a compact finite difference method for the

A Compact Finite Difference Method for theSolution of the Generalized Burgers–FisherEquationMurat Sari,1 Gürhan Gürarslan,2 Idris Dag3

1Department of Mathematics, Faculty of Art and Science, Pamukkale University,20070 Denizli,Turkey

2Department of Civil Engineering, Faculty of Engineering, Pamukkale University,20070 Denizli,Turkey

3Department of Mathematics, Faculty of Art and Science, Eskisehir OsmangaziUniversity, 26480 Eskisehir,Turkey

Received 31 March 2008; accepted 23 October 2008Published online 22 January 2009 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/num.20421

In this article, numerical solutions of the generalized Burgers–Fisher equation are obtained using a compactfinite difference method with minimal computational effort. To verify this, a combination of a sixth-ordercompact finite difference scheme in space and a low-storage third-order total variation diminishing Runge–Kutta scheme in time have been used. The computed results with the use of this technique have beencompared with the exact solution to show the accuracy of it. The approximate solutions to the equation havebeen computed without transforming the equation and without using linearization. Comparisons indicate thatthere is a very good agreement between the numerical solutions and the exact solutions in terms of accuracy.The present method is seen to be a very good alternative to some existing techniques for realistic problems.© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 26: 125–134, 2010

Keywords: compact finite difference method; generalized Burgers–Fisher equation; nonlinear PDE; Fisherequation

I. INTRODUCTION

Nonlinear partial differential equations are encountered in various fields of science. GeneralizedBurgers–Fisher equation is of high importance for describing different mechanisms. Fisher [1]first proposed the well-known equation, encountered in various disciplines, as a model for thepropagation of a mutant gene with u(x, t) displaying the density of advantage. Later, the equationhas been used as a basis for a wide variety of models for different problems. The most generalform of the Fisher equation is called as the generalized Burgers–Fisher equation.

Correspondence to: Murat Sari, Department of Mathematics, Faculty of Art and Science, Pamukkale University,20070 Denizli, Turkey (e-mail: [email protected])

© 2009 Wiley Periodicals, Inc.

126 SARI, GÜRARSLAN, AND DAG

Many researchers have spent a great deal of effort to compute the solution of the Burgers–Fisherequation using various numerical methods. A numerical simulation and explicit solutions of thegeneralized Burgers–Fisher equation were presented by Kaya and El-Sayed [2]. A restrictivePadé approximation for the solution of the generalized Burgers–Fisher equation was introducedby Ismail and Rabboh [3]. With the use of Adomian decomposition method, solutions of Burger–Huxley and Burgers–Fisher equations were obtained by Ismail et al. [4]. Recently, some variouspowerful mathematical methods such as tanh function methods [5–7], tanh–coth method [8], vari-ational iteration method [9], factorization method [10] and spectral collocation method [11, 12]have also been used in attempting to solve the equation.

A number of versions of the compact finite difference schemes were analyzed and implementedsuccessfully by some researchers [13–26] to deal with their own problems. Unlike some previ-ous techniques, using various transformations to reduce the equation into more simple equationand then solve it, the nonlinear equations are solved easily without transforming the equation byusing the current method. This method has also additional advantages over some rival techniques,mainly, avoidance of linearization, ease in use, and computationally cost effective to find solutionsof the given nonlinear equations.

In this study, a sixth-order compact finite difference scheme (CFD6) in space and a low-storage third-order total variation diminishing Runge–Kutta (TVD-RK3) [27] scheme in timewere implemented for obtaining explicit solution of the generalized Burgers–Fisher equation.The combination of the CFD6 with the TVD-RK3 provides an efficient explicit solution withhigh accuracy and minimal computational effort for realistic problems. The present method isuseful for obtaining numerical approximations of linear or nonlinear differential equations, and itis also quite straightforward to write codes in any programming languages. To the best knowledgeof the authors, this method has not been implemented for the problems represented by the gener-alized Burgers–Fisher equation so far. The results obtained by this way have been compared withthe exact solution to show the accuracy of it. The present method is of a general nature and cantherefore be used for solving the nonlinear partial differential equations arising in various areas.

Behaviors of many physical systems encountered in models of various mechanisms lead to thegeneralized Burgers–Fisher equation. The following one-dimensional generalized Burgers–Fisherequation [4], arising in various fields of science, of the form

ut + αuδux − uxx = βu(1 − uδ), 0 ≤ x ≤ 1, t ≥ 0 (1)

with the initial condition

u(x, 0) =[

1

2+ 1

2tanh(a1x)

]1/δ

(2)

and the boundary conditions

u(0, t) =[

1

2− 1

2tanh(a1a2t)

]1/δ

, t ≥ 0 (3)

and

u(1, t) =(

1

2+ 1

2tanh[a1(1 − a2t)]

)1/δ

, t ≥ 0. (4)

The exact solution of Eq. (1) is

u(x, t) =(

1

2+ 1

2tanh[a1(x − a2t)]

)1/δ

, t ≥ 0 (5)

Numerical Methods for Partial Differential Equations DOI 10.1002/num

CFD6 SCHEME FOR BURGERS-FISHER EQUATION 127

where

a1 = −αδ

2(δ + 1), a2 = α

δ + 1+ β(δ + 1)

α, (6)

where α, β, and δ are parameters.In this work, the generalized Burgers–Fisher equation was solved by the present method. The

numerical results are compared with the exact solutions to verify the accuracy of the currentmethod.

II. THE COMPACT FINITE DIFFERENCE SCHEME

The compact finite difference schemes can be dealt with in two essential categories: explicit com-pact and implicit compact approaches. While the first category computes the numerical derivativesdirectly at each grid by using large stencils, the second ones obtain all the numerical derivativesalong a grid line using smaller stencils and solving a linear system of equations. Because of thereasons given previously, the present work uses the second approach. To gain the solution of theBurgers–Fisher equation, discretizations are needed in both space and time.

A. Spatial and Temporal Discretization

Spatial derivatives are evaluated by the compact finite difference scheme [28]. For any scalarpointwise value u, the derivatives of u is obtained by solving a tridiagonal or pentadiagonal sys-tem. Much work has been done in deriving such formulae [28, 29]. More details on the formulaejust summarized in the following can be found in [28, 29]. A uniform one-dimensional mesh isconsidered, consisting of N points: x1, x2, . . . , xi−1, xi , xi+1, . . . , xN . The mesh size is denoted byh = xi+1 − xi . The first derivatives can be given at internal nodes as follows [28, 29]:

αu′i−1 + u′

i + αu′i+1 = b

ui+2 − ui−2

4h+ a

ui+1 − ui−1

2h, (7)

which gives rise to an α-family of fourth-order tridiagonal schemes with

a = 2

3(α + 2), b = 1

3(4α − 1),

where α = 0 leads to the explicit fourth-order scheme for the first derivative. A sixth-ordertridiagonal scheme is obtained by α = 1/3,

1

3u′

i−1 + u′i + 1

3u′

i+1 = 1

9

ui+2 − ui−2

4h+ 14

9

ui+1 − ui−1

2h.

For the nodes near the boundary, approximation formulae for the derivatives of nonperiodic prob-lems can be derived with the consideration of one-sided schemes. More details on the derivations



for the first- and second-order derivatives can be found in [28, 29]. The derived formulae atboundary points 1, 2, N − 1, and N , respectively, are:

u′i + 5u′

i+1 = 1

h

(−197

60ui − 5

12ui+1 + 5ui+2 − 5

3ui+3 + 5

12ui+4 − 1

20ui+5

),

2

11u′

i−1 + u′i + 2

11u′

i+1 = 1

h

(−20

33ui−1 − 35

132ui + 34

33ui+1 − 7

33ui+2 + 2

33ui+3 − 1

132ui+4

),

2

11u′

i−1 + u′i + 2

11u′

i+1 = 1

h

(20

33ui+1 + 35

132ui − 34

33ui−1 + 7

33ui−2 − 2

33ui−3 + 1

132ui−4

),

5u′i−1 + u′

i = 1

h

(197

60ui + 5

12ui−1 − 5ui−2 + 5

3ui−3 − 5

12ui−4 + 1

20ui−5

).

In order to obtain the formulae presented earlier, the procedure of Gaitonde and Visbal [29] wasfollowed. The formulae can be reexpressed as

BU ′ = AU

where U = (u1, . . . , un)T. The second-order derivative terms are obtained by applying the

first-order operator twice, i.e.,

BU ′′ = AU ′,

where

A = 1

h

−197

60

−5

125

−5

3

5

12

−1

20−20

33

−35

132

34

33

−7

33

2

33

−1

132−1

36

−7

90

7

9

1

36. . .

. . .. . .

. . .. . .

−1

36

−7

90

7

9

1

361

132

−2

33

7

33

−34

33

35

132

20

331

20

−5

12

5

3−5

5

12

197

60

N×N

,

B =

1 52

111

2

111

31

1

3. . .

. . .. . .

1

31

1

32

111

2

115 1

N×N

.



In the current work, the equations are integrated in time with consideration of the low-storageTVD-RK3 scheme. Assuming that the governing equation is

∂u

∂t= Lu,

where L stands for a spatial nonlinear differential operator. The low-storage TVD-RK3 schemeintegrates from time t0 (step k) to t0 + �t (step k + 1) through the operations

u(1) = uk + �tLuk ,

u(2) = 3

4uk + 1

4u(1) + 1

4�tLu(1),

uk+1 = 1

3uk + 2

3u(2) + 2

3�tLu(2).

III. APPLICATIONS TO THE BURGERS–FISHER EQUATION

For the approximate solution of the boundary value problem (1) with the boundary condi-tions given by Eq. (2) using the CFD6 method, first the interval [0, 1] is discretized such that0 = x1 < x2 < . . . < xN = 1, where N is the number of grid points. Application of the CFD6technique to Eq. (1) leads to

∂u

∂t

∣∣∣∣i

= (Lu)i . (8)

First- and second-order spatial derivatives in Eq. (1) are obtained with the use of the sixth-ordercompact schemes and then substituted in Eq. (8). Thus, the corresponding semi-discrete equationcan be solved with the TVD-RK3 scheme.

IV. NUMERICAL ILLUSTRATIONS

To see whether the present methodology leads to accurate solutions, the CFD6 solutions will beevaluated for some examples of the generalized Burger–Fisher equations given earlier.

Now, numerical solutions of the Burgers–Fisher equation are obtained for some values of theparameters to validate the current numerical scheme. To verify the efficiency, measure its accuracyand the versatility of the present scheme for our problem in comparison with the exact solution,absolute errors for different values of α, β and δ, are reported, which are defined by

|u(xi , tj ) − U(xi , tj )|in the point (xi , tj ), where u(xi , tj ) is the solution obtained by Eq. (8) solved by the combinationsuggested here, and U(xi , tj ) is the exact solution.

Consider the generalized Burgers–Fisher equation in the form (1) with the initial condition(2) and boundary conditions (3), (4) and the exact solution (5). For the comparison purposes, thefollowing examples [4,9] are selected. The results are compared with the analytical solutions, andthose obtained using various methods [4, 9]. The numerical computations were performed usinguniform grids. All computations were carried out using some codes produced in Visual Basic 6.0.



TABLE I. The absolute errors for various values of x, t , and δ with α = 0.001 and β = 0.001.

δ = 1 δ = 4

x t Exact Ismail et al. [4] CFD6 Absolute error Exact CFD6 Absolute error

0.1 0.001 0.499988 0.499986 0.499988 1.01 E − 07 0.840888 0.840888 1.75 E − 080.005 0.499989 0.499979 0.499988 4.38 E − 07 0.840890 0.840889 7.37 E − 070.010 0.499990 0.499971 0.499989 7.53 E − 07 0.840892 0.840891 1.27 E − 06

0.5 0.001 0.499938 0.499936 0.499938 1.04 E − 07 0.840854 0.840854 1.75 E − 080.005 0.499939 0.499929 0.499938 5.21 E − 07 0.840856 0.840856 8.77 E − 070.010 0.499940 0.499921 0.499939 1.04 E − 06 0.840859 0.840857 1.75 E − 06

0.9 0.001 0.499888 0.499886 0.499888 1.01 E − 07 0.840821 0.840821 1.75 E − 080.005 0.499889 0.499879 0.499888 4.38 E − 07 0.840823 0.840822 7.38 E − 070.010 0.499890 0.499871 0.499889 7.53 E − 07 0.840825 0.840824 1.27 E − 06

N and �t are taken to be 11 and 0.0001, respectively, in all examples. The differences between thecomputed solution and the exact solution for some values of the constants δ, α, and β are shownin Tables I–VII. As various problems of science were modeled by nonlinear partial differentialequations and since the generalized Burgers–Fisher equation is of high importance, various valuesof δ have been considered in Examples 1–7. For the computational work, the following examplesare selected.

Example 1 ([4]). In Table I, the absolute errors were shown for various values of x, t , and δ

with α = 0.001 and β = 0.001. Considering the values of the parameters, a comparison has beenmade between the computed results and the results of Ismail et al. [4].

Example 2 ([4]). In Table II, the absolute errors were shown for various values of x, t , and δ

with α = 1 and β = 1. A comparison between the CFD6 solution and the exact solution is givenin Table II. An additional comparison between the results of the current study and the results ofIsmail et al. [4] has been made. The obtained results are seen to be very reliable and accurate.

Example 3 ([4]). For the computational work in this example, α, β, and δ are taken to be1, 0, and 1, respectively. The absolute error has been shown for various values of x and t . Thecorresponding results of the parameters have been presented in Table III. Additional comparisonsbetween the computed results and Ismail et al. [4] are also given in this example. Again efficiencyand accuracy of the CFD6 results are clearly seen for the values of the parameters in Table III.

TABLE II. The absolute errors for various values of x, t , and δ with α = 1 and β = 1.

δ = 2 δ = 8


0.1 0.0001 0.69527 0.69499 0.69525 1.55 E − 05 0.911859 0.911839 2.07 E − 050.0005 0.69543 0.69402 0.69535 7.62 E − 05 0.912052 0.911950 1.02 E − 040.0010 0.69563 0.69282 0.69548 1.50 E − 04 0.912292 0.912092 2.00 E − 04

0.5 0.0001 0.64613 0.64586 0.64611 1.83 E − 05 0.889211 0.889183 2.75 E − 050.0005 0.64630 0.64495 0.64621 9.14 E − 05 0.889430 0.889293 1.37 E − 040.0010 0.64651 0.64382 0.64632 1.83 E − 04 0.889704 0.889429 2.74 E − 04

0.9 0.0001 0.59531 0.59506 0.59529 2.07 E − 05 0.863883 0.863849 3.43 E − 050.0005 0.59548 0.59420 0.59538 1.02 E − 04 0.864124 0.863956 1.69 E − 040.0010 0.59569 0.59314 0.59549 2.00 E − 04 0.864426 0.864095 3.31 E − 04



TABLE III. The absolute errors for various values of xand t with β = 0, α = 1, and δ = 1.

x t Exact Ismail et al. [4] CFD6 Absolute error

0.1 0.50 0.518741 0.518741 0.518741 1.68 E − 111.00 0.549834 0.549832 0.549834 1.79 E − 112.00 0.610639 0.610575 0.610639 1.46 E − 11

0.5 0.50 0.468791 0.468791 0.468791 3.40 E − 121.00 0.500000 0.499998 0.500000 3.72 E − 122.00 0.562177 0.562116 0.562177 3.13 E − 12

0.9 0.50 0.419458 0.419458 0.419458 1.31 E − 111.00 0.450166 0.450165 0.450166 1.37 E − 112.00 0.512497 0.512450 0.512497 1.07 E − 11

Example 4 ([4]). Table IV shows the absolute errors for various values of x, t , and δ withβ = 0 and α = 1. The results of the current method have been presented in Table IV. Comparisonsof the current results with the exact solution as well as the results of Ismail et al. [4] showed thatthe presented results are very accurate.

Example 5. Table V shows the absolute error for various values of x, t , and δ with β = 0and α = 1. The computed results in this example for the aforementioned values of the parametersare shown in Table V and are compared with the results of Ismail et al. [4]. It is important to seethat the computed solutions in Table V are very accurate.

Example 6 ([2]). The absolute error was shown in Table VI for various values of δ, x, andt with α = 0.1 and β = −0.0025. As in the previous examples, the results of the combination ofthe CFD6 with the low-storage TVD-RK3 have been presented in Table VI. Comparison of thecurrent results with the exact solution showed that the presented results are very accurate.

Example 7 ([9]). The absolute errors were shown in Table VII for various values of x and t

with δ = 1, β = 0, and α = 0.01. In the table, VIM and ADM stand for the variational iterationmethod and Adomian decomposition method, respectively. Comparison of the computed resultswith the exact solution showed that the presented results are still very accurate for some largevalues of time.

TABLE IV. The absolute errors for various values of x, t , and δ with β = 0 and α = 1.

δ = 2 δ = 8


0.1 0.50 0.714919 0.714919 0.714919 4.49 E − 11 0.914720 0.914720 4.60 E − 111.00 0.734037 0.734037 0.734037 4.19 E − 11 0.917569 0.917569 4.39 E − 112.00 0.770284 0.770272 0.770284 2.70 E − 11 0.923082 0.923082 3.78 E − 11

0.5 0.50 0.666837 0.666837 0.666837 8.13 E − 12 0.892474 0.892474 7.03 E − 121.00 0.687205 0.687205 0.687205 7.72 E − 12 0.895739 0.895739 6.75 E − 122.00 0.726464 0.726449 0.726464 4.77 E − 12 0.902104 0.902104 5.67 E − 12

0.9 0.50 0.616567 0.616567 0.616567 3.55 E − 11 0.867481 0.867481 3.69 E − 111.00 0.637701 0.637701 0.637701 3.23 E − 11 0.871097 0.871097 3.45 E − 112.00 0.679109 0.679095 0.679109 1.98 E − 11 0.878196 0.878196 2.84 E − 11



TABLE V. The absolute errors for various values of x, t , and δ with β = 0 and α = 1.

δ = 3 δ = 8


0.1 0.0001 0.783660 0.784115 0.783660 6.43 E − 13 0.911812 0.911812 4.80 E − 130.0005 0.783670 0.784106 0.783670 3.14 E − 12 0.911814 0.911814 2.34 E − 120.0010 0.783683 0.784127 0.783683 6.07 E − 12 0.911817 0.911817 4.55 E − 12

0.5 0.0001 0.741285 0.743150 0.741285 2.11 E − 15 0.889157 0.889157 1.03 E − 140.0005 0.741296 0.743145 0.741296 1.07 E − 14 0.889159 0.889159 4.81 E − 140.0010 0.741309 0.743157 0.741309 2.18 E − 14 0.889163 0.889163 8.94 E − 14

0.9 0.0001 0.696157 0.697089 0.696157 6.29 E − 13 0.863824 0.863824 6.19 E − 130.0005 0.696169 0.697089 0.696169 3.05 E − 12 0.863827 0.863827 2.99 E − 120.0010 0.696183 0.697088 0.696183 5.87 E − 12 0.863830 0.863830 5.73 E − 12

TABLE VI. The absolute errors for various values of δ, x, and t with α = 0.1 and β = −0.0025.

x

t 0.1 0.5 0.9

δ = 20.1 1.12133 E − 05 2.90376 E − 05 1.15442 E − 050.2 1.46989 E − 05 4.04291 E − 05 1.51006 E − 050.3 1.60003 E − 05 4.46809 E − 05 1.64278 E − 050.4 1.64875 E − 05 4.62715 E − 05 1.69246 E − 050.5 1.66712 E − 05 4.68703 E − 05 1.71119 E − 05



TABLE VII. The absolute errors for various values of x and t with δ = 1, β = 0, and α = 0.01.

x t VIM [9] ADM [9] CFD6

0.1 1 1.92 E − 14 1.92 E − 14 5.77 E − 150.5 9.73 E − 14 9.73 E − 14 4.19 E − 140.9 1.75 E − 13 1.75 E − 13 1.29 E − 14

0.1 10 1.63 E − 12 1.63 E − 12 8.49 E − 150.5 9.44 E − 12 9.44 E − 12 4.26 E − 140.9 1.73 E − 11 1.73 E − 11 1.82 E − 14

0.1 50 8.14 E − 12 8.14 E − 12 2.11 E − 150.5 2.03 E − 10 2.03 E − 10 2.46 E − 140.9 3.99 E − 10 3.99 E − 10 8.05 E − 15



In the examples mentioned earlier, although very few number of grids are used even when δ istaken to be very high, the CFD6 results are seen to be very accurate. Tables I–VII show that a verygood approximation to the actual solution of the equations was achieved by using the method.

It is evident that the overall errors can be made smaller by increasing the order of the com-pact finite difference scheme. A very good agreement between the results of the CFD6 schemeand exact solution was observed, which confirms the validity of the present method. The presentmethod is a very good alternative to the rival methods that face the well-known difficulties.

V. CONCLUSIONS

In this article, a combination of a CFD6 scheme in space and a low-storage TVD-RK3 methodin time has been proposed for the generalized Burgers–Fisher equation with a very small error.Comparisons of the computed results with exact solutions showed that the method is capableof solving the generalized Burgers–Fisher equation and is also capable of producing accuratesolutions with minimal computational effort for both time and space. The performance of thetechnique for the considered problems was measured by comparing with the exact solutions.It was seen that the compact finite difference technique approximates the exact solution verywell. This shows the efficiency and high accuracy of the method. As can be realized in TablesI–VII, the technique is seen to be a very reliable alternative technique to some rival methodsfor realistic problems. The avoidance of linearization and transformation and its computationallycost-effectiveness have made the current method an efficient alternative to some rival methods insolving physical problems modeled by the nonlinear partial differential equations.

The authors thank the anonymous referees of the journal of Numerical Methods For PartialDifferential Equations for their valuable comments and suggestions to improve the article.

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a compact finite difference method for the

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