Applied Mathematics and Computation 218 (2012) 7711–7716

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Numerical solution of one-dimensional Sine–Gordon equation usinghigh accuracy multiquadric quasi-interpolation q

Zi-Wu Jiang ⇑, Ren-Hong WangSchool of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China

a r t i c l e i n f o

Keywords:Multiquadric quasi-interpolationInverse multiquadric interpolationSine–Gordon equationsCollocationMeshfree method

0096-3003/$ - see front matter � 2012 Elsevier Incdoi:10.1016/j.amc.2011.12.095

q This work was supported by the National Natur⇑ Corresponding author.

E-mail address: zwjiang@yahoo.cn (Z.-W. Jiang).

a b s t r a c t

In this paper, we propose a numerical scheme to solve one-dimensional Sine–Gordonequation related to many scientific research topics by using high accuracy multiquadricquasi-interpolation. We use the derivatives of a multiquadric quasi-interpolant toapproximate the spatial derivatives, and a finite difference to approximate the temporalderivative. The advantages of the scheme are that it is meshfree, and in each time step onlya multiquadric quasi-interpolant is employed, so that the algorithm is very easy to imple-ment. The accuracy of our scheme is demonstrated by some test problems.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

The nonlinear one-dimensional Sine–Gordon (SG) equation

@2u@t2 ¼

@2u@x2 � sinðuÞ; a 6 x 6 b; t P 0; ð1Þ

which appear in differential geometry gained its significance because of the collisional behaviors of solitons that arise fromthese equations. This equation appeared in many scientific fields such as the propagation of fluxion in Josephson junctions[15] between two superconductors, the motion of rigid pendular attached to a stretched wire, solid state physics, nonlinearoptics, and dislocations in metals [19]. This paper is devoted to the numerical solution of equation (1) with the initialconditions

uðx;0Þ ¼ f ðxÞ; a 6 x 6 b; ð2Þ@u@tðx;0Þ ¼ gðxÞ; a 6 x 6 b; ð3Þ

and the boundary conditions

uða; tÞ ¼ h1ðtÞ; uðb; tÞ ¼ h2ðtÞ; t P 0: ð4Þ

Since the exact solution of the SG equation can only be obtained in special situations, many numerical schemes are con-structed to solve the SG equation. To mention some of them: finite difference schemes [3,4,8,11], finite elements method[1], and B-spline collocation method [16] etc. Although the previous methods have been widely used, they usually requirethe construction and update of a mesh and hence bring inconvenience during computation. To avoid the mesh generation, in

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al Science Foundation of China (U0935004, 110 71031, 11001037 and 10801024).

7712 Z.-W. Jiang, R.-H. Wang / Applied Mathematics and Computation 218 (2012) 7711–7716

recent years, meshless techniques have attracted attention of researchers. Dehghan and Shokri [7] proposed a numericalscheme to solve one-dimensional SG equation using collocation points and approximating directly the solution using thethin plate splines radial basis function (RBF). Unfortunately, this scheme need to solve a ill-conditional system of linear equa-tions. Ma and Wu [13] presented a meshless scheme by using a multiquadric (MQ) quasi-interpolation named LE withoutsolving a large-scale linear system of equations, but a polynomial pðxÞ was needed to improved the accuracy of the scheme.In this paper we developed a meshless approach by directly using high accuracy MQ quasi-interpolation defined on ½a; b�without using any polynomial.

The remainder of this paper is organized as follows: In Section 2, some elementary knowledge about the RBF interpolationand the MQ quasi-interpolation are introduced. In Section 3, we present the numerical techniques by using the high accuracyMQ quasi-interpolation to solve the SG equation. The computational results of the SG equation for some test problems areillustrated, and compared with those obtained with previous results in Section 4.

2. RBF interpolation and MQ quasi-interpolation

2.1. RBF interpolation

RBF was introduced by Krige [10] in 1951 to deal with geological problem. Hitherto, RBFs were widely used in many fields([6,18], etc). The process of interpolation by using RBF is as follows.

For a given region X 2 Rd and a set of finite distinct interpolation points

X ¼ fx1; . . . ; xNg � X;

if we are supplied with a function f : X ! R, we can construct an interpolant to f in the form:

Sf ;XðxÞ ¼XN

i¼1

ajuðkx� xjkÞ; for x 2 X; ð5Þ

where k � k denote Euclidean norm, and u is a certain RBF, say u : RP0 ! R. The undetermined coefficients fa1; . . . ;aNg canbe obtained by the linear system

XN

j¼1

ajuðkxi � xjkÞ ¼ f ðxiÞ; 1 6 i 6 N: ð6Þ

In this paper, inverse multiquadric (IMQ) RBF is chosen as the RBF mentioned in (5), and the adopted IMQ-RBF has the form:

uðrÞ ¼ s2

ðs2 þ r2Þ3=2 ; ð7Þ

where s is a shape parameter. From [14], we know that the linear system (6) have unique solution.

2.2. MQ quasi-interpolation

Quasi-interpolation of a univariate function f : ½a; b� ! R with MQs on the scattered points

a ¼ x0 < x1 < � � � < xn ¼ b; h :¼ max16i6nðxi � xi�1Þ; ð8Þ

usually takes the form

ðMf ÞðxÞ ¼Xn

i¼0

f ðxiÞwiðxÞ; ð9Þ

where wiðxÞ is linear combinations of the MQs

/iðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 þ ðx� xiÞ2

q; c 2 Rþis a shape parameter: ð10Þ

In 1988, Buhmann [5] studied the accuracy of quasi-interpolation operator M of a function f : R # R at infinite regularpoints in R and showed that this scheme reproduces linear polynomials. Beatson and Powell [2] proposed three univariateMQ quasi-interpolations, namely, LA;LB , and LC on interval [a,b]. Wu and Schaback [20] proposed the univariate quasi-inter-polation LD, and proved that the scheme possess preserving monotonicity and linear reproducing property as well as Oðh2Þerror of quasi-interpolant LDf in the case of c2j log cj ¼ Oðh2Þ. Based on Wu and Schaback’s work, Ling [12] proposed a mul-tilevel univariate quasi-interpolation LR, and this scheme can converge with a rate of Oðh2:5 log hÞ as c ¼ OðhÞ. To solve SGequation numerically, in this paper we adopt a MQ quasi-interpolation scheme constructed in [9] using multilevel method,named LW2 .

Z.-W. Jiang, R.-H. Wang / Applied Mathematics and Computation 218 (2012) 7711–7716 7713

Next we recall the construction of LW2 based on the IMQ-RBF interpolation and the LD operator. Given data fðxi; fiÞgni¼0,

Wu–Schaback’s LD operator is

ðLDÞf ðxÞ ¼Xn

j¼0

fjwjðxÞ; ð11Þ

where

w0ðxÞ ¼12þ /1ðxÞ � ðx� x0Þ

2ðx1 � x0Þ;

w1ðxÞ ¼/2ðxÞ � /1ðxÞ

2ðx2 � x1Þ� /1ðxÞ � ðx� x0Þ

2ðx1 � x0Þ;

wn�1ðxÞ ¼ðxn � xÞ � /n�1ðxÞ

2ðxn � xn�1Þ� /n�1ðxÞ � /n�2ðxÞ

2ðxn�1 � xn�2Þ;

wnðxÞ ¼12þ /n�1ðxÞ � ðxn � xÞ

2ðxn � xn�1Þ;

wjðxÞ ¼/jþ1ðxÞ � /jðxÞ

2ðxjþ1 � xjÞ�

/jðxÞ � /j�1ðxÞ2ðxj � xj�1Þ

; j ¼ 2; . . . ;n� 2:

ð12Þ

Suppose that fxigni¼0 are given scattered points, and N is a positive integer satisfying N < n, then we define an index

0 < k1 < k2 < � � � < kN < n, and pick a smaller set denoted by fxkjgN

j¼1 from the given points fxigni¼0. At very point xkj

of thesubset, we define the following value

f 00xkj¼

2 ðxkj� xkj�1Þf ðxkjþ1Þ � ðxkjþ1 � xkj�1Þf ðxkj

Þ þ ðxkjþ1 � xkjÞf ðxkj�1Þ

h iðxkj� xkj�1Þðxkjþ1 � xkj

Þðxkjþ1 � xkj�1Þ:

Based on the smaller set of data

fðxkj; f 00xkjÞgN

j¼1; h2 :¼ max26j6N

ðxkj� xkj�1

Þ; ð13Þ

by using the IMQ-RBF given in (7), a RBF interpolant Sf 00;N is obtained, and satisfying

Sf 00 ;NðxkjÞ ¼ f 00xkj

; j ¼ 1; . . . ;N:

From the above subsection, the interpolant Sf 00 ;N can be expressed as follows

Sf 00 ðxÞ ¼XN

j¼1

ajuðkx� xkjkÞ; ð14Þ

and the coefficients fajgNj¼1 are uniquely determined by the interpolation condition

Sf 00 ðxkiÞ ¼

XN

j¼1

ajuðkxki� xkj

kÞ ¼ f 00ðxkiÞ; i ¼ 1; . . . ;N: ð15Þ

By using f ðxÞ and the coefficients faigNi¼1 mentioned above, a function is constructed in the form

eðxÞ ¼ f ðxÞ �XN

j¼1

aj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ ðx� xkj

Þ2q

: ð16Þ

Then a MQ quasi-interpolation can be obtained by using LD defined by (11) and (12) on the data fxi; eðxiÞgni¼0, with the shape

parameter c. Furthermore, the MQ quasi-interpolation operator LW2 is presented as

LW2 f ðxÞ ¼XN

j¼1

aj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ ðx� xkj

Þ2q

þ LDeðxÞ: ð17Þ

Generally speaking, the shape parameters c and s should not be the same constant in (17). Moreover, the linear reproducingproperty and the high convergence rate of LW2 were also studied in [17]. Apparently, in application one can use the secondderivative of ðLW2 f ÞðxÞ

ðLW2 f Þ00ðxÞ ¼XN

j¼1

ajuðx� xkjÞ þ

Xn

i¼0

f ðxiÞ �XN

j¼1

aj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ ðxi � xkj

Þ2q !

w00i ðxÞ;

where x 2 ½x0; xn�, to approximation f 00ðxÞ.

7714 Z.-W. Jiang, R.-H. Wang / Applied Mathematics and Computation 218 (2012) 7711–7716

3. Numerical scheme using MQ quasi-interpolation LW2

Now, we present the numerical scheme for solving the SG equation by using the MQ quasi-interpolation LW2 . In our meth-od, we use the derivative of the MQ quasi-interpolation to approximate the spatial derivative of the differential equationsand employ a first order accurate forward difference to approximate the temporal derivative.

Discretizing the SG equation

@2u@t2 ¼

@2u@x2 � sinðuÞ

in time, we get

unþ1k ¼ 2un

k � un�1k þ s2ðuxxÞnk � s2 sinðun

kÞ;

where unk is the approximation of the value uðx; tÞ at ðxk; tnÞ; tn ¼ ns, and s is the time step. Because equation (1) is a devel-

opmental equation, it is reasonable to employ the finite difference method in the time domain. Then, we use the secondderivative of the MQ quasi-interpolation LW2 uðx; tnÞ to approximate uxx. Thus, the resulting numerical scheme is

unþ1k ¼ 2un

k � un�1k � s2 sinðun

kÞ þ s2XN

j¼1

ajuðxk � xkjÞ þ

Xn

i¼0

uni �

XN

j¼1

aj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ ðxi � xkj

Þ2q !

w00i ðxkÞ" #

; ð18Þ

where u and w are defined in (7) and (12) respectively, and fajgNj¼1 is determined by the uniquely solvable linear system

XN

j¼1

ajuðkxki� xkj

kÞ ¼2un

kiþ1

ðxkiþ1 � xkiÞðxkiþ1 � xki�1Þ

þ2un

ki�1

ðxki� xki�1Þðxkiþ1 � xki�1Þ

�2un

ki

ðxki� xki�1Þðxkiþ1 � xki

Þ ; 1 6 i 6 N:

ð19Þ

We iterate this scheme to achieve the numerical solution of (1).

4. Numerical example

In this section, we test the method by two classical examples. For the sake of simplification, we set

xi ¼ aþ iðb� aÞn

; i ¼ 0; . . . ;n;

and N ¼ nK, where K is a positive integer. The choice of the interpolation centers fxkj

gNj¼1 as follows

xk1¼ x1; xkN

¼ xn�1;

xkj¼ aþ ðj� 1Þðb� aÞ

N; j ¼ 2; . . . ;N:

The accuracy of the proposed method is measured using the root-mean-square error (RMS) and L1 error norms defined as

RMS ¼ 1nþ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i¼0

juexacti � unum

i j2vuut ;

and

L1 ¼max06i6n

juexacti � unum

i j:

Example 1. In this example, we consider SG equation (1) without nonlinear term sinðuÞ in the region �1 6 x 6 1. The initialconditions are given by

uðx;0Þ ¼ f ðxÞ ¼ sinðpxÞ;utðx;0Þ ¼ gðxÞ ¼ 0;

with the boundary conditions

uð�1; tÞ ¼ uð1; tÞ ¼ 0:

The analytical solution is given in [17] as

uðx; tÞ ¼ 12

sinðpðxþ tÞÞ þ sinðpðx� tÞÞð Þ:

The L1 errors and RMS errors are obtained in Table 1 for t ¼ 0:25; 0:5; 0:75 and 1. The graph of the error functioneuðx; tÞ � uðx; tÞ for t ¼ 1 is given in Fig. 1, where euðx; tÞ is the estimated function and uðx; tÞ is the analytical solution.

Table 1Computational results for Example 1.

t L1-error RMS Time (s)

0.25 1:24� 10�2 6:46� 10�4 0.61

0.50 5:47� 10�2 2:71� 10�3 0.64

0.75 6:14� 10�2 3:04� 10�3 0.66

1.00 7:59� 10�3 3:82� 10�4 0.67

L1 and RMS errors, with c ¼ 0:027; s ¼ 0:8; s ¼ h ¼ 0:01; K ¼ 8.

−1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 1−8

−6

−4

−2

0

2

4

6

8x 10−3

t=1.00

Fig. 1. Error function euðx; tÞ � uðx; tÞ in t ¼ 1, with dt ¼ dx ¼ 0:01 and K ¼ 8, for Example 1.

Table 2Comparison of L1 and RMS errors between LE and LW2 schemes.

t L1 error RMS

LE L�W2LE L�W2

0.1 1:54� 10�6 1:31� 10�5 7:43� 10�6 3:39� 10�7

0.2 4:25� 10�5 2:50� 10�5 1:76� 10�5 6:55� 10�7

0.3 9:02� 10�5 3:47� 10�5 3:60� 10�5 9:29� 10�7

0.4 1:62� 10�4 4:20� 10�5 1:62� 10�4 1:15� 10�6

0.5 2:58� 10�4 4:68� 10�5 1:10� 10�4 1:34� 10�6

0.6 3:73� 10�4 6:54� 10�5 1:65� 10�4 1:55� 10�6

0.7 4:98� 10�4 1:74� 10�4 2:29� 10�4 2:14� 10�6

0.8 6:24� 10�4 4:01� 10�4 2:98� 10�4 3:92� 10�6

0.9 7:44� 10�4 8:22� 10�4 3:69� 10�4 7:98� 10�6

1.0 8:49� 10�4 1:53� 10�3 4:37� 10�4 1:56� 10�5

⁄ denotes that we set K ¼ 10, s = 1.

Z.-W. Jiang, R.-H. Wang / Applied Mathematics and Computation 218 (2012) 7711–7716 7715

Example 2. In this example, we compare the accuracy of the MQ quasi-interpolation LW2 proposed in this paper to thescheme LE used by Ma and Wu, thus we utilize the same example adopted in [13].

We consider SG equation (1) in the domain �2 6 x 6 2, and the initial conditions are given by

uðx;0Þ ¼ gðxÞ ¼ 0;utðx;0Þ ¼ hðxÞ ¼ 4sechðxÞ:

The analytical solution is

uðx; tÞ ¼ 4 arctanðsechðxÞtÞ:

For comparison, the numerical results are commutated with parameters: time step s ¼ 0:01, space step h = 0.01 and shapeparameter c ¼ 0:1

ffiffiffih3p

. Table 2 show the values of L1 and RMS errors obtained by using LE and LW2 quasi-interpolationschemes, at t ¼ 0:1; 0:2; . . . ; 1:00 respectively.

7716 Z.-W. Jiang, R.-H. Wang / Applied Mathematics and Computation 218 (2012) 7711–7716

From Table 2, we see that the solutions found with LW2 scheme is in good agreement with the results obtained by LE

scheme in the sense of L1 errors, and better than it in the sense of RMS errors.

5. Conclusions

In this paper, a numerical scheme for the one-dimensional nonlinear Sine–Gordon equation by high accuracy MQ quasi-interpolation LW2 is presented. Form the examples, we can say that the LW2 scheme is feasible and the accuracy is better thanMa and Wu’s LE scheme in the sense of RMS errors.

For comparison, our technique is used for the non-equidistant grids though we use equidistant grids in our numericalexperiments. We see that the present technique requires calculating wiðxÞ; w00iðxÞ ði ¼ 0; . . . ; nÞ, and the inverse matrix ofðuðkxki

� xkjkÞÞN�N only once.

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