In this paper, we construct a numerical method to solve singularly perturbed onedimensional parabolic convection–diffusion problems. We use Euler method with uniform step size for temporal discretization and exponentialspline scheme on spatial uniform mesh of Shishkin type for full discretization. We show that the resulting method is uniformly convergent with respect to diffusion parameter. An extensive amount of analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter. The obtained numerical results show that the method is efficient, stable and reliable for solving convection–diffusion problem accurately even involving diffusion parameter.
AMS Mathematics Subject Classification : 65H05.
1. Introduction
In the present paper, we are concerned with the numerical approximation of the partial differential equation:
Here
Q_{T}
= Ω ×
I
, Ω ≡ (
a
,
b
), ∂Ω ≡ {
a
,
b
} ,
I
≡ (0,
T
), and
a
and
b
are real positive constants. We consider equation (1) associated with the Dirichlet boundary conditions
and the initial condition
where
u
(
x
,
t
) is the dependent variable,
x
and
t
are the independent variables and
f
,
p
,
q
and
u
_{0}
are given functions of their arguments. It is usually assumed that 0 <
ε_{c}
≪ 1 and 0 <
ε_{d}
≪ 1.
This problem encloses both the reaction–diffusion problem when
ε_{c}
= 0 and the convection–diffusion problem when
ε_{c}
= 1. These problems occur naturally in various fields of science and engineering, such as, heat transfer with large Peclet numbers, combustion, nuclear engineering, control theory, elasticity, fluid mechanics, aerodynamics, quantum mechanics, optimal control, chemicalreactor theory, convectiondiffusion process and geophysics; see
[9
,
10
,
18]
.
There has been a lot of effort in developing numerical methods for the solution of singular perturbation problems that are uniformly convergent. Clavero et al.
[1]
and Kadalbajoo et al.
[4]
gives a uniform convergent numerical method with respect to the diffusion parameter to solve the onedimensional time–dependent convection–diffusion problem. They used the implicit Euler method for the time discretization and the simple upwind finite difference scheme on a Shishkin mesh for the spatial discretization. Ramos
[7]
presented an exponentially fitted method for singularly perturbed, onedimensional (convection–diffusion) parabolic problems, and showed its uniform convergence in the perturbation parameter. Surla and Jerkovic
[13]
considered a singularly perturbed boundary value problem using a spline collocation method. Sakai and Usmani
[11
,
12]
gave a new concept of Bspline in terms of hyperbolic and trigonometric splines which are different from the earlier ones. It is proved that the hyperbolic and trigonometric Bsplines are characterized by a convolution of some special exponential functions and a characteristic function on the interval [0,1]. Sharma and Kaushik
[14]
presented a numerical approximation of a singularly perturbed time delayed convection diffusion problem on a rectangular domain. Finally, Zahra and El Mhlawy
[18]
developed a numerical method to solve twoparameter singularly perturbed semiLinear boundary value problems. This method is based on exponential spline with a piecewise uniform Shishkin mesh which is shown to be uniformly convergent independent of mesh parameters and perturbation parameter
ε_{c}
and
ε_{d}
.
Our contribution in this paper is to introduce a full discretization scheme based on Rothe’s approximation in time discretization and on the exponential spline approximation in the spatial discretization for solving singularly perturbed, onedimensional timedependent convection–diffusion problem associated with homogeneous Dirichlet boundary conditions. It starts with the discretization in time by the 2point Euler backward finite difference method in the time variable. After that we deal with the exponential spline method for the solution of the time discretized problem to compute the unknown function and the obtained system of algebraic equations is solved by iterative methods. Moreover, we obtain error estimates for the approximation process. The proposed technique is programmed using
Matlab 7
, and used in solving the proposed problem.
The rest of the paper is organized as follows: In section 2, we give some notations, assumptions and definitions, In §3, a time discretization scheme for the continuous problem is proposed in addition to the derivation of some a priori estimates. In §4, we prove the convergence results and error estimates of the semidiscretization scheme described in §3. We describe in §5 a full discretization scheme based on Rothe exponential spline methods and the error estimates of the full discretized solution are derived. Finally, numerical results and discussions are presented and comparisons are made with other solutions in §6.
2. Notations, assumptions and definitions
In the sequel, we will denote by (. , .) either the standard inner product in
L
_{2}
=
L
_{2}
(Ω) or the pairing between
and
V
^{∗}
≡
H
^{−1}
(see e.g.
[11]
). we use the symbols ., ∥.∥ and ∥.∥
_{∗}
as the norms in
L
_{2}
(Ω),
V
,
V
^{∗}
, respectively. By
we mean the strong and weak convergence. Also, we introduce some notations concerning the time discretization of our problem.
for any given family
The following elementary relations will be used in the following analysis:
and Young’s inequality
where
ε
is a small constant. We will assume, throughout this work, the following hypotheses on the given data.
(
H1
) The function
f
: Ω ×
I
→
R
is Lipschitz continuous in the sense of
with
f
(0, 0) = 0.
(
H2
) The functions
p
(
x
) and
q
(
x
) are smooth functions and satisfy
(
H3
)
Under these assumptions, we can define the weak solution of problem (1)(2).
Definition 2.1.
The measurable function
u
∈
C
(
I
;
L
_{2}
(Ω))∩
L
_{2}
(
I
;
V
) with ∂
_{t}u
∈
L
_{2}
(
I
;
V
^{∗}
) and
u
(
x
, 0) =
u
_{0}
(
x
) is said to be a weak (variational) solution of (1)−(3) if and only if the integral identity
holds for all
φ
∈
V
and a.e.
t
∈
I
.
Remark.
It is evident that for any functions
α
,
β
∈
H
^{1}
(Ω) and
γ
∈
L
_{2}
(Ω)
3. The temporal discretization scheme, A prior estimate
Our main goal is to approximate (1)−(3) from a numerical point of view and to prove its convergence. The scheme starts with the discretization in time by the 2point Euler backward finite difference method in the time variable, so that the problem is converted into a linear system of differential equations that easily solved numerically at each subsequent time level. Let
m
be a positive integer and subdivide the time interval
I
by the points
t_{j}
, where
t_{j}
=
jτ
,
τ
=
T/m
,
j
= 0, 1, · · · ,
m
. The suggested discretization scheme of problem (8) consists of the following problem (in the weak sense):
where ∂
_{x}
denotes to the derivative with respect to
x
and
f_{j}
=
f
(
x
,
t_{j}
).
The existence of a weak solution
w_{j}
∈
V
is proved in
[2]
. By means of
w_{j}
, (
j
= 0, 1, · · · ,
m
) determined by the proposed scheme 10 11 in each time step
t_{j}
, we introduce the following piecewise linear functions (Rothe functions)
and the corresponding step function
Using the notation of Rothe function and its corresponding step function, a piecewise constant interpolation of equation 11 over
I
for each
φ
∈
V
yields
where
t
_{j − 1}
≤
t
≤
t_{j}
and
with
In order to show the stability of the discrete solution and prove the convergence results, we shall derive some a priori estimates.
Lemma 3.1.
Under the assumptions (H1)(H3), there exists a positive constant C such that for any s
,
Proof
. Let us choose
φ
=
τ δw_{j}
in 11 and summing over
j
from 1 to
s
, we obtain
Taking into consideration (4), the second and the fourth terms are estimated by
With the aid of 11, Younge’s inequality (5) and (9), we estimate the third term by
The last term is estimated by the use of (6) as follows
Collecting (16)(20), using (
H3
), and choosing ε sufficiently small, we get
Applying Gronwall’s inequality, we conclude the proof.
Lemma 3.2.
Uniformly with respect to n one has
Proof
. The estimates (22) and (23) are a consequence of (15). By the use of the identity
the estimates (22)
_{3}
, and (23) are a consequence of (15)
_{2}
, (15)
_{3}
and the definitions of
w^{m}
and
and thus the proof completes.
4. Convergence results
This section is devoted to proving the convergence of the proposed scheme and estimating its accuracy. Before we are able to prove convergence we need to prove the compactness of
in
L
_{2}
(
I
;
L
_{2}
(Ω)) which is a consequence of the following assertion.
Lemma 4.1.
The estimate
holds uniformly for
0 <
z
<
z
_{0}
and n
.
Proof
.
[3]
We sum up (12) for
i
=
s
+1, · · · ,
s
+
k
considering
φ
=
w_{s}
_{+}
_{k}
−
w_{s}
)
τ
. Then we sum it up for
s
= 1, · · · ,
m
−
k
and obtain the estimate
Hence for
kτ
≤
z
≤ (
k
+ 1)
τ
we conclude the desired estimate.
Theorem 4.1.
There exists u
∈
L
_{2}
(
I
;
L
_{2}
(Ω)) ∩
H
^{1}
(
I
;
L
_{2}
(Ω))
such that
(in the sense of subsequences). Moreover, we have
Proof
. The estimate (22)
_{2}
implies
Hence, from lemma 4.1,
is compact in
L
_{2}
(
I
;
L
_{2}
(Ω)) because of Kolmogorov’s compactness argument. So we can conclude that
w^{n}
→
u
and
in
L
_{2}
(
I
;
L
_{2}
(Ω)) and also pointwise in
Q
. Also by the fact that
we obtain that
is weakly convergent in
L
_{2}
(
I
;
V
); i.e.
and this implies that
It remains to prove that ∂
_{t}w^{m}
= ∂
_{t}u
. For each
t
∈
I
, by lemma 3.3, ∂
_{t}w^{m}
is uniformly bounded in the reflexive Banach space
L
_{2}
(
I
;
L
_{2}
(Ω)) and hence has a subsequence which converges weakly to an element
ξ
∈
V
^{∗}
(EberlinSmulian theorem
[6]
). Thus
Using Fubini theorem, we get
Taking the limit as
n
→ ∞, we obtain
which implies
Therefore, we have ∂
_{t}u
=
ξ
. Using the above discussions and the fact that and hence
and passing with
n
→ ∞ in (14) the proof is complete.
To obtain an error estimate, let us start by introducing the following additional notation
We now take the difference between (8) and (14) we easily obtain the equality for every
φ
∈
V
Take
φ
=
e_{u}
and write the new equation as
I
+
II
+
III
+
IV
=
V
. To begin with, we split the parabolic term into two parts
I
=
I
_{1}
+
I
_{2}
, where
The second term of equation (37) is bounded by
The third and fourth terms of equation (37) are bounded by
In view of (
H1
) and Younge’s inequality we can estimate the right hand by
Collecting all the previous bounds, choosing
ξ
,
ε
, and
κ
sufficiently small and applying Gronwall lemma we obtain (28).
5. A Full discretization scheme
In this section, we use the exponential spline for the solution of discretization of problem (13) which is written in the form (denoting
y
(
x
) ≡
w_{j}
(
x
)):
with boundary conditions
in the time step
t_{j}
,
j
= 1, 2, · · · ,
n
,
w_{j}
_{−1}
is known and we have to solve the two parameter singularly perturbed boundary value problem
associated with the boundary condition (43). Here we use
G
(
x
) and
H
(
x
) to denote (
q
(
x
) + 1/
τ
) and (
f_{j}
+
w_{j−1}
/
τ
) respectively. To this end we consider a uniform mesh Δ with nodal point
x_{i}
on [
a
,
b
] such that Δ :
a
=
x
_{1}
<
x
_{2}
< . . . <
x_{m}
_{−1}
<
x_{m}
=
b
where
Let
y
(
x
) be the exact solution of the problem (44)(43) and
S_{i}
be an approximation to
y_{i}
=
y
(
x_{i}
) obtained by the segment
Q_{i}
(
x
) passing through the points (
x_{i}
,
S_{i}
) and (
x_{i}
_{+1}
,
S_{i}
_{+1}
). Each mixed spline segment
Q_{i}
(
x
) has the form, (for more details see
[15

18]
):
where
a_{i}
,
b_{i}
,
c_{i}
and
d_{i}
are constants and
k
is a free parameter.
Following
[18]
, we get the following relation for
i
= 1, 2, · · · ,
m
− 1,
where:
θ
=
kh
,
α
= (sinh(
θ
)−
θ
)/
θ
^{2}
sinh(
θ
) and
β
= (2
θ
cosh(
θ
)−2 sinh(
θ
))/
θ
^{2}
sinh(
θ
), when
k
→ 0 that
θ
→ 0 then
and the relation defined by (47) reduces to an ordinary cubic spline relation:
at the point
x_{i}
the proposed singularly perturbed problem may discretized by:
where,
G_{i}
=
G
(
x_{i}
),
H_{i}
=
H
(
x_{i}
) and
p_{i}
=
p
(
x_{i}
).
Substituting (49) into (47), we get the following linear equation for every
i
= 1, 2, · · · ,
m
− 1
Equation (50) gives
n
−1 linear algebraic equations in
n
−1 unknowns
S_{i}
,where
A_{i}
=
α p_{i}
_{−1}
−
β p_{i}
−3
α p_{i}
_{+1}
,
D_{i}
= 3
α p_{i}
_{−1}
+
β p_{i}
−
α p_{i}
_{+1}
,
E_{i}
= 4
α
(
p_{i}
_{+1}
−
p_{i}
_{−1}
).
Now we investigate the convergence analysis of the suggested algorithm. The exponential spline solution of (1)(3) is based on the linear equation given by (50) Let
Y
= (
y_{i}
),
S
= (
s_{i}
),
C
= (
c_{i}
),
T
= (
t_{i}
) and
E
= (
e_{i}
) =
Y
−
S
be
m
− 1 dimensional column vectors. It is easily seen that the system of equation (50) gives
n
− 1 linear algebraic equations in the
n
− 1 unknowns
S_{i}
,
i
= 1, 2, · · · ,
m
− 1. Then we can rewrite equation (50) in the standard matrix equation as:
Now the matrix
B
may be written as
where
and
We have the local truncation error are
From Eq. (52), then the row sum of the matrix
B
satisfies: for 
p_{i}
 ≤
P
≥ 0,
for small values of
ε_{d}
and
ε_{c}
then the matrix
B
is irreducible and monotone and it follows that
B
^{−1}
exist thus the problem (51) has a unique solution.
Lemma 5.1
(
[5]
).
For any
0 <
d
< 1,
we have up to a certain order q that it depends on the smoothness of the data
 Shishkin Mesh Strategy
We form the piecewiseuniform grid in such a way that more points are generated in the boundary layer regions than outside of it. We divide the interval
into three subintervals:
where transition parameters are given by
where
the real solutions of characteristic equation
with
n
to be the number of subdivision points of the interval [0,1] and we place
n
/4,
n
/2 and
n
/4 mesh points, respectively, in [0,
γ
_{1}
], [
γ
_{1}
, 1−
γ
_{2}
] and [1−
γ
_{2}
, 1]. Denote the step sizes in each subinterval by
and
respectively.
Where the resulting piecewiseuniform Shishkin mesh may be represented by:
where
n
is the number of discretization points and set of mesh points
with:
Thus, a uniform mesh is placed on each of these subintervals.
Lemma 5.2
(
[5]
).
The solution y
(
x
)
of
(42)(43)
has the representation
Theorem 5.1.
Let S
(
x
)
be the approximate solution of the solution y
(
x
)
of
(42),
then
where C is a constant independent of ε_{d} and ε_{c}
.
Proof
. The estimate is obtained on each subinterval Ω
_{i}
= [0, 1] separately. Let any function
z
on Ω
_{i}
, see
[5]
and so it is obvious that, on Ω
_{i}
Taking maximum norm on both sides, we get
and by appropriate Taylor expansions it is easy to see that
From (58) and Lemma 5.1, on Ω
_{i}
,
Also, using Lemma 5.3 and equation (58) on Ω
_{i}
,
Case 1.
The argument now depends on whether
In this case
ψ
_{1}
≤
C
ln
n
and
ψ
_{2}
≤
C
ln
n
. Then the result follows at once from (58)
Case 2.
When
ln
n
and
ln
n
. Suppose that
i
satisfies
and
Then
and
respectively. Now from (58)
If
i
satisfied
Then
γ
_{1}
≤
x_{i}
and
x_{i}
≤ 1 −
γ
_{2}
or
γ
_{2}
≤ 1 −
x_{i}
and so
Using the above values in (59), we get the required result.
We are now in a position to formulate the main contribution of this work by combining (28) and (59) in the following theorem.
Theorem 5.2.
Let the assumptions (H1)(H2) be satisfied
.
Then u^{α}
→
u in u
∈
H
^{1}
(
I
;
L
_{2}
(Ω))∩
L
_{2}
(
I
;
V
)
for α
→ 0
where u is the variational solution of problem
(1)(3)
and u^{α} is constructed from the solution of problem
(42)(43).
Moreover, we have the error estimates
6. Numerical Results
We present a case study to illustrate the performance of the error estimation procedures of section 5. For purposes of comparison, contrast and performance, two cases with known solutions were chosen.
Case study 1:
Consider the following singularly perturbed parabolic partial differential equation, see
[1]
.
u_{t}
−
ε_{d}u_{xx}
+
u_{x}
=
f
(
x
,
t
),
T
= 1,in the interval [0, 1].
associated with the Dirichlet boundary conditions
u
(0,
t
) = 0,
u
(1,
t
) = 0. and the initial condition
u
_{0}
(
x
) =
e
^{−1/}
^{ε}
+(1−
e
^{−1/}
^{ε}
)
x
−
e
^{−(1−}
^{x}
^{)/}
^{ε}
, where
f
(
x
,
t
) =
e
^{−}
^{t}
(−
c
_{1}
+
c
_{2}
(1 −
x
) +
e
^{−(1−}
^{x}
^{)/}
^{ε}
) . The exact solution is given by
u
(
x
,
t
) =
e
^{−}
^{t}
(
c
_{1}
+
c
_{2}
x
−
e
^{−(1−}
^{x}
^{)/}
^{ε}
), where
c
_{1}
=
e
^{−1/}
^{ε}
,
c
_{2}
= 1 −
e
^{−1/}
^{ε}
.
Case study 2:
Consider the following singularly perturbed parabolic partial differential equation.
u_{t}
−
ε_{d}u_{xx}
+
ε_{c}u_{x}
=
f
(
x
,
t
),
T
= 1,in the interval [0, 1].
associated with the Dirichlet boundary conditions
u
(0,
t
) = 0,
u
(1,
t
) = 0. and the initial condition
where
Maximum pointwise errors for Case study1
Maximum pointwise errors for Case study1
Maximum pointwise errors and order of convergence for Case study1
Maximum pointwise errors and order of convergence for Case study1
The exact solution is given by
where
All the computations were performed by using MATLAB 7.
Comparison of maximum pointwise errors for Case study1
Comparison of maximum pointwise errors for Case study1
Maximum errors for Case study2n= 128 andτ= 1.6/(n+ 1)2
Maximum errors for Case study2 n = 128 and τ = 1.6/(n + 1)^{2}
To compute the experimental rates of convergence
Ord^{n}
for every fixed
ε
, we use the rate of convergence from:
The numerical results displayed in
Table 1

3
clearly indicate that the proposed method based on an exponentialspline function with Shishkin mesh is
ε
uniformly convergent. Numerical solution profiles for case study are given in
figure 1
, respectively, for
n
= 128,
τ
= 0.02and different values of
ε_{d}
= 1, 2
^{−6}
and 2
^{−12}
. For comparison, our results are better than the numerical solution presented in Clavero et al.
[1]
. Numerical solution profiles for case study2 are given in
table 4
and also
figure 2
and
3
for different values of
ε_{d}
and
ε_{c}
.
It is clearly that all solutions are in good harmony with each approximate and exact solution and the convergence obtained is parameter uniform and satisfies the theoretical predictions.
Exact and approximate solutions profiles of the case study1 for n = 128, τ = 0.02 and ε_{d} = 2^{−6}and ε_{d} = 2^{−12} respectively
Exact and approximate solutions profiles of the case study2 for n = 128, 256, τ = 1.6/(n + 1)^{2}, ε_{c} = 2^{−6}, and ε_{d} = 2^{−6}.
Exact and approximate solutions profiles of the case study2 for n = 64, 128, τ = 0.002, ε_{c} = 2^{−6}, and ε_{d} = 2^{−6}.
7. Conclusion
In this paper, we have investigated the application of Rothe’s approximation in time discretization and the exponential spline approximation in the spatial discretization to solve the convection–diffusion problems. The method is shown to be uniformly convergent i.e., independent of mesh parameters and perturbation parameters
ε_{d}
and
ε_{c}
. It has also been seen that the accuracy in the numerical results for the proposed scheme is comparable to that obtained by the uniformly convergent method with a uniform mesh
[1]
. It has been found that the proposed algorithm gives highly accurate numerical results and higher order of convergence than an upwind finite difference scheme. The present method provides a very simple and accurate method for solving convection–diffusion problems.
BIO
W. K. Zahra received M.Sc. and Ph.D. from Faculty of Engineering, Tanta University, Tanta, Egypt. He is currently working as an Assistant Professor at Faculty of Engineering, Tanta university,Tanta, Egypt. His research interests include Numerical Methods for Singularly Perturbed Differential Equations.
Department of Physics and Engineering Mathematics, Faculty of Engineering, Tanta Univ., Tanta, Egypt.
email: waheed zahra@yahoo.com, wzahra@feng.tanta.edu.eg
M. S. ElAzab received M.Sc. from Faculty of Engineering, AlMansurah Univ., Al Mansurah, Egypt and Ph.D. from Comenius University, Slovakia. He is currently working as a Professor at Faculty of Engineering, AlMansurah Univ., Al Mansurah, Egypt. His research interests include Numerical Methods for Singularly Perturbed Differential Equations.
Department of Mathematics, Faculty of Engineering, AlMansurah Univ., Al Mansurah, Egypt.
Ashraf M. El Mhlawy received M.Sc. from Faculty of Engineering, Tanta University, Tanta, Egypt. His research interests include Numerical Methods for Singularly Perturbed Differential Equations.
Department of Physics and Engineering Mathematics, Faculty of Engineering, Tanta Univ., Tanta, Egypt.
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