applying multiquadric quasi-interpolation to solve burgers’ equation

Applied Mathematics and Computation 172 (2006) 472–484

www.elsevier.com/locate/amc

Applying multiquadric quasi-interpolationto solve Burgers� equationq

Ronghua Chen a,b,*, Zongmin Wu c

a Institute of Mathematics, Fudan University, Shanghai 200433, PR Chinab School of Mathematics and Computational Science, Hunan University of Science and

Technology, Xiangtan, Hunan 411201, PR Chinac Department of Mathematics, Fudan University, Shanghai 200433, PR China

Abstract

In this paper, we develop a kind of univariate multiquadric (MQ) quasi-interpolation

and use it to solve Burgers� equation (with viscosity). At first we construct the MQ

quasi-interpolation, which possesses the properties of linear reproducing and preserving

monotonicity. Next we obtain the numerical scheme, by using the derivative of the

quasi-interpolation to approximate the spatial derivative of the dependent variable

and a low order forward difference to approximate the time derivative of the dependent

variable. Then, we verify our method for two examples with distinguishing initial value

condition. One example is tested for three Reynolds number, that is, R = 10, R = 100,

and R = 10,000. From the numerical experiments, we see that the presented method

in this paper is valid although the accuracy of the technique is not higher than Hon

and Mao�s one. Another example is used to examine the travelling of the shock. We

can improve the accuracy by selecting an appropriate shape parameter and using a

0096-3003/$ - see front matter � 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2005.02.027

q Supported by NSFC 19971017 and NOYS 10125102.* Corresponding author. Address: Institute of Mathematics, Fudan University, Shanghai 200433,

PR China.

E-mail addresses: [email protected] (R. Chen), [email protected] (Z. Wu).

mailto:[email protected]

mailto:[email protected]

R. Chen, Z. Wu / Appl. Math. Comput. 172 (2006) 472–484 473

higher accurate MQ quasi-interpolation. The advantage of the resulting scheme is that

the algorithm is very simple, so it is very easy to implement.

� 2005 Elsevier Inc. All rights reserved.

Keywords: Multiquadric quasi-interpolation; Burgers� equation; Shape parameter; Radial basis

function

1. Introduction

Since Hardy proposed in 1968, the multiquadric (MQ) which is a kind of

radial basis function (RBF) has been investigated thoroughly. Hardy [22] sum-

marized the achievement of study of MQ from 1968 to 1988 and showed that

MQ can be applied in hydrology, geodesy, photogrammetry, surveying and

mapping, geophysics and crustal movement, geology and mining, and so on.In Franke�s review paper [14], the MQ was rated as one of the best methods

among 29 scattered data interpolation schemes based on their accuracy, stabil-

ity, efficiency, memory requirement, and ease to implementation. As an inter-

polation method, the MQ always produces a minimal semi-norm error as

proven by Madych and Nelson [29]. Thanks to the investigation of Beatson,

Buhmann, Carlson, Chen, Cheng, Chui, Dyn, Fasshauer, Foley, Fornberg,

Jetter, Kansa, Larsson, Li, Madych, Micchelli, Narcowich, Powell, Schaback,

Wendland, Ward, Wright and Wu et al. for the theory of the radial basis func-tions (see [1–4,8,10,13,25,28,30–35,37,39,42,43] and so on), the usages of the

RBFs become more and more wider. Since Kansa [23,24] successfully modified

MQ for solving partial differential equation (PDE), more and more researchers

have been attracted by this meshless, scattered data approximation scheme

(see, e.g., [5,11,12,15–21,26,27,36,38,40,41]). In most of the known methods

of solving differential equations using multiquadric, one must resolve a linear

system of equation at each time step. Hon and Wu [21], Wu [41], and others

have provided some successful examples using MQ quasi-interpolation to solv-ing differential equations. In this paper, we still use MQ quasi-interpolation so

that we do not require to solve any linear system of equation that we do not

meet the question of the ill-condition of the matrix. Therefore, we can save the

computational time and decrease the numerical error.

Hon and Mao [20] developed an efficient numerical scheme for Burgers�equation (with viscosity). They apply the MQ as a spatial approximation

scheme and a low order explicit finite difference approximation to the time

derivative. The method requires to solve a linear system, by using Gaussianelimination with partial pivoting, in order to obtain the coefficients of the inter-

polation function. Then get the value of the given points at the given time by

using the interpolation function, and the interpolation function is the linear

combinations with the MQ and the linear function. The method is valid for

474 R. Chen, Z. Wu / Appl. Math. Comput. 172 (2006) 472–484

the various Reynolds number R whose scope from 0.1 to 10,000, namely, the

method, has very broad applicability. They find that the method offers better

accuracy than other numerical methods [20]. Again, the results of the method

are very close to the analytical solution obtained by Cole [9] and the accurate

solution given by Christie and Mitchell [7].

In our methods, we use the derivative of the MQ quasi-interpolation toapproximate the spatial derivative of the differential equations and employ a

first order accurate forward difference for the approach of the temporal deriv-

ative as Hon and Mao do [20]. In our methods, we follow the Wu and Scha-

back�s [43] idea to construct the MQ quasi-interpolation.

The organization of the rest of this paper is as follows: In Section 2, we con-

struct the univariate MQ quasi-interpolation respectively. In Section 3, we de-

velop the numerical techniques using MQ to solve Burgers� equation. In

Section 4, we give the numerical results for R = 10, R = 100, and R = 10,000firstly. The accuracy of the method is not higher than that of Hon and Mao�sone [20], but we can improve the accuracy by selecting the shape parameter and

employing the high-order quasi-interpolation as shown by Beatson and Dyn in

[1] or by Zhang and Wu in [44]. Then, we use the method for another initial

value condition. The results are also acceptable. So the technique is valid

too. In Section 5, we derive conclusion and give remarks for the resulting

scheme and the further work.

2. Multiquadric quasi-interpolation

Beatson and Powell [2] proposed three univariate multiquadric quasi-inter-

polations, namely, LA, LB, and LC, to approximate a function {f(x),x0 6

x 6 xm} from the space that is spanned by the multiquadrics f/jðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� xjÞ2 þ c2

q; x 2 R; j ¼ 0; . . . ;mg and linear function, where c is a positive

constant and the centers {xj: j = 0, . . .,m} being given distinct points in the

interval [x0,xm]. Afterward, Beatson and Dyn [1] studied the properties of

the W-splines, the combination of the MQs, and obtained the error estimates

for quasi-interpolation schemes involving MQ based on a finite number of cen-

ters. Wu and Schaback [43] have proposed the univariate multiquadric quasi-

interpolation LD on [a,b] and proven that the scheme is shape preserving and

convergent. In this section, we construct a kind of special MQ quasi-interpola-

tion, which generalized LD.Given that fðxj; fjÞgmj¼0, where x0 < x1 < � � � < xm, we construct the univari-

ate quasi-interpolation in the form of

f �ðxÞ ¼Xmj¼0

fjWjðxÞ; ð2:1Þ


where

WjðxÞ ¼/jþ1ðxÞ � /jðxÞ2ðxjþ1 � xjÞ

�/jðxÞ � /j�1ðxÞ2ðxj � xj�1Þ

; 0 6 j 6 m. ð2:2Þ

We give the definition of /j(x) in the posterior definition.

Now, for the sake of reading easily, we introduce some definitions related to

quasi-interpolation.

Definition 2.1. If the quasi-interpolation f �(x) possesses the property

f �ðxÞ � C if f 0 ¼ f1 ¼ � � � ¼ fm ¼ C; ð2:3Þ

where C is any real constant, we say that the quasi-interpolation is constant

reproducing on [x0,xm].

Definition 2.2. We say that the quasi-interpolation f �(x) possesses linear repro-

ducing property on [x0,xm], if f�(x) = px + q as fj = pxj + q, j = 0, . . .,m, for all

p; q 2 R.

Remark 2.1. It is obvious that if a quasi-interpolation f �(x) possesses linear

reproducing property on [x0,xm], then it must be constant reproducing.

Definition 2.3. If the quasi-interpolation f �(x) is monotone increasing

(decreasing) for monotone increasing (decreasing) data fj, j = 0, . . .,m, then we

say that it possesses preserving monotonicity on [x0,xm].

In this paper, we define the quasi-interpolation as follows:

Definition 2.4. For the initial data fðxj; fjÞgmj¼0, fj = f(xj), the univariate quasi-

interpolation on [x0,xm], f�(x), is defined by (2.1) and (2.2),

/mðxÞ ¼ /0ðxÞ � 2xþ xm þ x0; ð2:4Þ

/�1ðxÞ ¼ /0ðxÞ þ x0 � x�1;

/mþ1ðxÞ ¼ /mðxÞ þ xmþ1 � xm

�ð2:5Þ

and

/jðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� xjÞ2 þ k2

q; 0 6 j 6 m� 1; ð2:6Þ

where k 2 R.

We can prove the following theorem (see [6] for detail).


Theorem 2.1. The quasi-interpolation f �(x), defined by Definition 2.4, possesses

linear reproducing property and preserving monotonicity on [x0,xm]. Meantime,

on [x0,xm], f�(x) can be rewritten as follows:

f �ðxÞ ¼ 1

2

Xm�1

j¼1

/jþ1ðxÞ � /jðxÞxjþ1 � xj

�/jðxÞ � /j�1ðxÞ

xj � xj�1

� �fj

þ 1

21þ /1ðxÞ � /0ðxÞ

x1 � x0

� �f0 þ

1

21� /mðxÞ � /m�1ðxÞ

xm � xm�1

� �fm; ð2:7Þ

or

f �ðxÞ ¼ f0 þ fm2

þ 1

2

Xm�1

j¼0

/jðxÞ � /jþ1ðxÞxjþ1 � xj

ðfjþ1 � fjÞ; ð2:8Þ

or

f �ðxÞ ¼ 1

2

Xm�1

j¼1

fjþ1 � fjxjþ1 � xj

� fj � fj�1

xj � xj�1

� �/jðxÞ þ

f0 þ fm2

þ f1 � f02ðx1 � x0Þ

/0ðxÞ �fm � fm�1

2ðxm � xm�1Þ/mðxÞ. ð2:9Þ

Moreover, on [x0,xm], we have

ðf �ðxÞÞ0 ¼ 1

2

Xm�1

j¼0

/0jðxÞ � /0

jþ1ðxÞxjþ1 � xj

ðfjþ1 � fjÞ ð2:10Þ

and

ðf �ðxÞÞ00 ¼ 1

2

Xm�1

j¼0

/00j ðxÞ � /00


ðfjþ1 � fjÞ. ð2:11Þ

Remark 2.2. We note that the formulae (2.7)–(2.11) and the linear reproduc-

ing property of the quasi-interpolation f �(x) have no relation to the definition

of /j(x), j = 0, . . .,m � 1 i.e. (2.6). In other words, all quasi-interpolation f �(x)

defined by (2.1), (2.2), (2.4) and (2.5) satisfy with (2.7)–(2.11) and possess the

linear reproducing property.

Theorem 2.2. Denote h = max16i6m{xi � xi�1}. f�(x) is the univariate multiqua-

dric quasi-interpolation defined by Definition 2.4. For k > 0, and f(x) 2C2(x0,xm), we have,

kf �ðxÞ � f ðxÞk1 6 K0Ch þ K1h2 þ K2khþ K3k

2 log h; ð2:12Þ


where

Ch ¼ min k;k2

h

� �; ð2:13Þ

K0, K1, K2, and K3 are the positive constants independent of h and k.

Remark 2.3. As k = 0, f �(x) changes into L(x), and now kf �(x) � f(x)k1 6

Kh2, where K is a constant which is independent of h.

3. Numerical scheme using MQ quasi-interpolation

In this section, we present the numerical scheme for solving Burgers� equa-tion (with viscosity) [20] by using the multiquadric (MQ) quasi-interpolation.

Discretizing Burgers� equation

ut þ uux ¼1

Ruxx; ð3:1Þ

in time, we get

unþ1j ¼ unj � s � unj � ðuxÞ

nj þ s

1

RðuxxÞnj . ð3:2Þ

Then, we use the derivatives of the MQ quasi-interpolation to approximate uxand uxx. To dump the dispersion of the scheme, we define a switch function

g(x, t), whose values are 0 or 1 at discrete points (xj, tn), as follows:

gnj ¼ maxf0; 1þminf0; signððuxÞnj � ðuxÞnkÞgg; ð3:3Þ

where k ¼ j� signðunj Þ. Thus, the resulting numerical scheme is

unþ1j ¼ unj � s � unj � ðuxÞ

nj � gnj þ s

1

RðuxxÞnj ; ð3:4Þ

where

ðuxÞnj ¼1

2

Xm�1

k¼0

/0kðxjÞ � /0

kþ1ðxjÞxkþ1 � xk

ðunkþ1 � unkÞ; ð3:5Þ

ðuxxÞnj ¼1

2

Xm�1

k¼0

/00kðxjÞ � /00

kþ1ðxjÞxkþ1 � xk

ðunkþ1 � unkÞ. ð3:6Þ

/j(x), j = 0, . . .,m is defined in (2.6) and (2.4). unj is the approximation of the

value of u(x, t) at point (xj, tn), tn = ns; s is time step.


4. Numerical examples

We test the algorithm by two examples. The first example is comparing the

computational results to that of analytic solution (for R = 10, and R = 100) or

Christie accurate solution (R = 10,000) and Hon and Mao�s technique. The ini-tial conditions are

uðx; 0Þ ¼ u0ðxÞ ¼ sin px; 0 6 x 6 1 ð4:1Þand the boundary conditions are

uð0; tÞ ¼ 0 ¼ uð1; tÞ. ð4:2ÞWe denote this scheme by MQQI scheme. For the sake of simplification, we set

hi ¼ h ¼ 1

m.

Thus, xj = jh, j = 0, . . .,m. The comparison of the results is shown in Tables

1–3. In Tables 1 and 2, we also give the relative errors with respect to the

Table 1

Comparison of results for R = 10 at t = 1

x Analytic

solution

Hon and Mao�smethod s = 10�3

MQQI

scheme s = 10�3

Relative

error · 10�2

0.10 0.0663 0.0664 0.071238 7.448

0.20 0.1312 0.1313 0.13431 2.3679

0.30 0.1928 0.1928 0.19339 0.30347

0.40 0.2480 0.2481 0.24538 1.0572

0.50 0.2919 0.2919 0.28517 2.3052

0.60 0.3161 0.3159 0.30473 3.5977

0.70 0.3081 0.3079 0.29288 4.9416

0.80 0.2537 0.2534 0.23784 6.2509

0.90 0.1461 0.1459 0.13542 7.3088

Table 2

Comparison of results for R = 100 at t = 1

x Analytic

solution


MQQI

scheme s = 10�3

Relative

error · 10�2

0.10 0.0754 0.0755 0.078675 4.3439

0.20 0.1506 0.1507 0.15202 0.96341

0.30 0.2257 0.2257 0.22554 0.072147

0.40 0.3003 0.3003 0.29904 0.41805

0.50 0.3744 0.3744 0.37226 0.57245

0.60 0.4478 0.4478 0.44484 0.66202

0.70 0.5203 0.5202 0.51643 0.74341

0.80 0.5915 0.5913 0.58622 0.89348

0.90 0.6600 0.6607 0.62956 4.6118

Table 3

Comparison of results for R = 10,000 at t = 1

x Christie accurate

solution


MQQI scheme

s = 10�3

0.056(0.05) 0.0422 0.0424 0.042395

0.111(0.11) 0.0843 0.0843 0.084337

0.167(0.16) 0.1263 0.1263 0.12619

0.222(0.22) 0.1684 0.1684 0.16796

0.278(0.27) 0.2103 0.2103 0.20957

0.333(0.33) 0.2522 0.2522 0.25129

0.389(0.38) 0.2939 0.2939 0.29281

0.444(0.44) 0.3355 0.3355 0.3342

0.500(0.50) 0.3769 0.3769 0.37544

0.556(0.55) 0.4182 0.4182 0.41468

0.611(0.61) 0.4592 0.4592 0.45729

0.667(0.66) 0.5000 0.4999 0.49783

0.722(0.72) 0.5404 0.5404 0.53806

0.778(0.77) 0.5806 0.5805 0.57794

0.833(0.83) 0.6203 0.6201 0.61739

0.889(0.88) 0.6596 0.6600 0.65635

0.944(0.94) 0.6983 0.6957 0.69475


analytical solutions given by Cole [9], where s = 0.001, h = 0.01, and k = 0.0072

for R = 10, k = 0.0029 for R = 100, respectively. The numbers in the parenthe-

ses of the first column, the second column, and the third column in Table 3 arequoted from Ref. [20], where the Christie accurate solution is computed by

Christie and Mitchell [7] using the Galerkin method with fully upwind cubic

functions and a particularly small value of spatial step h, for MQQI scheme,

we set R = 10,000, s = 10�3, h ¼ 172, and k = 1.43 · 10�4. By the way, we give

the results with respect to k = 10�4 in Ref. [6]. The numerical results of the

MQQI scheme, for t = 0.2, 0.4, 0.6, 0.8, and 1.0 together with the initial data,

are given in Figs. 1–3, which correspond to R = 10, R = 100, and R = 10,000,

respectively.From the tables and figures above, we can say that the scheme is feasible

although the accuracy is not higher than that of Hon and Mao�s method.

We know that, at each time step, the complexity of our techniques is

only OðmÞ. Furthermore, the implementation of the present methods is very

easy.

For another example, the boundary conditions are same as above, while the

initial value conditions are uðx; 0Þ ¼ sin 2pxþ 0.5 sin px. The numerical results

of this example are shown in Fig. 4.From Fig. 4, we believe the results are acceptable. It means that this scheme

is valid.

Fig. 1. The results of the Burgers� equation for R = 10 by using MQQI scheme, where the spatial

step h ¼ 1100, the temporal step s = 10�3, and the shape parameter k = 0.0072.

Fig. 2. The results of the Burgers� equation for R = 100 by using MQQI scheme, where the spatial

step h ¼ 1100, the temporal step s = 10�3, and the shape parameter k = 0.0029.


Fig. 3. The results of the Burgers� equation for R = 10,000 by using MQQI scheme, where the

spatial step h ¼ 172, the temporal step s = 10�3, and the shape parameter k = 1.43 · 10�4.

Fig. 4. The results of the Burgers� equation for R = 10,000 by using MQQI scheme, where the

spatial step h ¼ 180, the temporal step s = 10�3, the shape parameter k = 1.2 · 10�4, and the initial

value conditions are uðx; 0Þ ¼ sin 2pxþ 0.5 sin px.



5. Conclusion

From the above figures and tables, all in all, we conclude that the methods

are feasible and valid. Moreover, we can improve the accuracy by selecting the

appropriate shape parameter and using higher accurate MQ quasi-

interpolation.The techniques can be used for the non-equidistant grids, for comparison,

although we are using equidistant grids in our numerical experiments. We

see that the present techniques require to calculate

/0jðxÞ � /0


;/00

j ðxÞ � /00jþ1ðxÞ

xjþ1 � xj; j ¼ 0; . . . ;m� 1

once for all.

The results have very close relation to the shape parameter k. In fact, the

choice of the shape parameter is still a pendent question.

In Tables 1–3, we see that the accuracy of the present scheme is higher as the

value of R is larger. Unfortunately, from our numerical experiment whose re-

sults are not given here, we see that the effects are very bad for R = 0.1 andR = 1. So the method must be rebuilt for R = 0.1 and R = 1.

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