solving hyperbolic conservation laws using multiquadric quasi-interpolation

Solving Hyperbolic Conservation Laws UsingMultiquadric Quasi-interpolationRonghua Chen,1,2 Zongmin Wu3

1Institute of Mathematics, Fudan University, Shanghai, 200433, PR China2School of Mathematics and Computational Science, Hunan University of Science andTechnology, Xiangtan, Hunan, 411201, PR China

3Department of Mathematics, Fudan University, Shanghai, 200433, PR China

Received 15 March 2004; accepted 6 July 2005Published online 27 September 2005 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/num.20115

In this article, we apply the univariate multiquadric (MQ) quasi-interpolation to solve the hyperbolic conser-vation laws. At first we construct the MQ quasi-interpolation corresponding to periodic and inflow-outflowboundary conditions respectively. Next we obtain the numerical schemes to solve the partial differentialequations, by using the derivative of the quasi-interpolation to approximate the spatial derivative of thedifferential equation and a low-order explicit difference to approximate the temporal derivative of the differ-ential equation. Then we verify our scheme for the one-dimensional Burgers’ equation (without viscosity).We can see that the numerical results are very close to the exact solution and the computational accuracy ofthe scheme is O(τ ), where τ is the temporal step. We can improve the accuracy by using the high-order quasi-interpolation. Moreover the methods can be generalized to the other equations. © 2005 Wiley Periodicals,Inc. Numer Methods Partial Differential Eq 22: 776–796, 2006

Keywords: multiquadric quasi-interpolation; hyperbolic conservation laws; radial basis function; meshlessmethod.

1. INTRODUCTION

There are a lot of the finite difference schemes, such as FCT (flux corrected transport),MUSCL (monotonic upstream centered schemes for conservation laws), PPM (piecewise par-abolic method), TVD (total variation diminishing), ENO (essentially nonoscillatory), and WENO(weighted ENO) to approximate the hyperbolic conservation laws. The ENO and WENO schemesare nearly perfect schemes not only in the theoretic sense but also the applied effect. But the

Correspondence to: Ronghua Chen, Institute of Mathematics, Fudan University, Shanghai, 200433, PR China (email:[email protected])Contract grant sponsor: National Science Foundation of China; contract grant numbers: NSFC 19971017 and NOYS10125102AMS subject classification: 65L05, 65M06, 65M55

© 2005 Wiley Periodicals, Inc.

HYPERBOLIC CONSERVATION LAWS 777

constructions of these schemes are very hard because they require very abundant professionalknowledge and practical experience. Moreover the effects are not as good as we expect when theseschemes apply to the complicated boundary conditions and the initial conditions with scattereddata.

Recently, the meshless methods using radial basis function (RBF) have become the topics ofmany researchers. For example, Chen [1, 2], Fasshauer [3, 4], Fedoseyev, et al. [5], Hon [6–10],Kansa [11–13], Li and Chen [14], Sharan [15], Wong [16–18], and Wu [19,20] applied the radialbasis functions to solve differential equations; Chen [21], Li [22, 23] applied the radial basisfunctions to neural networks and so on. These methods have no restrictions of the regularity ofthe domain and the uniformity of the mesh, so they can be used to solve more practical problems.

Since Hardy proposed in 1968, the multiquadric (MQ) have been investigated thoroughly.Hardy [24] summarized the achievement of study of MQ from 1968 to 1988 and showed that MQcan be apply in hydrology, geodesy, photogrammetry, surveying and mapping, geophysics andcrustal movement, geology and mining and so on. In Franke’s review article [25], the MQ was ratedone of the best methods among 29 scattered data interpolation schemes based on their accuracy,stability, efficiency, memory requirement, and ease to implementation. As an interpolation method,the MQ always produces a minimal semi-norm error as proven by Madych and Nelson [26].Thanks to the investigation of Beatson, Dyn, Powell, Wu and Schaback et al. for the theory ofthe radial basis functions (see [27–31]), the usages of the RBFs become more and more widely.Since Kansa [11, 12] successfully modified MQ for solving partial differential equation (PDE),more and more researchers have been attracted by this meshless, scattered data approximationscheme [4, 6–10, 13, 15, 18, 20]. In most of the known methods of solving differential equationsusing multiquadric, one must resolve a linear system of equation at each time step. Hon and Wu [10]and Wu [20] have provided some successful examples using MQ to solve the differential equations.

Hon and Mao [9] developed an efficient numerical scheme for Burgers’ equation (with vis-cosity),by applying the MQ as a spatial approximation scheme and a low-order explicit finitedifference approximation to the time derivative. This method requires to solve a linear system, byusing Gaussian elimination with partial pivoting, in order to obtain the coefficients of the interpo-lation function. Then get the value of the given points at the given time by using the interpolationfunction. And the interpolation function is the linear combinations with the MQ and the linearfunction. The method is valid for the various Reynolds number R whose scope from 0.1 to 10000,namely, the method has very broad applicability. They find that the method offers better accuracythan other numerical methods [9]. Again, the results of the method are very close to the analyticalsolution obtained by Cole [32] and the accurate solution given by Christie [33].

So, in this article, we develop a method, namely, applying a kind of univariate multiquadric(MQ) quasi-interpolation to solve the partial differential equations. The present method can beused to deal with the complicated boundary conditions and the initial conditions with scattereddata. We use still MQ quasi-interpolation so that we do not require to solve any linear systemsuch that we do not meet the question of the ill-condition of the matrix. Therefore we can savethe computational time and decrease the numerical error.

In our methods, we use the derivative of the MQ quasi-interpolation to approximate the spatialderivative of the differential equations and employ a first-order accurate forward difference forthe approach of the temporal derivative as Hon and Mao do [9]. And we follow the Wu andSchaback’s [31] idea to construct the MQ quasi-interpolation.

The organization of this article is as follows: In section 2, we construct two kinds of univari-ate MQ quasi-interpolation corresponding to periodic and inflow-outflow boundary conditions,respectively. In section 3, we develop two numerical techniques using MQ to solve the hyperbolicequation. In section 4, we use them to solve the Burgers’ equation (without viscosity) where the

Numerical Methods for Partial Differential Equations DOI 10.1002/num

778 CHEN AND WU

boundary conditions are either periodic or inflow-outflow. The numerical results are given in twofigures and four tables. The results of our methods are very close to the exact results of the partialdifferential equations. Furthermore, we obtain a scheme for the viscid Burgers’ equation (that is,with viscosity) which is a parabolic equation. Comparing the results the present scheme to that ofthe Christie accurate solution and Hon and Mao’s method [9], we know that the scheme is validtoo. We can assert that our methods can be generalized to high-order accuracy and high-resolutionscheme by employing the high-order quasi-interpolation as showed by Beatson and Dyn [28] orby Zhang and Wu [34] and rewriting the scheme. In section 5, we derive conclusion and giveremarks for the resulting scheme and the further work.

2. MULTIQUADRIC QUASI-INTERPOLATION

Beatson and Powell [27] proposed three univariate multiquadric quasi-interpolations, namely, LA,LB, and LC , to approximate a function {f (x), x0 ≤ x ≤ xm} from the space that is spanned by themultiquadrics {φj (x) = √

(x − xj )2 + c2, x ∈ R, j = 0, . . . , m} and linear function, where c isa positive constant and the centers {xj : j = 0, . . . , m} being given distinct points in the interval[x0, xm]. Afterward, Beatson and Dyn [28] have studied the properties of the �-splines, the com-bination of the MQs, and obtained the error estimates for quasi-interpolation schemes involvingMQ based on a finite number of centers. Wu and Schaback [31] have proposed the univariatemultiquadric quasi-interpolation LD on [a, b] and proven that the scheme is shape preservingand convergent. In this section, we construct two kinds of special MQ quasi-interpolation, whichgeneralized LD.

Given points {(xj , fj )}mj=0, where x0 < x1 < · · · < xm, we construct the univariate quasi-

interpolation in the form of

f ∗(x) =m∑

j=0

fj�j (x), (2.1)

where

�j(x) = φj+1(x) − φj (x)

2(xj+1 − xj )− φj (x) − φj−1(x)

2(xj − xj−1), 0 ≤ j ≤ m. (2.2)

We give the definition of φj (x) in the posterior Definition.Now we give some definitions as follows.

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

f ∗(x) ≡ C for f0 = f1 = · · · = fm = C, (2.3)

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 propertyon [x0, xm], if f ∗(x) = px + q as fj = pxj + q, j = 0, . . . , m, for all p, q ∈ R.

Remark 2.1. It is obvious that if a quasi-interpolation f ∗(x) possesses linear reproducingproperty on [x0, xm], then it must be constant reproducing.

Definition 2.3. If the quasi-interpolation f ∗(x) is monotone increasing (decreasing) for mono-tone increasing (decreasing) data fj , j = 0, . . . , m, then we say that it possesses preservingmonotonicity on [x0, xm].Numerical Methods for Partial Differential Equations DOI 10.1002/num


2.1. Univariate MQ Quasi-interpolation for Periodic Function

If the given data have period T = xm − x0, i.e., fj+km = fj , for xj+km = kT + xj , k ∈ Z, werequire that the quasi-interpolation f ∗(x) possesses the same period T . To do so, we require onlyto circumscribe x ∈ [x0, xm] for (2.1) and then extend f ∗(x) to be a periodic function with periodT . So we consider the quasi-interpolation within one period and with the conditions

x0 − x−1 = xm − xm−1, x1 − x0 = xm+1 − xm, f−1 = fm−1, and fm+1 = f1. (2.4)

We can obtain following propositions, definition and theorems.

Proposition 2.1. If the quasi-interpolation f ∗(x), defined by (2.1), (2.2), and (2.4), satisfieswith

fm = f0, (2.5)

and

φ′′−1(x) = φ′′

0 (x) = φ′′m(x) = φ′′

m+1(x), (2.6)

then

(f ∗(x))′′ = 1

2

m−1∑j=0

(fj+1 − fj

xj+1 − xj

− fj − fj−1

xj − xj−1

)φ′′

j (x). (2.7)

Proof. From (2.1) and (2.2), we have

(f ∗(x))′′ =m∑

j=0

fj�′′j (x)

= 1

2

m∑j=0

(φ′′

j+1(x) − φ′′j (x)

xj+1 − xj

− φ′′j (x) − φ′′

j−1(x)

xj − xj−1

)fj

= 1

2

m−1∑j=0

(fj+1 − fj

xj+1 − xj

− fj − fj−1

xj − xj−1

)φ′′

j (x) − φ′′0 (x)f−1

2(x0 − x−1)+ φ′′

m(x)fm−1

2(xm − xm−1)

+ φ′′m+1(x)fm

2(xm+1 − xm)− φ′′

m(x)fm

2(xm − xm−1)+ φ′′

−1(x)f0

2(xm − xm−1)− φ′′

m(x)fm

2(xm+1 − xm).

Then, employing (2.4), (2.5), and (2.6), we complete the proof.

Remark 2.2. From (2.6), we have

{φ−1(x) = φ0(x) + ax + b,

φm+1(x) = φm(x) + cx + d.(2.8)

where a, b, c, d ∈ R,


780 CHEN AND WU

Proposition 2.2. The quasi-interpolation f ∗(x), defined by (2.1), (2.2), (2.4), and (2.8), isconstant reproducing if and only if

a

x0 − x−1+ c

xm+1 − xm

= 0,

b

x0 − x−1+ d

xm+1 − xm

= 2,(2.9)

or

a

xm − xm−1+ c

x1 − x0= 0,

b

xm − xm−1+ d

x1 − x0= 2.

(2.10)

Proof. As f0 = f1 = · · · = fm = C, we have

f ∗(x) =m∑

j=0

fj�j (x) = C

m∑j=0

�j(x).

Thus, f ∗(x) ≡ C if and only ifm∑

j=0

�j(x) ≡ 1.

Due to

m∑j=0

�j(x) = ax + b

2(x0 − x−1)+ cx + d

2(xm+1 − xm).

So, f ∗(x) ≡ C if and only if

ax + b

2(x0 − x−1)+ cx + d

2(xm+1 − xm)≡ 1.

That is to say, f ∗(x) ≡ C if and only if (2.9) hold. Due to x0 − x−1 = xm − xm−1, xm+1 − xm =x1 − x0, we finish the proof of the proposition.

Proposition 2.3. If the quasi-interpolation f ∗(x), defined by (2.1), (2.2), (2.4) and (2.8), meetswith either (2.5) and (2.10) or

a = c = 0, (2.11)

then

(f ∗(x))′ = 1

2

m−1∑j=0

φ′j (x) − φ′

j+1(x)

xj+1 − xj

(fj+1 − fj ). (2.12)



Proof. From (2.1) , (2.2), and (2.8), we have

(f ∗(x))′ =m∑

j=0

f (xj )�′j (x)

= 1

2

m−1∑j=0

φ′j (x) − φ′

j+1(x)

xj+1 − xj

(fj+1 − fj ) + 1

2fm

φ′m+1(x) − φ′

m(x)

xm+1 − xm

− 1

2f0

φ′0(x) − φ′

−1(x)

x0 − x−1

= 1

2

m−1∑j=0

φ′j (x) − φ′

j+1(x)

xj+1 − xj

(fj+1 − fj ) + 1

2fm

c

x1 − x0+ 1

2f0

a

xm − xm−1.

From (2.11), we have (2.12). From (2.5) and (2.10), we have (2.12) too.

Proposition 2.4. If the quasi-interpolation f ∗(x), defined by (2.1), (2.2), (2.4), and (2.8),possesses linear reproducing property on [x0, xm] and meets with (2.9) or (2.11), then we have

φ′0(x) − φ′

m(x) = 2. (2.13)

Proof. From Proposition 2.3, we have (2.12). Since f ∗(x) possesses linear reproducingproperty, we obtain

m−1∑j=0

(φ′j (x) − φ′

j+1(x)) = 2.

Because

m−1∑j=0

(φ′j (x) − φ′

j+1(x)) =m−1∑j=0

φ′j (x) −

m∑j=1

φ′j (x) = φ′

0(x) − φ′m(x).

Thus, we have (2.13).

Proposition 2.5. If the quasi-interpolationf ∗(x), defined by (2.1), (2.2), (2.4), and (2.8), satisfieswith (2.9), (2.11), {

b = x0 − x−1 = xm − xm−1,

d = xm+1 − xm = x1 − x0,(2.14)

andφm(x) = φ0(x) − 2x + xm + x0, (2.15)

then f ∗(x) possesses linear reproducing property on [x0, xm].Proof. From Proposition 2.3, we have (2.12). Hence, for fj = pxj + q, from (2.15), we

obtain

(f ∗(x))′ = 1

2p

m−1∑j=0

(φ′j (x) − φ′

j+1(x)) = p.


782 CHEN AND WU

Thus, we have f ∗(x) = px + r , where r is some real constant. Now, we must to prove r = q.Substitution fj = pxj + q into (2.1), we get

f ∗(x) =m∑

j=0

(pxj + q)�j(x) = p

m∑j=0

xj�j (x) + q

m∑j=0

�j(x).

Again, from (2.8), (2.9), (2.11), and (2.14), we have

m∑j=0

�j(x) ≡ 1.

So, we complete the proof of the proposition provided that

m∑j=0

xj�j (x) = x.

Because (2.1) can be rewritten as

f ∗(x) = 1

2

m∑j=0

(fj+1 − fj

xj+1 − xj

− fj − fj−1

xj − xj−1

)φj (x)

+ φm+1fm

2(xm+1 − xm)− φ0f−1

2(x0 − x−1)− φmfm+1

2(xm+1 − xm)+ φ−1f0

2(x0 − x−1).

Due to (2.8), (2.11), and (2.14), we have{φ−1(x) = φ0(x) + x0 − x−1,

φm+1(x) = φm(x) + xm+1 − xm.(2.16)

Therefore, for fj = xj , we obtain

f ∗(x) = φm(x) + xm+1 − xm

2(xm+1 − xm)xm − φ0(x)x−1

2(x0 − x−1)− φm(x)xm+1

2(xm+1 − xm)+ φ0(x) + x0 − x−1

2(x0 − x−1)x0

= φ0(x) − φm(x) + xm + x0

2,

From (2.15), we have f ∗(x) = x, namely,

m∑j=0

xj�j (x) = x.

Remark 2.3. (2.13) together with (2.8) ensure (2.6) valid.

From the above propositions, we have the following.



Theorem 2.1. The quasi-interpolation f ∗(x), defined by (2.1), (2.2), (2.4), (2.15), and (2.16),possesses linear reproducing property on [x0, xm]. Meantime, on [x0, xm], f ∗(x) can be rewrittenas follows:

f ∗(x) = 1

2

m−1∑j=1

(φj+1(x) − φj (x)

xj+1 − xj

− φj (x) − φj−1(x)

xj − xj−1

)fj

+ 1

2

(1 + φ1(x) − φ0(x)

x1 − x0

)f0 + 1

2

(1 − φm(x) − φm−1(x)

xm − xm−1

)fm, (2.17)

or

f ∗(x) = f0 + fm

2+ 1

2

m−1∑j=0

φj (x) − φj+1(x)

xj+1 − xj

(fj+1 − fj ), (2.18)

or

f ∗(x) = 1

2

m−1∑j=1

(fj+1 − fj

xj+1 − xj

− fj − fj−1

xj − xj−1

)φj (x)

+ f0 + fm

2+ f1 − f0

2(x1 − x0)φ0(x) − fm − fm−1

2(xm − xm−1)φm(x). (2.19)

Moreover, on [x0, xm], we have (2.12) and

(f ∗(x))′′ = 1

2

m−1∑j=0

φ′′j (x) − φ′′

j+1(x)

xj+1 − xj

(fj+1 − fj ). (2.20)

Definition 2.4. For the initial data stem from a periodic function f (x) with period T = xm −x0,the quasi-interpolation on [x0, xm], f ∗(x), is defined by (2.1), (2.2), (2.4):

φm(x) = φ0(x) − 2x + xm + x0, (2.21){φ−1(x) = φ0(x) + x0 − x−1,

φm+1(x) = φm(x) + xm+1 − xm.(2.22)

and

φj (x) =√

(x − xj )2 + λ2, 0 ≤ j ≤ m − 1, (2.23)

where λ ∈ R, then the extension of f ∗(x) with period T , we denote it as f ∗(x) still, is call theunivariate multiquadric quasi-interpolation with period T .

Theorem 2.2. The quasi-interpolation f ∗(x), defined by Definition 2.4, possesses linear repro-ducing property and preserving monotonicity on [x0, xm]. Meantime, on [x0, xm], f ∗(x) can berewritten as follows:

f ∗(x) = 1

2

m−1∑j=1

(φj+1(x) − φj (x)

xj+1 − xj

− φj (x) − φj−1(x)

xj − xj−1

)fj

+ 1

2

(1 + φ1(x) − φ0(x)

x1 − x0

)f0 + 1

2

(1 − φm(x) − φm−1(x)

xm − xm−1

)fm, (2.24)


784 CHEN AND WU

or

f ∗(x) = f0 + fm

2+ 1

2

m−1∑j=0

φj (x) − φj+1(x)

xj+1 − xj

(fj+1 − fj ), (2.25)

or

f ∗(x) = 1

2

m−1∑j=1

(fj+1 − fj

xj+1 − xj

− fj − fj−1

xj − xj−1

)φj (x)

+ f0 + fm

2+ f1 − f0

2(x1 − x0)φ0(x) − fm − fm−1

2(xm − xm−1)φm(x). (2.26)

Moreover, on [x0, xm], we have

(f ∗(x))′ = 1

2

m−1∑j=0

φ′j (x) − φ′

j+1(x)

xj+1 − xj

(fj+1 − fj ) (2.27)

and

(f ∗(x))′′ = 1

2

m−1∑j=0

φ′′j (x) − φ′′

j+1(x)

xj+1 − xj

(fj+1 − fj ). (2.28)

Proof. We obtain (2.24), (2.25), and (2.26) by computing straightforward. Differentiating(2.25), we have (2.27) and (2.28). We complete the proof of the linear reproducing propertyprovided that

f ∗(x) ≡ 1, for fj = 1 (2.29)

andf ∗(x) ≡ x, for fj = xj , (2.30)

where j = 0, . . . , m. Because (2.1) can be rewritten as

f ∗(x) = 1

2

m∑j=0

(fj+1 − fj

xj+1 − xj

− fj − fj−1

xj − xj−1

)φj (x)

+ φm+1(x)fm

2(xm+1 − xm)− φ0(x)f−1

2(x0 − x−1)− φm(x)fm+1

2(xm+1 − xm)+ φ−1(x)f0

2(x0 − x−1).

From (2.22), we have (2.29) for fj = 1 and

f ∗(x) = φm(x) + xm+1 − xm

2(xm+1 − xm)xm − φ0(x)x−1

2(x0 − x−1)− φm(x)xm+1

2(xm+1 − xm)+ φ0(x) + x0 − x−1

2(x0 − x−1)x0

= φ0(x) − φm(x) + xm + x0

2,

for fj = xj . Due to (2.21), we have (2.30). Now we prove that the quasi-interpolation f ∗(x),possesses preserving monotonicity on [x0, xm]. Wu and Schaback [31] have proven that −1 ≤Numerical Methods for Partial Differential Equations DOI 10.1002/num


φ′j (x) ≤ φ′

j−1(x) ≤ 1, for j = 1, . . . , m − 1. From (2.21), we have φ′m(x) = φ′

0(x) − 2 ≤ −1 ≤φ′

m−1(x). Therefore, φ′j (x) − φ′

j+1(x) ≥ 0, for j = 0, . . . , m − 1. From (2.27), we complete theproof of the preserving monotonicity.

Remark 2.4. We note that the formulae (2.24), (2.25), (2.26), (2.27), (2.28) and the lin-ear reproducing property of the quasi-interpolation f ∗(x) have no relation to the definition ofφj (x), j = 0, . . . , m − 1, i.e., (2.23). In other words, all quasi-interpolation f ∗(x) defined by(2.1), (2.2), (2.21), and (2.22) satisfy with (2.24), (2.25), (2.26), (2.27), (2.28) and possess thelinear reproducing property.

Using Wu and Schaback’s idea [31], we have:

Theorem 2.3. Denote h = max1≤i≤m{xi − xi−1}. f (x) is a periodic function with period T =xm − x0. f ∗(x), defined by Definition 2.4, is the univariate multiquadric quasi-interpolation withperiod T . If λ > 0, then

(1) for f (x) ∈ C2(x0, xm),

‖f ∗(x) − f (x)‖∞≤ K0Ch + K1h2 + K2λh + K3λ

2 log h, (2.31)

where

Ch = min

{λ,

λ2

h

}, (2.32)

(2) for f (x) ∈ C2[A, B],

‖f ∗(x) − f (x)‖∞≤ K1h2 + K2λh + K3λ

2 log h, (2.33)

where [A, B] ⊃ [x0 − h, xm + h], K0, K1, K2, and K3 are the positive constants independent of h

and λ.

Proof. Due tof (x) is a periodic function andf ∗(x), defined by Definition 2.4, is the extensionof the univariate multiquadric quasi-interpolation, we require only to prove the conclusion asx ∈ [x0, xm]. From (2.26), we have

2f ∗(x) = f0 + fm + φ0(x)f1 − f0

x1 − x0− φm(x)

fm − fm−1

xm − xm−1

+m−1∑j=1

φj (x)

(fj+1 − fj

xj+1 − xj

− fj − fj−1

xj − xj−1

)

= f0 + fm + φ0(x)�1(x0, x1)f − φm(x)�1(xm−1, xm)f

+m−1∑j=1

[φj (x)(xj+1 − xj−1)�

2(xj−1, xj , xj+1)f]

,

where �1 and �2 is the divided difference of the first- and second-order, respectively. Denotethe piecewise linear interpolation of f (x) with respect to the same initial data as L(x). For


786 CHEN AND WU

x ∈ [x0, xm], we have |x − x−1| = x − x−1, and |x − x0| = x − x0, |x − xm| = xm − x,|x − xm+1| = xm+1 − x. Hence we have

2L(x) =m∑

j=0

fj

( |x − xj+1| − |x − xj |xj+1 − xj

− |x − xj | − |x − xj−1|xj − xj−1

)

= f0 + fm + (x − x0)�1(x0, x1)f − (xm − x)�1(xm−1, xm)f

+m−1∑j=1

|x − xj |(xj+1 − xj−1)�2(xj−1, xj , xj+1)f .

Thus, we have2(f ∗(x) − L(x)) = M(x) + N(x),

where

M(x) =m−1∑j=1

{(φj (x) − |x − xj |)(xj+1 − xj−1)�

2(xj−1, xj , xj+1)f}

,

N(x) = (φ0(x) − (x − x0))�1(x0, x1)f − (φm(x) − (xm − x))�1(xm−1, xm)f

= (φ0(x) − (x − x0))(�1(x0, x1)f − �1(xm−1, xm)f ).

From the supposition of the theorem, the above-mentioned first- and second-order divideddifferences are bounded. Therefore,(1) for f (x) ∈ C2(x0, xm),

‖f ∗(x) − f (x)‖∞ ≤ 1

2‖N(x)‖∞ +

∥∥∥∥1

2M(x) + L(x) − f (x)

∥∥∥∥∞

.

Wu and Schaback have proven (see [31] for detail)∥∥∥∥1

2M(x) + L(x) − f (x)

∥∥∥∥∞

≤ K1h2 + K2λh + K3λ

2 log h

and √λ2 + y2− | y |≤ λ, for λ ≥ 0,

√λ2 + y2− | y |≤ λ2

2 | y | , for λ ≥ 0, and y = 0.

Hence

1

2‖ N(x) ‖∞≤ K0Ch

and (2.31) hold, where Ch defined by (2.32).



(2) for f (x) ∈ C2[A, B], since �1(xm−1, xm)f = �1(x−1, x0)f , we have

f ∗(x) − L(x) = 1

2

m−1∑j=0

{(φj (x) − |x − xj |)(xj+1 − xj−1)�

2(xj−1, xj , xj+1)f}

.

The proof is similar to that of [31] given, we have (2.33).

Remark 2.5. As λ = 0, f ∗(x) change into L(x), and now ‖ f ∗(x) − f (x) ‖∞≤ Kh2, whereK is a constant which independent of h.

From f (x) ∈ C2[A, B] is a periodic function with period T = xm − x0 and [A, B] ⊃ [x0 − h,xm + h], we know that f (x) ∈ C2(−∞, +∞).

2.2. MQ Quasi-interpolation for Inflow-Outflow Boundary Conditions

In this subsection, for the given data {(xj , fj )}mj=0 attached the conditions

f−1 = f0, fm+1 = fm, (2.34)

we construct the quasi-interpolation, where the conditions (2.34) are called the inflow-outflowconditions.

Corresponding to the above subsection, we get the following propositions, definition, andtheorems.

Proposition 2.6. If the quasi-interpolation f ∗(x), defined by (2.1), (2.2), and (2.34), satisfieswith

φ′′−1(x) = φ′′

0 (x), φ′′m(x) = φ′′

m+1(x), (2.35)

then

(f ∗(x))′′ = 1

2

m∑j=0

(fj+1 − fj

xj+1 − xj

− fj − fj−1

xj − xj−1

)φ′′

j (x). (2.36)

That is to say (2.7) holds too.

Proof. From (2.1) and (2.2), we have

(f ∗(x))′′ = 1

2

m∑j=0

(fj+1 − fj

xj+1 − xj

− fj − fj−1

xj − xj−1

)φ′′

j (x) − f−1φ′′0 (x)

2(x0 − x−1)

+ fmφ′′m+1(x)

2(xm+1 − xm)− fm+1φ

′′m(x)

2(xm+1 − xm)+ f0φ

′′−1(x)

2(x0 − x−1).

Then, employing (2.34) and (2.35), we obtain (2.36). Rewriting (2.36) as

(f ∗(x))′′ = 1

2

m−1∑j=0

φ′′j (x) − φ′′

j+1(x)

xj+1 − xj

(fj+1 − fj ) + fm+1 − fm

2(xm+1 − xm)φ′′

m(x) − f0 − f−1

2(x0 − x−1)φ′′

0 (x),

from (2.34) we complete the proof.

From (2.35), we have (2.8) too.


788 CHEN AND WU

Proposition 2.7. The univariate quasi-interpolation f ∗(x), defined by (2.1), (2.2), (2.8), and(2.34), is constant reproducing if and only if (2.9) holds.

Proof. The proof is same as that of the Proposition 2.2.

If we set x−1, xm such that

x0 − x−1 = xm − xm−1, xm+1 − xm = x1 − x0,

then (2.9) is equivalent to (2.10).

Proposition 2.8. If the the quasi-interpolation f ∗(x), defined by (2.1), (2.2), (2.8), and (2.34),meets with (2.11), then (2.12) holds.

Proof. The proof is same as that of Proposition 2.3.

Proposition 2.9. If the the quasi-interpolation f ∗(x), defined by (2.1), (2.2), (2.8), and (2.34),possesses linear reproducing property on [x0, xm] and meets with (2.11), then we have (2.13).

Proof. The proof is similar to that of Proposition 2.4.

Proposition 2.10. If the the quasi-interpolation f ∗(x), defined by (2.1), (2.2), (2.8), and (2.34),satisfies with (2.11), (2.21) and {

b = x0 − x−1,

d = xm+1 − xm,(2.37)

then f ∗(x) possesses linear reproducing property on [x0, xm] .

Proof. The proof is similar to that of Proposition 2.5.

From Proposition 2.6 to Proposition 2.10, we have the following.

Theorem 2.4. The quasi-interpolation f ∗(x), defined by (2.1), (2.2), (2.21), (2.22), and (2.34),possesses linear reproducing property on [x0, xm]. Meantime, on [x0, xm], f ∗(x) can be rewrittenas (2.24) or (2.25) or (2.26). Moreover, (2.27) and (2.28) hold.

Definition 2.5. For the initial data stem from a real function f (x) attached with (2.34), theunivariate quasi-interpolation on [x0, xm], f ∗(x), which is defined by (2.1), (2.2), (2.21), (2.22),and (2.23), is call the univariate quasi-interpolation with inflow-outflow on [x0, xm].

Theorem 2.5. The quasi-interpolation f ∗(x), defined by Definition 2.5, possesses linear repro-ducing property and preserving monotonicity on [x0, xm]. Meantime, on [x0, xm], f ∗(x) can berewritten as (2.24) or (2.25) or (2.26). Moreover, (2.27) and (2.28) hold.

Proof. The proof is similar to that of Theorem 2.2.

Using Wu and Schaback’s idea [31], we have also the following.



Theorem 2.6. Define h as ahead. f ∗(x), defined by Definition 2.5, is the univariate multiquadricquasi-interpolation with inflow-outflow on [x0, xm]. Then, for λ > 0 and f (x) ∈ C2(x0, xm), wehave (2.31).

Proof. The proof is similar to that of Theorem 2.2.

Remark 2.6. As λ = 0, f ∗(x) change into L(x), and now ‖ f ∗(x) − f (x) ‖∞≤ Kh2, whereK is a constant which independent of h.

3. SCHEMES USING MQ QUASI-INTERPOLATION TO SOLVE PDE

In this article, we solve mainly the one-dimensional scalar hyperbolic conservation laws by usingthe multiquadric (MQ) quasi-interpolation. For MQ quasi-interpolation application to vectorial,multi-dimensional hyperbolic conservation laws and others kind differential equations, they arethe open questions.

For the one-dimensional scalar hyperbolic conservation laws

ut + (f (u))x = 0, (3.1)

with initial conditionu(x, 0) = u0(x). (3.2)

We can rewrite (3.1) asut + a(u) · ux = 0, (3.3)

and discretize (3.3) in time as

un+1j = un

j − τ · (a(u))nj · (ux)

nj , (3.4)

where, a(u) = (f (u))u, unj is the approximation of the value of u(x, t) at point (xj , tn), tn = nτ ;

τ is the length of time step; (a(u))nj is the value of the function a(u) at u = un

j ; and the (ux)nj is

computed by (2.12), i.e.,

(ux)nj = 1

2

m−1∑i=0

φ′i (xj ) − φ′

i+1(xj )

xi+1 − xi

(uni+1 − un

i ), (3.5)

where φi(x), i = 0, . . . , m is defined in (2.23) and (2.21).To dump the dispersion of the scheme, we define the function g(x, t) as follows:

gnj =

{0, if (ux)

nj · (ux)

nk < 0 and min{|(ux)

nj |, |(ux)

nk |} > k1;

1, otherwise,(3.6)

orgn

j = max{0, 1 + min{0, sign((ux)nj · (ux)

nk)}}, (3.7)

where k = j − sign((a(u))nj ), k1 is a given positive constant.

Hence, we change (3.4) into

un+1j = un

j − τ · (a(u))nj · (ux)

nj · gn

j . (3.8)


790 CHEN AND WU

Thus we obtain two algorithms as follows:

Algorithm 1: This algorithm consists of (3.8), (3.5), and (3.6).

Algorithm 2: This algorithm consists of (3.8), (3.5), and (3.7).

4. NUMERICAL EXAMPLES

In this section, we verify our schemes by using two numerical examples. These are solvingthe Burgers’ equation (without viscosity) with periodic and inflow-outflow boundary conditions,respectively. We know that the magnitudes of the shape parameter λ is the key factor for attaininghigh accuracy. We set λ = 4h2 and k1 = 0.5, unless otherwise stated, which is the better choice webelieved. Of course, the choice is not always the best one. And the choice of the shape parameteris yet a pendent question. As for the choice of the time step, we follow the rule proposed byWu [20]. In our given figures, “Exact" imply the exact solutions.

Again, we obtain the numerical scheme for the Burgers’ equation (with viscosity), which is aparabolic equation. Comparing the results to the Christie accurate solution and that of the Honand Mao’s method [9], we see that the present method can be generalized.

Now, we use the above Algorithms to slove the Burgers’ equation (without viscosity)

ut + uux = 0, (4.1)

i.e., a(u) = u, subject to the initial conditions

u0(x) = α + β sin(π(x + γ )). (4.2)

For the sake of simplification, we set

hi = h = 2

m.

Our numerical experiment have two examples. One associates with periodic boundary condi-tions and α = 1.0, β = 0.5, γ = 0.0. The results at t = 0.3 and t = 0.636 ≈ 2/π , which isthe time of the formation of the shock, are shown in Fig. 1, where m = 80, τ = 10−3 . Anothercorresponds to α = 0.0, β = 1.0, γ = 1.0 and inflow-outflow boundary conditions, namely,x = −1 is treated as an inflow boundary and specified u(−1, t) = 0; x = 1 is treated as an out-flow boundary. Here, the boundaries x = ±1 are characteristic, and a stationary shock developsat x = 0 at t = 1/π . At t = 0.6, the solution has already started to decay considerably due to theinteraction of the shock with the expansion waves. The results at t = 0.3 and t = 0.6 are shownin Fig. 2, where m = 80, τ = 10−3 . In these examples, we use Algorithm 2. See [35] for theexact solutions.

Moreover, for the various combinations between λ and τ and for the various spatial step h, wecompute the L∞-norms and L1-norms of the errors between the numerical solution and the exactsolution and the computational orders of accuracy by using Algorithm 1, where the boundaryconditions are periodic ones, α = 1.0, β = 0.5, γ = 0.0, and t = 0.3. The results are showed inTables I–IV, where, ‖y‖, the L1-norm of the vector y = (y0 y1 · · · ym) is defined as

‖y‖ = 1

m + 1

m∑i=0

|yi |. (4.3)



FIG. 1. The results of (4.1), (4.2) with periodic boundary by using Algorithm 2. The exact solution iscomputed by Newton Raphson’s iterations with tolerance 10−6. For the exact solution, the spatial steph = (1/200); for the numerical results, h = (1/40). From the figure, we see that the method is valid. [Colorfigure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

The values of rc in Tables I–IV are the “computational order of accuracy,” which are calculated byassuming the error to be a constant times hrc ; this definition is meaningful only for h sufficientlysmall.

Figures 1 and 2 show that the methods above, for the Burgers’ equation (without viscosity),are effective. From Tables I–IV, we can conclude that the accuracy of the methods are O(τ ), andthere is a close relation between the accuracy of the resulting numerical solutions and the shapeparameter λ.

We turn now to consider the viscid Burgers’ equation [9]:

ut + uux = 1

Ruxx , (4.4)

subject to the initial conditions

u(x, 0) = u0(x) = sin πx, (4.5)

and the boundary conditions

u(0, t) = 0 = u(1, t). (4.6)

Now, discretization (4.4) in time, we get

un+1j = un

j − τ · unj · (ux)

nj · gn

j + τ1

R(uxx)

nj , (4.7)


792 CHEN AND WU

FIG. 2. The results of (4.1), (4.2) with inflow-outflow boundary by using Algorithm 2. The exact solutionis computed by Newton Raphson iterations with tolerance 10−6. For the exact solution, the spatial steph = (1/200); for the numerical results, h = (1/40). From the figure, we see that the method is valid. [Colorfigure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

where

(uxx)nj = 1

2

m−1∑i=0

φ′′i (xj ) − φ′′

i+1(xj )

xi+1 − xi

(uni+1 − un

i ). (4.8)

and Algorithm 3:

Algorithm 3: This algorithm consists of (4.7), (3.5), (3.7), and (4.8).

We test the algorithm by comparing the computational results to that of Christie accuratesolution, where, the Christie accurate solution is computed by Christie and Mitchell [33] using

TABLE I. The L∞ error of the solution of the (4.1), (4.2) with periodic boundary by using Algorithm 1and λ = 10τ at t = 0.3 and the computational order of accuracy of the algorithm at that time.

m τ = 0.3h rc τ = 0.6h rc τ = 0.3h2 rc τ = 0.6h2 rc

8 0.3954 0.6520 0.2294 0.31110.4080 0.6093 1.6033 1.3636

16 0.2980 0.4274 0.0755 0.12090.4994 0.3916 2.0685 2.2117

32 0.2108 0.3258 0.0180 0.02610.8457 0.4410 1.9683 2.1699

64 0.1173 0.2400 0.0046 0.00581.2389 0.9035 2.2016 2.1575

128 0.0496 0.1283 0.0011 0.0013



TABLE II. The L1 error of the solution of the (4.1), (4.2) with periodic boundary by using Algorithm 1and λ = 10τ at t = 0.3 and the computational order of accuracy of the algorithm at that time.


8 0.2348 0.3354 0.0691 0.13610.8551 0.5212 1.9568 2.0976

16 0.1298 0.2337 0.0178 0.03180.9391 0.7885 2.3058 2.5580

32 0.0677 0.1353 0.0036 0.00541.0021 0.7690 2.1919 2.4330

64 0.0338 0.0794 7.9 × 10−4 0.00101.6195 1.0746 2.0672 2.1597

128 0.0110 0.0377 1.9 × 10−4 2.2 × 10−4

the Galerkin method with fully upwind cubic functions and a particularly small value of spatialstep h. The numerical results of the Algorithm 3, for t = 0.2, 0.4, 0.6, 0.8, and 1.0 together withthe initial data, are given in Fig. 3, where R = 10000, τ = 10−3, h = (1/72) and λ = 10−4.

The maximum relative difference between Christie accurate solution and Algorithm 3 is0.99241% and the minimum relative difference is only 0.016588%.

From the comparison and Fig. 3, we can say that the scheme is feasible for Burgers’ equation(with viscosity) too. We know that, at each time step, the complexity of our techniques is onlyO(m). Furthermore, the implementation of the present methods are very easily.

5. CONCLUSION

From the above figures and tables, all in all, for solving one-dimensional hyperbolic conservationlaws, we conclude that the methods are feasible and valid.

The techniques can be use for the nonequidistant grids, for comparison, although we useequidistant grids in our numerical experiments. When using the techniques for the nonequidistantgrids, we suggest λ = 4h2

0 instead of λ = 4h2, where

h0 = min1≤i≤m

{xi − xi−1}.

As for the technique of the moving knots, we can refer to [20].

TABLE III. The L∞ error of the solution of the (4.1), (4.2) with periodic boundary by using Algorithm 1and λ = 4h2 at t = 0.3 and the computational order of accuracy of the algorithm at that time.


8 0.2776 0.3007 0.2609 0.26631.1499 0.8624 1.5696 1.5177

16 0.1251 0.1654 0.0879 0.09301.3700 0.8400 2.1504 2.1264

32 0.0484 0.0924 0.0198 0.02131.1180 1.0902 2.0444 2.0068

64 0.0223 0.0434 0.0048 0.00531.1005 1.1177 2.1255 2.0275

128 0.0104 0.0200 0.0011 0.0013


794 CHEN AND WU

TABLE IV. The L1 error of the solution of the (4.1), (4.2) with periodic boundary by using Algorithm 1and λ = 4h2 at t = 0.3 and the computational order of accuracy of the algorithm at that time.


8 0.0996 0.1156 0.0912 0.09351.4522 0.9058 2.0847 2.0046

16 0.0364 0.0617 0.0215 0.02331.2040 1.0117 2.4628 2.4048

32 0.0158 0.0306 0.0039 0.00441.1540 1.1076 2.2468 2.2472

64 0.0071 0.0142 8.2 × 10−4 9.3 × 10−4

1.0623 1.0837 2.1074 2.0992128 0.0034 0.0067 1.9 × 10−4 2.2 × 10−4

We see that, in the present techniques, for given x we require only to calculate

φ′j (x) − φ′

j+1(x)

xj+1 − xj

,φ′′

j (x) − φ′′j+1(x)

xj+1 − xj

, j = 0, . . . , m − 1

once for all.The results showed in the above figures except Fig. 3 are based upon λ = 4h2. Of course,

this is not always the best choice. In fact, the choice of the shape parameter λ is still a pendentquestion.

We can assert that (3.6) and (3.7) can be improved, but the two formulae are the simplestexpressions.

FIG. 3. The results of the Burgers’ equation (with viscosity) by using Algorithm 3, where, the spatial steph = (1/72), the temporal step τ = 10−3, the shape parameter λ = 10−4. [Color figure can be viewed in theonline issue, which is available at www.interscience.wiley.com.]



The methods can be generalized to the hyperbolic systems of the one-dimension and multi-dimension and other ordinary or partial differential equations.

The second author have extended the multiquadric (MQ) quasi-interpolation to higher-dimensional case (see [36] for detail). As for the approach using MQ quasi-interpolation to solvehigher-dimensional partial differential equations, the second author have published on severalinternational conferences recently and the paper are being preprinted.

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