PRAMANA

c Indian Academy of Sciences



journal of
physics

Vol. 81, No. 4

October 2013
pp. 547556

Numerical solution of the one-dimensional Burgers

equation: Implicit and fully implicit exponential finite
difference methods
BILGE INAN and AHMET REFIK BAHADIR
Department of Mathematics, Faculty of Arts and Science, Inonu University,
44280 Malatya, Turkey
Corresponding author. E-mail: bilge.inan@inonu.edu.tr
MS received 14 February 2013; revised 24 June 2013; accepted 4 July 2013
DOI: 10.1007/s12043-013-0599-z; ePublication: 21 September 2013
Abstract. This paper describes two new techniques which give improved exponential finite difference solutions of Burgers equation. These techniques are called implicit exponential finite
difference method and fully implicit exponential finite difference method for solving Burgers equation. As the Burgers equation is nonlinear, the scheme leads to a system of nonlinear equations. At
each time-step, Newtons method is used to solve this nonlinear system. The results are compared
with exact values and it is clearly shown that results obtained using both the methods are precise
and reliable.
Keywords. Burgers equation; exponential finite difference method; implicit exponential finite
difference method; fully implicit exponential finite difference method.
PACS Nos 02.60.x; 02.30.Jr; 02.70.Bf

1. Introduction
In this paper, we consider the one-dimensional non-linear Burgers equation
u
2u
u
+u
v 2 = 0,
t
x
x
with the initial condition
u (x, 0) = g(x),

a < x < b,

(1)

a<x <b

and the boundary conditions

u (a, t) = h 1(t)

and

u(b, t) = h 2(t),

t > 0,

where v is the positive coefficient of kinematic viscosity and g, h 1 and h 2 are the
prescribed functions of the variables.
Pramana J. Phys., Vol. 81, No. 4, October 2013

547

Bilge Inan and Ahmet Refik Bahadir

Burgers equation is found to describe various kinds of phenomena such as mathematical model of turbulence and the approximate theory of flow through a shock wave
travelling in a viscous fluid [1].
In literature, many numerical methods have been proposed and implemented for
approximating solution of the Burgers equation. Many authors have used numerical
techniques based on finite difference [18], finite element [913] and boundary element
[14] methods in attempting to solve the equation. Kadalbajoo et al [15] used a parameteruniform implicit difference scheme for solving time-dependent Burgers equation. The
explicit exponential finite difference method was originally developed by Bhattacharya
[16] for solving heat equation. Bhattacharya [17] and Handschuh and Keith [18] used
exponential finite difference method for solving Burgers equation. Bahadr solved the
KdV equation by using the exponential finite difference technique [19].
In this paper, we design two new schemes for solving the Burgers equation. Some
examples are presented to show the ability of these methods to solve the equation. It is
clearly seen that both numerical methods are reasonably in good agreement with the exact
solution.
2. Methods of solution
We obtain numerical solutions of the Burgers equation by implicit exponential finite difference method and fully implicit exponential finite difference method for three standard
problems. The accuracy of the proposed methods are measured using the L 2 and L error
norms defined by

L 2 = u U 2 = h

N


1/2
|u i Ui |

i=0

L = u U  = max |u i Ui | .
0iN

(2)

The solution domain is discretized into cells described by the node set (xi , tn ) in which
xi = i h (i = 0, 1, 2, ..., N ) and tn = nk (n = 0, 1, 2, ...), h = x is the spatial mesh
size and k = t is the time-step.
2.1 Implicit exponential finite difference scheme
The implicit exponential finite difference method (I-EFDM) for eq. (1) takes the following
nonlinear form:



 n+1
n+1
Ui1
xUin Ui+1
vt
n+1
n
= Ui exp

Ui
2v
Uin
(x)2
 n+1


n+1
Ui1 2Uin+1 + Ui+1
+
(3)
Uin
which is valid for values of i lying in the interval 1 i N 1.
548

Pramana J. Phys., Vol. 81, No. 4, October 2013

Numerical solution of the one-dimensional Burgers equation

2.2 Fully implicit exponential finite difference scheme
The fully implicit exponential finite difference method (FI-EFDM) for eq. (1) takes the
following nonlinear form:

Uin+1

Uin

exp

vt
(x)2


 n+1
n+1
Ui1
xUin+1 Ui+1

2v
Uin


 n+1
n+1
Ui1 2Uin+1 + Ui+1
+
Uin

(4)

which is valid for values of i lying in the interval 1 i N 1.

Here Uin denotes the exponential finite difference approximation to the exact solution u(x, t). Equations (3) and (4) are systems of nonlinear difference equations. Let
us consider these nonlinear systems of equations in the form
F(V) = 0,

(5)

T
where F = [ f 1 , f 2 , ..., f N 1 ]T and V = [U1n+1 , U2n+1 , ..., U Nn+1
1 ] . Newtons method
applied to eq. (5) results in the following iteration:

(1) Set V(0) , an initial guess.

(2) For m = 0, 1, 2, ... until convergence do:
Solve J (V(m) ) (m) = F(V(m) );
Set V(m+1) = V(m) + (m) ,
where J (V(m) ) is the Jacobian matrix which is evaluated analytically. The solution at the
previous time-step is taken as the initial estimate. The Newtons iteration at each timestep is stopped when F(V(m) ) 105 . The convergence is generally obtained in two
or three iterations.

3. Numerical results
Problem 1.
We first solve the Burgers equation (1) and the initial condition
u(x, 0) = sin( x) ,

0<x <1

with the boundary conditions

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

t >0

and the exact solution given by

 2 2 

2 v
n=1 an exp n vt n sin(n x)


u (x, t) =

2 2
a0 +
n=1 an exp n vt cos(n x)
Pramana J. Phys., Vol. 81, No. 4, October 2013

(6)

549

Bilge Inan and Ahmet Refik Bahadir

Table 1. Comparison of the I-EFDM solutions with the exact solution at t = 0.1 for
v = 1 and k = 105 using various mesh sizes.
x

h = 0.05

h = 0.025

h = 0.0125

h = 0.01

Exact

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9

0.109737
0.210184
0.292464
0.348637
0.372384
0.359872
0.310656
0.228393
0.121000

0.109595
0.209905
0.292059
0.348127
0.371806
0.359279
0.310116
0.227979
0.120774

0.109560
0.209835
0.291958
0.348000
0.371662
0.359131
0.309981
0.227875
0.120718

0.109556
0.209826
0.291945
0.347984
0.371644
0.359113
0.309965
0.227817
0.120687

0.109538
0.209792
0.291896
0.347924
0.371577
0.359046
0.309905
0.227817
0.120687

L2
L

0.000579
0.000827

0.000164
0.000234

0.000060
0.000086

0.000048
0.000068

with

a0 =



exp (2 v)1 [1 cos( x)] dx

an = 2



exp (2 v)1 [1 cos( x)] cos(n x) dx,

n = 1, 2, 3, . . . .

The results for Problem 1 are displayed in tables 13 and figure 1. The numerical
solution obtained by implicit exponential finite difference method and the exact solution
for different values of h are presented in table 1. Table 2 compares the numerical results
obtained by fully implicit exponential finite difference method and the exact solutions
Table 2. Comparison of the FI-EFDM solutions with the exact solution at t = 0.1 for
v = 1 and k = 105 using various mesh sizes.

550

h = 0.05

h = 0.025

h = 0.0125

h = 0.01

Exact

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9

0.109738
0.210185
0.292465
0.348638
0.372385
0.359871
0.310655
0.228392
0.120999

0.109596
0.209906
0.292060
0.348128
0.371807
0.359278
0.310115
0.227978
0.120774

0.109561
0.209836
0.291959
0.348000
0.371662
0.359130
0.309981
0.227874
0.120717

0.109556
0.209827
0.291946
0.347985
0.371645
0.359113
0.309964
0.227862
0.120711

0.109538
0.209792
0.291896
0.347924
0.371577
0.359046
0.309905
0.227817
0.120687

L2
L

0.000579
0.000827

0.000164
0.000234

0.000060
0.000086

0.000048
0.000068

Pramana J. Phys., Vol. 81, No. 4, October 2013

Numerical solution of the one-dimensional Burgers equation

Table 3. Comparison of the numerical solutions with the exact solution at different
times for v = 1.0, v = 0.01, h = 0.0125 and k = 105 .
v = 1.0
x

v = 0.01

I-EFDM

FI-EFDM

Exact

I-EFDM

FI-EFDM

Exact

0.25

0.10
0.15
0.20
0.25

0.253690
0.156651
0.096484
0.059251

0.253691
0.156651
0.096484
0.059251

0.253638
0.156601
0.096442
0.059218

0.566353
0.512175
0.466611
0.428021

0.566355
0.512179
0.466614
0.428024

0.566328
0.512148
0.466583
0.427995

0.50

0.10
0.15
0.20
0.25

0.371662
0.226901
0.138536
0.084585

0.371662
0.226901
0.138536
0.084585

0.371577
0.226824
0.138473
0.084538

0.947453
0.900157
0.848433
0.796831

0.947454
0.900159
0.848436
0.796835

0.947414
0.900098
0.848365
0.796762

0.75

0.10
0.15
0.20
0.25

0.272650
0.164429
0.099482
0.060382

0.272649
0.164429
0.099482
0.060382

0.272582
0.164369
0.099435
0.060347

0.860116
0.922814
0.962051
0.974916

0.860119
0.922817
0.962053
0.974916

0.860134
0.922756
0.961891
0.974689

at t = 0.1 for v = 1 and k = 105 using various mesh sizes. It is observed from
tables 1 and 2 that the values of L 2 and L decrease with decrease of h. Comparison
of both numerical solutions with exact solution at different times for v = 1.0, v = 0.01,
h = 0.0125 and k = 105 are given in table 3. The obtained solutions by I-EFDM
and FI-EFDM are compared with other methods [3,5,11,20] in table 4. All comparisons
show that the present methods offer better results than the others. In order to show how the
numerical solutions of Problem 1 obtained with fully implicit exponential finite difference
method, we give the graph in figure 1.

0.8

0.6

0.4

0.2

0
0.3
0.2
0.1
0

0.2

0.4

0.6

0.8

Figure 1. Solution with FI-EFDM at different times for v = 1, h = 0.025, k = 104 .

Pramana J. Phys., Vol. 81, No. 4, October 2013

551

Bilge Inan and Ahmet Refik Bahadir

Table 4. Comparison of the results for Problem 1 at different times for v = 0.1,
h = 0.0125 and k = 104 .
x

RHC [3]

RPA [5]

[11]

NM [20]

I-EFDM

FI-EFDM

Exact

0.25

0.4
0.6
0.8
1.0

0.317062
0.248472
0.202953
0.169527

0.308776
0.240654
0.195579
0.162513

0.31215
0.24360
0.19815
0.16473

0.30415
0.23629
0.19150
0.15861

0.308936
0.240775
0.195709
0.162599

0.308962
0.240795
0.195725
0.162612

0.308894
0.240739
0.195676
0.162565

0.50

0.4
0.6
0.8
1.0

0.583408
0.461714
0.373800
0.306184

0.569527
0.447117
0.359161
0.291843

0.57293
0.40588
0.36286
0.29532

0.56711
0.44360
0.35486
0.28710

0.569727
0.447307
0.359343
0.292026

0.569762
0.447337
0.359368
0.292046

0.569632
0.447206
0.359236
0.291916

0.75

0.4
0.6
0.8
1.0

0.638847
0.506429
0.393565
0.305862

0.625341
0.487089
0.373827
0.029726

0.63038
0.49268
0.37912
0.03038

0.61874
0.47855
0.36467
0.27860

0.625659
0.487495
0.374187
0.287700

0.625676
0.487513
0.374203
0.287714

0.625438
0.487215
0.373922
0.287474

RHC Restrictive HopfCole method.

RPA Restrictive Pade approximation.

Problem 2.
The initial condition for the current problem is
u(x, 0) = 4x(1 x) ,

0<x <1

Table 5. Comparison of the numerical solutions with the exact solution at different
times for v = 1.0, v = 0.01, h = 0.0125 and k = 105 .
v = 1.0
x

v = 0.01

I-EFDM

FI-EFDM

Exact

I-EFDM

FI-EFDM

Exact

0.25

0.10
0.15
0.20
0.25

0.261534
0.161529
0.099513
0.061121

0.261535
0.161529
0.099513
0.061121

0.261480
0.161478
0.099470
0.061088

0.607370
0.549431
0.499841
0.457427

0.607373
0.549435
0.499845
0.457431

0.607363
0.549421
0.499828
0.457413

0.50

0.10
0.15
0.20
0.25

0.383509
0.234135
0.142953
0.087282

0.383510
0.234135
0.142953
0.087282

0.383422
0.234055
0.142888
0.087233

0.956022
0.914453
0.867170
0.818373

0.956023
0.914454
0.867172
0.818376

0.956007
0.914426
0.867136
0.818337

0.75

0.10
0.15
0.20
0.25

0.281643
0.169800
0.102704
0.062326

0.281643
0.169800
0.102704
0.062326

0.281573
0.169738
0.102655
0.062290

0.886728
0.938493
0.969863
0.979625

0.886730
0.938495
0.969864
0.979625

0.886767
0.938437
0.969741
0.979469

552

Pramana J. Phys., Vol. 81, No. 4, October 2013

Numerical solution of the one-dimensional Burgers equation

Table 6. Comparison of the results for Problem 2 at different times for v = 0.1 and
h = 0.0125.
k = 105
x

k = 104

RHC [3]

RPA [5]

[11]

NM [20]

I-EFDM

FI-EFDM

Exact

0.25

0.4
0.6
0.8
1.0

0.306529
0.236051
0.190181
0.156646

0.317399
0.246058
0.199437
0.165529

0.32091
0.24910
0.20211
0.16782

0.31247
0.24148
0.19524
0.16153

0.317567
0.246175
0.199589
0.165633

0.317595
0.246196
0.199606
0.165647

0.317523
0.246138
0.199555
0.165599

0.50

0.4
0.6
0.8
1.0

0.565994
0.438926
0.348328
0.280038

0.584429
0.457888
0.367320
0.298271

0.58788
0.46174
0.37111
0.30183

0.58176
0.45414
0.36283
0.29336

0.584627
0.458077
0.367507
0.298455

0.584664
0.458110
0.367533
0.298476

0.584537
0.457976
0.367398
0.298343

0.75

0.4
0.6
0.8
1.0

0.626990
0.477908
0.360630
0.272623

0.645527
0.502564
0.385232
0.295779

0.65054
0.50825
0.39068
0.30057

0.63858
0.49362
0.37570
0.28663

0.645850
0.502969
0.385613
0.296092

0.645866
0.502987
0.385630
0.296106

0.645616
0.502676
0.385336
0.295857

and the boundary conditions

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

t >0

with the exact solution also given by eq. (6) but with the following coefficients:
1
a0 =
exp[x 2(3v)1 (3 2x)]dx
0

1
0.8

0.6
0.4
0.2
0
3
2
1
t

0.2

0.4

0.6

0.8

Figure 2. Solution with FI-EFDM at different times for v = 0.1, h = 0.01 and
k = 104 .

Pramana J. Phys., Vol. 81, No. 4, October 2013

553

554

0.153286
0.265791
0.304138
0.261417
0.172157
0.088064
0.035822
0.011863
0.003249
0.000742

0.000022
0.000019

0.153285
0.265789
0.304137
0.261417
0.172156
0.088063
0.035822
0.011863
0.003249
0.000742

0.000021
0.000018

0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0

L2
L

FI-EFDM

I-EFDM

t = 1.5

0.153273
0.265771
0.304125
0.261421
0.172169
0.088070
0.035820
0.011859
0.003246
0.000741

Exact

0.000022
0.000038

0.064268
0.118814
0.155098
0.167631
0.156298
0.127378
0.091319
0.057971
0.032844
0.016737

I-EFDM

0.000023
0.000038

0.064268
0.118815
0.155099
0.167632
0.156299
0.127379
0.091320
0.057972
0.032844
0.016737

FI-EFDM

t = 3.0

0.064262
0.118804
0.155087
0.167623
0.156296
0.127382
0.091325
0.057975
0.032844
0.016735

Exact

0.000408
0.000743

0.037993
0.071874
0.097937
0.113393
0.116989
0.109492
0.093685
0.073603
0.053298
0.035714

I-EFDM

0.000408
0.000743

0.037993
0.071875
0.097938
0.113394
0.116989
0.109493
0.093685
0.073604
0.053298
0.035714

FI-EFDM

t = 4.5

0.037989
0.071869
0.097931
0.113387
0.116984
0.109491
0.093685
0.073605
0.053300
0.035717

Exact

Table 7. Comparison of the numerical solutions with the exact solution at different times for a = 0, b = 8, v = 0.5, h = 0.025 and
k = 104 .

Bilge Inan and Ahmet Refik Bahadir

Pramana J. Phys., Vol. 81, No. 4, October 2013

Numerical solution of the one-dimensional Burgers equation

an = 2

exp[x 2 (3v)1 (3 2x)] cos(n x) dx,

n = 1, 2, 3, . . . .

In table 5, we compare the numerical results of Problem 2 obtained from both new
methods (I-EFDM and FI-EFDM) with the exact solutions for both v = 1.0 and 0.01. In
table 6, we compare the numerical results of our methods (I-EFDM and FI-EFDM) with
the methods proposed in [3,5,11,20]. The comparisons showed that the present methods
offer better results than the others. For v = 0.1, the computed solution of Problem 2 by
FI-EFDM are displayed in figure 2.
Problem 3.
The initial condition for the current problem is
x
 , a < x < b


u(x, 1) =
1 + exp 1/4v x 2 14
and the boundary conditions
u(a, t) = u(b, t) = 0,

t >0

with the analytical solution

u(x, t) =

x/t
.
1 + [t/exp(1/8v)]1/2 exp(x 2 /4vt)

The numerical results of Problem 3 are displayed in table 7 for v = 0.5, a = 0, b = 8

with h = 0.025 and k = 104 . It is observed from the table that the values of L 2 and
L are small enough. Figure 3 illustrates the fully implicit exponential finite difference
solutions of Problem 3 at different values of t for a = 0, b = 1.2, v = 0.005, h = 0.01
and k = 104 . The figures for solutions from both methods are not drawn since they
are very close to each other for the three problems. It is clearly seen from all the tables
that the obtained numerical results with both methods present in this paper are in good
agreement with the exact solution.
0.5
0.4

0.3
0.2
0.1
0
4

1.2
1
3

0.8
0.6
2

0.4
0.2

Figure 3. Solution with FI-EFDM at different times for a = 0, b = 1.2, v = 0.005,

h = 0.01 and k = 104 .

Pramana J. Phys., Vol. 81, No. 4, October 2013

555

Bilge Inan and Ahmet Refik Bahadir

4. Conclusion
In this paper, we defined two exponential finite difference methods for solving
Burgers equation. Numerical solutions for three different test problems were given. The
results showed that both implicit exponential finite difference method and fully implicit
exponential finite difference method offer high accuracy in the numerical solution of the
one-dimensional Burgers equation.
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Pramana J. Phys., Vol. 81, No. 4, October 2013