Uniformly Convergent Numerical Method for Two-parametric ......If µ=0, the problem becomes a...

J. Appl. Comput. Mech., 7(2) (2021) 535-545 DOI: 10.22055/JACM.2020.35193.2596

ISSN: 2383-4536 jacm.scu.ac.ir

Published online: November 21 2020

Uniformly Convergent Numerical Method for Two-parametric

Singularly Perturbed Parabolic Convection-diffusion Problems

Tariku Birabasa Mekonnen1 , Gemechis File Duressa2

1 Department of Mathematics, Wollega University, Nekemte, 395, Ethiopia, Email: [email protected]

2 Department of Mathematics, Jimma University, Jimma, 378, Ethiopia, Email: [email protected]

Received September 28 2020; Revised November 01 2020; Accepted for publication November 01 2020.

Corresponding author: Gemechis File Duressa ([email protected])

© 2020 Published by Shahid Chamran University of Ahvaz

Abstract. This paper deals with the numerical treatment of two-parametric singularly perturbed parabolic convection-diffusion problems. The scheme is developed through the Crank-Nicholson discretization method in the temporal dimension followed by fitting the B-spline collocation method in the spatial direction. The effect of the perturbation parameters on the solution profile of the problem is controlled by fitting a parameter. As a result, it has been observed that the method is a parameter-uniform convergent and its order of convergence is two. This is shown by the boundedness of the solution, its derivatives, and error estimation. The effectiveness of the proposed method is demonstrated by model numerical examples, and more accurate solutions are obtained as compared to previous findings available in the literature.

Keywords: Singularly perturbed, Parabolic convection-diffusion, B-spline collocation, Parameter-uniform.

1. Introduction

We present a numerical technique to solve the following singularly perturbed one-dimensional parabolic convection-diffusion

initial-boundary value problem (IBVP) on the domain = Ω× Ω(0, ], =(0,1)D T .

ε µ ε µ∂ ∂ ∂

≡ − − + = ∈∂ ∂ ∂

2

, 2( , ) ( , ) ( , ), (x,t) D,

u u uL u a x t b x t u f x t

t x x (1)

together with initial and boundary conditions

= ∈Ω( ,0) ( ), ,u x s x x

= = ∈(0, ) 0 (1, ), [0, ],u t u t t T

where ε and µ are two small positive parameters such that ε µ< ≪0 , 1. We assume that the coefficients ( , ), ( , )a x t b x t and ( , )f x t are sufficiently regular functions with the constraints α β≥ > ≥ >( , ) 0, ( , ) 0a x t b x t for all ∈( , )x t D . Moreover, we assume that the given data satisfy appropriate smoothness and suitability conditions on the corner of the domain (0,0) and (0,1) i.e. =(0) (0,0)s u and

=(1) (0,1)s u .These conditions ratify that the problem has a single solution. Additionally, the compatibility condition at the two corners assures that there exist constants 1C and 2C such that for all ∈( , )x t D [1],

− ≤ ≤ −1 2( , ) ( ) and ( , ) (1 )u x t s x C t u x t C x .

From these equations, if ( , )u x t is once differentiable in the time variable t , following the proof in [2] we can have ≤( , )tu x t C . Equation of the type (1) is categorized under singularly perturbed problems and its solution exhibits two boundary layers at the

two endpoints of the spatial domain. It has a multi-scale character, even for smooth data, with swift changes in the layer regions. The two boundary layers have different width relying on the relation between the two parameters ε and µ . In one case when µ ε→2 / 0 as ε→ 0, both have width ε( )O in the neighborhood of = 0x and = 1x . In the other case when ε µ →2/ 0 as µ→ 0 , each has a different width µ( )O and ε µ( / )O at = 0x and = 1x respectively. For more detailed information, one can refer [3]-[8]. Furthermore, the considered problem is a broad view of the extensively investigated convection-diffusion and reaction-diffusion problems [4]. That is in particular if µ = 1, it becomes a parabolic convection-diffusion problem and its solution exhibits a boundary layer of width ε( )O in the neighborhood of either the edge = 0x or = 1x depending on the sign of the convection coefficient ( , )a x t [9]. If µ = 0 , the problem becomes a parabolic reaction-diffusion type and two boundary layers each of

Tariku Birabasa Mekonnen and Gemechis File Duressa, Vol. 7, No. 2, 2021

Journal of Applied and Computational Mechanics, Vol. 7, No. 2, (2021), 535-545

536

width ε( )O appears at both ends = 0x and = 1x [10]. Many physical phenomena occur in nature, in numerous areas of science and engineering, such as modeling of water quality

problems in river networks, Navier-Stokes flows with large Reynolds numbers in the theory of hydrodynamic stability, electrical network, and vibration problems with large Peclet numbers, reaction-diffusion process, quantum mechanics, optimal control theory[10]. Moreover, the heat transfer in turbulent particle-laden channel flow [11], in a circular cylinder, when a drag coefficient (dimensionless parameter) is introduced [12], and problems in fluid mechanics and turbulent drag and heat transfer in high-speed gas flows [13] are some examples to list. The solution of the mentioned problems possess a boundary layer in its domain where it varies rapidly (inner layer), and in some other parts of the domain, it varies slowly (outer layer). Then, the spatial step size has to be refined considerably with respect to the singular perturbation parameter to maintain the rapid change of the solution within the boundary layers. To alleviate this, the advancement of the methods ranges from the ‘strained co-ordinate technique’ to the robust ones [13], [14]. Several fitted numerical methods with piecewise uniform mesh and fitted operator have been developed and compared for the numerical solution to the problems having the boundary layers at one or both ends of the interval.

Different degrees, either polynomial or non-polynomial, B-spline collocation methods have been proposed for the numerical treatment of different singular or regular perturbation problems. The details of these are given in [10], [15]-[21]. These methods were not fitted. But singularly perturbed problems with one parameter have been treated in [2], [22], [23] by using fitted numerical methods. As far as our paramount information is concerned, no scheme has been developed, where the Crank-Nicholson method is applied for temporal discretization followed by the B-spline collocation method for spatial discretization with a fitting factor, to solve the problem given in Eq. (1). In this manuscript, the aforesaid scheme has been developed to treat the solution of such a problem numerically. The boundedness of the solution and its derivatives for the considered problem has been established using the concept of the maximum principle.

The remaining part of the manuscript has been prearranged as follows. In section (2), the detailed explanation of the suggested method has been made and the theoretical order of convergence in the time variable is also shown. Error estimates of the proposed method and the parameter-convergence analysis have been presented in section (3). We have taken some numerical examples and presented their results in tabular and figure forms in section (4). Finally, in section (5), we have drawn conclusions based on discussions of the results.

Lemma 1: (Maximum principle) Let ∈ 2,1( , ) ( ).z x t C D If ≥( , ) 0z x t for all ∈ ∂( , )x t D (boundary of the domain) and ε µ ≥, ( , ) 0,L z x t for all

∈( , )x t D then ≥( , ) 0,z x t for all ∈( , ) .x t D

Proof: Assume to the opposite that there is a point ∈* *( , )x t D such that

∈= <* *

( , )( , ) min ( , ) 0.

x t Dz x t z x t

It is obvious that ∉ ∂* *( , )x t D and implies ∈* *( , ) .x t D Applying the differential operator ε µ,L in Eq. (1) on ( , )z x t at the point * *( , )x t

yields

ε µ ε µ∂ ∂ ∂

≡ − − + =∂ ∂ ∂

2* * * * * * * * * * * * * *

, 2( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ).

z z zL z x t x t a x t x t b x t z x t f x t

t x x

By the partial derivative test we have ∂ ∂ ≥2 2 * */ ( , ) 0,z x x t ∂ ∂ =* */ ( , ) 0,z t x t and ∂ ∂ =* */ ( , ) 0.z x x t Incorporating our assumptions,

β≥ >* *( , ) 0b x t with <* *( , ) 0,z x t we can arrive at ε µ ≤* *, ( , ) 0.L z x t This contradicts the given statement. Therefore, it can be

concluded that the minimum of ( , )z x t is non-negative.

Lemma 2: (Bounds of the solution) Let ( , )u x t be the solution of Eq. (1), then we have the estimation

β−≤ +1 max ( ) ,u f s x

where . is the maximum norm.

Proof: Let ±( , )z x t be two barrier functions defined by

β± −= + ±1( , ) max ( ) ( , ).z x t f s x u x t

Now evaluating the values of these barrier functions on the boundary of the domain ∂ ,D we get

β β± − −= + ± = + ± ≥1 1( ,0) max ( ) ( ,0) max ( ) ( ) 0,z x f s x u x f s x s x

β β± − −= + ± = + ≥1 1(0, ) max (0) (0, ) max (0) 0,z t f s u t f s

β β± − −= + ± = + ≥1 1(1, ) max (1) (1, ) max (1) 0.z t f s u t f s

Applying the differential operator in Eq. (1) on ±( , )z x t yields

ε µ ε µ± ± ±

± ±∂ ∂ ∂= − − +

∂ ∂ ∂

2

, 2

( , ) ( , ) ( , )( , ) ( , ) ( , ) ( , ),

z x t z x t z x tL z x t a x t b x t z x t

t x x

β ε β

µ β β

− −

− −

∂ ∂ = + ± − + ± − ∂ ∂

∂ + ± + + ± ∂

21 1

2

1 1

max ( ) ( , ) max ( ) ( , )

( , ) max ( ) ( , ) ( , ) max ( ) ( , ) ,

f s x u x t f s x u x tt x

a x t f s x u x t b x t f s x u x tx

( ) ε µβ−= + ± ±1,( , ) max ( ) ( , ) ( , ),b x t f s x u x t L u x t

( )β−= + ± ±1( , ) max ( ) ( , ) ( , ).b x t f s x u x t f x t

Uniformly convergent numerical method for two-parametric singularly perturbed parabolic convection-diffusion problems


537

From the constraints on the coefficients, we have β≥ >( , ) 0b x t which shows β− >1( , ) 1,b x t and the fact ≥ ( , )f f x t implies

that ε µ

± ≥, ( , ) 0L z x t . Hence, the maximum principle given by Lemma (1) confirms ± ≥( , ) 0z x t for all ∈( , )x t D . This immediately

completes the proof.

2. Description of the Method

2.1 Time variable discretization

Let k be a step size in the time direction and M be the number of subintervals of [0, ]T with equal length. This gives a time mesh

= = − = + =( 1) , 1,2,..., 1, .Mt m

TD t m k m M k

M

We discretize Eq.(1) by the Crank-Nicholson method to obtain

ε µ− − − −

− − −

− + +− − +

+ +=

1 1 1 1

1 1 1

( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

2 2( , ) ( , ) ( , ) ( , ) ( , ) ( , )

,2 2

m m xx m xx m m x m m x m

m m m m m m

u x t u x t u x t u x t a x t u x t a x t u x t

kb x t u x t b x t u x t f x t f x t

(2)

subject to the conditions

= ∀ ∈Ω( ,0) ( ), u x s x x and = = = +(0, ) 0, (1, ) 0, 1,2,..., 1,m mu t u t m M

which is a system of linear equation in space at each time. Now, simplifying and collecting the ( )thm and −( 1)thm time levels we

obtain

ε µ ε µ− − − − − −

− + + + = + − − + + 1 1 1 1 1 1

2 2( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ).xx m m x m m m xx m m x m m m m mu x t a x t u x t b x t u x t u x t a x t u x t b x t u x t f x t f x t

k k

This can be written as a differential equation in the space variable x ,

=( ,0) ( ),u x s x

ε µ ε µ≡− + + =ɶ ɶ ɶ, ( ) ( ) ( ) ( ) ( ) ( ),M

xx xL u u x a x u x q x u x g x

= =ɶ ɶ(0) 0 (1).u u

(3)

where ( )ε µ− − − − − −= + − − + +1 1 1 1 1 1( ) ( , ) ( , ) ( , ) ( , ) 2 / ( , ) ( , ) ( , ),xx m m x m m m m mg x u x t a x t u x t b x t k u x t f x t f x t ( )= +( ) ( , ) 2 / ,mq x b x t k =( ) ( , ),ma x a x t and

=ɶ( ) ( , ).mu x u x t

Lemma 3: (Discrete maximum principle) Let ɶ( )z x be a smooth solution of the discrete Eq. (3) such that ≥ɶ(0) 0,z ≥ɶ (1) 0 z and

ε µ ≥ɶ, ( ) 0ML z x for all ∈Ω,x then ≥ɶ( ) 0z x for all ∈Ω.x

Proof: Again to use proof by contradiction, suppose there is ∈ ∈Ω* : x x x such that∈Ω

= <ɶ ɶ*( ) min ( ) 0.x

z x z x This implies ∉ 0,1.x

From elementary calculus we have =ɶ *( ) 0xz x and ≥ɶ *( ) 0.xxz x Furthermore, from the assumption of the coefficient term we

have = +* *( ) ( ( , ) 2 / ).mq x b x t k Then ε µ ε µ= − + + ≤ɶ ɶ ɶ ɶ* * * * *, ( ) ( ) ( ) ( ) ( ) ( ) 0.M

xx xL z x z x a x z x q x z x This contradicts the original statement ε µ ≥ɶ, ( ) 0ML z x

for all ∈Ω.x So our supposition is wrong and it follows ≥ɶ( ) 0z x for all ∈Ω.x

Lemma 4: (Bounds on the derivative) [24], [25] For a fixed number < <0 1p and a certain order δ, the solution ɶ( )u x of Eq. (3)

satisfies the following derivative bound

( )ηηη η δ− −−∂≤ + + ≤ ≤

∂

ɶ10 (1 )

0 11 , for 0 ,ii

ip xp xi i

i

uC e e i

x

where C is a constant independent of ε µ, .

Lemma 5: (Semi-discrete error estimate) The error estimate in the temporal direction is given by

≤ 21 ( ).TE C k (4)

Proof: Now, we can express

− + −= = +1 /2 1 /2 1 /2( , ) ( , ) ( , 1 / 2),m m mu x t u x t u x t (5)

and

− − − −= = −1 1 /2 1 /2 1 /2( , ) ( , ) ( , 1 / 2).m m mu x t u x t u x t (6)

Then Eq. (2) becomes

ε µ− − − −

− − −

− + += + −

+ ++

1 1 1 1

1 1 1

( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

2 2( , ) ( , ) ( , ) ( , ) ( , ) ( , )

.2 2

m m xx m xx m m x m m x m

m m m m m m

u x t u x t u x t u x t a x t u x t a x t u x t

kb x t u x t b x t u x t f x t f x t

(7)

From Taylor's series expansion about the point −1 /2( , ),mx t we have



538

− − − − −

∂ ∂ ∂ ∂= + + + + +

∂ ∂ ∂ ∂

2 3 42 3 4

1 /2 1 /2 1 /2 1 /2 1/22 3 4

( / 2) ( / 2) ( / 2) ( / 2)( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ...

1! 2! 3! 4!m m m m m m

k k k ku u u uu x t u x t x t x t x t x t

x x x x (8)

− − − − − −

∂ ∂ ∂ ∂= − + − + −

∂ ∂ ∂ ∂

2 3 42 3 4

1 1 /2 1/2 1 /2 1 /2 1 /22 3 4

( / 2) ( / 2) ( / 2) ( / 2)( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ...

1! 2! 3! 4!m m m m m m

k k k ku u u uu x t u x t x t x t x t x t

x x x x (9)

Subtracting Eq. (9) from Eq. (8) gives

τ−−

−∂= +

∂1

11/2

( , ) ( , )( , ) ,m m

m

u x t u x tux t

t k (10)

where

τ −

∂= − ≡

∂

2 32

1 1 /23( , ) ( ).

24 m

k ux t O k

x

Under Lemma (3), we have shown that the semi-discrete solution satisfies the semi-discrete maximum principle. This concept and the result obtained in Eq. (10) indicate that the error estimate of the time semi-discretization is

≤ 21 ( ),TE C k

where ∈

= −∂ ∂ ≡3 3 2

, [0,1]max / ( , ) ( )x t

C u x x t O k is a positive constant independent of the perturbation parameters ε , µ and the time mesh

size .k That is the rate of convergence in the temporal direction is two.

2.2 Spatial variable discretization

Let h be a uniform spacing in the spatial direction and N be the number of subintervals of [0,1]. This gives a space mesh

+= = − = + = = −1

1( 1) , 1,2,..., 1, .N

x n n nD x n h n N h x xN

To find the approximate solution of the boundary value problem (3), consider the thr order forward difference operator of a given

function ( )F x at 1x which is defined by the recurrence relation

∆ = −1 2 1( ) ( ) ( ),F x F x F x

− −∆ = ∆ −∆1 11 2 1( ) ( ) ( ).r r rF x F x F x

From this recurrence relation, it is immediate that the fourth-order forward difference operator on ( )F x at −2nx is given by

− + + − −∆ = − + − +42 2 1 1 2( ) ( ) 4 ( ) 6 ( ) 4 ( ) ( ).n n n n n nF x F x F x F x F x F x

In analogous procedures as in [26], to derive a third-degree B-spline, we compute

−∆=

42

3

( )( ) ,x n

n

F xB x

h

where − −

− − +

− ≤= − =

2 232 2

( ), when ( ) ( ) .

0, otherwisen n

x n n

x x x xF x x x

From the definition of − +− 32( )nx x , it is straight forward that ≡( ) 0nB x for all +≥ 2.nx x Moreover, any fourth order difference

operator annihilates polynomial of degree three and less with evenly spaced knots when − ≥2 .nx x Then, we can obtain

+ + − − −

+ + −

+ +

− − − + − − − ∈

− − − + − ∈

− − −=

3 3 3 32 1 1 2 1

3 3 32 1 1

3 32 13

( ) 4( ) 6( ) 4( ) , [ , ),

( ) 4( ) 6( ) , [ , ),1

( ) 4( ) ( )

n n n n n n

n n n n n

n nn

x x x x x x x x x x x

x x x x x x x x x

x x x xB xh

+

+ + +

∈

− ∈

1

32 1 2

, [ , ),

( ) , [ , ),

0 , ot

n n

n n n

x x x

x x x x x

herwise.

Using the relations = +1nx x nh and + = +nn jx x jh like + +− = + −2 1n nx x x h x , + +− = − +2 1(( ) ),n nx x x x h

+ − −− = + − = − −2 1 13 (3 ( )),n n nx x x h x h x x and doing a bit of arithmetic manipulation we arrive at the third-degree B-splines (basis

functions) defined over the interval [0,1] by the following relation [18]:

− − −

− − − −

+ + + +

+

− ∈

+ − + − − − ∈

+ − + − − − ∈=

−

32 2 1

3 2 2 31 1 1 1

3 2 2 31 1 1 13

2

( ) , [ , ),

3 ( ) 3 ( ) 3( ) , [ , ),1

3 ( ) 3 ( ) 3( ) , [ , ),( )

(

n n n

n n n n n

n n n n nn

n

x x x x x

h h x x h x x x x x x x

h h x x h x x x x x x xB xh

x + +

∈

31 2) , [ , ),

0, otherwise.n nx x x x

(11)



539

Table 1. Values of the cubic B-spline basis functions at the nodal points

x −2nx −1n

x nx +1n

x +2n

x

( )n

B x 0 1 4 1 0

' ( )n

B x 0 3 / h 0 −3 / h 0

'' ( )n

B x 0 26 / h −212 / h

26 / h 0

At the nodal points − − + +2 1 1 2, , , and n n n n nx x x x x we can find the values of '' ( ), ' ( ) and ( )n n nB x B x B x from Eq. (11) as shown in Table (1).

Thus define an approximate solution ( )U x to the exact solution ɶ( )u x of the boundary value problem (3) in terms of B-splines as

+

=

= ∑2

0

( ) ( )N

n nn

U x E B x (12)

where 'snE are unknown time dependent parameters to be determined from the boundary conditions and collocation of the

differential equation. The set of cubic B-splines, π +=3 0 1 2 2( ) , , ,..., ,NB B B B B are linearly independent and span an +( 3)N -

dimensional subspace Ω( )NX [26]. To solve the IBVP (1), the spline functions are evaluated at nodal points ( , ).n mx t

At = ,nx x undertaking the notation =( , ) ,mn m nU x t U the total discrete scheme from Eq. (3) can be obtained as

ε µ− − + = = + = +( ) ( ) , 1,2,..., 1, 2,3,..., 1,m m m m m mxx n n x n n n nU a U q U g n N m M

for

+= = + = = = +11 1( ), 1,2,..., 1 and 0 , 1,2,..., 1.m m

n n NU s x n N U U m M

Now introducing a fitting factor σ we obtain

ε µ εσ µ≡− − + = = + = +,, ( ) ( ) , 1,2,..., 1, 2,3,..., 1.N M m m m m m m m

n xx n n x n n n nL U U a U q U g n N m M (13)

From the cubic B-spline and the values in Table (1), we have

( )

( )

− +

+ −

− +

= + +

= −

= − +

1 1

1 1

1 12

4 ,

3( ) ,

6( ) 2 .

m m m mn n n n

m m mx n n n

m m m mxx n n n n

U E E E

U E Eh

U E E Eh

(14)

In the corresponding time level, substituting Eq. (14) into Eq. (13) and simplifying we arrive at

− + − − − + −− + − ++ + = + + + = + = +0 1 0 1 1

1 1 1 1 , for 1,2,..., 1, 2,3,..., 1,m m m m m m mn n n n n n n n n n n n nA E A E A E B E B E B E F n N m M (15)

where

( ) ( ) ( )

( ) ( ) ( )

( )

εσ µ εσ εσ µ

εσ µ εσ εσ µ

− +

− − − − + − −

−

= − + + + = + + =− − + +

= − − − = − − − = + − −

+

2 0 2 2

1 2 1 0 2 1 1 2 1

2 1

6 3 2 / , 12 4 2 / , 6 3 2 / ,

6 3 2 / , 12 4 2 / , 6 3 2 / ,

= .

m m m m mn n n n n n n n

m m m m mn n n n n n n n

m m mn n n

A h a h b k A h b k A h a h b k

B h a h b k B h b k B h a h b k

F h f f

At each ( )thm time level, this gives +( 1)N equations with +( 3)N unknowns +0 1 2 2( , , ,..., ).m m m mNE E E E So, for = 1n and = + 1n N

eliminating 0mE and +2

mNE respectively from Eqs. (14) and (15), we get

−= +1 ,m mAE BE F (16)

where

− + −

− +

− +

+−−− −

++−

− + ++ + + +

− − = = − −

⋯

⋯

⋯

⋯ ⋮⋮ ⋮ ⋮ ⋮⋮

⋯

⋯

⋯

01 1 1 1

02 2 2

103 3 3

2

011 1

10

01 1 1 1

4 0 0 0

0 0

0 0

,

00

0 0

40 0 0

m

mm

NN N mN

N NN

N N N N

A A A A

A A AE

A A AE

A E

AA AE

A AA

A A A A

−

−−

−+

=

⋮

11

121

11

, ,

m

mm

mN

E

EE

E

− + −

−− +

− +

+− −−− −

+−

+

− + ++ + + +

− − + + = = + + − −

⋯

⋯

⋯

⋮⋯ ⋮⋮ ⋮ ⋮ ⋮

⋯

⋯

⋯

01 1 1 1

101 12 2 2

02 23 3 3

2

0 111 1

0

10

1 1 1 1

4 0 0 0

0 0

0 0

B ,

00

0 0

40 0 0

m m

m m

m mNN N N N

mN NN N

N N N N

B B B B

f fB B B

f fB B B

F h

BB B f f

B BB f

B B B B

− − −

+ − +−+ + + ++

− + + − +

⋮

11 1 1 1

111 1 1 11

0

,

0

m m

m mmN N N NN

U A U B

U A U Bf



540

which is a diagonally dominant and nonsingular collocation matrix as ( )− +− + >0 0n n nA A A and ( )− +− + >0 0.n n nB B B

Consequently, we get a uniquely solvable system of equations [26]. But to get the initial vector i.e. for = 1,m we use the initial

condition as

− +

+ +

+ = +

+ + = =

+ = −

1 1 11 2 1

1 1 1 11 1

1 1 11 1

4 2 '(0),3

4 , 2,3,..., ,

2 4 '(1).3

n n n n

N N N

hE E U s

E E E U n N

hE E U s

(17)

The fitting factor, which is a vital constituent, controls the large gradients in the solution over layer regions of the domain. It is given by [2] and [3] as

µ ρ µ ρσ

= coth ,

2 2

m mm n nn

a a (18)

where ρ ε= / .h

3. Convergence Analysis

First, let us see the following lemma which shows the property of the set of cubic B-spline π3( ).B

Lemma 6 The set of 'snB defined in Eq. (11) makes the following inequality true,

+ +

= =

≤ ≤ ∈∑ ∑2 2

0 0

( ) ( ) 10, [0,1].N N

n nn n

B x B x x

Proof: For any node ,nx from Table (1), we get

+

− +

=

= + + ≤ + + =∑2

1 10

( ) ( ) ( ) ( ) 1 4 1 6.N

n n n nn

B x B x B x B x

and then for −∈ 1[ , ],n nx x x we have − − +≤ ≤ ≤ ≤2 1 1( ) 1, ( ) 4, ( ) 4, ( ) 1.n n n nB x B x B x B x This implies for any −∈ 1[ , ],n nx x x +

=

≤∑2

0

( ) 10.N

nn

B x

Theorem 1: (Error estimate of the collocation) Let ( )U x be the collocation approximation in Ω( )NX to the solution ɶ( )u x of Eq.

(3). If ( )g x is twice continuously differentiable, then

∞− ≤ɶ 2( ) ( ) ,u x U x Ch

for h is sufficiently small and C is a positive constant independent of ε µ, .

Proof: Let ( )W x be a unique spline interpolate from Ω( )NX to the solution ɶ( )u x of the boundary value problem (3). Then it can be

expressed as

+

=

= ∑ ɶ2

0

( ) ( ) ( ),N

n nn

W x E x B x (19)

Adopting the De Boor and Hall error estimates [26] for this case, we have

∞ ∞

∞ ∞

∞ ∞

− ≤

− ≤

− ≤

ɶ ɶ

ɶ ɶ

ɶ ɶ

(4) 40

(4) 31

(4) 20

( ) ( ) ,

'( ) '( ) ,

''( ) ''( ) .

u x W x C u h

u x W x C u h

u x W x C u h

(20)

But using triangle inequality, we have

∞ ∞ ∞− ≤ − + −ɶ ɶ( ) ( ) ( ) ( ) ( ) ( ) .u x U x u x W x W x U x (21)

The first term ∞

−ɶ( ) ( )u x W x in Eq. (21) is given in Eq. (20).

To show the remaining part, we use the collocating conditions ε µ ε µ= =ɶ,, ,( ) ( ) ( ).N M M

n n nL U x L u x g x Also let ε µ = = +ɶ, ( ) ( ), 1,2,..., 1M

n nL W x g x n N

and it satisfies the boundary conditions +=1 1( ) ( ).NW x W x Then,

ε µ ε µ ε µ ε µ ε σ µ− = − = − − − − + −ɶ ɶ ɶ ɶ, , ,, , , ,( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )N M N M M N M

n n n n xx xx x xL U x L W x L u x L W x u W a x u W q x u W

ε σ εσ εσ µ

ε σ εσ µ

= − − + − − − + −

= − − − − − − + −

ɶ ɶ ɶ ɶ ɶ

ɶ ɶ ɶ ɶ

( ) ( )( ) ( )( ) ,

( 1) ( ) ( )( ) ( )( ) ,

xx xx xx xx x x

xx xx xx x x

u W u u a x u W q x u W

u u W a x u W q x u W

ε σ ε σ µ∞ ∞ ∞ ∞∞ ∞

≤ − + − + − + −ɶ ɶ ɶ ɶ1 ( ) ( ) .xx xx xx x xu u W a x u W q x u W



541

From the power series expansion of hyperbolic cotangent function we have + − +≃2 4coth 1 / 3 / 45 ...x x x x and then

− ≤ 2coth 1 .x x Cx This and Eq. (18) indicate σ− ≤ 21 .Ch Lemma (4) gives bounds on the derivatives of ɶ( )u x and using Eq. (20) we

obtain

ε µ ε µ∈Ω

− ≤, , 2, ,max ( ) ( ) .N M N M

xL U x L W x Ch

From Eq. (16), ε µ ε µ−, ,, ,( ) ( )N M N ML U x L W x yields

− −− = − + −ɶ ɶ ɶ1 1( ) ( ) ( ).m m m mA E E B E E F F

At each ( )thm time level since the matrices are invertible, and the initial and boundary values are bounded, so is

∞− ≤ɶ 2.m mE E Ch Equations (12) and (19) lead us to

+

∞ ∞=

− = − ≤∑ɶ2

2

0

( ) ( ) ( ) .N

m mn

n

W x U x E E B x Ch

This completes the required proof, showing that the theoretical rate of convergence in the spatial discretization is also two. We can see that the solution and its derivatives are bounded, and errors can be estimated. This tells us that the solution obtained from the developed scheme is stable. Now it is immediate from Lemma (5) and Theorem (1) to have the following theorem which shows parameter-uniform convergence.

Theorem 2: (Parameter-uniform convergence) Let ( , )n mx t be any point in the mesh grid ×N Mx tD D and ( , )n mu x t be the solution of

problem (1) at each grid point. If ( , )n mU x t is an approximate solution obtained by the collocation scheme in Eq. (16), then the error

estimate of the total discrete scheme is given by

∞− ≤ +2 2( , ) ( , ) ( ).n m n mu x t U x t C h k

Then, the order of convergence of the scheme obtained by the current method is two both in spatial and temporal directions. Furthermore, the convergence is independent of the parameter. We have measured the error in the maximum norm because we need it in a very small domain in which the boundary layer exists. To elaborate the theoretically obtained results, we take some examples.

4. Numerical Examples and Results

In this section, we have demonstrated the accuracy, maximum absolute errors, parameter uniform convergence, and the corresponding order of convergence of the proposed scheme with the help of MATLAB software to verify the theoretical estimates. Numerical experiments have been carried out using the following two examples. Example 1.1: As a first example, we have considered the problem in [4]

ε µ∂ ∂ ∂

− − + + = − −∂ ∂ ∂

22 2

2(1 ) 16 (1 ),

u u ux u x x

t x x

subject to = = =( ,0) 0, (0, ) 0 (1, ).u x u t u t

Example 1.2: The second example we have used to validate our theoretical is given in [25]

ε µ∂ ∂ ∂

− − + − + + + = − −∂ ∂ ∂

22 2 2

2(1 ) (1 5 ) ( )( 1),tu u u

x x t xt u x x et x x


Example 1.3: At last, one specific case of our problem is taken from [9]

πε π π

∂ ∂ ∂ − + + + + + + = − − ∂ ∂ ∂

22 2 3 3

2

1 11 sin( ) 1 sin( ) (1 ) (1 )sin( ),

2 2 2

u u ux x x t u x x t t t

t x x


We define the absolute maximum errors using the double mesh principle, as it is thought to get the analytical solution of the given examples, using the formula

ε µ≤ ≤ + ≤ ≤ +

= −, 2, 2

1 1,1 1max ,N M M M

N Nn N m M

E U U

where M and N are the number of mesh points in the t and x directions with k and h step sizes respectively. MNU is the approximate

solution computed using M and N number of meshes and 22

MNU is approximate solution calculated using a double number of

meshes 2M and 2N by bisecting the step sizes. As well, the corresponding rate of convergence is defined by

ε µ ε µ−=

, 2 ,2, ,log( ) log( )

.log(2)

N M N ME ER



542

Table 2. The maximum point-wise error and rate of convergence for Example 1.1 where ε −=

82 and different values of µ .

µ ↓ N=32 M=6

N=64 M=32

N=128 M=64

N=256 M=128

N=512 M=256

Current method −82 2.3129e-4 5.9651e-5 1.4905e-5 3.7393e-6 9.3673e-7

−102 2.2137e-4 5.7579e-5 1.4387e-5 3.6043e-6 9.0318e-7

−122 2.1907e-4 5.7105e-5 1.4269e-5 3.5746e-6 8.9550e-7

−142 2.1850e-4 5.6989e-5 1.4240e-5 3.5673e-6 8.9362e-7

−162 2.1836e-4 5.6960e-5 1.4233e-5 3.5655e-6 8.9315e-7

−182 2.1833e-4 5.6953e-5 1.4231e-5 3.5650e-6 8.9304e-7

−202 2.1832e-4 5.6951e-5 1.4230e-5 3.5649e-6 8.9301e-7

−222 2.1831e-4 5.6951e-5 1.4230e-5 3.5649e-6 8.9300e-7

−242 2.1831e-4 5.6951e-5 1.4230e-5 3.5649e-6 8.9300e-7

R 1.9386 2.0008 1.9970 1.9971 1.9986

Method in [4] −82 7.889e-03 3.922e-03 1.956e-03 9.768e-04 4.881e-04

−102 7.877e-03 3.916e-03 1.953e-03 9.752e-04 4.873e-04

−122 7.873e-03 3.915e-03 1.952e-03 9.748e-04 4.871e-04

−142 7.876e-03 3.914e-03 1.952e-03 9.747e-04 4.870e-04

−162 7.872e-03 3.914e-03 1.952e-03 9.747e-04 4.870e-04

−182 7.872e-03 3.914e-03 1.952e-03 9.746e-04 4.870e-04

−202 7.872e-03 3.914e-03 1.952e-03 9.746e-04 4.870e-04

−222 7.872e-03 3.914e-03 1.952e-03 9.746e-04 4.870e-04

−242 7.872e-03 3.914e-03 1.952e-03 9.746e-04 4.870e-04

R 1.0082 1.0039 1.0017 1.0009 0.9288

Table 3. The maximum point-wise error and rate of convergence for Example 1.2 where µ −=

710 and different values of ε .

ε ↓ N=64 M=16

N=128 M=32

N=256 M=64

N=512 M=128

Current method

−610 2.6377e-05 6.5877e-06 1.6474e-06 4.1182e-07

−710 2.6377e-05 6.5877e-06 1.6474e-06 4.1182e-07

−810 2.6377e-05 6.5877e-06 1.6474e-06 4.1182e-07

−910 2.6377e-05 6.5877e-06 1.6474e-06 4.1182e-07

R 2.0014 1.9996 2.0001 1.9999

Result in [7]

−610 3.8754e-5 1.0214e-5 2.6170e-6 6.6241e-7

−710 3.8753e-5 1.0214e-5 2.6170e-6 6.6241e-7

−810 3.8753e-5 1.0214e-5 2.6170e-6 6.6241e-7

−910 3.8753e-5 1.0214e-5 2.5461e-6 6.6241e-7

R 1.9238 1.9646 1.9821 1.9910

Result in [25]

−610 9.6949e-4 4.9906e-4 2.5231e-4 1.2824e-4

−710 9.8712e-4 5.0049e-4 2.5485e-4 1.2853e-4

−810 9.5128e-4 5.0026e-4 2.5237e-4 1.2781e-4

−910 9.6746e-4 5.0012e-4 2.5461e-4 1.2804e-4

R 0.95193 0.97394 0.99186 0.99188

5. Discussions and Conclusions

We have designed a numerical method to solve a one-dimensional two-parameter singularly perturbed parabolic convection-diffusion problems. This method comprises the Crank-Nicholson method for the temporal discretization and B-spline collocation method for the spatial discretization with a fitted operator for the system of ordinary differential equations yielded as a result of the time discretization. The theoretical assertions are attested by taking some purely hypothetical examples which are chosen to fulfill all the necessities of the corresponding model problem (1). The results of these examples, which are presented in tables and figures, have been discussed as follows.

For Example 1.1, Table (2) displays the absolute maximum point-wise errors and the corresponding rates of convergence for the numerical solution computed with the proposed cubic B-spline collocation method. We can observe that the method is parameter-uniform with better results than in [4] and the rate of convergence is two. The physical behavior of its solution is demonstrated by Fig. (1) indicating that it has a dual boundary layer. Apart from the mathematical concepts, as Navier-Stokes predicted, such figures can represent the steady-state incompressible laminar flow (simpler case) over a flat plate indicating uniform velocity above the plate and zero velocity at the surface of the plate. This non-uniform velocity profile exhibits a thin boundary layer close to the surface of the plate. It is also true for complicated fluid flows (turbulence, compressible, inside circular or cylindrical material, etc.).



543

Table 4. The maximum point-wise error for Example 1.3 with and without fitting factor where µ = 1 and different values of ε.

ε ↓ N=32 M=16

N=64 M=32

N=128 M=64

N=256 M=128

N=512 M=256

Current method without using fitting factor 010 5.7915e-7 1.4199e-7 3.5398e-8 8.8434e-9 2.2105e-9

−110 4.9971e-6 1.2128e-6 3.0142e-7 7.5292e-8 1.8815e-8

−210 3.1154e-4 1.3585e-4 3.9150e-5 8.5034e-6 1.9950e-6

−310 9.6238e-4 8.3741e-4 6.3948e-4 3.9681e-4 1.8384e-4

−410 1.1093e-3 1.1007e-3 1.0705e-3 1.0082e-3 8.9518e-4

−510 1.1256e-3 1.1329e-3 1.1334e-3 1.1283e-3 1.1151e-3

Current method using fitting factor 010 5.9148e-7 1.4804e-7 3.6913e-8 9.2256e-9 2.3063e-9

−110 3.7660e-6 9.4219e-7 2.3590e-7 5.8994e-8 1.4750e-8

−210 1.7243e-5 5.9951e-6 1.6882e-6 4.3637e-7 1.1003e-7

−310 1.8817e-5 9.1973e-6 4.5437e-6 2.1630e-6 8.3111e-7

−410 1.8817e-5 9.1973e-6 4.5465e-6 2.1630e-6 1.1272e-6

−510 1.8817e-5 9.1973e-6 4.5465e-6 2.1630e-6 1.1272e-6

Fig. 1. Numerical solution profile for the test Example 1.1 for

= =92 ,N M ε

−=

242 and µ −=

82 .

Fig. 2. Numerical solution profile for the test Example 1.2 for

= =9 82 , 2 ,N M ε

−=

910 and µ −=

710 .

Fig. 3. Numerical solution profile for the test Example 1.3 without using

fitting factor for = =9 82 , 2 ,N M ε

−=

310 .

Fig. 4. Numerical solution profile for the test Example 1.3 using fitting

factor for = =9 82 , 2 ,N M ε

−=

310 .

One can also note from Table (3) and Fig. (2) that the numerical solution of Example 1.2 obtained by our method confirms the theoretical results and gives more accurate results than in [7] and [25]. Example 1.3 is a particular example of the problem under investigation, usually known as singularly perturbed parabolic convection-diffusion with one-parameter. As displayed in Table (4),



544

when the value of the singular perturbation parameter µ goes to zero, the absolute maximum point-wise error goes constant if

the fitting factor is used, and varies if not. Furthermore, when the fitting factor is not used, the solution profile demonstrated by Fig. (3) shows a rapid change (or oscillation) at the right end of the spatial domain where the boundary layer occurs. On the other hand, we observe from Fig. (4) that there is no such oscillation when the fitting factor is used. This tells us the fitting factor controls the effect of the singularity due to the singular perturbation parameter in the solution.

In general, the present method is shown to give a convergent solution independent of mesh parameters and perturbation parameters. The proposed method gives more accurate results than some previous findings in the literature.

Author Contributions

Conceptualization: Tariku Birabasa Mekonnen & Gemechis File Duressa; Investigation and formal analysis: Tariku Birabasa Mekonnen & Gemechis File Duressa; Software programming: Tariku Birabasa Mekonnen; Visualization: Tariku Birabasa Mekonnen & Gemechis File Duressa; Writing- original draft: Tariku Birabasa Mekonnen; Writing- review & editing: Gemechis File Duressa. Both the authors discussed the results, reviewed, and approved the final version of the manuscript.

Acknowledgments

The authors would like to thank the editor and reviewers for their constructive comments.

Conflict of Interest

The authors of this manuscript, would like to declare that there is no known conflicts of interest with respect to the research, authorship, and publication.

Funding

The authors received no financial support for the research, authorship, and publication of this article.

References

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ORCID iD

Tariku Birabasa Mekonnen https://orcid.org/0000-0002-9963-6775



545

Gemechis File Duressa https://orcid.org/0000-0003-1889-4690

© 2020 by the authors. Licensee SCU, Ahvaz, Iran. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0 license) (http://creativecommons.org/licenses/by-nc/4.0/).

How to cite this article: Mekonnen T.B., Duressa G.F. Uniformly Convergent Numerical Method for Two-parametric Singularly Perturbed Parabolic Convection-diffusion Problems, J. Appl. Comput. Mech., 7(2), 2021, 535–545. https://doi.org/10.22055/JACM.2020.35193.2596

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