Applied Mathematics and Computation 186 (2007) 1502–1510
www.elsevier.com/locate/amc
Solving partial diﬀerential equation by using
multiquadric quasi-interpolation q
Ronghua Chen
a
a,*
, Zongmin Wu
b
School of Mathematics and Computational Science, Hunan University of Science and Technology,
Xiangtan, Hunan 411201, PR China
b
School of Mathematics, Fudan University, Shanghai 200433, PR China
Abstract
In this paper, we use a kind of univariate multiquadric (MQ) quasi-interpolation to solve partial diﬀerential equation
(PDE). 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 temporal derivative of the
dependent variable. The advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement. Our numerical experiment includes two examples. One is solving viscid Burgers’ equation for initial trapezoidal conditions. Another is simulating the interaction of two waves travelling in opposite direction. From the numerical
experiment, we can see that the present scheme is valid.
2006 Elsevier Inc. All rights reserved.
Keywords: Multiquadric quasi-interpolation; Radial basis function; Burgers’ equation; Two wave travelling in opposite direction;
Interaction
1. Introduction
Since Hardy proposed in 1968, the multiquadric (MQ) which is a kind of radial basis function (RBF) have
been investigated thoroughly. Hardy [16] summarized the achievement of study of MQ from 1968 to 1988
and showed that MQ can be apply in hydrology, geodesy, photogrammetry, surveying and mapping, geophysics and crustal movement, geology and mining and so on. Since Kansa [17,18] successfully modiﬁed
MQ for solving partial diﬀerential equation (PDE), more and more researchers have been attracted by this
meshless, scattered data approximation scheme (see, for example, [3,6–8,10–15,19–24]). 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 [15], Wu [24] and others have provided some successful examples
q
A project supported by Scientiﬁc Research Fund of Hunan Provincial Education Department (R. Chen). Wu was supported by NSFC
19971017 and NOYS 10125102.
*
Corresponding author.
E-mail addresses: [email protected], [email protected] (R. Chen), [email protected] (Z. Wu).
0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2006.07.160
R. Chen, Z. Wu / Applied Mathematics and Computation 186 (2007) 1502–1510
1503
using MQ quasi-interpolation to solving diﬀerential equations. In this paper, we use still 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 [14] 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 Gaussian elimination with partial
pivoting, in order to obtain the coeﬃcients of the interpolation 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 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. Again, the results of the method are very close to the analytical solution obtained by Cole and the accurate solution given by Christie (see [14] and the reference therein).
In our methods, we use the derivative of the MQ quasi-interpolation to approximate the spatial derivative
of the diﬀerential equations and employ a ﬁrst order accurate forward diﬀerence for the approach of the temporal derivative as Hon and Mao do [14].
The organization of the rest of this paper is as follows: In Section 2, we introduce a univariate MQ quasiinterpolation and the numerical scheme using this quasi-interpolation to solve Burgers’ equation. In Section 3,
we give two examples. One is to solve Burgers’ equation and another is to simulate the interaction of two
waves travelling in opposite direction. The results are also acceptable. So the technique is valid. In Section
4, we derive conclusion and give remarks for the resulting scheme and the further work.
2. Multiquadric quasi-interpolation and the construction of the numerical scheme
Beatson and Powell [2] proposed three univariate multiquadric quasi-interpolations, namely, LA ; LB , and
L
a function {f(x), x0 6 x 6 xm} from the space that is spanned by the multiquadrics
C
, to approximate
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
2
/j ðxÞ ¼ ðx xj Þ þ c ; x 2 R; j ¼ 0; . . . ; m 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] have 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 centers. Wu and Schaback [25] 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 kinds of special MQ quasi-interpolation, which generalized LD .
m
Given points fðxj ; fj Þgj¼0 , where x0 < x1 < < xm, we construct the univariate quasi-interpolation in the
form of
m
X
f ðxÞ ¼
fj Wj ðxÞ;
ð2:1Þ
j¼0
where
Wj ðxÞ ¼
/jþ1 ðxÞ /j ðxÞ /j ðxÞ /j1 ðxÞ
;
2ðxjþ1 xj Þ
2ðxj xj1 Þ
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 an 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 reproducing property on [x0, xm], if
f *(x) = px + q as fj = pxj + q, j = 0, . . . , m, for all p; q 2 R.
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R. Chen, Z. Wu / Applied Mathematics and Computation 186 (2007) 1502–1510
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 ; fj Þgm
j¼0 , fj = f(xj), the univariate quasi-interpolation on [x0, xm], f (x), is
deﬁned by (2.1) and (2.2):
/m ðxÞ ¼ /0 ðxÞ 2x þ xm þ x0 ;
/1 ðxÞ ¼ /0 ðxÞ þ x0 x1 ;
/mþ1 ðxÞ ¼ /m ðxÞ þ xmþ1 xm :
ð2:4Þ
ð2:5Þ
and
/j ðxÞ ¼
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ðx xj Þ2 þ k2 ;
0 6 j 6 m 1;
ð2:6Þ
where k 2 R.
We can prove the following theorem (see [5] 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 following:
m1 /jþ1 ðxÞ /j ðxÞ /j ðxÞ /j1 ðxÞ
1X
1
/1 ðxÞ /0 ðxÞ
1þ
f ðxÞ ¼
fj þ
f0
2 j¼1
2
x1 x0
xjþ1 xj
xj xj1
1
/m ðxÞ /m1 ðxÞ
1
þ
ð2:7Þ
fm ;
2
xm xm1
or
f ðxÞ ¼
m1
/j ðxÞ /jþ1 ðxÞ
f0 þ fm 1 X
þ
ðfjþ1 fj Þ;
2 j¼0
xjþ1 xj
2
ð2:8Þ
or
m1 1X
fjþ1 fj fj fj1
f0 þ fm
f1 f0
fm fm1
þ
/0 ðxÞ / ðxÞ:
f ðxÞ ¼
/j ðxÞ þ
2 j¼1 xjþ1 xj xj xj1
2
2ðx1 x0 Þ
2ðxm xm1 Þ m
ð2:9Þ
Moreover, on [x0, xm], we have
0
ðf ðxÞÞ ¼
m1
/0j ðxÞ /0jþ1 ðxÞ
1X
ðfjþ1 fj Þ:
2 j¼0
xjþ1 xj
ð2:10Þ
m1
/00j ðxÞ /00jþ1 ðxÞ
1X
ðfjþ1 fj Þ:
2 j¼0
xjþ1 xj
ð2:11Þ
and
00
ðf ðxÞÞ ¼
Remark 2.2. We note that the formulae (2.7)–(2.11) and the linear reproducing 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.
R. Chen, Z. Wu / Applied Mathematics and Computation 186 (2007) 1502–1510
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Theorem 2.2. Denote h ¼ max16i6m fxi xi1 g. f *(x) is the univariate multiquadric quasi-interpolation defined by
Definition 2.4. For k > 0 and f(x) 2 C2(x0, xm), we have
kf ðxÞ f ðxÞk1 6 K 0 C h þ K 1 h2 þ K 2 kh þ K 3 k2 log h;
ð2:12Þ
where
k2
C h ¼ min k;
;
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) change into L(x), and now k f *(x) f(x)k1 6 Kh2, where K is a constant which
independent of h.
Now, we present the numerical scheme for solving viscid Burgers’ equation by using the multiquadric (MQ)
quasi-interpolation developed by us.
Discretizing Burgers’ equation
ut þ uux ¼
1
uxx
R
ð2:14Þ
in time, we get
1
n
n
ujnþ1 ¼ unj s unj ðux Þj þ s ðuxx Þj :
R
ð2:15Þ
Then, we use the derivatives of the MQ quasi-interpolation to approximate ux and 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
following:
n
n
gnj ¼ maxf0; 1 þ minf0; signððux Þj ðux Þk Þgg
ð2:16Þ
where k ¼ j signðunj Þ. Thus the resulting numerical scheme is
1
n
n
ujnþ1 ¼ unj s unj ðux Þj gnj þ s ðuxx Þj
R
ð2:17Þ
where
n
ðux Þj ¼
n
m1
/0k ðxj Þ /0kþ1 ðxj Þ n
1X
ðukþ1 unk Þ;
2 k¼0
xkþ1 xk
ðuxx Þj ¼
m1
/00k ðxj Þ /00kþ1 ðxj Þ n
1X
ðukþ1 unk Þ:
2 k¼0
xkþ1 xk
ð2:18Þ
ð2:19Þ
/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.
The content above can be found in [4], here, for the sake of integrality of the paper, we iterate them.
3. Numerical examples
In this section, we give two examples to test the technique. The ﬁrst example is solving viscid Burgers’ equation with R = 10 000. Another example is simulating the interaction of two waves travelling in opposite
direction.
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R. Chen, Z. Wu / Applied Mathematics and Computation 186 (2007) 1502–1510
3.1. Solving viscid Burgers’ equation
For this example, the initial conditions are
8
< 103 x; 0 6 x 6 0:3;
uðx; 0Þ ¼ 1; 0:3 6 x 6 0:6;
:
0; 0:6 < x 6 1:0
ð3:1Þ
and the boundary conditions are
uð0; tÞ ¼ 0 ¼ uð1; tÞ:
ð3:2Þ
We 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 numerical results of the MQQI scheme, for t = 0.1, 0.2, 0.3, 0.4, and 0.5 together with the initial data, are given in Fig. 1.
From Fig. 1, we can say that the scheme is feasible. We know that, at each time step, the complexity of our
techniques is only OðmÞ. Furthermore, the implementation of the present methods are very easily.
3.2. Simulating two waves travelling in opposite direction
Another example is a two-component, quasi-nonlinear hyperbolic system, the solution of which is composed of two waves travelling in opposite directions and located initial at x = 0.2 and x = 0.2. The system
is given by
ut ¼ ux 100uv; 0:5 < x < 0:5; t > 0;
ð3:3Þ
vt ¼ vx 100uv; 0:5 < x < 0:5; t > 0:
with the initial conditions
8
0:5½1 þ cosð10pxÞ;
>
>
>
< uðx; 0Þ ¼ 0;
>
0:5½1 þ cosð10pxÞ;
>
>
: vðx; 0Þ ¼
0;
x 2 ½0:3; 0:1;
otherwise;
x 2 ½0:1; 0:3;
ð3:4Þ
otherwise
1
0.9
0.8
0.7
0.6
0.5
0.4
t=0
t=0.1
t=0.2
t=0.3
t=0.4
t=0.5
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
Fig. 1. The results of the viscid Burgers’ equation with R = 10 000 by using MQQI scheme, where the spatial step h ¼ 100
, the temporal step
s = 103, the shape parameter k = 1.5 · 104.
R. Chen, Z. Wu / Applied Mathematics and Computation 186 (2007) 1502–1510
and the boundary conditions
uð0:5; tÞ ¼ uð0:5; tÞ ¼ 0;
1507
ð3:5Þ
vð0:5; tÞ ¼ vð0:5; tÞ ¼ 0:
Note that u(x, 0) and v(x, 0) are functions with mere C1 continuity, which represent wave pulse located at
x = 0.2 and x = 0.2, respectively. Initially, the nonlinear term 100uv vanishes, so that for t > 0 these waves
start to move without change of shape and with speed 1, u to the right and v to the left. At t = 0.1 they collide
at x = 0 and the nonlinear term becomes positive, resulting in a nonlinear interaction leading to changes in the
shapes and speeds of the waves. Speciﬁcally, the crests of waves collide a little beyond t = 0.25 and they have
separated again at approximately t = 0.3, so that from this time on the solution behavior is again dictated by
the linear terms. At the nonlinear interaction, the pulses lose their symmetry and experience a decrease in
amplitude (see [9]).
1
t=0.1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
–0.5
–0.4
–0.3
–0.2
–0.1
0
0.1
0.2
0.3
0.4
0.5
1
Fig. 2. The results at t = 0.1 by using MQQI scheme, where the spatial step h ¼ 140
, the temporal step s = 103, the shape parameter
k = 0.001.
1
t=0.2
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
–0.5
–0.4
–0.3
–0.2
–0.1
0
0.1
0.2
Fig. 3. The results at t = 0.2 by using MQQI scheme, where the spatial step h ¼
k = 0.001.
0.3
1
,
140
0.4
0.5
the temporal step s = 103, the shape parameter
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R. Chen, Z. Wu / Applied Mathematics and Computation 186 (2007) 1502–1510
For this example, the numerical scheme is as follows:
( nþ1
uj ¼ unj sððux Þnj gnj þ 100unj vnj Þ; j ¼ 1; . . . ; m 1;
¼ vnj þ sððvx Þnj snj 100unj vnj Þ;
vnþ1
j
ð3:6Þ
j ¼ 1; . . . ; m 1;
where, the computation of ux and vx is same as above, namely
n
ðux Þj ¼
n
ðvx Þj
m1
/0k ðxj Þ /0kþ1 ðxj Þ n
1X
ðukþ1 unk Þ;
2 k¼0
xkþ1 xk
ð3:7Þ
m1
/0k ðxj Þ /0kþ1 ðxj Þ n
1X
¼
ðvkþ1 vnk Þ:
2 k¼0
xkþ1 xk
1
0.9
t=0.25
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
–0.5
–0.4
–0.3
–0.2
–0.1
0
0.1
0.2
0.3
0.4
0.5
1
Fig. 4. The results at t = 0.25 by using MQQI scheme, where the spatial step h ¼ 140
, the temporal step s = 103, the shape parameter
k = 0.001.
1
0.9
t=0.3
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
–0.5
–0.4
–0.3
–0.2
–0.1
0
0.1
0.2
Fig. 5. The results at t = 0.3 by using MQQI scheme, where the spatial step h ¼
k = 0.001.
0.3
1
,
140
0.4
0.5
the temporal step s = 103, the shape parameter
R. Chen, Z. Wu / Applied Mathematics and Computation 186 (2007) 1502–1510
1509
1
0.9
t=0.5
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
–0.5
–0.4
–0.3
–0.2
–0.1
0
0.1
0.2
0.3
0.4
0.5
1
Fig. 6. The results at t = 0.5 by using MQQI scheme, where the spatial step h ¼ 140
, the temporal step s = 103, the shape parameter
k = 0.001.
The switch functions g and s are also same as above, viz.
gnj ¼ maxf0; 1 þ minf0; signððux Þnj ðux Þnj1 Þgg;
n
n
snj ¼ maxf0; 1 þ minf0; signððvx Þj ðvx Þjþ1 Þgg:
ð3:8Þ
x0 = 0.5, xm = 0.5, un0 ¼ unm ¼ 0; vn0 ¼ vnm ¼ 0.
The numerical results of this example at t = 0.1, 0.2, 0.25, 0.3, and 0.5 are shown in ﬁve ﬁgures as follows.
From Figs. 2–6, we believe the results are acceptable. It means that this scheme is also valid.
4. Conclusion
From the ﬁgures above, we conclude that the methods are feasible and valid.
The techniques can be use for the nonequidistant grids although we using equidistant grids in our numerical
experiments. We see that, for given x, the present techniques requires only to calculate
/0j ðxÞ /0jþ1 ðxÞ
;
xjþ1 xj
/00j ðxÞ /00jþ1 ðxÞ
;
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.
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