the laplace transform method for burgers' equation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids 2010; 63:1060–1076Published online 23 June 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld.2116

The Laplace transform method for Burgers’ equation

Suqin Chen∗,†, Xionghua Wu, Yingwei Wang and Weibin Kong

Department of Mathematics, Tongji University, Siping Road No. 1239, Shanghai 200092,People’s Republic of China

SUMMARY

The Laplace transform method (LTM) is introduced to solve Burgers’ equation. Because of the nonlinearterm in Burgers’ equation, one cannot directly apply the LTM. Increment linearization technique isintroduced to deal with the situation. This is a key idea in this paper. The increment linearizationtechnique is the following: In time level t , we divide the solution u(x, t) into two parts: u(x, tk) andw(x, t), tk�t�tk+1, and obtain a time-dependent linear partial differential equation (PDE) for w(x, t).For this PDE, the LTM is applied to eliminate time dependency. The subsequent boundary value problemis solved by rational collocation method on transformed Chebyshev points. To face the well-knowncomputational challenge represented by the numerical inversion of the Laplace transform, Talbot’s methodis applied, consisting of numerically integrating the Bromwich integral on a special contour by means oftrapezoidal or midpoint rules. Numerical experiments illustrate that the present method is effective andcompetitive. Copyright q 2009 John Wiley & Sons, Ltd.

Received 25 June 2008; Revised 14 May 2009; Accepted 16 May 2009

KEY WORDS: Burgers’ equation; increment linearization technique; rational collocation method;the Laplace transform; Talbot’s method and numerical inversion

1. INTRODUCTION

In this paper, we construct a numerical method for solving the one-dimensional Burgers’ equation.The one-dimensional Burgers’ equation with initial and boundary value problem (BVP) is the

∗Correspondence to: Suqin Chen, Department of Mathematics, Tongji University, Siping Road No. 1239, Shanghai200092, People’s Republic of China.

†E-mail: [email protected]

Contract/grant sponsor: National Nature Science Foundation of China; contract/grant numbers: 10671146, 50678122

Copyright q 2009 John Wiley & Sons, Ltd.

THE LAPLACE TRANSFORM METHOD FOR BURGERS’ EQUATION 1061

following:

�u�t

+u�u�x

= ��2u�x2

, x ∈�, t ∈(0,T ]u(a, t) = ua,u(b, t)=ub, t ∈(0,T ]u(x,0) = �(x), x ∈�

(1)

where �=(a,b), and ua , ub are given constants. Burgers’ equation is a very important fluiddynamic model; the study of this equation has been considered by many authors both for theconceptual understanding of a class of physical flows and for testing various numerical methods.Benton and Platzman [1] surveyed exact solutions of the one-dimensional Burgers’ equation. Inmany cases these solutions involve infinite series, which may converge very slowly for smallvalues of �>0 [2]. Various numerical techniques such as finite difference, finite element, boundaryelement, and spectral methods [3–10] are used to solve the equation numerically.

To obtain accurate numerical solutions of Burgers’ equation, it is desirable to use high-orderapproximations in space and time. For example, in [8], Shu-Sen Xie et al. presented a numericalmethod for Burgers’ equation by using piecewise quadratic polynomial basis functions and thereproducing kernel function (the quadratic reproducing kernel function method, QRKM). Foranother example, in [10], Tee and Trefethen used the adaptive Runge–Kutta 5(4) method ofDormand and Prince [11] to discretize the time part of Burgers’ equation, and used rational spectralcollocation method to discretize the spatial part of the partial differential equation (PDE). Yetbecause of the difficulties introduced by the combination of nonlinearity and small values of �>0,most computations so far have been limited to the use of small time steps.

In the present paper, we propose a new time-marching method in which one can use relativelylarge time steps and obtain highly accurate numerical solutions for Burgers’ equation. The newtime-marching method is related to the Laplace transform method (LTM). The Laplace transformis a technique proposed early, which removes the time dependency and converts the problem intoa BVP. However, this approach never really became popular in computational work. The mainreason may be that the Laplace transform, particularly its numerical inversion, has a reputation forbeing a computational challenge. The numerical inversion of the Laplace transform is an ill-posedproblem when the transform is known only as a real-valued function. Recently, Talbot’s method[12] has become a very popular technique for numerical inversion of the Laplace transform. Thetechnique combines the trapezoid rule or the midpoint rule with contour integration by usingcomplex arithmetic. The convergence rates for these optimized quadrature formulas are very fast:roughly O(3−M ) [13], where M is the number of sample points. Additionally, unlike the traditionaltime-marching method, an interesting advantage of using the Laplace transform is that it is easilyparallelizable. The nonlinear term in Burgers’ equation is a stumbling block when one uses theLTM to solve Burgers’ equation. In this paper, a key idea is to use the increment linearizationtechnique to linearize the nonlinear term in Burgers’ equation, and overcome the drawback. Let�t denote the time step, tk =k�t . In each time step, we define

u(x, t)=u(x, tk)+w(x, t), tk�t�tk+1 (2)

where u(x, tk) is the solution at tk . Using (2) and omitting the nonlinear term, which is O(�t2)we can eliminate the nonlinear term in Burgers’ equation. In the space discretization, we workwith the rational collocation method based on sinh function developed by Tee and Trefethen [10].

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2010; 63:1060–1076DOI: 10.1002/fld

1062 S. CHEN ET AL.

The layout of this paper is as follows. In Section 2, we use Equation (2) to obtain a linearizedequation. In Section 3, the Laplace transform is used to convert a class of time-dependent linearPDE to BVPs. A short review of the rational collocation method based on sinh function is presentedin Section 4. In Section 5, we employ Talbot’s method for the numerical inversion of the Laplacetransform. Three examples involving shock-like wave formation are given in Section 6. The lastsection gives the conclusions.

2. THE LINEARIZATION METHOD FOR BURGERS’ EQUATION

Generally, we assume that the computational interval in space is [−1,1]. At the time level t ,substituting (2) into (1), we find that

�w(x, t)

�t+u(x, tk)

�w(x, t)

�x+ �u(x, tk)

�xw(x, t)−�

�2w(x, t)

�x2+w(x, t)

�w(x, t)

�x

=��2u(x, tk)

�x2−u(x, tk)

�u(x, tk)

�xw(x, tk)=0

w(−1, t)=0, w(1, t)=0, tk�t�tk+1

(3)

Omitting the nonlinear term

w(x, t)�w(x, t)

�x

which is O(�t2), we obtain the following linear equation:

�w(x, t)

�t+u(x, tk)

�w(x, t)

�x+�u(x, tk)

�xw(x, t)−�

�2w(x, t)

�x2= �

�2u(x, tk)

�x2−u(x, tk)

�u(x, tk)

�xw(x, tk) = 0

w(−1, t)=0, w(1, t) = 0, tk�t�tk+1

(4)

In each time step, the method described in Sections 3–5 will be applied to obtain the numericalsolution w(x, tk+1) of (4), and further we have the solution at the time level tk+1:

u(x, tk+1)=u(x, tk)+w(x, tk+1) (5)

3. THE LAPLACE TRANSFORM FOR TIME-DEPENDENT LINEAR IBVPs (4)

Let w(x, t) be a piecewise continuous function. The Laplace transform w(x,s) of w(x, t) isdefined by

w(x,s)=∫ ∞

0e−stw(x, t)dt



To solve the IBVPs (4), we take the Laplace transform of w(x, t) giving w(x,s) as the solution to

sw(x,s)+u(x, tk)�w(x,s)

�x+ �u(x, tk)

�xw(x,s)−�

�2w(x,s)

�x2

=(

��2u(x, tk)

�x2−u(x, tk)

�u(x, tk)

�x

)/s

(6)

w(−1,s)=0, w(1,s)=0

For numerical purposes, (6) is solved for a sequence of value of {sn}, and then a numerical inversionLaplace transform to {w(x,sn)} is applied.

4. THE RATIONAL COLLOCATION METHOD BASED ON SINHFUNCTION FOR BVPs (6)

We now introduce the rational collocation method based on the sinh function [10]. The barycentricform of a rational function pN (x),which interpolates dataw0,w1, . . . ,wN at points x0, x1, . . . , xN is

pN (x)=∑N

k=0�k

x−xkwk∑N

k=0�k

x−xk

(7)

where �k , k=0, . . . ,N are called barycentric weights. An advantage of representing a rationalinterpolation in barycentric form is the simplicity of its derivative formulae at x j . The nth derivative

of pN (x) at x j can be written as p(n)N (x j )=∑N

k=0 D(n)jk wk . The elements of D(1) and D(2), that

is, first- and second-order differential matrices are given by

D(1)jk =

⎧⎪⎪⎨⎪⎪⎩

�k

� j (x j −xk)if j �=k

−∑i �= j

D(1)ji if j =k

(8)

D(2)jk =

⎧⎪⎪⎨⎪⎪⎩2D(1)

jk

(D(1)jj − 1

x j −xk

)if j �=k

−∑i �= j

D(2)ji if j =k

(9)

When xk =cos(k�/N ) are Chebyshev points of the second kind, �0= 12 , �k =(−1)k , k=1, . . . ,

N−1, �N =(−1)N/2, the above formulae are the differential matrices in the Chebyshev collocationmethod [10].

Lagrange polynomial interpolations can be represented in barycentric form [10], which arestable and have a special, beautiful symmetry. And a rational interpolation can be made frompolynomial interpolation in barycentric form by modifying its points and leaving its barycentricweights unchanged.

The rational interpolation based on barycentric form with transformed Chebyshev points hasthe following convergence analysis.


1064 S. CHEN ET AL.

Theorem 4.1 (Berrut et al. [14])Let D1 and D2 be domains in C containing J =[−1,1] and a real interval I , respectively. Letg :D1→D2 be a conformal map such that g(J )= I . If w :D2→C is a function such that thecomposition w◦g :D1→C is analytic inside and on an ellipse E�, �>1, with foci at ±1 and thesum of its semi-major and semi-minor axes equal to �. Let PN (x) be the rational function (7)interpolating w between the transformed Chebyshev points xk =g(cos(k�/N )) with barycentricweights

�0= 12 , �k =(−1)k, k=1, . . . ,N−1, �N =(−1)N/2

Then, for every x ∈[−1,1],|pN (x)−w(x)|=O(�−N ) (10)

It can be seen from (Theorem 4.1) that the rational interpolation, which interpolates at transformedChebyshev points, preserves exponential convergence.

The convergence rate depends on the size of the largest ellipse with foci ±1 of which w isanalytic, simply called ellipse of analyticity of w [10]. Theorem 4.1 suggests that a conformalmap g should be chosen so that the ellipse of analytic w◦g is larger than the ellipse of w.

If we consider the case where the function w has singularities at point z=a+bi for a∈ R andb>0, the sinh transform can be defined as

g(z)=a+b sinh(�(a,b)z−�(a,b)) (11)

where � and � are chosen so that the interval z∈{−1�Re(z)�1, Im(z)=0} corresponds tox ∈{−1�Re(x)�1, Im(x)=0} with z=±1 corresponding to x=±1, respectively. Then we have

�(a,b)= 1

2

[sinh−1

(1+a

b

)+sinh−1

(1−a

b

)](12)

�(a,b)= 1

2

[sinh−1

(1+a

b

)−sinh−1

(1−a

b

)](13)

Then g−1 transform z=a+ib in the z-plane to x=s+i t in the x-plane. After some calculations,we can obtain that the singular point z=a+ib is transformed by g−1 to

x j = �(a,b)

�(a,b)+i

(2 j+1)�

2�(a,b), j ∈ Z (14)

We choose the one closest to the interval [−1,1], i.e.

s(a,b)+ t (a,b)i= �(a,b)

�(a,b)+i

�

2�(a,b)(15)

Then the following theorem given in [15] shows that the transformed point x=s+ ti is furtheraway from [−1,1] than the point z=a+bi .

Theorem 4.2 (Elliott and Johnston [15])Suppose a�0 and b>0 are given. The point (a,b) in the z-plane is closer to the basic interval [−1,1]than the point (s, t) as defined in (15), in the x-plane.



For BVPs (6), let the collocation points be

X ={xk =g

(cos

(k�

N

)),k=0, . . . ,N

}

In the conformal map g(z)=a+b sinh(�(a,b)z−�(a,b)), a is equal to the location of shock-likewave, b=� while the width of the shock-like wave is O(�). Discretizing the BVPs (6) at pointsxk for k=0, . . . ,N , yields

sn I w(sn)+diag(uk)D(1)w(sn)+diag(D(1)uk)w(sn)−�D(2)w(sn)

=(�D(2)uk−diag(uk)(D(1)uk))/sn (16)

where w(sn)=[w0(sn), . . . , wN (sn)]T with w j (sn) evaluating w(x j ,sn), uk =[u(x0, tk), . . . ,u(xN , tk)]T, diag(uk) is a diagonal matrix with the elements of uk on the main diagonal, matricesD(1), D(2) and I are the first-order differential matrix, second-order differential matrix, and theidentity matrix, respectively. Combining the boundary conditions and solving (16), one can obtainthe numerical solution w(sn)=[w0(sn), . . . , wN (sn)]T.

5. NUMERICAL INVERSION OF THE LAPLACE TRANSFORM

The Laplace transform represents a very effective tool for solving several problems in the fieldsof science and engineering; however, the numerical inversion of the Laplace transforms was adifficult problem at one time. In the recent years, many results have been obtained in the fieldsof numerical inversion. Among them, a very accurate and general method is due to Talbot [12].Talbot’s method has proved to be applicable to both a wide range of Laplace transform functionsand to a wide range of values of t where we need to compute f (t). The method has exponentialconvergence rate under certain circumstances [13]. In this section, we will apply Talbot’s methodto obtain the approximation of w(x, t) based on the sequence {w(x,sn)}.

The inversion Laplace transform of w is defined by

w(x, t)= 1

2�i

∫Bestw(x,s)ds (17)

which is called the Bromwich integral, with B as the Bromwich line Re(s)=>0. 0 is themaximum value of the real part of all singularities of w.

In Talbot’s method, the Bromwich line is deformed into a curve � that begins and ends in theleft half-plane, such that Re(s)→−∞ on the contour. Owing to the exponential factor est, theintegrand decays rapidly on such a contour. Talbot’s contour is parameterized by

� :s()=+�(cot+�i), −�� (18)

where , �, and � are real parameters that determine the geometry of the curve. Both � and �are positive. We assume that � lies in the region of analyticity of w. Using the Cauchy integraltheorem and (18), the Bromwich integral (17) can be expressed as

w(x, t)= 1

2�i

∫Bestw(x,s)ds= 1

2�i

∫ �

−�es()t w(x,s())s′()d (19)


1066 S. CHEN ET AL.

where s′()=�(cot−csc2 +�i). The above integral can be approximated by the trapezoidalrule or the midpoint rule. Here, we use the midpoint rule with an even number of intervals, say2M , as used in [13].

The grid is defined as

n =(2n+1)�

2M, n=−M, . . . ,M−1 (20)

Denote the approximation to (19) by

wM (x, t)= 1

2Mi

M−1∑n=−M

es(n)t s′(n)w(x,s(n)) (21)

or

wM (x, t)= Im

(1

2M

M−1∑n=−M

es(n)t s′(n)w(x,s(n))

)(22)

If symmetry is used, allowing sn =s(n), s′n =s′(n), we can get the approximation of w(x, t), i.e.

wM (x, t)= Im

(1

M

M−1∑n=0

esnt s′nw(x,sn)

)(23)

Here, w(x,sn) are solved from (16).It is well known that Talbot’s method (18) with fixed parameters converges at a subgeometric

rate of O(e−c√M ); see [12]. Weideman [13] let both and � be proportional to the ratio M/t ,

and a geometric rate O(e−cM) as M→∞ can be obtained. Moreover, Weideman finds optimalparameters for Talbot’s method for some problems: when the following Talbot’s contour

� :s()= M

t(−0.2407+0.2378cot(0.7409)+0.1349i) (24)

is used as the Bromwich line, the convergence rate is given by O(e−2.56M ).

6. NUMERICAL EXAMPLE

In this section, we present three examples involving the development of a sharp front at an interiorpoint and near a boundary point to test the accuracy and efficiency of the present method.

The accuracy of our method will be measured using discrete L2- and L∞-error norms defined by

‖ek‖L2 =(N−1∑j=1

|ek(x j )|2 1

N−1

)1/2

, ‖ek‖L∞ = max1� j�N−1

|ek(x j )|

where ek(x j )=ue(x j , tk)−un(x j , tk), ue(x j , tk) is the exact solution of the considered problem,un(x j , tk) is the numerical solution. In the following three examples, �t denotes the time step,M denotes interval number in (23), N denotes collocation point number in (16).



Example 1We consider the following Burgers’ equation with initial and boundary conditions

�u�t

+u�u�x

= ��2u�x2

, 0<x<1, t ∈(0,T ]

u(0, t) = 0, u(1, t)=0, t ∈(0,T ]u(x,0) = sin(�x), 0<x<1

(25)

The exact solution of the problem is given by

u(x, t)=2��

∑∞n=1 an exp(−n2�2�t)n sin(n�x)

a0+∑∞n=1 an exp(−n2�2�t)cos(n�x)

(26)

where

a0=∫ 1

0exp(−(2��)−1(1−cos(�x)))dx

an =2∫ 1

0exp(−(2��)−1(1−cos(�x)))cos(n�x)dx, n=1,2, . . .

In this experiment, the time step �t=0.001, M=16, and N =12 are used. Figure 1 illustratesthe graphs of the numerical and corresponding exact solution curves of Example 1 for �=1. Inthe figure, the exact (solid line) and numerical (dashed line) solution curves are drawn on theleft of the figure and they are in considerably good agreement. In Table I, we compare errors indiscrete L2- and L∞-norms of Example 1 for �=1 at t=0.1,0.2, and 0.4 generated by using ourLTM and the QRKM [8]. In addition, Table I also lists CPU times of LTM and QRKM in solvingExample 1. From Table I, it is observed that LTM gives better accuracy and lower computationalcost is required, which shows that LTM is effective and competitive.

To test the rate of convergence, some numerical solutions and error norms computed by usingLTM for Example 1 with �=1 at t=0.1 are presented in Tables II–IV. Error norms with differenttime step sizes �t for fixed M=16 and fixed N =12 are presented in Table II. Table III listserror norms with different M for fixed time step size �t=0.001 and fixed N =12. Table IVgives error norms with different N for fixed time step size �t=0.001 and fixed M=16. Onecan see that the accuracy measured in the discrete L2- and L∞-norms increases monotonously asthe time step sizes get smaller, as interval number gets larger, and as collocation point numbergets larger. The rate of convergence is about O(�t2.2), O(3−M ), and O(3.16−N ) for both norms.Figures 2 and 3 show a comparison between numerical and exact solutions for viscosity coefficients�=0.1(�t=0.001,M=16,N =30) and �=0.01 (�t=0.001, M=16, N =40).

It is known that the Fourier solutions for �<0.01 fail to converge because of the slow convergenceof the infinite series [2]. The numerical solutions for �=0.001 and �=1e−4 at different timesare drawn in Figures 4 and 5, which show the correct physical behavior. The curves in Figures 4and 5 are similar to the ones in [8].


1068 S. CHEN ET AL.

0 0.2 0.4 0.6 0.8 1–0.2

0

0.2

0.4

0.6

0.8

1

1.2

t=0.0

t=0.1

t=0.2

t=0.4

Exact and approximate solutionsat different times

0 0.2 0.4 0.6 0.8 1–3

–2

–1

0

1

2

3

4x 10–7

t=0.1

t=0.2

t=0.4

Distribution of absolute errorsat different times

Figure 1. Exact, numerical solutions (left), and distribution of absolute errors(right) for �=1 at different times.

Table I. Error norms and CPU times for Example 1 using LTM and QRKM at different times with �=1.

t

0.1 0.2 0.4

LTM (N =12, �t=0.001)‖e‖L2 1.9561×10−7 6.4277×10−8 1.5843×10−8

‖e‖L∞ 3.7601×10−7 1.3052×10−7 2.7478×10−8

CPU time 0.015 0.015 0.016

QRKM (N =1000, �t=0.001)‖e‖L2 4.56483×10−5 3.39538×10−5 0.94262×10−5

‖e‖L∞ 6.45697×10−5 4.80322×10−5 1.33307×10−5

CPU time 2.375 4.1880 7.7970

Example 2As another example, we consider Burgers’ equation with initial and boundary conditions

u(x,0) = sin(2�x), 0<x<1

u(0, t) = 0, u(1, t)=0, t ∈(0,T ] (27)



Table II. Error norms for Example 1 with different time step sizes �t .

�t=0.05 �t=0.025 �t=0.0125 �t=0.0625 �t=0.003125

‖e‖L2 8.9661×10−4 1.6333×10−4 3.4823×10−5 8.0398×10−6 1.9280×10−6

‖e‖L∞ 1.5×10−3 2.7628×10−4 5.8283×10−5 1.3356×10−5 3.1546×10−6

Table III. Error norms for Example 1 with different point numbers M .

M=8 M=10 M=12 M=14 M=16

‖e‖L2 1.3×10−3 1.1625×10−4 1.0230×10−5 8.7392×10−7 2.0089×10−7

‖e‖L∞ 2.1×10−3 1.9670×10−4 1.7331×10−5 1.5708×10−6 3.9341×10−7

Table IV. Error norms for Example 1 with different point number N .

N ‖e‖L2 ‖e‖L∞

4 2.3×10−3 4.0×10−3

5 3.6055×10−4 6.1175×10−4

6 1.0067×10−5 1.8030×10−5

7 2.0890×10−6 4.3574×10−6

8 4.8157×10−7 1.0094×10−6

9 2.5452×10−7 6.2130×10−7

10 2.1171×10−7 3.7856×10−7

11 1.9933×10−7 3.9406×10−7

12 1.9561×10−7 3.7601×10−7

This example is usually used for simulating the shock-like wave formation. The exact solution ofthe problem is given by

u(x, t)=2��

∑∞n=1 an exp(−n2�2�t)n sin(n�x)

a0+∑∞n=1 an exp(−n2�2�t)cos(n�x)

(28)

where

a0=∫ 1

0exp(−(4��)−1(1−cos(2�x)))dx

an =2∫ 1

0exp(−(4��)−1(1−cos(2�x)))cos(n�x)dx, n=1,2, . . .

The numerical experiments for Example 2 are done in the same pattern as Example 1. Figure 6shows a comparison between numerical and exact solutions of Example 2 at different times for�=0.005(�t=0.01, M=16, N =80). The curves for distribution of absolute errors at differenttimes are also shown on the right of Figure 6. It is clearly seen that the present solutions arein very good agreement with the exact ones. In Table V, we compared CPU times and errors indiscrete L2- and L∞-norms of Example 2 for �=0.005 at t=1.4, 2.0, and 2.6 generated by using


1070 S. CHEN ET AL.

0 0.2 0.4 0.6 0.8 1–0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1–5

0

5

10

15

20x 10–8


Distribution of absoluteerrors at different times

t=0.0

t=0.6

t=0.8

t=1.0

t=3.0

t=3.0

t=1.0

t=0.8 t=0.6

Figure 2. Exact, numerical solutions (left), and distribution of absolute errors(right) for �=0.1 at different times.

our LTM and the QRKM [8]. Table V shows that lower computational cost is required and betteraccuracy is obtained for LTM.

The numerical solutions at different times for �=0.001(�t=0.01, M=16, N =80) aredrawn in Figure 7, which shows the correct physical behavior. Fourier solutions do notconverge for this small value �.

Example 3We consider the following Burgers’ equation with initial and boundary conditions:

�u�t

−u�u�x

= ��2u�x2

, −1<x<1, t ∈(0,T ]u(−1, t) = −1, u(1, t)=1, t ∈(0,T ]

u(x,0) =sinh

( x

2�

)cosh

( x

2�

)+exp

(− 1

4�

) , −1<x<1

(29)



0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1–3

–2.5

–2

–1.5

–1

–0.5

0

0.5

1x 10–6



t=0.0 t=0.6

t=0.8

t=1.0

t=3.0

t=0.6

t=0.8

t=1.0 t=3.0


The exact solution of the problem is given by

u(x, t)=sinh

( x

2�

)cosh

( x

2�

)+exp

(− t+1

4�

) , −1<x<1 (30)

In order to further assess the effectiveness and accuracy of LTM, we compared CPU times anderrors in discrete L2- and L∞-norms generated by using the Crank–Nicolson method (CN) [16]and the exponential time-differencing fourth-order Runge–Kutta (ETDRK4) [17] for Example 3with viscosity �=0.00075 at t=3. Figure 8 illustrates the numerical and exact solutions using LTMwith the time step �t=0.01, collocation points number N =100, and M=16. Figure 9 shows thatCN exhibits undesirable oscillation at t=3 due to the sharp change in the value around x=0 whenthe time interval h=0.05 and the space size k=0.05 are used. To capture this sharp transitionlayer while avoiding oscillation, extremely small time and space intervals are required. Table VI


1072 S. CHEN ET AL.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0.0

t=1.0

t=2.0

t=3.0 t=4.0

Figure 4. Numerical solutions for �=0.001 at different times.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=0.0

t=1.0

t=2.0

t=3.0

t=4.0

Figure 5. Numerical solutions for �=10−4 at different times.

shows that the computation cost of using LTM is less than one of using CN and ETDRK4, andLTM gives better accuracy, which shows that LTM is effective and competitive.

Remark 6.1All numerical results in this paper were obtained on Pentium(R) 4 at a clock speed of 1.8Ghz.



0 0.2 0.4 0.6 0.8 1–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1–6

–4

–2

0

2

4

6x 10–5

Exact and approximatesolutions at different times


t=0.0

t=0.9

t=1.4

t=2.0 t=2.6

t=0.9

t=1.4

t=2.0

t=2.6


Table V. Error norms and CPU time for Example 2 using LTM and QRKMat different times with �=0.005.

t

1.4 2.0 2.6

LTM (N =80, �t=0.01)‖e‖L2 1.8967×10−5 1.0392×10−5 6.3852×10−6

‖e‖L∞ 3.3991×10−5 1.9320×10−5 1.2088×10−5

CPU time 0.4840 0.5000 0.5160

QRKM (N =1000, �t=0.01)‖e‖L2 5.54924×10−4 4.12911×10−4 3.30783×10−4

‖e‖L∞ 1.08571×10−3 7.70085×10−4 6.00762×10−4

CPU time 3.1720 4.1720 5.2500


1074 S. CHEN ET AL.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

t=0.0

t=0.2

t=0.8

t=2.0 t=3.0

Figure 7. Numerical solutions for �=0.001 at different times.

–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1–1.5

–1

–0.5

0

0.5

1

1.5numerical solutionexact solution

Figure 8. Exact, numerical solutions from LTM for �=7.5e−4 at t=3.

7. CONCLUSION

We have presented a highly accurate and highly efficient method, LTM, that greatly outperformsQRKM, CN, and ETDRK4 for solving Burgers’ equation for wide range of viscosity. Numericalexperiments show that the present method is effective and competitive. It combines ideas from theexisting increment linearization method with numerical inversion technique of Laplace transformusing Talbot’s method. This is the first time that the Laplace transform method has been applied



–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1–1.5

–1

–0.5

0

0.5

1

1.5Numerical solutionExact solution

Figure 9. Exact, numerical solutions from CN for �=7.5e−4 at t=3.

Table VI. Error norms and CPU time for Example 3 using LTM, CN, and ETDRK4 at t=3.

‖e‖L2 ‖e‖L∞ CPU time

LTM (N =100, �t=0.01) 5.7777×10−10 9.6230×10−10 1.094

CN (N =2000, �t=0.001) 1.000×10−3 2.120×10−2 743.8440

ETDRK4 (N =100, �t=0.001) 3.1978×10−4 1.1699×10−4 28.5160

to solve nonlinear PDE. It is believed that this approach will also provide useful hints for solvingmore general problems in fluid dynamics.

ACKNOWLEDGEMENTS

The support from the National Nature Science Foundation of China (No.10671146, 50678122) is fullyacknowledged. The authors thank the referees for their useful comments and suggestions.

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the laplace transform method for burgers' equation

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