A numerical solution of Burgers’ equation based on least squares approximation

Applied Mathematics and Computation 176 (2006) 270–279 www.elsevier.com/locate/amc

A numerical solution of BurgersÕ equation based on least squares approximation E.N. Aksan a

a,*

¨ zdesß a, T. O ¨ zisß , A. O

b

_ ¨ n€ Department of Mathematics, Faculty of Arts and Science, Ino u University, 44069 Malatya, Turkey b _ Department of Mathematics, Faculty of Science, Ege University, Izmir, Turkey

Abstract BurgersÕ equation which is one-dimensional non-linear partial differential equation was converted to p non-linear ordinary differential equations by using the method of discretization in time. Each of them was solved by the least squares method. For various values of viscosity at different time steps, the numerical solutions obtained were compared with the exact solutions. It was seen that both of them were in excellent agreement. While the exact solution was not available for viscosity smaller than 0.01, it was shown that mathematical structure of the equation for the obtained numerical solutions did not decay. Ó 2005 Elsevier Inc. All rights reserved. Keywords: BurgersÕ equation; Method of discretization in time; Least squares method

1. Introduction The one-dimensional non-linear partial equation oU oU o2 U þU ¼e 2 ot ox ox

ð1Þ

is known as BurgersÕ equation. BurgersÕ model of turbulence is a very important fluid dynamic model and the study of this model and the theory of shock waves has been considered by many authors both for conceptual understanding of a class of physical flows and for testing various numerical methods. The distinctive feature of Eq. (1) is that it is the simplest mathematical formulation of the competition between non-linear advection and the viscous diffusion. It contains the simplest form of non-linear advection term UUx and dissipation term eUxx where e = 1/Re (e: kinematics viscosity and Re: Reynolds number) for simulating the physical phenomena of wave motion and thus determines the behavior of the solution. The mathematical properties of Eq. (1) have been studied by Cole [1]. Particularly, the detailed relationship between Eq. (1) and both turbulence

*

Corresponding author. E-mail address: [email protected] (E.N. Aksan).

0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.09.045

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271

theory and shock wave theory was described by Cole. He also gave an exact solution of BurgersÕ equation. Benton and Platzman [2] have demonstrated about 35 distinct exact solutions of Burgers-like equations and their classifications. It is well known that the exact solution of BurgersÕ equation can only be computed for restricted values of e which represent the kinematics viscosity of the fluid motion. Because of this fact, various numerical methods was employed to obtain the solution of BurgersÕ equation with small e values. It is not our purpose to exhaust all existing numerical schemes for BurgersÕ equation but to mention some of them especially which use variational approaches and finite-elements. For example, Jamet and Bonnerot [3] solved BurgersÕ equation by using isoparametric rectangular space–time finite-element and this required an enormous number of evenly space grid points to describe the numerical values in the boundary layer region. Varoglu and Finn [4] presented an isoparametric space–time finite-element approach for solving BurgersÕ equation, utilizing the hyperbolic differential equation associated with BurgersÕ equation. Another approach which has been used by Caldwell et al. [5] is the finite-element method such that by altering the size of the element at each stage using information from the previous steps. Caldwell et al. [6] give an indication of how complementary variational principles can be applied to BurgersÕ equation. Later, Sauders et al. [7] have demonstrated how a variational-iterative scheme based on complementary variational principles can be applied to non-linear partial differential equations and the test problem chosen is the steady-state version of BurgersÕ equation. Caldwell and Smith [8] further developed the finite-element method to allow for different sized ¨ zisß and O ¨ zdesß [9] apelements at each stage of the calculation based on feedback from the previous step. O ¨ plied a direct variational method to generate limiting form of the solution of BurgersÕ equation. Ozisß et al. [10] applied a simple finite-element approach with linear elements to BurgersÕ equation reduced by Hopf–Cole ¨ zdesß [11] have reduced BurgersÕ equation to the system of non-linear ordinary transformation. Aksan and O differential equations by discretization in time and solved each non-linear ordinary differential equation by Galerkin method in each time step. As they claimed, for moderately small kinematics viscosity, their approach can provide high accuracy while using a small number of grid points (i.e. N = 5) and this makes the approach very economical computational wise. In the case where the kinematics viscosity is small enough i.e. e = 0.0001, the exact solution is not available and a discrepancy exists in the literature, their results clarify the behavior of the solution for small times, i.e. T = tmax 6 0.15. Also it is demonstrated that the parabolic structure of the equation decayed for tmax = 0.5. In this paper, again, the reduced BurgersÕ equation by the method of discretization in time has been solved by the least squares method in each time step. It is, now, aimed that establishing the existence of the solution in this case, it can be made some improvement on the values of kinematics viscosity and maximum time values. In other words, we can get solutions with moderately small e values for moderately large times. Because, besides being the best approximation the highly attractive feature of the method of least squares, from the viewpoint of computation, is that the coefficient matrix of minimized system is more effectively formed than the matrix associated with finite-element method. 2. The method of solution The problems in the mathematical physics and engineering etc. usually lead to partial differential equations which are hard to solve analytically. The method of discretization in time converts such a partial differential equation to a set of ordinary differential equations. In this method, T being total time, for the variable t we divide interval [0, T] into p subintervals I1, I2, . . . , Ip of lengths Dt = T/p such that Ij = [tj1, tj] with t0 = 0 and replace derivative oU/ot by the difference quotient (zj(x)  zj1(x))/Dt at t = tj [12]. Let us consider BurgersÕ equation (1) with the following initial and boundary conditions: U ðx; 0Þ ¼ sin px

ð2Þ

in X;

U ð0; tÞ ¼ U ð1; tÞ ¼ 0;

t > 0;

where X is the interval (0, 1). The exact solution of Eq. (1) with conditions (2) and (3) was given by Cole [1] as P1 n2 p2 et n sin npx n¼1 an e P1 U ðx; tÞ ¼ 2pe ; 2 p2 et n cos npx a0 þ n¼1 an e

ð3Þ

272

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where a0 ¼

Z

1

h i 1 exp ð2peÞ ð1  cos pxÞ dx

0

and an ¼ 2

Z

1

h i 1 exp ð2peÞ ð1  cos pxÞ cos npx dx;

n P 1.

0

As applied to initial boundary value problem (1)–(3), the method of discretization in time involves successively solving the equation for j = 1, 2, . . . , p 1 1 zj ¼ zj1 ; Dt Dt zj ð1Þ ¼ 0

 ez00j þ zj z0j þ

ð4Þ

zj ð0Þ ¼ 0;

ð5Þ

with z0 ðxÞ ¼ U ðx; 0Þ ¼ sin px. Eq. (4) is to be solved successively for z1, z2, . . . , zp. The solution z1is used to determine z2, and so on. Thus, the method of discretization in time allowed us to reduce the initial boundary value problem (1)–(3) to p boundary value problems (4) and (5). Each of these problems can be solved either exactly or approximately. Since Eq. (4) with (5) is a non-linear boundary value problem and its exact solution becomes more difficult with increasing j. Therefore, in order to solve it we choose the least squares method which is a variational method. In the least squares method, an approximate solution of operator equation given by Au ¼ f

in X

is constructed as the function uN 2 DA, uN ¼

N X

cn / n ;

n¼1

which minimizes 2 kAuN  f k ;

ð6Þ

where A : DA ! H, f 2 H, the linear set DA is assumed to be dense in the Hilbert space H and /n 2 DA are basis functions which satisfy all of the boundary conditions of the problem. Condition (6) leads to a system of equations in the unknowns c1, c2, . . . , cN: okAuN  f k2 ¼ 0. ock Eq. (7) is equivalent to   o ðAuN  f Þ; ðAuN  f Þ ¼ 0 ock

ð7Þ

ð8Þ

[13]. Left-hand side of Eq. (8) represents the inner product in Hilbert space H by defined   Z o ðAuN  f Þ ðAuN  f Þ dx. ock X If Eq. (4) is written as the operator equation Azj ¼

1 f Dt

ð9Þ

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273

it is obvious that Azj is in the form Azj ¼ ez00j þ zj z0j þ

1 zj ; Dt

where f = zj1. Let us assume the approximate solution zNj of Eq. (9) in the form zNj ¼

N X

cjn sin npx.

ð10Þ

n¼1

Thus, for our problem the least squares system is    ! 1 o 1 N N ¼0 Azj  f ; j Azj  f Dt Dt ock

ð11Þ

for k = 1, 2, . . . , N. Substituting Eq. (10) into Eq. (9) and after a simple arrangement, we obtain ep2

N X

n2 cjn sin npx þ p

n¼1

N X

cjn sin npx

n¼1

N X

mcjm cos mpx þ

m¼1

N 1 X 1 cjn sin npx  f ¼ 0. Dt n¼1 Dt

Clearly, left-hand side of the last equation represents AzNj  Dt1 f , i.e. AzNj 

N N N X X X 1 f ¼ ep2 n2 cjn sin npx þ p cjn sin npx mcjm cos mpx Dt n¼1 n¼1 m¼1

þ

N 1 X 1 cj sin npx  f . Dt n¼1 n Dt

ð12Þ

Therefore, derivative of Eq. (12) with respect to unknowns cjk ; k ¼ 1; 2; . . . ; N ; is found to be     N X o 1 1 N 2 2 f ¼ ep Az  k þ mcjm cos mpx sin kpx þ p sin kpx j Dt Dt ocjk m¼1 þ pk cos kpx

N X

cjn sin npx.

ð13Þ

n¼1

Substituting Eqs. (12) and (13) into Eq. (11) we obtain ep2

N X

n2 cjn sin npx þ p

n¼1

N X N X

mcjn cjm sin npx cos mpx

n¼1 m¼1

  N N X 1 X 1 1 j 2 2 þ cn sin npx  f ; ep k þ icji cos ipx sin kpx þ p sin kpx Dt n¼1 Dt Dt i¼1 þ pk cos kpx

N X

! cji

sin ipx

¼ 0;

i¼1

where k = 1, 2, . . . , N. Using the properties of inner product, from Eq. (14) we get

ð14Þ

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e2 p4 k 2 þ þ kep3

 N N X ep2 X n2 cjn ðsin npx; sin kpxÞ þ ep3 n2 icjn cji ðsin npx; sin kpx cos ipxÞ Dt n¼1 n;i¼1

N X

n2 cjn cji ðsin npx; cos kpx sin ipxÞ

n;i¼1 N  p X mcj cj ðsin npx cos mpx; sin kpxÞ þ ep3 k 2 þ Dt n;m¼1 n m N X

þ p2

micjn cjm cji ðsin npx cos mpx; sin kpx cos ipxÞ

n;m;i¼1 N X 2

mcjn cjm cji ðsin npx cos mpx; cos kpx sin ipxÞ n;m;i¼1 ! N N X 1 p X 2 2 j ð Þ c sin npx; sin kpx þ icjn cji ðsin npx; sin kpx cos ipxÞ þ ep k þ n 2 Dt ðDtÞ n¼1 n;i¼1 ! N 2 2 X kp ep k 1 j j þ c c ðsin npx; cos kpx sin ipxÞ  þ ðf ; sin kpxÞ 2 Dt n;i¼1 n i Dt ðDtÞ

þ kp



N N p X kp X icji ðf ; sin kpx cos ipxÞ  cj ðf ; cos kpx sin ipxÞ ¼ 0; Dt i¼1 Dt i¼1 i

where  ðsin npx; sin kpxÞ ¼

1=2;

n ¼ k;

n 6¼ k. 8 1=4; n ¼ k þ i; > > > > < 1=4; n ¼ i  k; ðsin npx; sin kpx cos ipxÞ ¼ > 1=4; n ¼ k  i; > > > : 0; otherwise. 8 1=4; n ¼ k þ i; > > > > < 1=4; n ¼ k  i; ðsin npx; cos kpx sin ipxÞ ¼ > 1=4; n ¼ i  k; > > > : 0; otherwise. 8 1=4; n þ m ¼ k; > > > > < 1=4; m  n ¼ k; ðsin npx cos mpx; sin kpxÞ ¼ > n  m ¼ k; > 1=4; > > : 0; otherwise. 8 1=8; n þ m ¼ k þ i; > > > > > 1=8; n þ m ¼ i  k; > > > > > 1=8; n þ m ¼ k  i; > > > > < 1=8; m  n ¼ k þ i; ðsin npx cos mpx; sin kpx cos ipxÞ ¼ > 1=8; n  m ¼ k þ i; > > > > > 1=8; n  m ¼ i  k; > > > > > > 1=8; n  m ¼ k  i; > > : 0; otherwise. 0;

ð15Þ

E.N. Aksan et al. / Applied Mathematics and Computation 176 (2006) 270–279

8 1=8; > > > > > 1=8; > > > > > 1=8; > > > < 1=8; ðsin npx cos mpx; cos kpx sin ipxÞ ¼ > 1=8; > > > > > 1=8; > > > > > 1=8; > > : 0;

275

n þ m ¼ k þ i; n þ m ¼ k  i; n þ m ¼ i  k; m  n ¼ k þ i; n  m ¼ k þ i; n  m ¼ k  i; n  m ¼ i  k; otherwise

and f is in the form f ¼

N X

cj1 n sin npx

n¼1

for j = 1, 2, . . . , p and c01 ¼ 1; c02 ¼ 0; c03 ¼ 0; . . . ; c0N ¼ 0 for j = 1. Thus the last three terms in Eq. (15) can be obtained easily. For each j, Eq. (15) is a non-linear system of equations. Thus, each of the obtained systems was solved by the Newton method for the non-linear system of equations. 3. Numerical results and discussion In order to demonstrate the adoptability and accuracy of the present approach, we have applied it to the problem given by Eqs. (1)–(3) whose exact solution is exists and is given by Cole [1] in terms of infinite series. The resulting system of non-linear equations which consist of N equations and N unknowns is solved by well known Newton method. The reason for choosing Newton method is that this method converges very rapidly due to the Jacobian matrix which is symmetrically structured in the solution algorithm. To start the Newton method, though the choice of initial vector is very important and cumbersome, the initial solution vector is taken to be c1,(0) = (0, 0, . . . , 0)t. For consecutive steps, the initial vectors  are taken to be the solutions of  j;ðkÞ   j;ðkÞ j;ðk1Þ  j;ðk1Þ   c ¼ max16i6N ci  ci the previous steps. For stopping criteria; c  6 0:00001 is required 1 and in most cases, only two iterations were good enough to obtain required stopping criteria. The results were obtained for N = 10 and for various e values. To emphasize the accuracy of the method for moderate size viscosity values, we have given the comparisons with analytical solutions obtained from the infinite series of Cole [1] for e = 1.0, 0.1 and 0.05. Both Tables 1–4 and Figs. 1, 2 show that solutions are in good agreement with analytical solutions and it was observed that the numerical solutions coincides with the exact solutions as Dt is decreased, Table 3. The effect of increasing of the number of basis elements on the numerical solutions is given Table 4. It is clear that the numerical solutions converge the exact solutions as N is increased. In the case e is smaller than 0.01, the exact solution is not available and a discrepancy exists in the literature. Also, it is not

Table 1 Comparison of the numerical solutions obtained for e = 1, Dt = 0.0001 at different times with the exact solutions x

t = 0.001

t = 0.01

t = 0.1

Numerical

Exact

Numerical

Exact

Numerical

Exact

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.30509 0.58057 0.79963 0.94082 0.99018 0.94261 0.80252 0.58346 0.30688

0.30509 0.58057 0.79962 0.94082 0.99018 0.94261 0.80252 0.58347 0.30688

0.27327 0.52162 0.72192 0.85465 0.90576 0.86836 0.74411 0.54382 0.28700

0.27324 0.52156 0.72185 0.85459 0.90571 0.86833 0.74410 0.54382 0.28700

0.10959 0.20990 0.29204 0.34809 0.37176 0.35922 0.31006 0.22793 0.12075

0.10954 0.20979 0.29190 0.34792 0.37158 0.35905 0.30991 0.22782 0.12069

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Table 2 Comparison of the numerical solutions obtained for e = 0.05, Dt = 0.0001 at different times with the exact solutions x

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

t = 0.001

t = 0.01

t = 0.1

Numerical

Exact

Numerical

Exact

Numerical

Exact

0.30795 0.58601 0.80713 0.94966 0.99950 0.95151 0.81011 0.58899 0.30979

0.30795 0.58601 0.80713 0.94966 0.99950 0.95151 0.81011 0.58899 0.30979

0.29865 0.57045 0.79034 0.93696 0.99459 0.95513 0.81975 0.59988 0.31685

0.29865 0.57044 0.79034 0.93696 0.99460 0.95513 0.81976 0.59988 0.31686

0.23021 0.44993 0.64720 0.80781 0.91578 0.95125 0.88966 0.70719 0.39763

0.23041 0.44996 0.64692 0.80776 0.91602 0.95133 0.88949 0.70714 0.39788

Table 3 Comparison of the numerical solutions obtained with various values of Dt for e = 1 at t = 0.1 with the exact solution x

Numerical solutions

Exact solution

Dt = 0.01

Dt = 0.0001

Dt = 0.00001

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.11008 0.21083 0.29334 0.34964 0.37340 0.36080 0.31141 0.22892 0.12127

0.10959 0.20990 0.29204 0.34809 0.37176 0.35922 0.31006 0.22793 0.12075

0.10954 0.20980 0.29191 0.34794 0.37160 0.35906 0.30992 0.22783 0.12069

kek1

0.00487

0.00048

0.00003

0.10954 0.20979 0.29190 0.34792 0.37158 0.35905 0.30991 0.22782 0.12069

Table 4 Comparison of the numerical solutions obtained with different number of basis elements for e = 0.1, Dt = 0.0001 at t = 0.1 with the exact solutions x

Numerical solutions

Exact solution

N=2

N=3

N=5

N = 10

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.25769 0.49640 0.69663 0.83877 0.90505 0.88273 0.76777 0.56754 0.30166

0.22705 0.43862 0.62362 0.77369 0.87597 0.90770 0.84045 0.65545 0.36105

0.22365 0.43575 0.62501 0.77790 0.87734 0.90402 0.83696 0.65744 0.36553

0.22334 0.43581 0.62529 0.77777 0.87715 0.90420 0.83701 0.65730 0.36560

kek1

0.10512

0.00614

0.00030

0.00017

0.22345 0.43580 0.62512 0.77772 0.87728 0.90425 0.83692 0.65731 0.36575

practical to evaluate the analytical solution at these values due to slow convergence of the infinite series and thus the exact solution in this regime is unknown. On the other hand, the development of a decent numerical method for seeking accurate and efficient numerical solutions of BurgersÕ equation with small e values, still, has been challenging task. For example, Kakuda and Tosaka [14] proposed a general boundary element approach. For very small viscosity values (e ffi 105) these authors found significantly different behavior for the early time solutions (t = 0.2) from those of Varoglu and Finn [4] and Nguyen and Reynen [15]. Zhang

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277

Fig. 1. Comparison of the numerical solutions obtained for e = 1 at different times with the exact solutions.

Fig. 2. Comparison of the numerical solutions obtained for e = 0.1 at different times with the exact solutions.

et al. [16] numerically solved BurgersÕ equation involving very small viscosity by using a new approach based on the distributed approximating functional for representing spatial derivatives of the velocity field. As they claimed, for moderately small viscosities, their approach can provide very high accuracy while using a small ¨ zisß and Aslan [17] presented the solution of BurgersÕ equation involving number of points. More recently O very small viscosity value (e 6 105) by using semi-implicit, time-difference scheme to reduce BurgersÕ equation to two-point, non-linear ordinary differential equation and solving it by matched asymptotic expansion method. In their solutions, qualitative findings obtained from the asymptotic studies of the exact solution for small viscosity values noted by Cole are in good agreement. Keeping in mind all these discrepancy, present results are given in the form of figures in Figs. 3 and 4 for e = 102 and e = 105. Since the problem involves the decay of a sinusoidal disturbance with time, it is expected that as e is decreased in value the solution curves tend to skew more and more to the right as time proceeds, Fig. 4. As it can be seen from Figs. 3, 4 and Table 5 our solutions demonstrate the development of steep fronts at early times and are close to those Varoglu and Finn [4] and Nguyen and Reynen [15] for t = 0.2. At the later times our solutions show some wiggles, but general trend of the solution is preserved. This means that our approach needs some modifications to improve solutions for later times. But, this may be obtained by increasing the number of equations in the solution system as well as in the iteration numbers. Since our solution, as it stands, preserves the trends of the solution, namely shows the decay of sinusoidal disturbance with time for small viscosities, we believed that the increment in the number of equations and also in the number of iterations will give better results for later time

278

E.N. Aksan et al. / Applied Mathematics and Computation 176 (2006) 270–279

Fig. 3. Comparison of the numerical solution obtained for e = 0.01 at t = 0.15 with the exact solution.

Fig. 4. The numerical solutions obtained at different times for e = 0.00001.

Table 5 The numerical solutions obtained for various values of e and Dt = 0.0001 at different times x

t

Numerical solutions e = 0.01

e = 0.001

e = 0.0001

e = 0.00001

0.25

0.01 0.05 0.10 0.15 0.20

0.69093 0.63092 0.56655 0.51274 0.46660

0.69151 0.63316 0.56976 0.51622 0.46895

0.69157 0.63338 0.57008 0.51657 0.46913

0.69158 0.63340 0.57012 0.51660 0.46914

0.5

0.01 0.05 0.10 0.15 0.20

0.99852 0.98330 0.94702 0.89836 0.84305

0.99940 0.98747 0.95407 0.90650 0.84997

0.99949 0.98789 0.95478 0.90731 0.85061

0.99950 0.98794 0.95485 0.90739 0.85067

0.75

0.01 0.05 0.10 0.15 0.20

0.72224 0.78437 0.86042 0.92388 0.96356

0.72291 0.78865 0.87117 0.94155 0.98491

0.72298 0.78908 0.87226 0.94332 0.98700

0.72299 0.78912 0.87236 0.94350 0.98721

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solutions. But all these operations will increase computational times very rapidly, because, Zhang et al. [16] have used very large mapping parameter (A = 1.56068) to shift most of 200 grid points into the boundary layer region to obtain oscillation free solutions and small time increment is chosen to ensure high accuracy. The convergence of the solution for e = 105 is also confirmed by repeating the calculation with 300 grids and second order approximation. On the other hand, Kakuda and Tosaka [14], and Varoglu and Finn [4] have used 200 elements and 6 iterations and 200 elements and 5 iterations respectively for similar results. Therefore, we have not felt it worthwhile to make comparison with other solutions, partly because most of them only gives graphical representations of the solution without giving more information about it, partly make their comparison with ColeÕs series solution as we did. References [1] J.D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9 (1951) 225–236. [2] E.R. Benton, G.W. Platzman, A table of solution of the one-dimensional BurgersÕ equation, Quart. Appl. Math. 9 (1972) 195–212. [3] P. Jamet, R. Bonnerot, Numerical solution of the Eulerian equations of compressible flow by finite element method which follows the free boundary and interfaces, J. Comput. Phys. 18 (1975) 21–45. [4] E. Varoglu, W.D.L. Finn, Space–time finite elements incorporating characteristics for the BurgersÕ equation, Int. J. Numer. Methods Eng. 16 (1980) 171–184. [5] J. Caldwell, P. Wanless, A.E. Cook, A finite element approach to BurgersÕ equation, Appl. Math. Model. 5 (1981) 189–193. [6] J. Caldwell, P. Wanless, B.L. Borrows, A practical application of variational-iterative schemes, J. Phys. D: Appl. Phys. 13 (1980) 1117–1119. [7] R. Saunders, J. Caldwell, P. Wanless, A variational-iterative scheme applied to BurgersÕ equation, IMA J. Numer. Anal. 4 (1984) 349– 362. [8] J. Caldwell, P. Smith, Solution of BurgersÕ equation with a large ReynoldÕs number, Appl. Math. Model. 6 (1982) 381–385. ¨ zisß, A. O ¨ zdesß, A direct variational methods applied to BurgersÕ equation, J. Comput. Appl. Math. 71 (1996) 163–175. [9] T. O ¨ zisß, E.N. Aksan, A. O ¨ zdesß, A finite element approach for solution of BurgersÕ equation, Appl. Math. Comput. 139 (2003) 417– [10] T. O 428. ¨ zdesß, A numerical solution of BurgersÕ equation, Appl. Math. Comput. 156 (2004) 395–402. [11] E.N. Aksan, A. O [12] K. Rektorys, The Method of Discretization in Time and Partial Differential Equations, D. Reidel Publishing Company, Holland, 1982, p. 1. [13] K. Rektorys, Variational Methods in Mathematics, Science and Engineering, D. Reidel Publishing Company, Holland, 1980, p. 161. [14] K. Kakuda, N. Tosaka, The generalized boundary element approach to BurgersÕ equation, Int. J. Numer. Methods Eng. 29 (1990) 245–261. [15] H. Nguyen, J. Reynen, A space–time finite element approach to BurgersÕ equation, in: C. Taylor et al. (Eds.), Numerical Methods for Non-linear Problems, vol. 2, Pineridge Press, 1984, pp. 718–728. [16] D.S. Zhang, G.W. Wie, D.J. Kouri, Burgers equation with high Reynolds number, Phys. Fluids 9 (6) (1997) 1853–1855. ¨ zisß, Y. Aslan, The semi-approximate approach for solving BurgersÕ equation with high Reynolds number, Appl. Math. Comput. [17] T. O 163 (2005) 131–145.