119877 119905) = 0

119906119909119909

(119909119871 119905) = 119906

119909119909(119909119877 119905) = 0

0 le 119905 le 119879

(3)

where 120572 and 120574 are any real numberWhen minus119909119871≫ 0 and 119909

119877≫

0 the initial-boundary value problem (1)ndash(3) is consistent sothe boundary condition (3) is reasonable By assumptions thesolitary wave solution and its derivatives have the followingasymptotic values 119906 rarr 0 as 119909 rarr plusmninfin and for 119899 ge 1120597119899119906120597119909119899

rarr 0 as 119909 rarr plusmninfin Moreover we obtain the solutionproperties as follows [19]

1198681= int

119909119877

119909119871

119906 (119909 119905) 119889119909

1198682= int

119909119877

119909119871

119906 (119909 119905)2119889119909

1198683= int

119909119877

119909119871

[2120574119906 (119909 119905)3minus 3120572 [119906 (119909 119905)

119909]2

]

(4)

The content of this paper is organized as follows In thenext section we create fourth-order finite difference schemesfor the KdV equation with the initial and boundary condi-tions The stability of finite difference schemes is discussedand the conservative approximations are also given Theresults on validation of finite difference schemes are pre-sented in Section 3 where we make a detailed comparisonwith available data to confirm and illustrate our theoreticalanalysis Finally we finish our paper by conclusions in the lastsection

2 Difference Schemes

We start the discussion of finite difference schemes bydefining a grid of points in the (119909 119905) plane For simplicity we

use a uniform grid for a discrete process with states identifiedby 119909119895= 119909119871+119895ℎwhich the grid size is ℎ = (119909

119877minus119909119871)119872 where

119872 is the number of grid pointsTherefore the grid will be thepoints (119909

119895 119905119899) = (119909

119871+ 119895ℎ 119899120591) for arbitrary integers 119895 and 119899

Here 120591 is a time increment (time step length) We write thenotation 119906

119899

119895for a value of a function 119906 at the grid point

(119909119871+ 119895ℎ 119899120591)In this paper we give a complete description of our finite

difference schemes and an algorithm for the formulationof the problem (1)ndash(3) We use the following notations forsimplicity

119906119899

119895=

119906119899+1

119895+ 119906119899minus1

119895

2 (119906

119899

119895)=

119906119899+1

119895minus 119906119899minus1

119895

2120591

(119906119899

119895)119909=

119906119899

119895minus 119906119899

119895minus1

ℎ (119906

119899

119895)119909=

119906119899

119895+1minus 119906119899

119895

ℎ

(119906119899

119895)119909=

119906119899

119895+1minus 119906119899

119895minus1

2ℎ (119906

119899

119895)119909=

119906119899

119895+2minus 119906119899

119895minus2

4ℎ

(119906119899 V119899) = ℎ

119872minus1

sum

119895=1

119906119899

119895V119899119895

10038171003817100381710038171199061198991003817100381710038171003817

2

= (119906119899 119906119899)

10038171003817100381710038171199061198991003817100381710038171003817infin = max

1le119895le119872minus1

10038161003816100381610038161003816119906119899

119895

10038161003816100381610038161003816

(5)

As introduced in the following subsections the tech-niques for determining the value of numerical solution to (1)are used

21 Compact Fourth-Order Finite Difference Scheme By set-ting119908 = minus120572119906

119909119909119909minus120574(1199062)119909 (1) can be written as119908 = 119906

119905 By the

Taylor expansion we obtain

119908119899

119895= (120597119905119906)119899

119895= (119906119899

119895)+ 119874 (120591

2) (6)

119908119899

119895= minus120572[(119906

119899

119895)119909119909119909

minusℎ2

4(1205975

119909119906)119899

119895]

minus 120574 [[(119906119899

119895)2

]119909

minusℎ2

6(1205973

1199091199062)119899

119895] + 119874 (ℎ

4)

(7)

From (6) we have

120572 (1205975

119909119906)119899

119895= minus120574 (120597

3

1199091199062)119899

119895minus (1205972

119909119908)119899

119895 (8)

Substituting (8) into (7) we get

119908119899

119895= minus120572 (119906

119899

119895)119909119909119909

minusℎ2

4(1205972

119909119908)119899

119895minus 120574 [(119906

119899

119895)2

]119909

minusℎ2

12120574 (1205973

1199091199062)119899

119895+ 119874 (ℎ

4)

(9)

Using second-order accuracy for approximation we obtain

(1205973

1199091199062)119899

119895= [(119906

119899

119895)2

]119909119909119909

+ 119874 (ℎ2)

(1205972

119909119908)119899

119895= (119908119899

119895)119909119909

+ 119874 (ℎ2)

(10)

Mathematical Problems in Engineering 3

The following method is the proposed compact finite differ-ence scheme to solve the problem (1)ndash(3)

(119906119899

119895)+

ℎ2

4(119906119899

119895)119909119909

+ 120572 (119906119899

119895)119909119909119909

+ 120574 [(119906119899

119895) (119906119899

119895)]119909

+120574ℎ2

12[(119906119899

119895) (119906119899

119895)]119909119909119909

= 0

(11)

where

1199060

119895= 1199060(119909119895) 0 le 119895 le 119872 (12)

Since the boundary conditions are homogeneous they give

119906119899

0= 119906119899

119872= 0 (119906

119899

0)119909= (119906119899

119872)119909= 0 1 le 119899 le 119873 (13)

At this time let 119890119899

119895= V119899119895minus 119906119899

119895where V119899

119895and 119906

119899

119895are the

solution of (1)ndash(3) and (11)ndash(13) respectivelyThen we obtainthe following error equation

119903119899

119895= (119890119899

119895)+

ℎ2

4(119890119899

119895)119909119909

+ 120572 (119890119899

119895)119909119909119909

+ 120574 [(V119899119895) (V119899119895)]119909

minus 120574 [(119906119899

119895) (119906119899

119895)]119909+

120574ℎ2

12[(V119899119895) (V119899119895)]119909119909119909

minus120574ℎ2

12[(119906119899

119895) (119906119899

119895)]119909119909119909

(14)

where 119903119899

119895denotes the truncation error By using the Taylor

expansion it is easy to see that 119903119899119895= 119874(120591

2+ℎ4) holds as 120591 ℎ rarr

0The Von Neumann stability analysis of (11) with 119906

119899

119895=

120585119899119890119894119896119895ℎ where 119894

2= minus1 and 119896 is a wave number gives the

following the amplification factor

1205852=

119860 minus 119894120591119861

119860 + 119894120591119861 (15)

where

119860 = 6ℎ3(cos (119896ℎ) + 1)

119861 = 12120572 (sin (2119896ℎ) minus 2 sin (119896ℎ))

+ 120574ℎ2(119906119899

119895) (sin (4119896ℎ) + 10 sin (2119896ℎ))

(16)

The amplification factor which is a complex number has itsmodulus equal to one therefore the compact finite differencescheme is unconditionally stable

Theorem 1 Suppose 119906(119909 119905) is smooth enough then the scheme(11)ndash(13) is conservative in a sense

119868119899

1=

ℎ

2

119872minus1

sum

119895=1

(119906119899+1

119895+ 119906119899

119895) = 119868119899minus1

1= sdot sdot sdot = 119868

0

1 (17)

under assumptions 1199061= 119906119872minus1

= 0

Proof By multiplying (11) by ℎ summing up for 119895 from 1 to119872minus1 and considering the boundary condition and assuming1199061= 119906119872minus1

= 0 we get

ℎ

2120591

119872minus1

sum

119895=1

(119906119899+1

119895minus 119906119899minus1

119895) = 0 (18)

Then this gives (17)

22 Standard Fourth-Order Finite Difference Scheme By thefact (1199062)

119909= (23)[119906119906

119909+(1199062)119909] and using an implicit finite dif-

ference method we propose a standard seven-point implicitdifference scheme for the problem (1)ndash(3)

(119906119899

119895)+ 120572(

3

2(119906119899

119895)119909119909119909

minus1

2(119906119899

119895)119909119909 119909

)

+ 2120574 [4

9((119906119899

119895119906119899

119895)119909+ 119906119899

119895(119906119899

119895)119909) minus

1

9((119906119899

119895119906119899

119895)119909+ 119906119899

119895(119906119899

119895)119909)]

= 0

(19)

where

1199060

119895= 1199060(119909119895) 0 le 119895 le 119872 (20)

Since the boundary conditions are homogeneous we obtain

119906119899

0= 119906119899

119872= 0 (21)

4 (119906119899

0)119909minus (119906119899

0)119909= 4 (119906

119899

119872)119909minus (119906119899

119872)119909= 0 (22)

minus (119906119899

minus1)119909119909

+ 14 (119906119899

0)119909119909

minus (119906119899

1)119909119909

= minus (119906119899

119872minus1)119909119909

14 (119906119899

119872)119909119909

minus (119906119899

119872+1)119909119909

= 0 1 le 119899 le 119873

(23)

119906 119906119909 and 119906

119909119909are required by the standard fourth-order

technique to be zero at the upstream and downstreambound-aries because the method utilizes a seven-point finite differ-ence scheme for the approximation of solution 119906 Throughthe analytical technique of contrasting (11) requires twohomogeneous boundary conditions only

Now let 119890119899119895= V119899119895minus 119906119899

119895where V119899

119895and 119906

119899

119895are the solution of

(1)ndash(3) and (19)ndash(22) respectively Then we obtain the fol-lowing error equation

(119890119899

119895)+ 120572

3

2(119890119899

119895)119909119909119909

minus 1205721

2(119890119899

119895)119909119909 119909

+8120574

9[((V119899119895V119899119895)119909+ V119899119895(V119899119895)119909) minus ((119906

119899

119895119906119899

119895)119909+ 119906119899

119895(119906119899

119895)119909)]

minus2120574

9[((V119899119895V119899119895)119909+ V119899119895(V119899119895)119909) minus ((119906

119899

119895119906119899

119895)119909+ 119906119899

119895(119906119899

119895)119909)] = 0

(24)

where 119903119899

119895denotes the truncation error By using the Taylor

expansion it is easy to see that 119903119899119895= 119874(120591

2+ℎ4) holds as 120591 ℎ rarr

0

4 Mathematical Problems in Engineering

The Von Neumann stability analysis of (19) with 119906119899

119895=

120585119899119890119894119896119895ℎ gives the following amplification factor

1205852=

36ℎ3minus 119894120591119860

36ℎ3 + 119894120591119860 (25)

where

119860 = 4120574ℎ2(119906119899

119895) (minus sin (4119896ℎ) + 7 sin (2119896ℎ) + 8 sin (119896ℎ))

+ 9120572 (minus sin (3119896ℎ) + 8 sin (2119896ℎ) minus 13 sin (119896ℎ))

(26)

The amplification factor which is a complex numberhas its modulus equal to one therefore the finite differencescheme is unconditionally stable

Theorem2 Suppose119906(119909 119905) is smooth enough then the scheme(11)ndash(13) is conservative in a sense

119868119899

1=

ℎ

2

119872minus1

sum

119895=1

(119906119899+1

119895+ 119906119899

119895)

+ 120591ℎ120574

119872minus1

sum

119895=1

[4

9119906119899

119895(119906119899+1

119895)119909minus

1

9119906119899

119895(119906119899+1

119895)119909]

= 119868119899minus1

1= sdot sdot sdot = 119868

0

1

(27)

under assumptions 1199061= 1199062= 119906119872minus2

= 119906119872minus1

= 0 Moreoverthe scheme (19)ndash(22) is conservative in a sense

119868119899

2=

1

2

10038171003817100381710038171199061198991003817100381710038171003817

2

+1

2

10038171003817100381710038171003817119906119899+110038171003817100381710038171003817

2

= 119868119899minus1

2= sdot sdot sdot = 119868

0

2 (28)

Proof By multiplying (11) by ℎ summing up for 119895 from 1 to119872minus1 and considering the boundary condition and assuming1199061= 1199062= 119906119872minus2

= 119906119872minus1

= 0 we have

120591ℎ

119872minus1

sum

119895=1

[8

9(119906119899

119895(119906119899

119895)119909) minus

2

9(119906119899

119895(119906119899+1

119895)119909)]

= 120591ℎ

119872minus1

sum

119895=1

[4

9(119906119899

119895(119906119899+1

119895)119909minus 119906119899minus1

119895(119906119899

119895)119909)

minus1

9(119906119899

119895(119906119899+1

119895)119909minus 119906119899minus1

119895(119906119899

119895)119909)]

(29)

As a result we have

ℎ

2

119872minus1

sum

119895=1

(119906119899+1

119895minus 119906119899minus1

119895)

+ 120574120591ℎ

119872minus1

sum

119895=1

[4

9(119906119899

119895(119906119899+1

119895)119909minus 119906119899minus1

119895(119906119899

119895)119909)

minus1

9(119906119899

119895(119906119899+1

119895)119909minus 119906119899minus1

119895(119906119899

119895)119909)] = 0

(30)

Then this gives (27) We then take an inner product between(19) and 2119906

119899 We obtain

1

2120591(10038171003817100381710038171003817119906119899+110038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899minus110038171003817100381710038171003817

2

) +3120572

2((119906119899)119909119909119909

(119906119899))

minus120572

2((119906119899)119909119909 119909

(119906119899)) + 2120574 (120593

119899(119906119899 119906119899) 119906119899) = 0

(31)

where

120593119899(119906119899

119895 119906119899

119895) =

4

9((119906119899

119895119906119899

119895)119909+ 119906119899

119895(119906119899

119895)119909)

minus1

9((119906119899

119895119906119899

119894)119909+ 119906119899

119895(119906119899

119895)119909)

(32)

by considering the boundary condition (13) According to

(119906119899

119909119909119909 119906119899) = 0

(119906119899

119909119909 119909 119906119899) = 0

(33)

indeed

(120593119899(119906119899 119906119899) 119906119899)

=4ℎ

9

119872minus1

sum

119895=1

[119906119899

119895(119906119899+1

119895)119909+ (119906119899

119895119906119899+1

119895)119909] 119906119899+1

119895

minusℎ

9

119872minus1

sum

119895=1

[119906119899

119895(119906119899+1

119895)119909+ (119906119899

119895119906119899+1

119895)119909] 119906119899+1

119895

=2

9

119872minus1

sum

119895=1

[(119906119899

119895119906119899+1

119895119906119899+1

119895+1minus 119906119899

119895minus1119906119899+1

119895minus1119906119899+1

119895)

+ (119906119899

119895+1119906119899+1

119895119906119899+1

119895+1minus 119906119899

119895119906119899+1

119895minus1119906119899+1

119895)]

minus1

36

119872minus1

sum

119895=1

[(119906119899

119895119906119899+1

119895119906119899+1

119895+2minus 119906119899

119895minus2119906119899+1

119895minus2119906119899+1

119895)

+ (119906119899

119895+2119906119899+1

119895119906119899+1

119895+2minus 119906119899

119895119906119899+1

119895minus2119906119899+1

119895)]

= 0

(34)

Therefore

1

2120591(10038171003817100381710038171003817119906119899+110038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899minus110038171003817100381710038171003817

2

) = 0 (35)

Then this gives (28)

A conservative approximation confirms that the energywould not increase in time which allows making the schemestable

3 Numerical Experiments

In this section we present numerical experiments on theclassical KdV equation when 120572 = 1 and 120574 = 3 with both dif-ference schemes The accuracy of the methods is measured

Mathematical Problems in Engineering 5

Table 1 Error and convergence rate of the compact finite difference scheme (11) at 119905 = 60 ℎ = 05 and 120591 = 025

120591 ℎ 1205914 ℎ2 12059116 ℎ4119890 139538 times 10

minus2715872 times 10

minus4449013 times 10

minus5

Rate mdash 428481 399487

119890infin

764991 times 10minus3

332024 times 10minus4

208869 times 10minus5

Rate mdash 452608 399062

Table 2 Error and convergence rate of the standard fourth-order finite difference scheme (19) at 119905 = 60 ℎ = 05 and 120591 = 025

120591 ℎ 1205914 ℎ2 12059116 ℎ4119890 159924 times 10

minus1979739 times 10

minus3609352 times 10

minus4

Rate mdash 402885 400705

119890infin

863999 times 10minus2

533149 times 10minus3

333067 times 10minus4

Rate mdash 401842 400066

Table 3 Invariants of 1198681 1198682 and 119868

3of the compact fourth-order finite difference scheme (11)

119905 1198681

1198682

1198683

0 20000000000 06666666667 1205883634610 19999449243 06666680888 1205920147320 20001106778 06666680896 1205918697830 19999055324 06666679386 1205915516740 20001880153 06666680804 1205919379150 19999670401 06666680255 1205926253860 19998768932 06666679688 12059162036

by the comparison of numerical solutions with the exactsolutions as well as other numerical solutions from methodsin the literatures by using sdot and sdot

infinnorm The initial

conditions for each problem are chosen in such a way that theexact solutions can be explicitly computed In case 120572 = 1 and120574 = 3 the KdV equation has the analytical solution as

119906 (119909 119905) = 05 sech2 (05 (119909 minus 119905)) (36)

Therefore the initial condition of (1) takes the form

1199060(119909) = 05 sech2 (05 (119909)) (37)

For these particular experiments we set 119909119871

= minus40119909119877

= 100 and 119879 = 60 We make a comparison between thecompact fourth-order finite difference scheme (11) and thestandard fourth-order finite difference scheme (19) So theresults on this experiment in terms of errors at the time 119905 = 60

is reported in Tables 1 and 2 respectively It is clear that theresults obtained by the compact fourth-order differencescheme (11) are more accurate than the ones obtained by thestandard fourth-order difference scheme but the estimationof the rate of convergence for both schemes is close to the the-oretically predicted fourth-order rate of convergence It canbe seen that the computational efficiency of the scheme (11) isbetter than that of the scheme (19) in terms of error

Conservative approximation that is a supplementaryconstraint is essential for a suitable difference equation tomake a discrete analogue effective to the fundamental con-servation properties of the governing equationThen we can

calculate three conservative approximations by using discreteforms as follows

1198681asymp

ℎ

2

119872

sum

119895=1

(119906119899+1

119895+ 119906119899

119895)

1198682asymp

ℎ

2

119872

sum

119895=1

[(119906119899+1

119895)2

+ (119906119899

119895)2

]

1198683asymp ℎ

119872

sum

119895=1

[

[

2120574(

(119906119899+1

119895)3

+ (119906119899

119895)3

2)

minus3120572(

(119906119899+1

119895)2

119909+ (119906119899

119895)2

119909

2)]

]

(38)

Here we take ℎ = 025 and 120591 = ℎ2 at 119905 isin [0 60] for the com-

pact fourth-order finite difference scheme (11) and the stan-dard fourth-order finite difference scheme (19) and resultsare presented in Tables 3 and 4 respectively The numericalresults show that both two schemes can preserve the discreteconservation properties

The second-order explicit scheme (Z-K scheme) and thesecond-order implicit scheme (Goda scheme) are used fortesting the numerical performance of the new schemes InFigure 1 we see that the Z-K scheme computes reasonablesolutions using ℎ = 01 and 120591 = 001 except that the approx-imate solution at 119905 = 01 does not maintain the shape ofthe exact solution Similar calculations at 119905 = 01 and 119905 = 011

6 Mathematical Problems in Engineering

Table 4 Invariants of 1198681 1198682 and 119868

3of the standard fourth-order finite difference scheme (19)

119905 1198681

1198682

1198683

0 20000000000 06666666667 1205883634610 20000527573 06666666667 1205911524120 20000219448 06666666667 1205912578330 19999931738 06666666667 1205910591540 20001264687 06666666667 1205909947750 19999456225 06666666667 1205911628160 19998875333 06666666667 12059106816

0002

004006

00801

020

0010203040506

t

x

u(xt)

minus20

Figure 1 Explicit solutions using the Z-K scheme at 119905 isin [0 01]119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

0

01

02

03

04

05

06

07

minus01minus15 minus10 minus5 0 5 10 15

Figure 2 Explicit solution using the Z-K scheme at 10 time steps119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

are demonstrated in Figures 2 and 3 respectively The figuresshow that numerical waveforms begin to oscillate at 119905 = 01

and show a blowup when 119905 = 011 According to the resultsthe Z-K scheme is numerically unstable regardless of howsmall time increment is

As shown in Figure 2 the results of the Z-K scheme aregreatly fluctuating at 10 time steps Therefore It can not beused to predict the behavior of the solution at long timeFigures 4 and 5 present the numerical solutions by using the

10

10 15

5

5

0

0

minus5

minus5

minus10

minus10minus15

Figure 3 Explicit solution using the Z-K scheme at 11 time steps119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

02

46

810

020

0

01

02

03

04

05

06

tx

u(xt)

minus20

Figure 4 Implicit solutions using the Goda scheme at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

Goda schemeWe see that the Goda scheme can run very wellat ℎ = 05 and 120591 = 025 However the result is still slightlyoscillate at the left side of the solution

Using the same parameters as the Goda scheme Figures 6and 7 present waveforms with 119905 isin [0 10] The result obtainedby the fourth-order difference schemes is greatly improvedcompared to that obtained by the second-order schemes

Figure 8 shows the numerical solution at 119905 = 200 Theresult from the compact fourth-order difference scheme (11)is almost perfectly sharp From the point of view for the long

Mathematical Problems in Engineering 7

0

005

01

015

02

025

03

035

04

045

minus40minus005

minus30 minus20 minus10 0 2010 30 40 50

Figure 5 Implicit solution using the Goda scheme at 119905 = 10 119909119871=

minus40 119909119877= 100 ℎ = 05 and 120591 = 025

02

46

810

0

20

0

01

02

03

04

05

06

tx minus20

u(xt)

Figure 6 Numerical solutions using the scheme (11) at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

time behavior of the resolution the compact fourth-orderdifference scheme (11) can be seen to be much better than thestandard implicit fourth-order scheme (19)

The results of this section suffice to claim that bothnumerical implementations offer a valid approach toward thenumerical investigation of a solution of the KdV equationespecially for the compact finite difference method

4 Conclusion

Two conservative finite difference schemes for the KdV equa-tion are introduced and analyzed The construction of thecompact finite difference scheme (11) requires only a regularfive-point stencil at higher time level which is similar to thestandard second-order Crank-Nicolson scheme the explicitscheme [16] and the implicit scheme [18] However the con-struction of the standard fourth-order scheme (19) requires aseven-point stencil at higher time levelThe accuracy and sta-bility of the numerical schemes for the solutions of the KdV

02

46

810

0

20

0

01

02

03

04

05

06

tx minus20

u(xt)

Figure 7 Numerical solutions using the scheme (19) at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

190 195 200 205 210

0

01

02

03

04

05

06

Scheme (11)Scheme (19)Exact solution

minus01

Figure 8 Numerical solutions at 119905 = 200 119909119871

= minus40 119909119877

= 300ℎ = 05 and 120591 = 025

equation can be tested by using the exact solution In thepaper the numerical experiments show that the presentmethods support the analysis of convergence rate The per-formance of the fourth-order schemes is well efficient at longtime by comparing with the second-order schemes [16 18]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was supported by Chiang Mai University

References

[1] D J Korteweg and G de Vries ldquoOn the change of form of longwaves advancing in a rectangular canal and on a new type of

8 Mathematical Problems in Engineering

long stationary waverdquo Philosophical Magazine vol 39 pp 422ndash449 1895

[2] D Pavoni ldquoSingle and multidomain Chebyshev collocationmethods for the Korteweg-de Vries equationrdquo Calcolo vol 25no 4 pp 311ndash346 1988

[3] A A Soliman ldquoCollocation solution of the Korteweg-de Vriesequation using septic splinesrdquo International Journal of ComputerMathematics vol 81 no 3 pp 325ndash331 2004

[4] H Kalisch and X Raynaud ldquoOn the rate of convergence ofa collocation projection of the KdV equationrdquo MathematicalModelling andNumerical Analysis vol 41 no 1 pp 95ndash110 2007

[5] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[6] L R T Gardner G A Gardner and A H A Ali ldquoSimulationsof solitons using quadratic spline finite elementsrdquo ComputerMethods in Applied Mechanics and Engineering vol 92 no 2pp 231ndash243 1991

[7] M E Alexander and J L Morris ldquoGalerkin methods applied tosomemodel equations for non-linear dispersive wavesrdquo Journalof Computational Physics vol 30 no 3 pp 428ndash451 1979

[8] S R Barros and J W Cardenas ldquoA nonlinear Galerkin methodfor the shallow-water equations on periodic domainsrdquo Journalof Computational Physics vol 172 no 2 pp 592ndash608 2001

[9] H Ma and W Sun ldquoA Legendre-Petrov-Galerkin and Cheby-shev collocation method for third-order differential equationsrdquoSIAM Journal on Numerical Analysis vol 38 no 5 pp 1425ndash1438 2000

[10] J Shen ldquoA new dual-Petrov-Galerkin method for third andhigher odd-order differential equations application to the KDVequationrdquo SIAM Journal onNumerical Analysis vol 41 no 5 pp1595ndash1619 2003

[11] W Heinrichs ldquoSpectral approximation of third-order prob-lemsrdquo Journal of Scientific Computing vol 14 no 3 pp 275ndash2891999

[12] YMaday andAQuarteroni ldquoError analysis for spectral approx-imation of the Korteweg-de Vries equationrdquo ModelisationMathematique et Analyse numerique vol 22 no 3 pp 499ndash5291988

[13] S Zhu ldquoA scheme with a higher-order discrete invariant for theKdV equationrdquo Applied Mathematics Letters vol 14 no 1 pp17ndash20 2001

[14] F-l Qu andW-q Wang ldquoAlternating segment explicit-implicitscheme for nonlinear third-orderKdV equationrdquoAppliedMath-ematics and Mechanics English Edition vol 28 no 7 pp 973ndash980 2007

[15] H-P Wang Y-S Wang and Y-Y Hu ldquoAn explicit scheme forthe KdV equationrdquo Chinese Physics Letters vol 25 no 7 pp2335ndash2338 2008

[16] N J Zabusky and M D Kruskal ldquoInteraction of ldquosolitonsrdquo in acollisionless plasma and the recurrence of initial statesrdquoPhysicalReview Letters vol 15 no 6 pp 240ndash243 1965

[17] O Kolebaje and O Oyewande ldquoNumerical solution of theKorteweg-de Vries equation by finite differenece an d adomaindecomposition methodrdquo International Journal of Basic andApplied Sciences vol 1 no 3 pp 321ndash335 2012

[18] K Goda ldquoOn stability of some finite difference schemes for theKorteweg-de Vries equationrdquo Journal of the Physical Society ofJapan vol 39 no 1 pp 229ndash236 1975

[19] S Hamdi W H Enright W E Schiesser and J J GottliebldquoExact solutions and conservation laws for coupled generalized

Korteweg-de Vries and quintic regularized long wave equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 63 no 5ndash7 pp e1425ndashe1434 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

The following method is the proposed compact finite differ-ence scheme to solve the problem (1)ndash(3)

(119906119899

119895)+

ℎ2

4(119906119899

119895)119909119909

+ 120572 (119906119899

119895)119909119909119909

+ 120574 [(119906119899

119895) (119906119899

119895)]119909

+120574ℎ2

12[(119906119899

119895) (119906119899

119895)]119909119909119909

= 0

(11)

where

1199060

119895= 1199060(119909119895) 0 le 119895 le 119872 (12)

Since the boundary conditions are homogeneous they give

119906119899

0= 119906119899

119872= 0 (119906

119899

0)119909= (119906119899

119872)119909= 0 1 le 119899 le 119873 (13)

At this time let 119890119899

119895= V119899119895minus 119906119899

119895where V119899

119895and 119906

119899

119895are the

solution of (1)ndash(3) and (11)ndash(13) respectivelyThen we obtainthe following error equation

119903119899

119895= (119890119899

119895)+

ℎ2

4(119890119899

119895)119909119909

+ 120572 (119890119899

119895)119909119909119909

+ 120574 [(V119899119895) (V119899119895)]119909

minus 120574 [(119906119899

119895) (119906119899

119895)]119909+

120574ℎ2

12[(V119899119895) (V119899119895)]119909119909119909

minus120574ℎ2

12[(119906119899

119895) (119906119899

119895)]119909119909119909

(14)

where 119903119899

119895denotes the truncation error By using the Taylor

expansion it is easy to see that 119903119899119895= 119874(120591

2+ℎ4) holds as 120591 ℎ rarr

0The Von Neumann stability analysis of (11) with 119906

119899

119895=

120585119899119890119894119896119895ℎ where 119894

2= minus1 and 119896 is a wave number gives the

following the amplification factor

1205852=

119860 minus 119894120591119861

119860 + 119894120591119861 (15)

where

119860 = 6ℎ3(cos (119896ℎ) + 1)

119861 = 12120572 (sin (2119896ℎ) minus 2 sin (119896ℎ))

+ 120574ℎ2(119906119899

119895) (sin (4119896ℎ) + 10 sin (2119896ℎ))

(16)

The amplification factor which is a complex number has itsmodulus equal to one therefore the compact finite differencescheme is unconditionally stable

Theorem 1 Suppose 119906(119909 119905) is smooth enough then the scheme(11)ndash(13) is conservative in a sense

119868119899

1=

ℎ

2

119872minus1

sum

119895=1

(119906119899+1

119895+ 119906119899

119895) = 119868119899minus1

1= sdot sdot sdot = 119868

0

1 (17)

under assumptions 1199061= 119906119872minus1

= 0

Proof By multiplying (11) by ℎ summing up for 119895 from 1 to119872minus1 and considering the boundary condition and assuming1199061= 119906119872minus1

= 0 we get

ℎ

2120591

119872minus1

sum

119895=1

(119906119899+1

119895minus 119906119899minus1

119895) = 0 (18)

Then this gives (17)

22 Standard Fourth-Order Finite Difference Scheme By thefact (1199062)

119909= (23)[119906119906

119909+(1199062)119909] and using an implicit finite dif-

ference method we propose a standard seven-point implicitdifference scheme for the problem (1)ndash(3)

(119906119899

119895)+ 120572(

3

2(119906119899

119895)119909119909119909

minus1

2(119906119899

119895)119909119909 119909

)

+ 2120574 [4

9((119906119899

119895119906119899

119895)119909+ 119906119899

119895(119906119899

119895)119909) minus

1

9((119906119899

119895119906119899

119895)119909+ 119906119899

119895(119906119899

119895)119909)]

= 0

(19)

where

1199060

119895= 1199060(119909119895) 0 le 119895 le 119872 (20)

Since the boundary conditions are homogeneous we obtain

119906119899

0= 119906119899

119872= 0 (21)

4 (119906119899

0)119909minus (119906119899

0)119909= 4 (119906

119899

119872)119909minus (119906119899

119872)119909= 0 (22)

minus (119906119899

minus1)119909119909

+ 14 (119906119899

0)119909119909

minus (119906119899

1)119909119909

= minus (119906119899

119872minus1)119909119909

14 (119906119899

119872)119909119909

minus (119906119899

119872+1)119909119909

= 0 1 le 119899 le 119873

(23)

119906 119906119909 and 119906

119909119909are required by the standard fourth-order

technique to be zero at the upstream and downstreambound-aries because the method utilizes a seven-point finite differ-ence scheme for the approximation of solution 119906 Throughthe analytical technique of contrasting (11) requires twohomogeneous boundary conditions only

Now let 119890119899119895= V119899119895minus 119906119899

119895where V119899

119895and 119906

119899

119895are the solution of

(1)ndash(3) and (19)ndash(22) respectively Then we obtain the fol-lowing error equation

(119890119899

119895)+ 120572

3

2(119890119899

119895)119909119909119909

minus 1205721

2(119890119899

119895)119909119909 119909

+8120574

9[((V119899119895V119899119895)119909+ V119899119895(V119899119895)119909) minus ((119906

119899

119895119906119899

119895)119909+ 119906119899

119895(119906119899

119895)119909)]

minus2120574

9[((V119899119895V119899119895)119909+ V119899119895(V119899119895)119909) minus ((119906

119899

119895119906119899

119895)119909+ 119906119899

119895(119906119899

119895)119909)] = 0

(24)

where 119903119899

119895denotes the truncation error By using the Taylor

expansion it is easy to see that 119903119899119895= 119874(120591

2+ℎ4) holds as 120591 ℎ rarr

0

4 Mathematical Problems in Engineering

The Von Neumann stability analysis of (19) with 119906119899

119895=

120585119899119890119894119896119895ℎ gives the following amplification factor

1205852=

36ℎ3minus 119894120591119860

36ℎ3 + 119894120591119860 (25)

where

119860 = 4120574ℎ2(119906119899

119895) (minus sin (4119896ℎ) + 7 sin (2119896ℎ) + 8 sin (119896ℎ))

+ 9120572 (minus sin (3119896ℎ) + 8 sin (2119896ℎ) minus 13 sin (119896ℎ))

(26)

The amplification factor which is a complex numberhas its modulus equal to one therefore the finite differencescheme is unconditionally stable

Theorem2 Suppose119906(119909 119905) is smooth enough then the scheme(11)ndash(13) is conservative in a sense

119868119899

1=

ℎ

2

119872minus1

sum

119895=1

(119906119899+1

119895+ 119906119899

119895)

+ 120591ℎ120574

119872minus1

sum

119895=1

[4

9119906119899

119895(119906119899+1

119895)119909minus

1

9119906119899

119895(119906119899+1

119895)119909]

= 119868119899minus1

1= sdot sdot sdot = 119868

0

1

(27)

under assumptions 1199061= 1199062= 119906119872minus2

= 119906119872minus1

= 0 Moreoverthe scheme (19)ndash(22) is conservative in a sense

119868119899

2=

1

2

10038171003817100381710038171199061198991003817100381710038171003817

2

+1

2

10038171003817100381710038171003817119906119899+110038171003817100381710038171003817

2

= 119868119899minus1

2= sdot sdot sdot = 119868

0

2 (28)

Proof By multiplying (11) by ℎ summing up for 119895 from 1 to119872minus1 and considering the boundary condition and assuming1199061= 1199062= 119906119872minus2

= 119906119872minus1

= 0 we have

120591ℎ

119872minus1

sum

119895=1

[8

9(119906119899

119895(119906119899

119895)119909) minus

2

9(119906119899

119895(119906119899+1

119895)119909)]

= 120591ℎ

119872minus1

sum

119895=1

[4

9(119906119899

119895(119906119899+1

119895)119909minus 119906119899minus1

119895(119906119899

119895)119909)

minus1

9(119906119899

119895(119906119899+1

119895)119909minus 119906119899minus1

119895(119906119899

119895)119909)]

(29)

As a result we have

ℎ

2

119872minus1

sum

119895=1

(119906119899+1

119895minus 119906119899minus1

119895)

+ 120574120591ℎ

119872minus1

sum

119895=1

[4

9(119906119899

119895(119906119899+1

119895)119909minus 119906119899minus1

119895(119906119899

119895)119909)

minus1

9(119906119899

119895(119906119899+1

119895)119909minus 119906119899minus1

119895(119906119899

119895)119909)] = 0

(30)

Then this gives (27) We then take an inner product between(19) and 2119906

119899 We obtain

1

2120591(10038171003817100381710038171003817119906119899+110038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899minus110038171003817100381710038171003817

2

) +3120572

2((119906119899)119909119909119909

(119906119899))

minus120572

2((119906119899)119909119909 119909

(119906119899)) + 2120574 (120593

119899(119906119899 119906119899) 119906119899) = 0

(31)

where

120593119899(119906119899

119895 119906119899

119895) =

4

9((119906119899

119895119906119899

119895)119909+ 119906119899

119895(119906119899

119895)119909)

minus1

9((119906119899

119895119906119899

119894)119909+ 119906119899

119895(119906119899

119895)119909)

(32)

by considering the boundary condition (13) According to

(119906119899

119909119909119909 119906119899) = 0

(119906119899

119909119909 119909 119906119899) = 0

(33)

indeed

(120593119899(119906119899 119906119899) 119906119899)

=4ℎ

9

119872minus1

sum

119895=1

[119906119899

119895(119906119899+1

119895)119909+ (119906119899

119895119906119899+1

119895)119909] 119906119899+1

119895

minusℎ

9

119872minus1

sum

119895=1

[119906119899

119895(119906119899+1

119895)119909+ (119906119899

119895119906119899+1

119895)119909] 119906119899+1

119895

=2

9

119872minus1

sum

119895=1

[(119906119899

119895119906119899+1

119895119906119899+1

119895+1minus 119906119899

119895minus1119906119899+1

119895minus1119906119899+1

119895)

+ (119906119899

119895+1119906119899+1

119895119906119899+1

119895+1minus 119906119899

119895119906119899+1

119895minus1119906119899+1

119895)]

minus1

36

119872minus1

sum

119895=1

[(119906119899

119895119906119899+1

119895119906119899+1

119895+2minus 119906119899

119895minus2119906119899+1

119895minus2119906119899+1

119895)

+ (119906119899

119895+2119906119899+1

119895119906119899+1

119895+2minus 119906119899

119895119906119899+1

119895minus2119906119899+1

119895)]

= 0

(34)

Therefore

1

2120591(10038171003817100381710038171003817119906119899+110038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899minus110038171003817100381710038171003817

2

) = 0 (35)

Then this gives (28)

A conservative approximation confirms that the energywould not increase in time which allows making the schemestable

3 Numerical Experiments

In this section we present numerical experiments on theclassical KdV equation when 120572 = 1 and 120574 = 3 with both dif-ference schemes The accuracy of the methods is measured

Mathematical Problems in Engineering 5

Table 1 Error and convergence rate of the compact finite difference scheme (11) at 119905 = 60 ℎ = 05 and 120591 = 025

120591 ℎ 1205914 ℎ2 12059116 ℎ4119890 139538 times 10

minus2715872 times 10

minus4449013 times 10

minus5

Rate mdash 428481 399487

119890infin

764991 times 10minus3

332024 times 10minus4

208869 times 10minus5

Rate mdash 452608 399062

Table 2 Error and convergence rate of the standard fourth-order finite difference scheme (19) at 119905 = 60 ℎ = 05 and 120591 = 025

120591 ℎ 1205914 ℎ2 12059116 ℎ4119890 159924 times 10

minus1979739 times 10

minus3609352 times 10

minus4

Rate mdash 402885 400705

119890infin

863999 times 10minus2

533149 times 10minus3

333067 times 10minus4

Rate mdash 401842 400066

Table 3 Invariants of 1198681 1198682 and 119868

3of the compact fourth-order finite difference scheme (11)

119905 1198681

1198682

1198683

0 20000000000 06666666667 1205883634610 19999449243 06666680888 1205920147320 20001106778 06666680896 1205918697830 19999055324 06666679386 1205915516740 20001880153 06666680804 1205919379150 19999670401 06666680255 1205926253860 19998768932 06666679688 12059162036

by the comparison of numerical solutions with the exactsolutions as well as other numerical solutions from methodsin the literatures by using sdot and sdot

infinnorm The initial

conditions for each problem are chosen in such a way that theexact solutions can be explicitly computed In case 120572 = 1 and120574 = 3 the KdV equation has the analytical solution as

119906 (119909 119905) = 05 sech2 (05 (119909 minus 119905)) (36)

Therefore the initial condition of (1) takes the form

1199060(119909) = 05 sech2 (05 (119909)) (37)

For these particular experiments we set 119909119871

= minus40119909119877

= 100 and 119879 = 60 We make a comparison between thecompact fourth-order finite difference scheme (11) and thestandard fourth-order finite difference scheme (19) So theresults on this experiment in terms of errors at the time 119905 = 60

is reported in Tables 1 and 2 respectively It is clear that theresults obtained by the compact fourth-order differencescheme (11) are more accurate than the ones obtained by thestandard fourth-order difference scheme but the estimationof the rate of convergence for both schemes is close to the the-oretically predicted fourth-order rate of convergence It canbe seen that the computational efficiency of the scheme (11) isbetter than that of the scheme (19) in terms of error

Conservative approximation that is a supplementaryconstraint is essential for a suitable difference equation tomake a discrete analogue effective to the fundamental con-servation properties of the governing equationThen we can

calculate three conservative approximations by using discreteforms as follows

1198681asymp

ℎ

2

119872

sum

119895=1

(119906119899+1

119895+ 119906119899

119895)

1198682asymp

ℎ

2

119872

sum

119895=1

[(119906119899+1

119895)2

+ (119906119899

119895)2

]

1198683asymp ℎ

119872

sum

119895=1

[

[

2120574(

(119906119899+1

119895)3

+ (119906119899

119895)3

2)

minus3120572(

(119906119899+1

119895)2

119909+ (119906119899

119895)2

119909

2)]

]

(38)

Here we take ℎ = 025 and 120591 = ℎ2 at 119905 isin [0 60] for the com-

pact fourth-order finite difference scheme (11) and the stan-dard fourth-order finite difference scheme (19) and resultsare presented in Tables 3 and 4 respectively The numericalresults show that both two schemes can preserve the discreteconservation properties

The second-order explicit scheme (Z-K scheme) and thesecond-order implicit scheme (Goda scheme) are used fortesting the numerical performance of the new schemes InFigure 1 we see that the Z-K scheme computes reasonablesolutions using ℎ = 01 and 120591 = 001 except that the approx-imate solution at 119905 = 01 does not maintain the shape ofthe exact solution Similar calculations at 119905 = 01 and 119905 = 011

6 Mathematical Problems in Engineering

Table 4 Invariants of 1198681 1198682 and 119868

3of the standard fourth-order finite difference scheme (19)

119905 1198681

1198682

1198683

0 20000000000 06666666667 1205883634610 20000527573 06666666667 1205911524120 20000219448 06666666667 1205912578330 19999931738 06666666667 1205910591540 20001264687 06666666667 1205909947750 19999456225 06666666667 1205911628160 19998875333 06666666667 12059106816

0002

004006

00801

020

0010203040506

t

x

u(xt)

minus20

Figure 1 Explicit solutions using the Z-K scheme at 119905 isin [0 01]119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

0

01

02

03

04

05

06

07

minus01minus15 minus10 minus5 0 5 10 15

Figure 2 Explicit solution using the Z-K scheme at 10 time steps119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

are demonstrated in Figures 2 and 3 respectively The figuresshow that numerical waveforms begin to oscillate at 119905 = 01

and show a blowup when 119905 = 011 According to the resultsthe Z-K scheme is numerically unstable regardless of howsmall time increment is

As shown in Figure 2 the results of the Z-K scheme aregreatly fluctuating at 10 time steps Therefore It can not beused to predict the behavior of the solution at long timeFigures 4 and 5 present the numerical solutions by using the

10

10 15

5

5

0

0

minus5

minus5

minus10

minus10minus15

Figure 3 Explicit solution using the Z-K scheme at 11 time steps119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

02

46

810

020

0

01

02

03

04

05

06

tx

u(xt)

minus20

Figure 4 Implicit solutions using the Goda scheme at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

Goda schemeWe see that the Goda scheme can run very wellat ℎ = 05 and 120591 = 025 However the result is still slightlyoscillate at the left side of the solution

Using the same parameters as the Goda scheme Figures 6and 7 present waveforms with 119905 isin [0 10] The result obtainedby the fourth-order difference schemes is greatly improvedcompared to that obtained by the second-order schemes

Figure 8 shows the numerical solution at 119905 = 200 Theresult from the compact fourth-order difference scheme (11)is almost perfectly sharp From the point of view for the long

Mathematical Problems in Engineering 7

0

005

01

015

02

025

03

035

04

045

minus40minus005

minus30 minus20 minus10 0 2010 30 40 50

Figure 5 Implicit solution using the Goda scheme at 119905 = 10 119909119871=

minus40 119909119877= 100 ℎ = 05 and 120591 = 025

02

46

810

0

20

0

01

02

03

04

05

06

tx minus20

u(xt)

Figure 6 Numerical solutions using the scheme (11) at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

time behavior of the resolution the compact fourth-orderdifference scheme (11) can be seen to be much better than thestandard implicit fourth-order scheme (19)

The results of this section suffice to claim that bothnumerical implementations offer a valid approach toward thenumerical investigation of a solution of the KdV equationespecially for the compact finite difference method

4 Conclusion

Two conservative finite difference schemes for the KdV equa-tion are introduced and analyzed The construction of thecompact finite difference scheme (11) requires only a regularfive-point stencil at higher time level which is similar to thestandard second-order Crank-Nicolson scheme the explicitscheme [16] and the implicit scheme [18] However the con-struction of the standard fourth-order scheme (19) requires aseven-point stencil at higher time levelThe accuracy and sta-bility of the numerical schemes for the solutions of the KdV

02

46

810

0

20

0

01

02

03

04

05

06

tx minus20

u(xt)

Figure 7 Numerical solutions using the scheme (19) at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

190 195 200 205 210

0

01

02

03

04

05

06

Scheme (11)Scheme (19)Exact solution

minus01

Figure 8 Numerical solutions at 119905 = 200 119909119871

= minus40 119909119877

= 300ℎ = 05 and 120591 = 025

equation can be tested by using the exact solution In thepaper the numerical experiments show that the presentmethods support the analysis of convergence rate The per-formance of the fourth-order schemes is well efficient at longtime by comparing with the second-order schemes [16 18]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was supported by Chiang Mai University

References

[1] D J Korteweg and G de Vries ldquoOn the change of form of longwaves advancing in a rectangular canal and on a new type of

8 Mathematical Problems in Engineering

long stationary waverdquo Philosophical Magazine vol 39 pp 422ndash449 1895

[2] D Pavoni ldquoSingle and multidomain Chebyshev collocationmethods for the Korteweg-de Vries equationrdquo Calcolo vol 25no 4 pp 311ndash346 1988

[3] A A Soliman ldquoCollocation solution of the Korteweg-de Vriesequation using septic splinesrdquo International Journal of ComputerMathematics vol 81 no 3 pp 325ndash331 2004

[4] H Kalisch and X Raynaud ldquoOn the rate of convergence ofa collocation projection of the KdV equationrdquo MathematicalModelling andNumerical Analysis vol 41 no 1 pp 95ndash110 2007

[5] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[6] L R T Gardner G A Gardner and A H A Ali ldquoSimulationsof solitons using quadratic spline finite elementsrdquo ComputerMethods in Applied Mechanics and Engineering vol 92 no 2pp 231ndash243 1991

[7] M E Alexander and J L Morris ldquoGalerkin methods applied tosomemodel equations for non-linear dispersive wavesrdquo Journalof Computational Physics vol 30 no 3 pp 428ndash451 1979

[8] S R Barros and J W Cardenas ldquoA nonlinear Galerkin methodfor the shallow-water equations on periodic domainsrdquo Journalof Computational Physics vol 172 no 2 pp 592ndash608 2001

[9] H Ma and W Sun ldquoA Legendre-Petrov-Galerkin and Cheby-shev collocation method for third-order differential equationsrdquoSIAM Journal on Numerical Analysis vol 38 no 5 pp 1425ndash1438 2000

[10] J Shen ldquoA new dual-Petrov-Galerkin method for third andhigher odd-order differential equations application to the KDVequationrdquo SIAM Journal onNumerical Analysis vol 41 no 5 pp1595ndash1619 2003

[11] W Heinrichs ldquoSpectral approximation of third-order prob-lemsrdquo Journal of Scientific Computing vol 14 no 3 pp 275ndash2891999

[12] YMaday andAQuarteroni ldquoError analysis for spectral approx-imation of the Korteweg-de Vries equationrdquo ModelisationMathematique et Analyse numerique vol 22 no 3 pp 499ndash5291988

[13] S Zhu ldquoA scheme with a higher-order discrete invariant for theKdV equationrdquo Applied Mathematics Letters vol 14 no 1 pp17ndash20 2001

[14] F-l Qu andW-q Wang ldquoAlternating segment explicit-implicitscheme for nonlinear third-orderKdV equationrdquoAppliedMath-ematics and Mechanics English Edition vol 28 no 7 pp 973ndash980 2007

[15] H-P Wang Y-S Wang and Y-Y Hu ldquoAn explicit scheme forthe KdV equationrdquo Chinese Physics Letters vol 25 no 7 pp2335ndash2338 2008

[16] N J Zabusky and M D Kruskal ldquoInteraction of ldquosolitonsrdquo in acollisionless plasma and the recurrence of initial statesrdquoPhysicalReview Letters vol 15 no 6 pp 240ndash243 1965

[17] O Kolebaje and O Oyewande ldquoNumerical solution of theKorteweg-de Vries equation by finite differenece an d adomaindecomposition methodrdquo International Journal of Basic andApplied Sciences vol 1 no 3 pp 321ndash335 2012

[18] K Goda ldquoOn stability of some finite difference schemes for theKorteweg-de Vries equationrdquo Journal of the Physical Society ofJapan vol 39 no 1 pp 229ndash236 1975

[19] S Hamdi W H Enright W E Schiesser and J J GottliebldquoExact solutions and conservation laws for coupled generalized

Korteweg-de Vries and quintic regularized long wave equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 63 no 5ndash7 pp e1425ndashe1434 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

The Von Neumann stability analysis of (19) with 119906119899

119895=

120585119899119890119894119896119895ℎ gives the following amplification factor

1205852=

36ℎ3minus 119894120591119860

36ℎ3 + 119894120591119860 (25)

where

119860 = 4120574ℎ2(119906119899

119895) (minus sin (4119896ℎ) + 7 sin (2119896ℎ) + 8 sin (119896ℎ))

+ 9120572 (minus sin (3119896ℎ) + 8 sin (2119896ℎ) minus 13 sin (119896ℎ))

(26)

The amplification factor which is a complex numberhas its modulus equal to one therefore the finite differencescheme is unconditionally stable

Theorem2 Suppose119906(119909 119905) is smooth enough then the scheme(11)ndash(13) is conservative in a sense

119868119899

1=

ℎ

2

119872minus1

sum

119895=1

(119906119899+1

119895+ 119906119899

119895)

+ 120591ℎ120574

119872minus1

sum

119895=1

[4

9119906119899

119895(119906119899+1

119895)119909minus

1

9119906119899

119895(119906119899+1

119895)119909]

= 119868119899minus1

1= sdot sdot sdot = 119868

0

1

(27)

under assumptions 1199061= 1199062= 119906119872minus2

= 119906119872minus1

= 0 Moreoverthe scheme (19)ndash(22) is conservative in a sense

119868119899

2=

1

2

10038171003817100381710038171199061198991003817100381710038171003817

2

+1

2

10038171003817100381710038171003817119906119899+110038171003817100381710038171003817

2

= 119868119899minus1

2= sdot sdot sdot = 119868

0

2 (28)

Proof By multiplying (11) by ℎ summing up for 119895 from 1 to119872minus1 and considering the boundary condition and assuming1199061= 1199062= 119906119872minus2

= 119906119872minus1

= 0 we have

120591ℎ

119872minus1

sum

119895=1

[8

9(119906119899

119895(119906119899

119895)119909) minus

2

9(119906119899

119895(119906119899+1

119895)119909)]

= 120591ℎ

119872minus1

sum

119895=1

[4

9(119906119899

119895(119906119899+1

119895)119909minus 119906119899minus1

119895(119906119899

119895)119909)

minus1

9(119906119899

119895(119906119899+1

119895)119909minus 119906119899minus1

119895(119906119899

119895)119909)]

(29)

As a result we have

ℎ

2

119872minus1

sum

119895=1

(119906119899+1

119895minus 119906119899minus1

119895)

+ 120574120591ℎ

119872minus1

sum

119895=1

[4

9(119906119899

119895(119906119899+1

119895)119909minus 119906119899minus1

119895(119906119899

119895)119909)

minus1

9(119906119899

119895(119906119899+1

119895)119909minus 119906119899minus1

119895(119906119899

119895)119909)] = 0

(30)

Then this gives (27) We then take an inner product between(19) and 2119906

119899 We obtain

1

2120591(10038171003817100381710038171003817119906119899+110038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899minus110038171003817100381710038171003817

2

) +3120572

2((119906119899)119909119909119909

(119906119899))

minus120572

2((119906119899)119909119909 119909

(119906119899)) + 2120574 (120593

119899(119906119899 119906119899) 119906119899) = 0

(31)

where

120593119899(119906119899

119895 119906119899

119895) =

4

9((119906119899

119895119906119899

119895)119909+ 119906119899

119895(119906119899

119895)119909)

minus1

9((119906119899

119895119906119899

119894)119909+ 119906119899

119895(119906119899

119895)119909)

(32)

by considering the boundary condition (13) According to

(119906119899

119909119909119909 119906119899) = 0

(119906119899

119909119909 119909 119906119899) = 0

(33)

indeed

(120593119899(119906119899 119906119899) 119906119899)

=4ℎ

9

119872minus1

sum

119895=1

[119906119899

119895(119906119899+1

119895)119909+ (119906119899

119895119906119899+1

119895)119909] 119906119899+1

119895

minusℎ

9

119872minus1

sum

119895=1

[119906119899

119895(119906119899+1

119895)119909+ (119906119899

119895119906119899+1

119895)119909] 119906119899+1

119895

=2

9

119872minus1

sum

119895=1

[(119906119899

119895119906119899+1

119895119906119899+1

119895+1minus 119906119899

119895minus1119906119899+1

119895minus1119906119899+1

119895)

+ (119906119899

119895+1119906119899+1

119895119906119899+1

119895+1minus 119906119899

119895119906119899+1

119895minus1119906119899+1

119895)]

minus1

36

119872minus1

sum

119895=1

[(119906119899

119895119906119899+1

119895119906119899+1

119895+2minus 119906119899

119895minus2119906119899+1

119895minus2119906119899+1

119895)

+ (119906119899

119895+2119906119899+1

119895119906119899+1

119895+2minus 119906119899

119895119906119899+1

119895minus2119906119899+1

119895)]

= 0

(34)

Therefore

1

2120591(10038171003817100381710038171003817119906119899+110038171003817100381710038171003817

2

minus10038171003817100381710038171003817119906119899minus110038171003817100381710038171003817

2

) = 0 (35)

Then this gives (28)

A conservative approximation confirms that the energywould not increase in time which allows making the schemestable

3 Numerical Experiments

In this section we present numerical experiments on theclassical KdV equation when 120572 = 1 and 120574 = 3 with both dif-ference schemes The accuracy of the methods is measured

Mathematical Problems in Engineering 5

Table 1 Error and convergence rate of the compact finite difference scheme (11) at 119905 = 60 ℎ = 05 and 120591 = 025

120591 ℎ 1205914 ℎ2 12059116 ℎ4119890 139538 times 10

minus2715872 times 10

minus4449013 times 10

minus5

Rate mdash 428481 399487

119890infin

764991 times 10minus3

332024 times 10minus4

208869 times 10minus5

Rate mdash 452608 399062

Table 2 Error and convergence rate of the standard fourth-order finite difference scheme (19) at 119905 = 60 ℎ = 05 and 120591 = 025

120591 ℎ 1205914 ℎ2 12059116 ℎ4119890 159924 times 10

minus1979739 times 10

minus3609352 times 10

minus4

Rate mdash 402885 400705

119890infin

863999 times 10minus2

533149 times 10minus3

333067 times 10minus4

Rate mdash 401842 400066

Table 3 Invariants of 1198681 1198682 and 119868

3of the compact fourth-order finite difference scheme (11)

119905 1198681

1198682

1198683

0 20000000000 06666666667 1205883634610 19999449243 06666680888 1205920147320 20001106778 06666680896 1205918697830 19999055324 06666679386 1205915516740 20001880153 06666680804 1205919379150 19999670401 06666680255 1205926253860 19998768932 06666679688 12059162036

by the comparison of numerical solutions with the exactsolutions as well as other numerical solutions from methodsin the literatures by using sdot and sdot

infinnorm The initial

conditions for each problem are chosen in such a way that theexact solutions can be explicitly computed In case 120572 = 1 and120574 = 3 the KdV equation has the analytical solution as

119906 (119909 119905) = 05 sech2 (05 (119909 minus 119905)) (36)

Therefore the initial condition of (1) takes the form

1199060(119909) = 05 sech2 (05 (119909)) (37)

For these particular experiments we set 119909119871

= minus40119909119877

= 100 and 119879 = 60 We make a comparison between thecompact fourth-order finite difference scheme (11) and thestandard fourth-order finite difference scheme (19) So theresults on this experiment in terms of errors at the time 119905 = 60

is reported in Tables 1 and 2 respectively It is clear that theresults obtained by the compact fourth-order differencescheme (11) are more accurate than the ones obtained by thestandard fourth-order difference scheme but the estimationof the rate of convergence for both schemes is close to the the-oretically predicted fourth-order rate of convergence It canbe seen that the computational efficiency of the scheme (11) isbetter than that of the scheme (19) in terms of error

Conservative approximation that is a supplementaryconstraint is essential for a suitable difference equation tomake a discrete analogue effective to the fundamental con-servation properties of the governing equationThen we can

calculate three conservative approximations by using discreteforms as follows

1198681asymp

ℎ

2

119872

sum

119895=1

(119906119899+1

119895+ 119906119899

119895)

1198682asymp

ℎ

2

119872

sum

119895=1

[(119906119899+1

119895)2

+ (119906119899

119895)2

]

1198683asymp ℎ

119872

sum

119895=1

[

[

2120574(

(119906119899+1

119895)3

+ (119906119899

119895)3

2)

minus3120572(

(119906119899+1

119895)2

119909+ (119906119899

119895)2

119909

2)]

]

(38)

Here we take ℎ = 025 and 120591 = ℎ2 at 119905 isin [0 60] for the com-

pact fourth-order finite difference scheme (11) and the stan-dard fourth-order finite difference scheme (19) and resultsare presented in Tables 3 and 4 respectively The numericalresults show that both two schemes can preserve the discreteconservation properties

The second-order explicit scheme (Z-K scheme) and thesecond-order implicit scheme (Goda scheme) are used fortesting the numerical performance of the new schemes InFigure 1 we see that the Z-K scheme computes reasonablesolutions using ℎ = 01 and 120591 = 001 except that the approx-imate solution at 119905 = 01 does not maintain the shape ofthe exact solution Similar calculations at 119905 = 01 and 119905 = 011

6 Mathematical Problems in Engineering

Table 4 Invariants of 1198681 1198682 and 119868

3of the standard fourth-order finite difference scheme (19)

119905 1198681

1198682

1198683

0 20000000000 06666666667 1205883634610 20000527573 06666666667 1205911524120 20000219448 06666666667 1205912578330 19999931738 06666666667 1205910591540 20001264687 06666666667 1205909947750 19999456225 06666666667 1205911628160 19998875333 06666666667 12059106816

0002

004006

00801

020

0010203040506

t

x

u(xt)

minus20

Figure 1 Explicit solutions using the Z-K scheme at 119905 isin [0 01]119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

0

01

02

03

04

05

06

07

minus01minus15 minus10 minus5 0 5 10 15

Figure 2 Explicit solution using the Z-K scheme at 10 time steps119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

are demonstrated in Figures 2 and 3 respectively The figuresshow that numerical waveforms begin to oscillate at 119905 = 01

and show a blowup when 119905 = 011 According to the resultsthe Z-K scheme is numerically unstable regardless of howsmall time increment is

As shown in Figure 2 the results of the Z-K scheme aregreatly fluctuating at 10 time steps Therefore It can not beused to predict the behavior of the solution at long timeFigures 4 and 5 present the numerical solutions by using the

10

10 15

5

5

0

0

minus5

minus5

minus10

minus10minus15

Figure 3 Explicit solution using the Z-K scheme at 11 time steps119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

02

46

810

020

0

01

02

03

04

05

06

tx

u(xt)

minus20

Figure 4 Implicit solutions using the Goda scheme at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

Goda schemeWe see that the Goda scheme can run very wellat ℎ = 05 and 120591 = 025 However the result is still slightlyoscillate at the left side of the solution

Using the same parameters as the Goda scheme Figures 6and 7 present waveforms with 119905 isin [0 10] The result obtainedby the fourth-order difference schemes is greatly improvedcompared to that obtained by the second-order schemes

Figure 8 shows the numerical solution at 119905 = 200 Theresult from the compact fourth-order difference scheme (11)is almost perfectly sharp From the point of view for the long

Mathematical Problems in Engineering 7

0

005

01

015

02

025

03

035

04

045

minus40minus005

minus30 minus20 minus10 0 2010 30 40 50

Figure 5 Implicit solution using the Goda scheme at 119905 = 10 119909119871=

minus40 119909119877= 100 ℎ = 05 and 120591 = 025

02

46

810

0

20

0

01

02

03

04

05

06

tx minus20

u(xt)

Figure 6 Numerical solutions using the scheme (11) at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

time behavior of the resolution the compact fourth-orderdifference scheme (11) can be seen to be much better than thestandard implicit fourth-order scheme (19)

The results of this section suffice to claim that bothnumerical implementations offer a valid approach toward thenumerical investigation of a solution of the KdV equationespecially for the compact finite difference method

4 Conclusion

Two conservative finite difference schemes for the KdV equa-tion are introduced and analyzed The construction of thecompact finite difference scheme (11) requires only a regularfive-point stencil at higher time level which is similar to thestandard second-order Crank-Nicolson scheme the explicitscheme [16] and the implicit scheme [18] However the con-struction of the standard fourth-order scheme (19) requires aseven-point stencil at higher time levelThe accuracy and sta-bility of the numerical schemes for the solutions of the KdV

02

46

810

0

20

0

01

02

03

04

05

06

tx minus20

u(xt)

Figure 7 Numerical solutions using the scheme (19) at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

190 195 200 205 210

0

01

02

03

04

05

06

Scheme (11)Scheme (19)Exact solution

minus01

Figure 8 Numerical solutions at 119905 = 200 119909119871

= minus40 119909119877

= 300ℎ = 05 and 120591 = 025

equation can be tested by using the exact solution In thepaper the numerical experiments show that the presentmethods support the analysis of convergence rate The per-formance of the fourth-order schemes is well efficient at longtime by comparing with the second-order schemes [16 18]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was supported by Chiang Mai University

References

[1] D J Korteweg and G de Vries ldquoOn the change of form of longwaves advancing in a rectangular canal and on a new type of

8 Mathematical Problems in Engineering

long stationary waverdquo Philosophical Magazine vol 39 pp 422ndash449 1895

[2] D Pavoni ldquoSingle and multidomain Chebyshev collocationmethods for the Korteweg-de Vries equationrdquo Calcolo vol 25no 4 pp 311ndash346 1988

[3] A A Soliman ldquoCollocation solution of the Korteweg-de Vriesequation using septic splinesrdquo International Journal of ComputerMathematics vol 81 no 3 pp 325ndash331 2004

[4] H Kalisch and X Raynaud ldquoOn the rate of convergence ofa collocation projection of the KdV equationrdquo MathematicalModelling andNumerical Analysis vol 41 no 1 pp 95ndash110 2007

[5] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[6] L R T Gardner G A Gardner and A H A Ali ldquoSimulationsof solitons using quadratic spline finite elementsrdquo ComputerMethods in Applied Mechanics and Engineering vol 92 no 2pp 231ndash243 1991

[7] M E Alexander and J L Morris ldquoGalerkin methods applied tosomemodel equations for non-linear dispersive wavesrdquo Journalof Computational Physics vol 30 no 3 pp 428ndash451 1979

[8] S R Barros and J W Cardenas ldquoA nonlinear Galerkin methodfor the shallow-water equations on periodic domainsrdquo Journalof Computational Physics vol 172 no 2 pp 592ndash608 2001

[9] H Ma and W Sun ldquoA Legendre-Petrov-Galerkin and Cheby-shev collocation method for third-order differential equationsrdquoSIAM Journal on Numerical Analysis vol 38 no 5 pp 1425ndash1438 2000

[10] J Shen ldquoA new dual-Petrov-Galerkin method for third andhigher odd-order differential equations application to the KDVequationrdquo SIAM Journal onNumerical Analysis vol 41 no 5 pp1595ndash1619 2003

[11] W Heinrichs ldquoSpectral approximation of third-order prob-lemsrdquo Journal of Scientific Computing vol 14 no 3 pp 275ndash2891999

[12] YMaday andAQuarteroni ldquoError analysis for spectral approx-imation of the Korteweg-de Vries equationrdquo ModelisationMathematique et Analyse numerique vol 22 no 3 pp 499ndash5291988

[13] S Zhu ldquoA scheme with a higher-order discrete invariant for theKdV equationrdquo Applied Mathematics Letters vol 14 no 1 pp17ndash20 2001

[14] F-l Qu andW-q Wang ldquoAlternating segment explicit-implicitscheme for nonlinear third-orderKdV equationrdquoAppliedMath-ematics and Mechanics English Edition vol 28 no 7 pp 973ndash980 2007

[15] H-P Wang Y-S Wang and Y-Y Hu ldquoAn explicit scheme forthe KdV equationrdquo Chinese Physics Letters vol 25 no 7 pp2335ndash2338 2008

[16] N J Zabusky and M D Kruskal ldquoInteraction of ldquosolitonsrdquo in acollisionless plasma and the recurrence of initial statesrdquoPhysicalReview Letters vol 15 no 6 pp 240ndash243 1965

[17] O Kolebaje and O Oyewande ldquoNumerical solution of theKorteweg-de Vries equation by finite differenece an d adomaindecomposition methodrdquo International Journal of Basic andApplied Sciences vol 1 no 3 pp 321ndash335 2012

[18] K Goda ldquoOn stability of some finite difference schemes for theKorteweg-de Vries equationrdquo Journal of the Physical Society ofJapan vol 39 no 1 pp 229ndash236 1975

[19] S Hamdi W H Enright W E Schiesser and J J GottliebldquoExact solutions and conservation laws for coupled generalized

Korteweg-de Vries and quintic regularized long wave equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 63 no 5ndash7 pp e1425ndashe1434 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Table 1 Error and convergence rate of the compact finite difference scheme (11) at 119905 = 60 ℎ = 05 and 120591 = 025

120591 ℎ 1205914 ℎ2 12059116 ℎ4119890 139538 times 10

minus2715872 times 10

minus4449013 times 10

minus5

Rate mdash 428481 399487

119890infin

764991 times 10minus3

332024 times 10minus4

208869 times 10minus5

Rate mdash 452608 399062

Table 2 Error and convergence rate of the standard fourth-order finite difference scheme (19) at 119905 = 60 ℎ = 05 and 120591 = 025

120591 ℎ 1205914 ℎ2 12059116 ℎ4119890 159924 times 10

minus1979739 times 10

minus3609352 times 10

minus4

Rate mdash 402885 400705

119890infin

863999 times 10minus2

533149 times 10minus3

333067 times 10minus4

Rate mdash 401842 400066

Table 3 Invariants of 1198681 1198682 and 119868

3of the compact fourth-order finite difference scheme (11)

119905 1198681

1198682

1198683

0 20000000000 06666666667 1205883634610 19999449243 06666680888 1205920147320 20001106778 06666680896 1205918697830 19999055324 06666679386 1205915516740 20001880153 06666680804 1205919379150 19999670401 06666680255 1205926253860 19998768932 06666679688 12059162036

by the comparison of numerical solutions with the exactsolutions as well as other numerical solutions from methodsin the literatures by using sdot and sdot

infinnorm The initial

conditions for each problem are chosen in such a way that theexact solutions can be explicitly computed In case 120572 = 1 and120574 = 3 the KdV equation has the analytical solution as

119906 (119909 119905) = 05 sech2 (05 (119909 minus 119905)) (36)

Therefore the initial condition of (1) takes the form

1199060(119909) = 05 sech2 (05 (119909)) (37)

For these particular experiments we set 119909119871

= minus40119909119877

= 100 and 119879 = 60 We make a comparison between thecompact fourth-order finite difference scheme (11) and thestandard fourth-order finite difference scheme (19) So theresults on this experiment in terms of errors at the time 119905 = 60

is reported in Tables 1 and 2 respectively It is clear that theresults obtained by the compact fourth-order differencescheme (11) are more accurate than the ones obtained by thestandard fourth-order difference scheme but the estimationof the rate of convergence for both schemes is close to the the-oretically predicted fourth-order rate of convergence It canbe seen that the computational efficiency of the scheme (11) isbetter than that of the scheme (19) in terms of error

Conservative approximation that is a supplementaryconstraint is essential for a suitable difference equation tomake a discrete analogue effective to the fundamental con-servation properties of the governing equationThen we can

calculate three conservative approximations by using discreteforms as follows

1198681asymp

ℎ

2

119872

sum

119895=1

(119906119899+1

119895+ 119906119899

119895)

1198682asymp

ℎ

2

119872

sum

119895=1

[(119906119899+1

119895)2

+ (119906119899

119895)2

]

1198683asymp ℎ

119872

sum

119895=1

[

[

2120574(

(119906119899+1

119895)3

+ (119906119899

119895)3

2)

minus3120572(

(119906119899+1

119895)2

119909+ (119906119899

119895)2

119909

2)]

]

(38)

Here we take ℎ = 025 and 120591 = ℎ2 at 119905 isin [0 60] for the com-

pact fourth-order finite difference scheme (11) and the stan-dard fourth-order finite difference scheme (19) and resultsare presented in Tables 3 and 4 respectively The numericalresults show that both two schemes can preserve the discreteconservation properties

The second-order explicit scheme (Z-K scheme) and thesecond-order implicit scheme (Goda scheme) are used fortesting the numerical performance of the new schemes InFigure 1 we see that the Z-K scheme computes reasonablesolutions using ℎ = 01 and 120591 = 001 except that the approx-imate solution at 119905 = 01 does not maintain the shape ofthe exact solution Similar calculations at 119905 = 01 and 119905 = 011

6 Mathematical Problems in Engineering

Table 4 Invariants of 1198681 1198682 and 119868

3of the standard fourth-order finite difference scheme (19)

119905 1198681

1198682

1198683

0 20000000000 06666666667 1205883634610 20000527573 06666666667 1205911524120 20000219448 06666666667 1205912578330 19999931738 06666666667 1205910591540 20001264687 06666666667 1205909947750 19999456225 06666666667 1205911628160 19998875333 06666666667 12059106816

0002

004006

00801

020

0010203040506

t

x

u(xt)

minus20

Figure 1 Explicit solutions using the Z-K scheme at 119905 isin [0 01]119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

0

01

02

03

04

05

06

07

minus01minus15 minus10 minus5 0 5 10 15

Figure 2 Explicit solution using the Z-K scheme at 10 time steps119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

are demonstrated in Figures 2 and 3 respectively The figuresshow that numerical waveforms begin to oscillate at 119905 = 01

and show a blowup when 119905 = 011 According to the resultsthe Z-K scheme is numerically unstable regardless of howsmall time increment is

As shown in Figure 2 the results of the Z-K scheme aregreatly fluctuating at 10 time steps Therefore It can not beused to predict the behavior of the solution at long timeFigures 4 and 5 present the numerical solutions by using the

10

10 15

5

5

0

0

minus5

minus5

minus10

minus10minus15

Figure 3 Explicit solution using the Z-K scheme at 11 time steps119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

02

46

810

020

0

01

02

03

04

05

06

tx

u(xt)

minus20

Figure 4 Implicit solutions using the Goda scheme at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

Goda schemeWe see that the Goda scheme can run very wellat ℎ = 05 and 120591 = 025 However the result is still slightlyoscillate at the left side of the solution

Using the same parameters as the Goda scheme Figures 6and 7 present waveforms with 119905 isin [0 10] The result obtainedby the fourth-order difference schemes is greatly improvedcompared to that obtained by the second-order schemes

Figure 8 shows the numerical solution at 119905 = 200 Theresult from the compact fourth-order difference scheme (11)is almost perfectly sharp From the point of view for the long

Mathematical Problems in Engineering 7

0

005

01

015

02

025

03

035

04

045

minus40minus005

minus30 minus20 minus10 0 2010 30 40 50

Figure 5 Implicit solution using the Goda scheme at 119905 = 10 119909119871=

minus40 119909119877= 100 ℎ = 05 and 120591 = 025

02

46

810

0

20

0

01

02

03

04

05

06

tx minus20

u(xt)

Figure 6 Numerical solutions using the scheme (11) at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

time behavior of the resolution the compact fourth-orderdifference scheme (11) can be seen to be much better than thestandard implicit fourth-order scheme (19)

The results of this section suffice to claim that bothnumerical implementations offer a valid approach toward thenumerical investigation of a solution of the KdV equationespecially for the compact finite difference method

4 Conclusion

Two conservative finite difference schemes for the KdV equa-tion are introduced and analyzed The construction of thecompact finite difference scheme (11) requires only a regularfive-point stencil at higher time level which is similar to thestandard second-order Crank-Nicolson scheme the explicitscheme [16] and the implicit scheme [18] However the con-struction of the standard fourth-order scheme (19) requires aseven-point stencil at higher time levelThe accuracy and sta-bility of the numerical schemes for the solutions of the KdV

02

46

810

0

20

0

01

02

03

04

05

06

tx minus20

u(xt)

Figure 7 Numerical solutions using the scheme (19) at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

190 195 200 205 210

0

01

02

03

04

05

06

Scheme (11)Scheme (19)Exact solution

minus01

Figure 8 Numerical solutions at 119905 = 200 119909119871

= minus40 119909119877

= 300ℎ = 05 and 120591 = 025

equation can be tested by using the exact solution In thepaper the numerical experiments show that the presentmethods support the analysis of convergence rate The per-formance of the fourth-order schemes is well efficient at longtime by comparing with the second-order schemes [16 18]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was supported by Chiang Mai University

References

[1] D J Korteweg and G de Vries ldquoOn the change of form of longwaves advancing in a rectangular canal and on a new type of

8 Mathematical Problems in Engineering

long stationary waverdquo Philosophical Magazine vol 39 pp 422ndash449 1895

[2] D Pavoni ldquoSingle and multidomain Chebyshev collocationmethods for the Korteweg-de Vries equationrdquo Calcolo vol 25no 4 pp 311ndash346 1988

[3] A A Soliman ldquoCollocation solution of the Korteweg-de Vriesequation using septic splinesrdquo International Journal of ComputerMathematics vol 81 no 3 pp 325ndash331 2004

[4] H Kalisch and X Raynaud ldquoOn the rate of convergence ofa collocation projection of the KdV equationrdquo MathematicalModelling andNumerical Analysis vol 41 no 1 pp 95ndash110 2007

[5] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[6] L R T Gardner G A Gardner and A H A Ali ldquoSimulationsof solitons using quadratic spline finite elementsrdquo ComputerMethods in Applied Mechanics and Engineering vol 92 no 2pp 231ndash243 1991

[7] M E Alexander and J L Morris ldquoGalerkin methods applied tosomemodel equations for non-linear dispersive wavesrdquo Journalof Computational Physics vol 30 no 3 pp 428ndash451 1979

[8] S R Barros and J W Cardenas ldquoA nonlinear Galerkin methodfor the shallow-water equations on periodic domainsrdquo Journalof Computational Physics vol 172 no 2 pp 592ndash608 2001

[9] H Ma and W Sun ldquoA Legendre-Petrov-Galerkin and Cheby-shev collocation method for third-order differential equationsrdquoSIAM Journal on Numerical Analysis vol 38 no 5 pp 1425ndash1438 2000

[10] J Shen ldquoA new dual-Petrov-Galerkin method for third andhigher odd-order differential equations application to the KDVequationrdquo SIAM Journal onNumerical Analysis vol 41 no 5 pp1595ndash1619 2003

[11] W Heinrichs ldquoSpectral approximation of third-order prob-lemsrdquo Journal of Scientific Computing vol 14 no 3 pp 275ndash2891999

[12] YMaday andAQuarteroni ldquoError analysis for spectral approx-imation of the Korteweg-de Vries equationrdquo ModelisationMathematique et Analyse numerique vol 22 no 3 pp 499ndash5291988

[13] S Zhu ldquoA scheme with a higher-order discrete invariant for theKdV equationrdquo Applied Mathematics Letters vol 14 no 1 pp17ndash20 2001

[14] F-l Qu andW-q Wang ldquoAlternating segment explicit-implicitscheme for nonlinear third-orderKdV equationrdquoAppliedMath-ematics and Mechanics English Edition vol 28 no 7 pp 973ndash980 2007

[15] H-P Wang Y-S Wang and Y-Y Hu ldquoAn explicit scheme forthe KdV equationrdquo Chinese Physics Letters vol 25 no 7 pp2335ndash2338 2008

[16] N J Zabusky and M D Kruskal ldquoInteraction of ldquosolitonsrdquo in acollisionless plasma and the recurrence of initial statesrdquoPhysicalReview Letters vol 15 no 6 pp 240ndash243 1965

[17] O Kolebaje and O Oyewande ldquoNumerical solution of theKorteweg-de Vries equation by finite differenece an d adomaindecomposition methodrdquo International Journal of Basic andApplied Sciences vol 1 no 3 pp 321ndash335 2012

[18] K Goda ldquoOn stability of some finite difference schemes for theKorteweg-de Vries equationrdquo Journal of the Physical Society ofJapan vol 39 no 1 pp 229ndash236 1975

[19] S Hamdi W H Enright W E Schiesser and J J GottliebldquoExact solutions and conservation laws for coupled generalized

Korteweg-de Vries and quintic regularized long wave equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 63 no 5ndash7 pp e1425ndashe1434 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Table 4 Invariants of 1198681 1198682 and 119868

3of the standard fourth-order finite difference scheme (19)

119905 1198681

1198682

1198683

0 20000000000 06666666667 1205883634610 20000527573 06666666667 1205911524120 20000219448 06666666667 1205912578330 19999931738 06666666667 1205910591540 20001264687 06666666667 1205909947750 19999456225 06666666667 1205911628160 19998875333 06666666667 12059106816

0002

004006

00801

020

0010203040506

t

x

u(xt)

minus20

Figure 1 Explicit solutions using the Z-K scheme at 119905 isin [0 01]119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

0

01

02

03

04

05

06

07

minus01minus15 minus10 minus5 0 5 10 15

Figure 2 Explicit solution using the Z-K scheme at 10 time steps119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

are demonstrated in Figures 2 and 3 respectively The figuresshow that numerical waveforms begin to oscillate at 119905 = 01

and show a blowup when 119905 = 011 According to the resultsthe Z-K scheme is numerically unstable regardless of howsmall time increment is

As shown in Figure 2 the results of the Z-K scheme aregreatly fluctuating at 10 time steps Therefore It can not beused to predict the behavior of the solution at long timeFigures 4 and 5 present the numerical solutions by using the

10

10 15

5

5

0

0

minus5

minus5

minus10

minus10minus15

Figure 3 Explicit solution using the Z-K scheme at 11 time steps119909119871= minus40 119909

119877= 100 ℎ = 01 and 120591 = 001

02

46

810

020

0

01

02

03

04

05

06

tx

u(xt)

minus20

Figure 4 Implicit solutions using the Goda scheme at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

Goda schemeWe see that the Goda scheme can run very wellat ℎ = 05 and 120591 = 025 However the result is still slightlyoscillate at the left side of the solution

Using the same parameters as the Goda scheme Figures 6and 7 present waveforms with 119905 isin [0 10] The result obtainedby the fourth-order difference schemes is greatly improvedcompared to that obtained by the second-order schemes

Figure 8 shows the numerical solution at 119905 = 200 Theresult from the compact fourth-order difference scheme (11)is almost perfectly sharp From the point of view for the long

Mathematical Problems in Engineering 7

0

005

01

015

02

025

03

035

04

045

minus40minus005

minus30 minus20 minus10 0 2010 30 40 50

Figure 5 Implicit solution using the Goda scheme at 119905 = 10 119909119871=

minus40 119909119877= 100 ℎ = 05 and 120591 = 025

02

46

810

0

20

0

01

02

03

04

05

06

tx minus20

u(xt)

Figure 6 Numerical solutions using the scheme (11) at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

time behavior of the resolution the compact fourth-orderdifference scheme (11) can be seen to be much better than thestandard implicit fourth-order scheme (19)

The results of this section suffice to claim that bothnumerical implementations offer a valid approach toward thenumerical investigation of a solution of the KdV equationespecially for the compact finite difference method

4 Conclusion

Two conservative finite difference schemes for the KdV equa-tion are introduced and analyzed The construction of thecompact finite difference scheme (11) requires only a regularfive-point stencil at higher time level which is similar to thestandard second-order Crank-Nicolson scheme the explicitscheme [16] and the implicit scheme [18] However the con-struction of the standard fourth-order scheme (19) requires aseven-point stencil at higher time levelThe accuracy and sta-bility of the numerical schemes for the solutions of the KdV

02

46

810

0

20

0

01

02

03

04

05

06

tx minus20

u(xt)

Figure 7 Numerical solutions using the scheme (19) at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

190 195 200 205 210

0

01

02

03

04

05

06

Scheme (11)Scheme (19)Exact solution

minus01

Figure 8 Numerical solutions at 119905 = 200 119909119871

= minus40 119909119877

= 300ℎ = 05 and 120591 = 025

equation can be tested by using the exact solution In thepaper the numerical experiments show that the presentmethods support the analysis of convergence rate The per-formance of the fourth-order schemes is well efficient at longtime by comparing with the second-order schemes [16 18]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was supported by Chiang Mai University

References

[1] D J Korteweg and G de Vries ldquoOn the change of form of longwaves advancing in a rectangular canal and on a new type of

8 Mathematical Problems in Engineering

long stationary waverdquo Philosophical Magazine vol 39 pp 422ndash449 1895

[2] D Pavoni ldquoSingle and multidomain Chebyshev collocationmethods for the Korteweg-de Vries equationrdquo Calcolo vol 25no 4 pp 311ndash346 1988

[3] A A Soliman ldquoCollocation solution of the Korteweg-de Vriesequation using septic splinesrdquo International Journal of ComputerMathematics vol 81 no 3 pp 325ndash331 2004

[4] H Kalisch and X Raynaud ldquoOn the rate of convergence ofa collocation projection of the KdV equationrdquo MathematicalModelling andNumerical Analysis vol 41 no 1 pp 95ndash110 2007

[5] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[6] L R T Gardner G A Gardner and A H A Ali ldquoSimulationsof solitons using quadratic spline finite elementsrdquo ComputerMethods in Applied Mechanics and Engineering vol 92 no 2pp 231ndash243 1991

[7] M E Alexander and J L Morris ldquoGalerkin methods applied tosomemodel equations for non-linear dispersive wavesrdquo Journalof Computational Physics vol 30 no 3 pp 428ndash451 1979

[8] S R Barros and J W Cardenas ldquoA nonlinear Galerkin methodfor the shallow-water equations on periodic domainsrdquo Journalof Computational Physics vol 172 no 2 pp 592ndash608 2001

[9] H Ma and W Sun ldquoA Legendre-Petrov-Galerkin and Cheby-shev collocation method for third-order differential equationsrdquoSIAM Journal on Numerical Analysis vol 38 no 5 pp 1425ndash1438 2000

[10] J Shen ldquoA new dual-Petrov-Galerkin method for third andhigher odd-order differential equations application to the KDVequationrdquo SIAM Journal onNumerical Analysis vol 41 no 5 pp1595ndash1619 2003

[11] W Heinrichs ldquoSpectral approximation of third-order prob-lemsrdquo Journal of Scientific Computing vol 14 no 3 pp 275ndash2891999

[12] YMaday andAQuarteroni ldquoError analysis for spectral approx-imation of the Korteweg-de Vries equationrdquo ModelisationMathematique et Analyse numerique vol 22 no 3 pp 499ndash5291988

[13] S Zhu ldquoA scheme with a higher-order discrete invariant for theKdV equationrdquo Applied Mathematics Letters vol 14 no 1 pp17ndash20 2001

[14] F-l Qu andW-q Wang ldquoAlternating segment explicit-implicitscheme for nonlinear third-orderKdV equationrdquoAppliedMath-ematics and Mechanics English Edition vol 28 no 7 pp 973ndash980 2007

[15] H-P Wang Y-S Wang and Y-Y Hu ldquoAn explicit scheme forthe KdV equationrdquo Chinese Physics Letters vol 25 no 7 pp2335ndash2338 2008

[16] N J Zabusky and M D Kruskal ldquoInteraction of ldquosolitonsrdquo in acollisionless plasma and the recurrence of initial statesrdquoPhysicalReview Letters vol 15 no 6 pp 240ndash243 1965

[17] O Kolebaje and O Oyewande ldquoNumerical solution of theKorteweg-de Vries equation by finite differenece an d adomaindecomposition methodrdquo International Journal of Basic andApplied Sciences vol 1 no 3 pp 321ndash335 2012

[18] K Goda ldquoOn stability of some finite difference schemes for theKorteweg-de Vries equationrdquo Journal of the Physical Society ofJapan vol 39 no 1 pp 229ndash236 1975

[19] S Hamdi W H Enright W E Schiesser and J J GottliebldquoExact solutions and conservation laws for coupled generalized

Korteweg-de Vries and quintic regularized long wave equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 63 no 5ndash7 pp e1425ndashe1434 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0

005

01

015

02

025

03

035

04

045

minus40minus005

minus30 minus20 minus10 0 2010 30 40 50

Figure 5 Implicit solution using the Goda scheme at 119905 = 10 119909119871=

minus40 119909119877= 100 ℎ = 05 and 120591 = 025

02

46

810

0

20

0

01

02

03

04

05

06

tx minus20

u(xt)

Figure 6 Numerical solutions using the scheme (11) at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

time behavior of the resolution the compact fourth-orderdifference scheme (11) can be seen to be much better than thestandard implicit fourth-order scheme (19)

The results of this section suffice to claim that bothnumerical implementations offer a valid approach toward thenumerical investigation of a solution of the KdV equationespecially for the compact finite difference method

4 Conclusion

Two conservative finite difference schemes for the KdV equa-tion are introduced and analyzed The construction of thecompact finite difference scheme (11) requires only a regularfive-point stencil at higher time level which is similar to thestandard second-order Crank-Nicolson scheme the explicitscheme [16] and the implicit scheme [18] However the con-struction of the standard fourth-order scheme (19) requires aseven-point stencil at higher time levelThe accuracy and sta-bility of the numerical schemes for the solutions of the KdV

02

46

810

0

20

0

01

02

03

04

05

06

tx minus20

u(xt)

Figure 7 Numerical solutions using the scheme (19) at 119905 isin [0 10]119909119871= minus40 119909

119877= 100 ℎ = 05 and 120591 = 025

190 195 200 205 210

0

01

02

03

04

05

06

Scheme (11)Scheme (19)Exact solution

minus01

Figure 8 Numerical solutions at 119905 = 200 119909119871

= minus40 119909119877

= 300ℎ = 05 and 120591 = 025

equation can be tested by using the exact solution In thepaper the numerical experiments show that the presentmethods support the analysis of convergence rate The per-formance of the fourth-order schemes is well efficient at longtime by comparing with the second-order schemes [16 18]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was supported by Chiang Mai University

References

[1] D J Korteweg and G de Vries ldquoOn the change of form of longwaves advancing in a rectangular canal and on a new type of

8 Mathematical Problems in Engineering

long stationary waverdquo Philosophical Magazine vol 39 pp 422ndash449 1895

[2] D Pavoni ldquoSingle and multidomain Chebyshev collocationmethods for the Korteweg-de Vries equationrdquo Calcolo vol 25no 4 pp 311ndash346 1988

[3] A A Soliman ldquoCollocation solution of the Korteweg-de Vriesequation using septic splinesrdquo International Journal of ComputerMathematics vol 81 no 3 pp 325ndash331 2004

[4] H Kalisch and X Raynaud ldquoOn the rate of convergence ofa collocation projection of the KdV equationrdquo MathematicalModelling andNumerical Analysis vol 41 no 1 pp 95ndash110 2007

[5] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[6] L R T Gardner G A Gardner and A H A Ali ldquoSimulationsof solitons using quadratic spline finite elementsrdquo ComputerMethods in Applied Mechanics and Engineering vol 92 no 2pp 231ndash243 1991

[7] M E Alexander and J L Morris ldquoGalerkin methods applied tosomemodel equations for non-linear dispersive wavesrdquo Journalof Computational Physics vol 30 no 3 pp 428ndash451 1979

[8] S R Barros and J W Cardenas ldquoA nonlinear Galerkin methodfor the shallow-water equations on periodic domainsrdquo Journalof Computational Physics vol 172 no 2 pp 592ndash608 2001

[9] H Ma and W Sun ldquoA Legendre-Petrov-Galerkin and Cheby-shev collocation method for third-order differential equationsrdquoSIAM Journal on Numerical Analysis vol 38 no 5 pp 1425ndash1438 2000

[10] J Shen ldquoA new dual-Petrov-Galerkin method for third andhigher odd-order differential equations application to the KDVequationrdquo SIAM Journal onNumerical Analysis vol 41 no 5 pp1595ndash1619 2003

[11] W Heinrichs ldquoSpectral approximation of third-order prob-lemsrdquo Journal of Scientific Computing vol 14 no 3 pp 275ndash2891999

[12] YMaday andAQuarteroni ldquoError analysis for spectral approx-imation of the Korteweg-de Vries equationrdquo ModelisationMathematique et Analyse numerique vol 22 no 3 pp 499ndash5291988

[13] S Zhu ldquoA scheme with a higher-order discrete invariant for theKdV equationrdquo Applied Mathematics Letters vol 14 no 1 pp17ndash20 2001

[14] F-l Qu andW-q Wang ldquoAlternating segment explicit-implicitscheme for nonlinear third-orderKdV equationrdquoAppliedMath-ematics and Mechanics English Edition vol 28 no 7 pp 973ndash980 2007

[15] H-P Wang Y-S Wang and Y-Y Hu ldquoAn explicit scheme forthe KdV equationrdquo Chinese Physics Letters vol 25 no 7 pp2335ndash2338 2008

[16] N J Zabusky and M D Kruskal ldquoInteraction of ldquosolitonsrdquo in acollisionless plasma and the recurrence of initial statesrdquoPhysicalReview Letters vol 15 no 6 pp 240ndash243 1965

[17] O Kolebaje and O Oyewande ldquoNumerical solution of theKorteweg-de Vries equation by finite differenece an d adomaindecomposition methodrdquo International Journal of Basic andApplied Sciences vol 1 no 3 pp 321ndash335 2012

[18] K Goda ldquoOn stability of some finite difference schemes for theKorteweg-de Vries equationrdquo Journal of the Physical Society ofJapan vol 39 no 1 pp 229ndash236 1975

[19] S Hamdi W H Enright W E Schiesser and J J GottliebldquoExact solutions and conservation laws for coupled generalized

Korteweg-de Vries and quintic regularized long wave equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 63 no 5ndash7 pp e1425ndashe1434 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

long stationary waverdquo Philosophical Magazine vol 39 pp 422ndash449 1895

[2] D Pavoni ldquoSingle and multidomain Chebyshev collocationmethods for the Korteweg-de Vries equationrdquo Calcolo vol 25no 4 pp 311ndash346 1988

[3] A A Soliman ldquoCollocation solution of the Korteweg-de Vriesequation using septic splinesrdquo International Journal of ComputerMathematics vol 81 no 3 pp 325ndash331 2004

[4] H Kalisch and X Raynaud ldquoOn the rate of convergence ofa collocation projection of the KdV equationrdquo MathematicalModelling andNumerical Analysis vol 41 no 1 pp 95ndash110 2007

[5] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[6] L R T Gardner G A Gardner and A H A Ali ldquoSimulationsof solitons using quadratic spline finite elementsrdquo ComputerMethods in Applied Mechanics and Engineering vol 92 no 2pp 231ndash243 1991

[7] M E Alexander and J L Morris ldquoGalerkin methods applied tosomemodel equations for non-linear dispersive wavesrdquo Journalof Computational Physics vol 30 no 3 pp 428ndash451 1979

[8] S R Barros and J W Cardenas ldquoA nonlinear Galerkin methodfor the shallow-water equations on periodic domainsrdquo Journalof Computational Physics vol 172 no 2 pp 592ndash608 2001

[9] H Ma and W Sun ldquoA Legendre-Petrov-Galerkin and Cheby-shev collocation method for third-order differential equationsrdquoSIAM Journal on Numerical Analysis vol 38 no 5 pp 1425ndash1438 2000

[10] J Shen ldquoA new dual-Petrov-Galerkin method for third andhigher odd-order differential equations application to the KDVequationrdquo SIAM Journal onNumerical Analysis vol 41 no 5 pp1595ndash1619 2003

[11] W Heinrichs ldquoSpectral approximation of third-order prob-lemsrdquo Journal of Scientific Computing vol 14 no 3 pp 275ndash2891999

[12] YMaday andAQuarteroni ldquoError analysis for spectral approx-imation of the Korteweg-de Vries equationrdquo ModelisationMathematique et Analyse numerique vol 22 no 3 pp 499ndash5291988

[13] S Zhu ldquoA scheme with a higher-order discrete invariant for theKdV equationrdquo Applied Mathematics Letters vol 14 no 1 pp17ndash20 2001

[14] F-l Qu andW-q Wang ldquoAlternating segment explicit-implicitscheme for nonlinear third-orderKdV equationrdquoAppliedMath-ematics and Mechanics English Edition vol 28 no 7 pp 973ndash980 2007

[15] H-P Wang Y-S Wang and Y-Y Hu ldquoAn explicit scheme forthe KdV equationrdquo Chinese Physics Letters vol 25 no 7 pp2335ndash2338 2008

[16] N J Zabusky and M D Kruskal ldquoInteraction of ldquosolitonsrdquo in acollisionless plasma and the recurrence of initial statesrdquoPhysicalReview Letters vol 15 no 6 pp 240ndash243 1965

[17] O Kolebaje and O Oyewande ldquoNumerical solution of theKorteweg-de Vries equation by finite differenece an d adomaindecomposition methodrdquo International Journal of Basic andApplied Sciences vol 1 no 3 pp 321ndash335 2012

[18] K Goda ldquoOn stability of some finite difference schemes for theKorteweg-de Vries equationrdquo Journal of the Physical Society ofJapan vol 39 no 1 pp 229ndash236 1975

[19] S Hamdi W H Enright W E Schiesser and J J GottliebldquoExact solutions and conservation laws for coupled generalized

Korteweg-de Vries and quintic regularized long wave equa-tionsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 63 no 5ndash7 pp e1425ndashe1434 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of