Applied Mathematics, 2011, 2, 883-889
doi:10.4236/am.2011.27118 Published Online July 2011 (http://www.SciRP.org/journal/am)
A Note on Crank-Nicolson Scheme for Burgers’ Equation
Kanti Pandey, Lajja Verma
Department of Mathematics and Astronomy, University of Lucknow, Lucknow, India
Received April 5, 2011; revised May 23, 2011; accepted May 26, 2011
Abstract
In this work we generate the numerical solutions of the Burgers’ equation by applying the Crank-Nicolson
method directly to the Burgers’ equation, i.e., we do not use Hopf-Cole transformation to reduce Burgers’
equation into the linear heat equation. Absolute error of the present method is compared to the absolute error
of the two existing methods for two test problems. The method is also analyzed for a third test problem, nu-
merical solutions as well as exact solutions for different values of viscosity are calculated and we find that
the numerical solutions are very close to exact solution.
Keywords: Hopf-Cole Transformation, Burgers’ Equation, Crank-Nicolson Scheme, Nonlinear Partial
Differential Equations
1. Introduction
Burgers’ equation is one of the most important nonlinear
partial differential equations governed by the following
equation
  
2
=,
22
,, ,0,10,.
d
tx
x
w
ww
ww xtxtT






(1)
This equation is nonlinear and can be considered as
nonlinear analog of the Navier-Stokes equations. It has a
convection term, a diffusion term and a time-dependent
term. It also has a large variety of applications in
modeling of water in unsaturated oil, dynamics of soil in
water, statics of flow problems, mixing and turbulent
diffusion, cosmology and seismology [1-3].
With viscous term the Burgers’ Equation (1) is
parabolic while without viscous term it is hyperbolic. In
the later case it possesses discontinuous solutions due to
the nonlinear term and even if smooth initial condition is
considered the solution may be discontinuous after finite
time. It also governs the phenomenon of shock waves
[4].
In the present work we consider the Burgers’ Equation
(1) with the initial condition
 
,0=, 01,wxf xx (2)
and the boundary conditions


 
1
2
0,=, 0,
0,=, 0,
wt gttT
wt gttT

 (4)
where >0
d

is a coefficient of viscous diffusion and
f
, 1
g
and 2
g
are the sufficiently smooth given func-
tions.
Several researchers have successfully used Burgers’
equation to develop new algorithms and to test various
existing algorithms. In most of the cases researchers used
Hopf-Cole [5,6] transformation to linearize the Burgers’
equations into parabolic partial differential equation.
Some of the researchers also tried to tackle the nonlinear
Burgers’ equation directly (without Hopf-Cole). Kadal-
bajoo et al. [7] applied Crank-Nicolson finite difference
method to the linearized Burgers’ equation by Hopf-Cole
transformation which is unconditionally stable and is
second order convergent in both space and time with no
restriction on mesh size. Gorguis [8] applied the Adomian
decomposition method on the Burgers’ equation directly
and compared the numerical result with the analytical
result. In another result due to Kutluay et al. [9] a direct
approach via least square quadratic B-spline finite
element method is discussed. Recently Pandey et al. [10]
discussed Douglas finite difference scheme on linearized
Burgers’ equation which is fourth order convergent in
space and second order convergent in time.
In this paper we first apply Crank-Nicolson finite
difference scheme directly on the nonlinear Equation (1)
and derive a nonlinear finite difference scheme, and then
use it to derive a system of linear equations which we

K. PANDEY ET AL. 884
solve by using Mathematica (version 7.0). For two test
problems we compare the absolute error of the numerical
solutions to the absolute error of the numerical results
established by Kadalbajoo et al. [7] and Pandey et al.
[10]. For the third test problem we compare our
numerical results with the analytical solution. The exact
solution for the third problem is calculated on
Mathematica (version 7.0).
In this paper we show that it is more appropriate to
consider the Burgers’ equation directly than reducing it
into linear parabolic problem and then discretize it.
Therefore this paper is an improvement over the results
in [7] and [10].
This paper is divided in 4 sections. In Section 2 we
give expression for exact solution by Hopf-Cole transfor-
mation which is used to calculate the exact solutions. In
the same section we collocate and discretize to get a
nonlinear finite difference equation and then through a
simple approximation we deduce linear finite difference
equation. In Section 3 we give three examples and in
Section 4 we demonstrate properties of the computed
numerical solutions in the form of Tables 1-11 and
Figures 1-4.
2. Description of the Method
2.1. Exact Solution
Hopf and Cole [5,6] suggested that (1) can be reduced to
linear heat equation by the non-linear transformation. Let


=log
d,



 (5)
and
=.
x
w

(6)
Putting in (1) we get
2
1=
22
d
tx . xx


  (7)
Next applying the transformation (5) we get
=
2
d
t.
x
x



(8)
The fourier series solution to the linearized heat equa-
tion is

22
0
=1
π
,= expcosπ
2
d
n
n
nt
x
tA An






x (9)
with fourier coefficients at as
=0t

 
1
00
1
0
=d,
=2cosπd,
n
AExx
A
Exnx x

 (11)
where




0=,ww
 
0
0
1
=exp d
x
d
Ex w








. Using the Hopf-Cole
transformation we have the exact solution
 

=1
0
=1
sin π
,=π
cos π
nn
n
d
nn
n
A
Htn nx
wxt
A
AHtn x





 (12)
where

22
π
=exp2
d
nnt
Ht





.
2.2. Discretization
First of all we divide the solution space into a uniform
mesh. For this we divide the interval


0, 1 into
equal subintervals and divide the interval

N

0, T into
M
equal subintervals. Let =1hN be the mesh width
in space and =
i
x
ih for . Let

=1 1iN=kTM be
the mesh width in time and for
=
j
tjk


=0 1jM.
Now Collocating the Burgers’ Equation (1) we get


 
22
,
,,
2
1=
22
=11, =0 1.
ij
ij ij
d
w
ww
tx x
iNjM




 ,
(13)
where


,=, ijij
wwxt. The Crank-Nicolson method
[11,12] gives the following system of nonlinear equations,
2
,1 ,
,1 ,
,1 , 2
2
1
22
=.
22
ij ij
ijijx x
ij ij
dx
ww
k
ww h
ww
k
h







 






(14)
where
x

and
x

are central difference operator and
averaging operator respectively. To linearize we put
,1 ,ij ij,j
v=ww
i

into (14) where ,,1 =
ij ijij
vw w
,

and
neglecting


2
,ij
Ov we get
1, 1,,
1, 1,
1, 1,,
1, 1,
11
422 2
1
422
1
=24
1
24
dd
ij ijij
d
ij ij
dij ijdij
dij ij
r
s
wvr
r
sw v
rrwwrw
rsww








v

 
 

 














(15)
where


=
s
kh and


2
=rkh.
3. Numerical Results and Discussions
0

and In this section we demonstrate the accuracy of the

K. PANDEY ET AL.885
present method by solving three test problems and
compare the results with the two existing results. The
computations are performed using Mathematica 7.0 and
Origin 7.5.
3.1. Problem 1
Consider Equation (1) with boundary conditions and
initial condition as
 
0,=1,=0, >0,wtwtt (16)

,0 =sinπ,wx x (17)
where

2
d
 is the coefficient of kinematic viscosity.
We substitute

,= dx wxt


 (18)
in Equation (1) and get =, 0<<1, >
2
d
txx
xt

 0 with
initial condition
 
1
,0=exp1cos π, 0<<1.
πd
xx 




x
(19)
and boundary condition
 
0,=1,=0, >0.
xx
ttt


The exact solution of the Burgers’ Equation (1) is (12)
with given Fourier coefficients:


1
00
1
0
1
=exp1cosπd,
π
1
=2exp1cos πcosπd.
π
d
nd
Axx
A
xnxx














3.2. Problem 2
As a second example consider (1) with the boundary
conditions (16) and initial condition
 
,0=41 , 0<<1.wxxxx (20)
The exact solution (12) can be obtained in the similar
fashion as in Problem 3.1 with the Fourier coefficients as
follows :


2
1
00
2
1
0
2
=exp32 d,
3
2
=2 exp3 2cosπd.
3
d
nd
x
Axx
x
A
xnx











x
3.3. Problem 3
Consider Equation (1) with boundary conditions (16) and
initial condition

2πsinπ
,0=, 0<<1.
2cosπ
x
wx x
x
 (21)
The exact solution can be calculated by using the
formula (12) where
2/
1
00
2/
1
0
2cosπ
=d,
2
2cosπ
=2cos πd.
2
d
d
n
x
Ax
x
A
nxx












(22)
4. Conclusions
We present Crank-Nicolson finite difference scheme for
Burgers’ equation without Hopf-Cole transformation.
We claim that it is better to solve the nonlinear Burgers’
equation directly, i.e., without reducing it to linear heat
equation by Hopf-Cole transformation. Our claim is very
well supported by the Tables 1 -10 and Graphs 1-3. From
Table 11 and Graph 4 it is also proved that numerical
results are in good agreement with the analytical solution.
The exact solution for the Problem 3.3 is calculated by
using Mathematica 7.0. Figure 4 also depicts the
physical behavior of the solutions and thus behavior of
any physical system governed by Burgers’ equation can
be studied by this method.
4.1. Tables
Computed results are displayed in Tables 1 to 11 at
different nodal points for different values of viscosity. In
Tables 1 to 6 we compare the absolute error with the
absolute error of [7,10] for the Problem 3.1. From Table
7 to Table 10 absolute error is compared to the absolute
Table 1. Comparison of the absolute error with the absolute
error of [7,10] for Problem 3.1 at T = 0.01, for vd = 20 and K
= 0.0001 for N = 40.
x
/Error Without
Hopf-Cole Kadalbajoo et al. Pandey et al.
0.1 5.78801E–05 6E–05 0.00016
0.2 0.000109996 0.00011 0.00031
0.3 0.000151182 0.00016 0.00044
0.4 0.000177403 0.00019 0.00051
0.5 0.000186149 0.0002 0.00054
0.6 0.000176667 0.00019 0.00051
0.7 0.000149991 0.00016 0.00044
0.8 0.000108805 0.00011 0.00031
0.9 5.71442E–05 6E–05 0.00016

K. PANDEY ET AL. 886
Table 2. Comparison of the absolute error with the absolute
error of [7,10] for Problem 3.1 at T = 0.01, for vd = 20 and K
= 0.0001 for N = 80.
x
/Error Without
Hopf-Cole Kadalbajoo et al. Pandey et al.
0.1 1.37807E–05 1E–05 7E–05
0.2 2.61869E–05 3E–05 0.00014
0.3 3.59876E–05 5E–05 0.00021
0.4 4.22225E–05 5E–05 0.00024
0.5 4.42961E–05 6E–05 0.00025
0.6 4.20319E–05 5E–05 0.00024
0.7 3.56792E–05 5E–05 0.00021
0.8 2.58784E–05 3E–05 0.00014
0.9 0.00001359 1E–05 8E-05
Table 3. Comparison of the absolute error with the absolute
error of [7,10] for Problem 3.1 at T = 0.1, for vd = 2 and K =
0.001 for N = 40.
x
/Error Without
Hopf-Cole
Kadalbajoo et
al.
Pandey et
al.
0.1 6.10384E–05 6E–05 0.00016
0.2 0.0001156 0.00011 0.0003
0.3 0.000157849 0.00016 0.00042
0.4 0.00018329 0.00018 0.0005
0.5 0.000189452 0.0002 0.00054
0.6 0.000176365 0.00019 0.00052
0.7 0.000146518 0.00017 0.00045
0.8 0.000104112 0.00048 0.00033
0.9 5.38605E–05 0.00026 0.00018
Table 4. Comparison of the absolute error with the absolute
error of [7,10] for Problem 3.1 at T = 0.1, for vd = 2 and K =
0.001 for N = 80.
x
/Error Without
Hopf-Cole Kadalbajoo et al. Pandey et al.
0.1 1.45914E–05 2E–05 7E–05
0.2 2.76247E–05 3E–05 0.00014
0.3 3.76962E–05 4E–05 0.0002
0.4 4.37257E–05 4E–05 0.00023
0.5 4.51272E–05 5E–05 0.00025
0.6 4.19278E–05 5E–05 0.00024
0.7 3.47543E–05 5E–05 0.00021
0.8 0.000024642 3E–05 0.00016
0.9 1.27275E–05 2E–05 8E–05
Table 5. Comparison of the absolute error with the absolute
error of [7,10] for Problem 3.1 for vd = 20, K = 0.01, N = 80
at different times.
x
/Error Without
Hopf-Cole Kadalbajoo et al. Pandey et al.
0.4 5.9E–05 8.00E–05 0.00023
0.6 4.4E–05 5.00E–05 0.00016
1 4E–05 2.00E–05 8E–05
3 1.48E–05 0 1E–05
0.4 0.000118 8.00E–05 0.00029
0.6 9.5E–05 7.00E–05 0.00024
1 7.1E–05 4.00E–05 0.00015
3 2.39E–05 1.00E–05 2E–05
0.4 3.7E–05 4.00E–05 0.00021
0.6 7.5E–05 6.00E–05 0.00021
1 7.2E–05 3.00E–05 0.00015
3 0.0010165 0.00101 0.0002
Table 6. Comparison of the absolute error with the absolute
error of [7,10] for Problem 3.1 for vd = 0.02, K = 0.01, N =
80 at different times.
x
/Error Without
Hopf-Cole
Kadalbajoo et
al. Pandey et al.
0.4 7.2E–05 0.00038 0.00076
0.6 5E–05 6E–05 0.00012
1 2.7E–05 2E–05 0.00013
3 8E–06 0 6E–05
0.4 0.000167 0.00726 0.01517
0.6 0.000106 0.00269 0.00736
1 5.2E–05 0.00058 0.00229
3 7E–06 0 4E-05
0.4 0.000384 0.02654 0.04398
0.6 0.000234 0.01 0.02528
1 9.6E–05 0.00228 0.0093
3 3.3E–05 0 0.00047
Table 7. Comparison of the absolute error with the absolute
error of [7,10] for Problem 3.2 at T = 0.1, for vd = 2 and K =
0.001 for N = 40.
x
/Error Without
Hopf-Cole
Kadalbajoo et
al.
Pandey et
al.
0.1 6.34389E–05 6E–05 0.00012
0.2 0.000119993 0.00011 0.00023
0.3 0.000163549 0.00016 0.00032
0.4 0.00018952 0.00019 0.00037
0.5 0.000195514 0.0002 0.00039
0.6 0.000181726 0.0002 0.00038
0.7 0.000150809 0.00017 0.00033
0.8 0.000107093 0.00012 0.00024
0.9 5.53858E–05 7E–05 0.00013

K. PANDEY ET AL.887
Table 8. Comparison of the absolute error with the absolute
error of [7,10] for Problem 3.2 at T = 0.1, for vd = 2 and K =
0.001 for N = 80.
x
/Error Without
Hopf-Cole
Kadalbajoo et
al.
Pandey et
al.
0.1 0.000015144 1E–05 3E–05
0.2 2.86478E–05 3E–05 6E–05
0.3 3.90448E–05 5E–05 9E–05
0.4 4.52233E–05 5E–05 9E–05
0.5 0.000046599 5E–05 1E–04
0.6 4.32271E–05 5E–05 1E–04
0.7 3.57791E–05 5E–05 9E–05
0.8 2.53375E–05 3E–05 6E–05
0.9 1.30758E–05 2E–05 3E–05
Table 9. Comparison of the absolute error with the absolute
error of [7,10] for Problem 3.2 for vd = 0.2, K = 0.01, N = 80
at different times.
x
/Error Without
Hopf-Cole Kadalbajoo et al. Pandey et al.
0.4 6.3E–05 9E–05 0.00017
0.6 4.6E–05 5E–05 0.00011
1 3.5E–05 2E–05 6E–05
3 1.16E–05 0 1E–05
0.4 0.000118 8E–05 0.00013
0.6 0.000101 7E–05 0.00012
1 8.1E–05 3E–05 8E–05
3 2.46E–05 1E–05 1E–05
0.4 3.6E–05 4E–05 8E–05
0.6 7.1E–05 7E–05 3E–05
1 6.8E–05 4E–05 6E–05
3 1.45E–05 0 1E–05
Table 10. Comparison of the absolute error with the
absolute error of [7,10] for Problem 3.2 for vd = 0.02, K =
0.01, N = 80 at different times.
x
/Error Without
Hopf-Cole
Kadalbajoo et
al. Pandey et al.
0.4 0.00012 0.00047 0.00113
0.6 8.6E–05 8E–05 0.00024
1 4.9E–05 2E–05 0.00011
3 1.21E–05 0 6E–05
0.4 0.000167 0.00818 0.0172
0.6 0.000128 0.00293 0.00839
1 7.7E–05 0.00059 0.00258
3 1.6E–05 0 5E–05
0.4 0.000286 0.0289 0.04617
0.6 0.000212 0.011 0.02718
1 0.000109 0.00238 0.0101
3 4.4E–05 4E–05 0.0005
Table 11. Comparison of the numerical solution with the
exact solution for Problem 3.3 at different space points at T
= 0.01, for vd = 20 and K = 0.0001 for different values of N.
x
=10N=20N=40N =80N Exact
0.10.368710.377309 0.370262 0.370173 0.370141
0.2 0.705991 0.720825 0.708758 0.708574 0.70851
0.3 0.9818120.998880.985219 0.984931 0.984834
0.41.16936 1.18429 1.17275 1.17236 1.17224
0.51.24751 1.25697 1.25035 1.24988 1.24973
0.61.20385 1.20667 1.20583 1.20532 1.20517
0.71.03763 1.0351 1.03875 1.03827 1.03813
0.8 0.761828 0.757042 0.762304 0.761929 0.761818
0.9 0.403226 0.399692 0.403361 0.403154 0.403094
(a)
(b)
Figure 1. Absolute errors of Problem 3.1 at different times
for vd = 20 and K = 0.0001, (a) N = 40, (b), N = 80.

K. PANDEY ET AL. 888
(a) (b)
Figure 2. Absolute errors of Problem 3.1 at different times for vd = 2 and K = 0.001, (a) N = 40, (b) N = 80.
(a) (b)
Figure 3. Absolute errors of Problem 3.2 at different times for vd = 2 and K = 0.001, (a) N = 40, (b) N = 80.
(a) (b)
Figure 4. Numerical solutions of Problem 3.3 at different times for vd = 2 and K = 0.001, (a) N = 40, (b) N = 80.

K. PANDEY ET AL. 889
error of [7,10] for the Problem 3.2. In Table 11 for
Problem 3.3 we compare the numerical solution to the
exact solution and it is observed that computed result
shows greater agreement with the exact solution as the
mesh size is refined.
4.2. Figures
In Figures 1 and 2 we compare the absolute error with
the absolute error of [7] for Problem 3.1 and in Figure 3
we compare the absolute error with the absolute error of
[7] for Problem 3.2. Finally in Figure 4 we compare the
exact solution with the numerical solution for Problem
3.3.
5. References
[1] N. Su, J. P. C. Watt, K. W. Vincent, M. E. Close and R.
Mao, “Analysis of Turbulent Flow Patterns of Soil Water
under Field Conditions Using Burgers’ Equation and Po-
rous Suction-Cup Samplers,” Australian Journal of Soil
Research, Vol. 42, No. 1, 2004, pp. 9-16.
doi:10.1071/SR02142
[2] N. J. Zabusky and M. D. Kruskal, “Interaction of Solitons
in a Collisionless Plasma and the Recurrence of Initial
States,” Physical Revi ew, Vol. 15, No. 6, 1965, pp. 240-243.
doi:10.1088/0305-4470/33/18/308
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