imls ritz.pdf

Research ArticleThe Improved Moving Least-Square Ritz Method forthe One-Dimensional Sine-Gordon Equation

Qi Wei and Rongjun Cheng

Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China

Correspondence should be addressed to Rongjun Cheng; [email protected]

Received 15 December 2013; Accepted 15 January 2014; Published 23 February 2014

Academic Editor: Zan Zhang

Copyright 2014 Q. Wei and R. Cheng.This is an open access article distributed under theCreativeCommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Analysis of the one-dimensional sine-Gordon equation is performed using the improved moving least-square Ritz method (IMLS-Ritzmethod).The improvedmoving least-square approximation is employed to approximate the 1D displacement field. A system ofdiscrete equations is obtained by application of the Ritz minimization procedure. The effectiveness and accuracy of the IMLS-Ritzmethod for the sine-Gordon equation are investigated by numerical examples in this paper.

1. Introduction

It is well known that many physical phenomena in oneor higher-dimensional space can be described by a solitonmodel. Many of these models are based on simple integrablemodels such as Korteweg-de Vries equation and the non-linear Schrodinger equation. Solitons have found to modelamong others shallow-water waves, optical fibres, Josephson-junction oscillators, and so forth. Equations which also leadto solitary waves are the sine-Gordon. The sine-Gordonequation arises in extended rectangular Josephson junctions,which consist of two layers of super conducting materialsseparated by an isolating barrier. A typical arrangementis a layer of lead and a layer of niobium separated by alayer of niobium oxide. A quantum particle has a nonzerosignificant probability of being able to penetrate to the otherside of a potential barrier that would be impenetrable to thecorresponding classical particle. This phenomenon is usuallyreferred to as quantum tunneling [13].

A numerical study for sine-Gordon equation has beenproposed including the finite difference schemes [46],the finite element methods [7, 8], the modified Adomiandecomposition method [9], the boundary integral equationapproach [10], radius basis function (RBF) [11], the meshlesslocal Petrov-Galerkin (MLPG) [12], the discrete singularconvolution [13], meshless local boundary integral equationmethod (LBIE) [14], the dual reciprocity boundary element

method (DRBEM) [15], and the mesh-free kp-Ritz method[16].

The meshless method is a new and interesting numericaltechnique. Important meshless methods have been devel-oped and proposed, such as smooth particle hydrodynamicsmethods (SPH) [17], radial basis function (RBF) [18], elementfree Galerkin method (EFG) [19], meshless local Petrov-Galerkin method (MLPG) [20], reproducing kernel particlemethod (RKPM) [2123], the boundary element free method(BEFM) [2427], the complex variablemeshlessmethod [2836], improved element free Galerkin method (IEFG) [3741],and the improved meshless local Petrov-Galerkin method[4246].

The moving least-square (MLS) technique was origi-nally used for data fitting. Nowadays, the MLS techniquehas been employed as the shape functions of the mesh-less method or element-free Galerkin (EFG) method [47].Though EFG method is now a very popular numericalcomputational method, a disadvantage of the method isthat the final algebraic equations system is sometimes ill-conditioned. Sometimes a poor solution will be obtained dueto the ill-conditioned system. The improved moving least-square (IMLS) approximation has been proposed [4852]to overcome this disadvantage. In the IMLS, the orthogonalfunction system with a weight function is chosen to be thebasis functions. The algebraic equations system in the IMLSapproximation will be no more ill-conditioned.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 383219, 10 pageshttp://dx.doi.org/10.1155/2014/383219

2 Mathematical Problems in Engineering

The Ritz [53] approximation technique is a generaliza-tion of the Rayleigh [54] method and it has been widelyused in computational mechanics. The element-free kp-Ritzmethod is firstly developed and implemented for the freevibration analysis of rotating cylindrical panels by Liewet al. [55]. The kp-Ritz method was widely applied andused in many kinds of problems, such as free vibration oftwo-side simply-supported laminated cylindrical panels [56],nonlinear analysis of laminated composite plates [57], Sine-Gordon equation [16], 3D wave equation [58], and biologicalpopulation problem [59].

A new numerical method which is named the IMLS-Ritzmethod for the sine-Gordon equation is presented in thispaper. In this paper, the unknown function is approximatedby these IMLS approximation; a system of nonlinear discreteequations is obtained by the Ritz minimization procedure,and the boundary conditions are enforced by the penaltymethod. Numerical examples are presented to validate theaccuracy and efficiency of the proposed method.

2. IMLS-Ritz Formulation forthe Sine-Gordon Equation

Consider the following one-dimensional sine-Gordon equa-tion:

2

2+

=

2

2 sin () , , 0 < (1a)

with initial conditions

(, 0) =

1() , (1b)

(, 0)

=

2()

(1c)

and boundary conditions

(, ) =

1() , (, ) =

2() , (1d)

where denotes the domain of , denotes the boundaries,

and 1() and

2() are wave modes or kinks and velocity,

respectively. Parameter is the so-called dissipative term,assumed to be a real number with 0.

The weighted integral form of (1a) is obtained as follows:

[

2

2 sin ()

2

2

] = 0. (2)

The weak form of (2) is

[

+ sin () +

2

2+

] = 0. (3)

The energy functional () can be written as

() =

1

2

+

[

2

2+

+ sin ()] .(4)

In the improvedmoving least-square approximation [19],define a local approximation by

uh (x, x) =m

j=1pj (x) aj (x) p

T(x) (x) . (5)

This defines the quadratic form

=

=1

(x x) [

(x, x) (x

)]

2

. (6)

Equation (6) can be rewritten in the vector form

= (pa u)W (x) (pa u) . (7)

To find the coefficients a(x), we obtain the extremum of by

Ja= A (x) a (x) B (x) u = 0 (8)

which results in the equation system

A (x) a (x) = B (x) u. (9)

If the functions 1(x),

2(x), . . . ,

(x) satisfy the condi-

tions

(

,

) =

=1

(x)

(x) = {

0 =

=

(, = 1, 2, . . . , ) ,

(10)

then 1(x),

2(x), . . . ,

(x) is called a weighted orthogonal

function set with a weight function {} about points {x

}.

The weighted orthogonal basis function set p = () can be

formed with the Schmidt method [3941],

1= 1,

=

1

1

=1

(

1,

)

(

,

)

, = 2, 3, . . . .

(11)

Equation (9) can be rewritten as

[

[

[

[

[

(

1,

1) 0 0

0 (

2,

2) 0

...... d

...0 0 (

,

)

]

]

]

]

]

[

[

[

[

[

1(x)

2(x)...

(x)

]

]

]

]

]

=

[

[

[

[

[

(

1,

)

(

2,

)

...(

,

)

]

]

]

]

]

.

(12)

The coefficients (x) can be directly obtained as follows:

(x) =(

,

)

(

,

)

; ( = 1, 2, . . . , ) ; (13)

that is,

a (x) = A (x)B (x) u, (14)

Mathematical Problems in Engineering 3

where

A (x) =

[

[

[

[

[

[

[

[

[

[

[

1

(

1,

1)

0 0

0

1

(

2,

2)

0

...... d

...0 0

1

(

,

)

]

]

]

]

]

]

]

]

]

]

]

. (15)

From (12), the approximation function (x) can berewritten as

(x) = (x) u =

=1

(x) , (16)

where(x) is the shape function and

(x) = (1(x) ,

2(x) , . . . ,

(x)) = p (x)A (x)B (x) .

(17)

Taking derivatives of (17), we can obtain the first deriva-tives of shape function

,(x) =

=1

[

,(AB)

+

(A,B + AB

,)

] . (18)

Imposing boundary conditions by penalty method, thetotal energy functional for this problem will be obtained:

() =

1

2

+

[

2

2+

+ sin ()]

+

2

( )

2.

(19)

By (16), we can derive the approximation function

(, ) =

=1

()

() = () T,

(, )

=

=1

()

()

=

=1

()

()

= () T,

2 (, )

2=

2

2

=1

()

()

=

=1

()

2

()

2= () T.

(20)

Substituting (20) into (19) and applying theRitzminimizationprocedure to the energy function (), we obtain

()

= 0, =

() ,

()

,

2

()

2,

= 1, 2, . . . , .

(21)

In the matrix form, the results can be expressed as

C T + C T + KT = F, (22)

where

=

()

() ,

=

(x)

(x) + (

|

=+

|

=) ,

=

() sin () + (

|

=+

|

=) .

(23)

Making time discretization of (22) by the center differ-ence method, we get

CT(+2) 2T(+1) + T()

2

+ CT(+1) T()

+ KT(+1)+ T()

2

= F,

(24)

where

T() = T () = (

1(

) ,

2(

) , . . . ,

(

)) ,

=

() sin () + (

|=+

|=) .

(25)

The numerical solution of the one-dimensional sine-Gordon equation will be obtained by solving the aboveiteration equation.

3. Numerical Examples and Analysis

To verify the efficiency and accuracy of the proposed IMLS-Ritz method for the sine-Gordon equation, two examples arestudied and the numerical results are presented. The weightfunction is chosen to be cubic spline and the bases are chosento be linear in all examples.

Example 1. Consider the sine-Gordon Equation (1a)(1d)without nonlinear term sin() over the region 1 1with initial conditions

(, 0) =

1() = sin () ,

(, 0) =

2() = 0

(26)

with boundary conditions

(1, ) = (1, ) = 0. (27)

The exact solution is

(, ) =

1

2

(sin ( + ) + sin ( )) . (28)


Table 1:The comparisons of exact solution with numerical solutions by IEFG and EFGmethods with 41 nodes at = 0.1with = 0.001 andmax = 3.0 (Example 1).

Node number Exact solution IMLS-Ritz method EFG method21 0.1488 0.1479 0.147322 0.2939 0.2918 0.290923 0.4318 0.4297 0.427524 0.5590 0.5583 0.553625 0.6725 0.6732 0.666126 0.7694 0.7687 0.765327 0.8474 0.8471 0.839928 0.9045 0.9032 0.898929 0.9393 0.9389 0.938030 0.8474 0.8478 0.848731 0.7694 0.7685 0.762532 0.6725 0.6732 0.6753

1.0 0.5 0.0 0.5 1.0 1.5 2.0

1.0

0.5

0.0

0.5

1.0

x

Exact solutiont = 0.1

t = 0.2

t = 0.3

t = 0.4

u(x,t

)

Figure 1: Numerical solution and exact solution of (, ) when =0.1, 0.2, 0.3, 0.4 (Example 1).

The IMLS-Ritz method is applied to solve the aboveequation with penalty factor = 104 and time step length = 0.001, max = 3.0. In Figure 1, the numerical solutionand exact solution are plotted at times = 0.1, 0.2, 0.3,and 0.4, respectively. In Figures 2, 3, and 4, the graphs oferror function (, ) (, ) are plotted at times = 0.1,0.2, and 0.3, respectively, where (, ) is the exact solutionand numerical solution (, ) is obtained by using theIMLS-Ritz method. Table 1 shows the comparison of exactsolutions and numerical solutions by IMLS-Ritz method andEFG method. From the results of Table 1, we can draw theconclusion that IMLS-Ritz method has higher accuracy thanthe EFGmethod.The surfaces of the numerical solution withthe IMLS-Ritz method and exact solutions are plotted inFigures 5 and 6.

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 18

6

4

2

0

2

4

6

810

3t = 0.1, dt = 0.001

Figure 2: Error function (, ) (, ), where (, ) is an exactsolution and numerical solution (, ) is obtained by using theIMLS-Ritz method with = 0.001 at = 0.1 (Example 1).

Example 2. Consider the case = 0 in (1a)(1d) over therectangular region 20 20 and initial condition

(, 0) =

1() = 0,

(, 0) =

2() = 4sech ()

(29)

which derives the analytic solution

(, ) = 4tan1 (sech ()) (30)

and the boundary conditions can be obtained from (30).

The IMLS-Ritz method is applied to solve the aboveequation with penalty factor = 105 and time step length = 0.01, max = 2.2. Figure 7 depicts the numerical andexact solution when = 1, 5, 10, 20, and 30, respectively. InFigures 8, 9, and 10, the graphs of error function (, ) (, ) are plotted at times = 1, 5, and 10, respectively, where


1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10.04

0.03

0.02

0.01

0

0.01

0.02

0.03

0.04t = 0.2, dt = 0.001


1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10.08

0.06

0.04

0.02

0

0.02

0.04

0.06

0.08t = 0.3, dt = 0.001


00.05

0.10.15

0.2 10.5

00.5

11

0.5

0

0.5

1

tx

u(x,t

)

Figure 5: The surface of exact solution (Example 1).

00.05

0.10.15

0.2 10.5

00.5

11

0.5

0

0.5

1

tx

u(x,t

)

Figure 6:The surface of numerical solutionwith IMLS-Ritzmethod(Example 1).

20 10 0 10 20 300.5

0.00.51.01.52.02.53.03.54.04.55.05.56.06.5

Exact solution

t = 1

t = 5

t = 20

t = 10

t = 30

u(x,t

)

Figure 7: Numerical solution and exact solution of (, )when =1, 5, 10, 20, 30 (Example 2).

(, ) is the exact solution and numerical solution (, )is obtained by using the IMLS-Ritz method. Table 2 showsthe comparison of exact solutions and numerical solutionsby IMLS-Ritz method and EFG method. From the resultsof Table 2, it is shown that IMLS-Ritz method has higheraccuracy than the EFGmethod.The surfaces of the numericalsolution with the IMLS-Ritz method and exact solution areplotted in Figures 11 and 12.

Example 3. Consider the case in (1a)(1d) over the rectangu-lar region 20 20 and initial condition

(, 0) =

1() = 4 arctan () ,

(, 0) =

2() =

4

1

2(1 +

2)

(31)


20 15 10 5 0 5 10 15 204

3

2

1

0

1

210

4t = 1, dt = 0.01

Figure 8: Error function (, ) (, ) with = 0.01 at = 1(Example 2).

20 15 10 5 0 5 10 15 202

1.5

1

0.5

0

0.5

110

3t = 5, dt = 0.01


and the boundary conditions can be derived from thefollowing exact solitary wave solution:

(, ) = 4 arctan (()/12

) , (32)

where = /1 2 and is the velocity of solitary wave.

The IMLS-Ritz method is applied to solve the aboveequation with penalty factor = 107 and time step length = 0.01, max = 2.7, = 0.5. Table 3 shows the comparisonof exact solutions and numerical solutions by IMLS-Ritzmethod and EFG method. From the results of Table 3, it isshown that IMLS-Ritz method has higher accuracy than theEFGmethod.The surfaces of the numerical solution with theIMLS-Ritz method and exact solution are plotted in Figures

20 15 10 5 0 5 10 15 202.5

2

1.5

1

0.5

0

0.5

1

103

t = 10, dt = 0.01


0

510

1520 20

10

010

200

2

4

6

8

xt

u(x,t

)

510

15 10

010

xt


05

1015

20 2010

010

200

2

4

6

8

xt

u(x,t

)

510

15 100

10

xt

Figure 12: The surface of numerical solution with IMLS-Ritzmethod (Example 2).


Table 2: The comparisons of exact solution with numerical solutions by IEFG and EFG methods with 40 nodes at = 1 with = 0.01 andmax = 2.2 (Example 2).


Table 3: The comparisons of exact solution with numerical solutions by IEFG and EFG methods with 81 nodes at = 1 with = 0.001 andmax = 2.7 (Example 3).


00.5

11.5

2

2010

010

200

2

4

6

8

tx 0.5

11.5

100

10

tx

u(x,t

)


13 and 14. In Figure 15, the graph of error function (, ) (, ) is plotted at time = 1, where (, ) is the exact

00.5

11.5

2

2010

010

200

2

4

6

8

t

x

u(x,t

)

0.51

1.510

010

t

x

Figure 14: The surface of numerical solution with IMLS-Ritzmethod (Example 3).

solution and the numerical solution (, ) is obtained byusing the IMLS-Ritz method.


20 15 10 5 0 5 10 15 200

0.005

0.01

0.015

0.02

0.025

0.03

0.035t = 2, dt = 0.01


From these figures, it is shown that numerical resultsobtained by the IMLS-Ritz method are in good agreementwith the exact solutions.

4. Conclusion

This paper presents a numerical method, named the IMLS-Ritz method, for the one-dimensional sine-Gordon equation.The IMLS approximation is employed to approximate the1D displacement field. A system of discrete equations isobtained through application of the Ritz minimization. Inthe IMLS approximation, the basis function is chosen asthe orthogonal function system with a weight function. TheIMLS approximation has greater computational efficiencyand precision than the MLS approximation, and it does notlead to an ill-conditioned system of equations.The numericalresults show that the technique is accurate and efficient.

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of Ningbo City (Grant nos. 2013A610067,2102A610023, and 2013A610103), theNatural Science Founda-tion of Zhejiang Province of China (Grant no. Y6110007), andthe National Natural Science of China (Grant no. 41305016).

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