Numerical Solution of Diffusion Equation by Finite Difference Method

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 6 Ver. IV (Nov. - Dec. 2015), PP 19-25
www.iosrjournals.org
DOI: 10.9790/5728-11641925 www.iosrjournals.org 19 | Page
Numerical Solution of Diffusion Equation by Finite Difference
Method
1
M. M. Rahaman, 2
M.M.H. Sikdar,3
M. B. Hossain, 4
M.A. Rahaman, 5
M.Jamal.
Hossain
1,3
Associate Professor, Department of Mathematics, Patuakhali Science and Technology University,
Bangladesh,
2
Associate Professor, Department of Statistics, Patuakhali Scienceand Technology University, Bangladesh
4
Assistant Professor, Department of Computer Science and information Technology, Patuakhali Science and
Technology University, Bangladesh
5
Associate Professor, Department of Computer Science and information Technology, Patuakhali Science and
Technology University, Bangladesh
Abstract: In this work error estimation for numerical solution of Diffusion equation by finite difference method
is done. The Explicit centered difference scheme is described to find the numerical approximation of the
Diffusion equation. The numerical scheme is implemented in order to perform the numerical features of error
estimation. To get analytic solution, we present the variable separation method. We develop a computer
program to implement the finite difference method in scientific programming language. An example is used for
comparison; the numerical results are compared with analytical solutions.
Keywords: Analytic solution, Diffusion equation, Finite difference scheme, Initial value problem (IVP),
Relative error.
I. Introduction
In Mathematics, the finite difference methods are numerical methods for approximating the
solutions to differential equations using finite difference equations to approximate derivatives. Our goal
is to approximate solutions to differential equations.i, e. to find a function (or some discrete
approximation to this functions) which satisfies a given relationship between several of its derivatives
on some given region of space /and or time, along with some boundary conditions along the edges of
this domain. A finite difference method proceeds by replacing the derivatives in the differential equation
by the finite difference approximations. This gives a large algebraic system of equations to be solved in
place of the differential equation, something that is easily solved on a computer.
In (A.N. Richmond, 2006), the authors develop the analytical solutions of non‐trivial examples of a
well‐known class of initial‐boundary value problems which, by the choice of parameters, can be reduced to
regular or singular Sturm‐Liouville problems. In (Sweilam et. al, 2012) the author presents the C-N-FDM to
solve the linear time fractional diffusion equation. They claimed that the C-N-FDM gives good results. The
authors studied the Spectral methods for solving the one dimensional parabolic heat equation (Juan- Gabriel et.
al 2006). In (Hikment Koyunbakan and Emrah Yilmaz, 2010), the Authors claimed that The ADM method is
more accurate. In (Subir et. al, 2011), the authors present the Adomian Decomposition method to solve the
nonlinear diffusion equation with fractional time derivatives. With the above discussion in view, our intention is
to investigate mathematical models, to establish the stability condition of the numerical scheme and to analyze
the error of the scheme.
In section 2, present a short discussion on the derivation of Diffusion equation as IBVP. In section 3,
the analytical solution of diffusion equation is illustrated by variable separation method. We describe an explicit
centered difference scheme for Diffusion equation as an IBVP with two sided boundary conditions in section 4.
In section 4, we also set up the stability condition of the numerical scheme. In section 5, we develop a computer
program in scientific programming language for the implementation of the numerical scheme and perform
numerical simulations in order to verify the behavior for various parameters. Finally the conclusions of the
paper are given in the last section.
II. Governing Equation And Its Derivation:
In this study we consider the governing equation as IBVP
2
2
x
c
D
t
c






Numerical Solution of Diffusion Equation by Finite Difference Method
c is the concentration at the point x at the time t , D is the diffusive constant in the x direction, t is the
time.
With appropriate initial and boundary condition
bxaxcxtc  );(),( 00
Ttttcatc a  0);(),(
)(),( tcbtc b
Consider the equation of mass conservation of the tracer. The continuity equation states that divergence of mass
flux equals change in mass in a control volume.
t
c
q





If we assume that  is constant in time and space, the continuity equation can be written as
t
c
q



Using Fick’s law for q , we have a general Diffusion equation
t
c
cD



If D is constant, the diffusion equation is given by as
t
c
cD


2
The diffusion coefficient theoretically is a tensor. However, for most cases, we assume it is a scalar. The
diffusion equation written in the Cartesian coordinate system in a one dimensional.
t
c
x
c
D





2
2
III. Analytical Solution Of The Governing Equation By The Method Of Variable Separation:
Consider XTc  be the solution of the diffusion equation
2
2
x
c
D
t
c




 Ttt 0 bxa  (1)
with the homogeneous boundary condition
Initial condition 0)0,( cxc  , and boundary condition 0),0( tc , 0),( tLc , Lx 0
Then TX
t
c



, TX
x
c



and TX
x
c



2
2
.
Now from the given equation, we have
X
X
DT
T 


(2)
Each side of (2) must be constant,
2




X
X
DT
T
(say)

Then 02
 TDT  and 02
 XX  whose solution are,
tD
eCT
2
1

 and xBxAX  sincos 11 
Thus a solution of the partial differential equation is
tD
eCxBxAtxc
2
111 )sincos(),( 
 
 )sincos(
2
xBxAe tD

 
(3)
Applying the boundary condition
Since 0),0( tc ,
tD
Ae
2
0 

0A , since 0
2
 tD
e 
.
Thus from (3), we have
xBetxc tD

sin),(
2

 (4)
Since 0),( tLc , LBe tD

sin0
2


If 0B the solution is identically zero, so we must have
0sin L since 0B , 0
2
 tD
e 
L
n
  , ,2,1,0 n ………
By the principle of superposition
The solution is
x
L
n
eBtxc
Dt
L
n
n
n


sin),(
2
22
1


 (5)
In order to satisfy the last condition, x
L
n
Bxc
n
n

sin)0,(
1




Using Fourier series, 
L
n xdx
L
n
xc
L
B
0
sin)0,(
2 
The solution of the governing equation can be written as follows
x
L
n
exdx
L
n
xc
L
txc
Dt
L
n
n
L


sin)sin)0,(
2
(),(
2
22
1 0


  (6)

IV. Formulation Of The Diffusion Equation
We would like to consider the diffusion equation as an initial and homogeneous boundary value
problem
2
2
x
c
D
t
c





, Ttt 0 , bxa 
Initial condition 0)0,( cxc  , and boundary condition 0),0( tc , 0),( tLc
In order to develop the scheme, we discretize the tx  plane by choosing a mesh width xh  space and a
time step tk  . The finite difference methods we will develop produce approximations
nn
i Rc  to the
solution ),( ni txc at the discrete points by
ihxi 
,
.....3,2,1,0i
nktn  ,
.....3,2,1,0n
Let the solution ),( ni txc be denoted by
n
iC and its approximate value by
n
ic .
Simple approximations to the first derivative in the time direction by forward difference can be obtained from
)(
1
to
t
CC
t
c n
i
n
i





 
Discretization of 2
2
x
c


is obtain from second order central difference in space.
)(
2 2
2
11
2
2
xo
x
CCC
x
c n
i
n
i
n
i





 
We obtain
)(
2 2
2
11
1
xto
x
CCC
D
t
CC n
i
n
i
n
i
n
i
n
i





 

(7)
The terms )( 2
xto  denote the order of the method. Neglecting the error terms and simplifying. We obtain
the difference methods
=>
n
i
n
i
n
i
n
i c
x
tD
c
x
tD
c
x
tD
c 12212
1
)21( 









 (8)
This is the required explicit centered difference scheme for the IBVP
n
i
n
i
n
i
n
i cccc 11
1
)21( 

  (9)
This scheme uses a second order central difference in space and the first order forward Euler scheme in time.
Where 2
x
tD


 Note that if
2
1
0   , then the solution at the new time is a weighted average of the
solution at the old time .This implies a discrete maximum principle, and therefore numerical stability. It also
implies monotonocity: if
n
i
n
i cc 1 for all i , then
0)())(21()( 1211
11
1  


n
i
n
i
n
i
n
i
n
i
n
i
n
i
n
i cccccccc 
However, we must choose the time step to be small: we must have
2
1
 , or equivalently that
D
x
t
2
2


This time step restriction typically requires an unacceptably large number of time steps, unless the diffusion
constant D is very small.
4.1 Stability of the explicit centered difference scheme (8) is given by the conditions
2
1
0 2




x
tD

Proof: The explicit centered difference scheme (8) takes the form
=>
n
i
n
i
n
i
n
i c
x
tD
c
x
tD
c
x
tD
c 12212
1
)21( 










n
i
n
i
n
i
n
i cccc 11
1
)21( 

  (10)
Where
2
x
tD



The equation (10) implies that if
2
1
0   , and then the solution at the new time is a weighted average of the
solution at the old time. This implies a discrete maximum principle. We can conclude that the explicit centered
difference scheme (10) is stable for
2
1
0 2




x
t
D
V. Error Estimation Of The Scheme:
In order to perform error estimation, we consider the exact solution of the model equation with initial condition
)1()()0,( 0 xxxcxc  and homogeneous boundary condition. We get
x
L
n
exdx
L
n
xc
L
txc
Dt
L
n
n
L


sin)sin)0,(
2
(),(
2
22
1 0


 
We compute the error defined by
e
Ne
C
CC
e


for all time where ec is the exact solution and Nc is the Numerical solution computed by the finite difference
scheme.
5.1Results And Discussion:
We solve the diffusion equation by implementing the centered difference scheme, while varying the
different parameter values.
Figure 1: The behavior of numerical solution at smD /001.0 2
 , sm /005.0 2
.

Concentration distribution for each diffusion rate at time t=24 min. In figure-1, the profile for varying
contaminant diffusion rate, we saw that the contaminant concentration with a higher diffusion rate decreases at a
higher rate than that with a lower diffusion rate. The curve marked by “star” shows the concentration profile for
diffusion rate smD /001.0 2
 and the curve visible by “dot line” represents the concentration profile for
diffusion rate smD /005.0 2

Figure 2: Analytic solution and Numerical solution at different time Analytical solution of diffusion equation is
compared with the numerical solution at different time in figure-2. The curve noticeable by “blue line” shows
the numerical solution, the curve visible by “red line” represents numerical solution. The results are very close.
Figure 3: The Numerical solution and Analytical solution in mesh.
Figure 4: The error in the numerical result is shown for ∆x=0.1

Figure 5: The error in the numerical result is shown for ∆x=0.01
Figure-4 and figure-5 shows the error in the numerical solution from each of the methods when
compared with the analytical solution, for the atwo cases N=10, N=100, corresponding to ∆x=0.1, 0.01
respectively. Comparisons are made for the solution at different time for smaller ∆x the errors reduce in size.
The errors for the central difference scheme decrease as the grid size decrease.
VI. Conclusion:
The study has presented the numerical and analytical solution of Diffusion equation. The explicit
centered difference scheme is used in order to perform the numerical features of error estimation. We have seen
that the contaminant concentration with a higher diffusion rate decreases at a higher rate than that with a lower
diffusion rate. In order to execute the numerical method we have developed a computer program in the language
of scientific computing that is a very good agreement of the finite difference method for Diffusion equation.
Reference
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Numerical Solution of Diffusion Equation by Finite Difference Method

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