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An efficient hybrid numerical schemefor convection-dominated boundary-value problemsRajesh K. Bawa a & Srinivasan Natesan ba Department of Computer Science, Punjabi University, Patiala,Indiab Department of Mathematics, Indian Institute of TechnologyGuwahati, Guwahati, IndiaVersion of record first published: 05 Dec 2008.

To cite this article: Rajesh K. Bawa & Srinivasan Natesan (2009): An efficient hybrid numericalscheme for convection-dominated boundary-value problems, International Journal of ComputerMathematics, 86:2, 261-273

To link to this article: http://dx.doi.org/10.1080/00207160801955678

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International Journal of Computer MathematicsVol. 86, No. 2, February 2009, 261–273

An efficient hybrid numerical scheme for convection-dominatedboundary-value problems

Rajesh K. Bawaa and Srinivasan Natesanb*

aDepartment of Computer Science, Punjabi University, Patiala, India; bDepartment of Mathematics,Indian Institute of Technology Guwahati, Guwahati, India

(Received 10 September 2007; revised version received 05 January 2008; second revision received 21 January 2008;accepted 27 January 2008 )

This article presents a numerical scheme for convection-dominated two-point boundary-value problems.The proposed scheme combines the cubic spline scheme and the midpoint scheme in an appropriatemanner. In the inner region, the convective term is approximated by three-point differences by splineapproximation of solution at the mesh points, whereas in the outer region the midpoint approximations areused for convective term, and the classical central difference scheme is used for the diffusive term. Thefirst-order derivative in the left boundary point is approximated by the cubic spline. This scheme is appliedon the boundary layer resolving Shishkin mesh. Truncation error is derived, and the proposed method isapplied to couple of examples to show its accuracy and efficiency.

Keywords: singular perturbation problems; numerical solution; cubic spline; midpoint scheme; piece-wiseuniform mesh

1. Introduction

In this article, we consider the following convection-dominated two-point boundary-valueproblem (BVP):

Lu(x) ≡ εu′′(x) + a(x)u′(x) − b(x)u(x) = f (x), x ∈ � = (0, 1), (1)

B0u(0) ≡ b(0)u(0) − a(0)u′(0) = −f (0), B1u(1) ≡ u(1) = β, (2)

where 0 < ε � 1 is the perturbation parameter, a, b and f are sufficiently smooth functions suchthat a(x) ≥ a > 0, b(x) ≥ b ≥ 0, x ∈ � = [0, 1]. Under these assumptions, the BVP (1) and (2)has a unique solution u(x) ∈ C2(�)

⋂C1(�) exhibiting a weak boundary layer at x = 0 (see, for

example, [1]).Convection-dominated BVPs often arise in fluid dynamics, where the convection coefficient

(velocity field) dominates the diffusion coefficient (viscosity term); these problems are alsoknown as singular perturbation problems (SPPs). Navier–Stokes equations are the well-known

*Corresponding author. Email: natesan@iitg.ernet.in

ISSN 0020-7160 print/ISSN 1029-0265 online© 2009 Taylor & FrancisDOI: 10.1080/00207160801955678http://www.informaworld.com

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262 R.K. Bawa and S. Natesan

example for SPPs, which model the flow of low viscous fluids. In addition, these types ofproblems occur in several other applied areas such as chemical reactions, quantum mechanics, etc.In general, the solution of SPPs exhibits boundary or interior layers, and the solution has steepgradients in the boundary layers and behaves smoothly outside the layer regions. This multiscalebehaviour creates difficulties in solving these problems numerically. Basically, the classical finitedifference/element schemes fail to capture the steep gradients in the boundary layer regions.Therefore, special attention is required for the numerical approximations of these problems. Fora detailed discussion on the analytical and numerical treatment of SPPs, we may refer the readerto the books of O’Malley [2], Doolan et al. [3], Roos et al. [1] and Miller et al. [4].

In this article, our main objective is to construct a hybrid numerical scheme for the SPP (1)and (2). The application of the second-order cubic spline difference scheme to the BVP (1) and(2) in � using the Shishkin mesh may result in oscillations in the outer region (where the meshesare coarse) due to involvement of three-point approximations of the convective term for smallervalues of ε. The use of a midpoint scheme for the BVP (1) and (2) in � results in an oscillationfree scheme with a first-order convergence rate. In order to retain the second-order convergence ofthe cubic spline scheme together with the non-oscillating behaviour of the midpoint scheme, wepropose a hybrid scheme for the convection-dominated BVP (1) and (2), which combines thesetwo schemes in an appropriate manner, that is, the cubic spline scheme is used in the inner region,and the midpoint scheme is used in the outer region.

The convection-diffusion BVP (1) and (2) has been studied earlier by Natesan et al. [5–7].In all these articles, the BVP is solved by domain decomposition methods. The domain of thedifferential equation is divided into non-overlapping subdomains and the differential equationis solved on each subdomain with suitable conditions at the interfaces of the domain. Thesemethods are suitable for parallel computers; indeed, in [7], the numerical scheme is implementedin a parallel machine. In [8], the authors devised a booster method for the convection–diffusionNeumann problem, where an asymptotic approximate solution is incorporated into any numericalmethod to improve the accuracy. The BVP (1) and (2) has been solved by an adaptive single-stepexponential scheme after converting it into a system of first-order ODEs in [9]. Bawa et al. [10]have derived difference schemes for SPPs using cubic splines. Natesan et al. [11] proposed aparameter-uniform numerical method for singularly perturbed turning point problems.

In the following sections, a, K and C denote generic positive constants independent of nodalpoints, mesh size and the perturbation parameter ε.

2. The continuous problem

In this section, we present the analytical behaviour of the solution of the following BVP, whichwill be used to derive error bounds for the derivatives of the solution:

Ly ≡ εy ′′(x) + a(x)y ′(x) − b(x)y(x) = g(x, ε), x ∈ �, (3)

B0u(0) ≡ b(0)y(0) − a(0)y ′(0) = −g(0, ε), B1y(1) ≡ y(1) = β. (4)

DEFINITION 2.1 A function g(x, ε) is said to be of Class(K, j), if the derivatives of g with respectto x satisfy

| g(i)(x, ε) |≤ K[1 + ε−i exp

(−ax

ε

)], 0 ≤ i ≤ j, x ∈ �.

LEMMA 2.2 [3] Let v be a smooth function satisfying B0v(0) ≥ 0, B1v(1) ≥ 0 and Lv(x) ≤ 0,∀x ∈ �. Then v(x) ≥ 0, ∀x ∈ �.

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LEMMA 2.3[3] Let v be a smooth function. Then, we have the following uniform stability estimate:

| v(x) |≤ C

[| B0v(0) | + | B1v(1) | + max

y∈�

| Lv(y) |]

∀x ∈ �.

LEMMA 2.4 Let g ∈ Class(K, 0). Then the solution y of (3) and (4) satisfies

| y(i)(x) | ≤ C, i = 0, 1.

Proof The proof can be seen in [6]. �

LEMMA 2.5 [6] Let g ∈ Class(K, j). Then the solution y of (3) and (4) satisfies

| y ′′(0) |≤ C and | y(i)(0) |≤ Cε−i+1, i = 3(1)j + 1.

THEOREM 2.1 Let g be of Class(K, j). Then the solution y of (3) and (4) satisfies

| y(i)(x) |≤ C[1 + ε−i+1 exp

(−ax

ε

)], i = 2(1)j + 1, x ∈ �.

Proof Refer to [6] for the proof. �

COROLLARY 2.6 If u(x) is the solution of (1) and (2) and a, b and f are in Cj (�), then u satisfies

| u(i)(x) |≤ C[1 + ε−i+1 exp

(−ax

ε

)], i = 1(1)j + 1, x ∈ �.

3. The discrete problem

In this section, first we derive the cubic spline scheme for the SPP (1) and (2) on variable meshes,and then we provide the piece-wise uniform Shishkin meshes. Finally, we construct the hybridscheme.

3.1 Cubic spline difference scheme

Let the mesh points of � = [0, 1] be

x0 = 0, xi =i−1∑k=0

hk, hk = xk+1 − xk, xN = 1, i = 1, 2, . . . , N − 1. (5)

We derive the difference scheme in the following.For given values u(x0), u(x1), . . . , u(xN) of a function u(x) at the nodal points x0, x1, . . . , xN ,

there exists an interpolating cubic spline S(x) with the following properties:

(i) S(x) coincides with a polynomial of degree 3 on each subinterval [xi, xi+1], i = 0, . . . ,

N − 1;(ii) S(x) ∈ C2(�);

(iii) S(xi) = u(xi), i = 0, . . . , N .

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264 R.K. Bawa and S. Natesan

The cubic spline can be given by

S(x) = (xi+1 − x)3

6hi

Mi + (x − xi)3

6hi

Mi+1 +(

u(xi) − h2i

6Mi

) (xi+1 − x

hi

)+

(u(xi+1) − h2

i

6Mi+1

) (x − xi

hi

), xi ≤ x ≤ xi+1, i = 0, . . . , N − 1, (6)

where Mi = S ′′(xi), i = 0, . . . , N .The first derivative of S(x) is given by

S ′(x) = −Mi

(xi+1 − x)2

2hi

+ Mi+1(x − xi)

2

2hi

+ u(xi+1) − u(xi)

hi

− (Mi+1 − Mi)

6hi, xi ≤ x ≤ xi+1, i = 0, . . . , N − 1, (7)

and the second derivative is

S ′′(x) = Mi

(xi+1 − x)

hi

+ Mi+1(x − xi)

hi

. (8)

For the one-sided limit of the first derivative, from Equation (7), we have

S ′(xi−) = hi−1

6Mi−1 + hi−1

3Mi + u(xi) − u(xi−1)

hi−1, (9)

and

S ′(xi+) = −hi

3Mi − hi

6Mi+1 + u(xi+1) − u(xi)

hi

. (10)

From Equations (6) and (8), the functions S(x) and S ′′(x) are continuous on � and for S ′(x)

to be continuous at the interior nodes xi , we have from Equations (9) and (10), the followingwell-known ‘continuity condition’:

hi−1

6Mi−1 +

(hi + hi−1

3

)Mi + hi

6Mi+1 =

(ui+1 − ui

hi

)−

(ui − ui−1

hi−1

), i = 1, . . . , N − 1.

(11)This equation ensures the continuity of the first-order derivative of the spline S(x) at the interiornodes.

To obtain the second-order approximations of the first-order derivatives at the nodal points interms of the approximate values ui of u(x) at xi , we do the following.

Taking the usual Taylor series expansion for u around xi , and neglecting the third- and fourth-order terms, we obtain the following approximations for ui+1 and ui−1:

ui+1 ui + hiu′i + h2

i

2u′′

i , (12)

ui−1 ui − hi−1u′i + h2

i−1

2u′′

i . (13)

Multiplying Equation (13) by h2i /h2

i−1 and subtracting from Equation (12), we have the followingapproximation for u′

i :

u′i 1

hihi−1(hi−1 + hi)[h2

i−1ui+1 + (h2i − h2

i−1)ui − h2i ui−1]. (14)

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Multiplying Equation (13) by hi/hi−1 and adding it to Equation (12), we obtain the followingapproximation for u′′

i :

u′′i 2

hihi−1(hi−1 + hi)[hi−1ui+1 − (hi−1 + hi)ui + hiui−1]. (15)

Also, we have

u′i+1 ≈ u′

i + hiu′′i , (16)

u′i−1 u′

i − hi−1u′′i . (17)

Using the expressions for u′i and u′′

i from Equations (14) and (15), respectively, and putting themin Equation (16), we have the following approximation for u′

i+1:

u′i+1 1

hihi−1(hi + hi−1)[(h2

i−1 + 2hihi−1)ui+1 − (hi−1 + hi)2ui + h2

i ui−1]. (18)

Similarly, using the expressions for u′i and u′′

i from Equations (14) and (15), respectively, andputting them in Equation (17), we obtain the following approximation for u′

i−1:

u′i−1 1

hihi−1(hi + hi−1)[−h2

i−1ui+1 + (hi−1 + hi)2ui − (h2

i + 2hihi−1)ui−1]. (19)

Substituting

εMj = −a(xj )u′j + b(xj )uj + f (xj ), j = i, i ± 1 (20)

in Equation (11) and using Equations (14), (18) and (19) for the first-order derivatives, we getthe following system which gives the approximations u1, u2, . . . , uN−1 of the solution u(x) atx1, x2, . . . , xN−1:[

3ε

hi−1(hi + hi−1)− 2hi−1 + hi

2(hi + hi−1)2ai−1 − hi

hi−1(hi + hi−1)ai

+ h2i

2hi−1(hi + hi−1)2ai+1 − hi−1

2(hi + hi−1)bi−1

]ui−1

+[ −3ε

hihi−1+ 1

2hi

ai−1 + hi − hi−1

hihi−1ai − 1

2hi−1ai+1 − bi

]ui

+[

3ε

hi(hi + hi−1)− h2

i−1

2hi(hi + hi−1)2ai−1 + hi−1

hi(hi + hi−1)ai

+ 2hi + hi−1

2(hi + hi−1)2ai+1 − hi

2(hi + hi−1)bi+1

]ui+1

=[

hi−1

2(hi + hi−1)

]fi−1 + fi +

[hi

2(hi + hi−1)

]fi+1. (21)

Now, using expressions (9) and (10) for the approximation of the first derivative at the leftboundary point, we obtain the following:[

3εa0

h20

− a20

h0− a0a1

2h0− a0b0 + 3εb0

h0

]u0 +

[−3εa0

h20

+ a20

h0+ a0a1

2h0+ a0b1

2

]u1

=[−3ε

h0− a0

]f0 − a0

2f1. (22)

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266 R.K. Bawa and S. Natesan

Finally, Equations (21) and (22) constitute the system of linear algebraic equations, which givesthe approximations u0, u1, . . . , uN of the solution u(x) at x0, x1, . . . , xN .

3.2 Piece-wise uniform Shishkin mesh

The cubic spline difference scheme derived in Section 3.1 is on variable meshes and it is a moregeneral one. For SPPs one needs finer mesh in the boundary layer regions and coarse mesh inthe regular region which can be easily obtained viz. the piece-wise uniform Shishkin mesh. Moreprecisely, the domain � is divided into two subintervals as � = [0, σ ) ∪ [σ, 1], for some σ suchthat 0 < σ ≤ 1/2. On the subinterval [0, σ ) a uniform mesh with N/2 mesh intervals is placed,where [σ, 1] has a uniform mesh with N/2 mesh intervals. It is obvious that the mesh is uniformwhen σ = 1/2, and it is fitted to the problem by choosing σ be the following function of N , ε

and σ0:

σ = min

{1

2, σ0ε ln N

}, (23)

where σ0 > 0 is a constant chosen in such a way that the resulting hybrid scheme is of second-orderconvergent. More precisely, we take σ0 = 2/α.

Further, we denote the mesh size in the regions [0, σ ) as h(1) = 2σ/N , and in [σ, 1] byh(2) = 2(1 − σ)/N .

3.3 Hybrid–numerical scheme

As pointed out in Section 1, the cubic spline scheme loses its stability in the outer region wherethe meshes are coarse. In order to retain its stability, we use the midpoint scheme [12] in the outerregion. This combination yields the following uniformly-stable second-order scheme:

LNui ≡ r−i ui−1 + rc

i ui + r+i ui+1 = q−

i fi−1 + qci fi + q+

i fi+1, i = 1, . . . , N − 1, (24)

along with the following equations corresponding to the boundary points:

rc0u0 + r+

0 u1 = qc0f0 + q+

0 f1,

uN = β, (25)

where for i = 1, . . . , N/2 − 1

r−i =

[ +3ε

hi−1(hi + hi−1)− 2hi−1 + hi

2(hi + hi−1)2ai−1 − hi

hi−1(hi + hi−1)ai

+ h2i

2hi−1(hi + hi−1)2ai+1 − hi−1

2(hi + hi−1)bi−1

],

rci =

[ −3ε

hihi−1+ 1

2hi

ai−1 + hi − hi−1

hihi−1ai − 1

2hi−1ai+1 − bi

],

r+i =

[+3ε

hi(hi + hi−1)− h2

i−1

2hi(hi + hi−1)2ai−1 + hi−1

hi(hi + hi−1)ai

+ 2hi + hi−1

2(hi + hi−1)2ai+1 − hi

2(hi + hi−1)bi+1

],

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q−i = hi−1

2(hi + hi−1),

qci = 1,

q+i = hi

2(hi + hi−1), (26)

and for i = N/2, . . . , N − 1

r−i = 2ε

hi−1(hi + hi−1),

rci = −2ε

hi−1(hi + hi−1)− 2ε

hi(hi + hi−1)− 1

2hi

(ai + ai+1) − 1

2(bi + bi+1),

r+i = 2ε

hi(hi + hi−1)+ 1

2hi

(ai + ai+1) − 1

2(bi + bi+1),

q−i = 0, qc

i = 1

2, q+

i = 1

2, (27)

and

rc0 = 3εa0

h20

− a20

h0− a0a1

2h0− a0b0 + 3εb0

h0,

r+0 = −3εa0

h20

+ a20

h0+ a0a1

2h0+ a0b1

2,

qc0 = −3ε

h0− a0,

q+0 = −a0

2. (28)

The tri-diagonal system of linear algebraic equations (24) and (25) can be solved by any existingcodes.

4. Error analysis

In this section, we derive the truncation error for the proposed scheme. The discrete stabilityanalysis and error estimates will be carried out in our forthcoming article [13].

For i = 1, . . . , N/2 − 1, the truncation error of the hybrid scheme is given by

τi,u = [r−i u(xi−1) + rc

i u(xi) + r+i u(xi+1)] − [q−

i f (xi−1) + qci f (xi) + q+

i f (xi+1)]. (29)

Using the differential equation (1) for f in the above expression, we obtain

τi,u =[r−i u(xi−1) + rc

i u(xi) + r+i u(xi+1)]

− [q−i (εu′′(xi−1) + ai−1u

′(xi−1) − bi−1u(xi−1))

+ qci (εu

′′(xi) + aiu′(xi) − biu(xi)) + q+

i (εu′′(xi+1)

+ ai+1u′(xi+1) − bi+1u(xi+1))]. (30)

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268 R.K. Bawa and S. Natesan

Now, making use of the Taylor series expansion, we have

u(xi−1) = u(xi) − hi−1u′(xi) + h2

i−1

2! u′′(xi) − h3i−1

3! u(iii)(xi) + h4i−1

4! u(iv)(xi) + · · ·

and

u(xi+1) = u(xi) + hiu′(xi) + h2

i

2! u′′(xi) + h3

i

3! u(iii)(xi) + h4

i

4! u(iv)(xi) + · · · .

Using the values of u(xi−1) and u(xi+1) in Equation (30), we have

τi,u = T0,iu(xi) + T1,iu′(xi) + T2,iu

′′(xi) + T3,iu(iii)(xi) + T4,iu

(iv)(xi) + h.o.t., (31)

where

T0,i = r−i + rc

i + r+i − (q−

i bi−1 + qci bi + q+

i bi+1),

T1,i = −hi−1r−i + hir

+i − (q−

i ai−1 + qci ai + q+

i ai+1) − (hi−1q−i bi−1 + hi+1q

+i bi+1),

T2,i = h2i−1

2! r−i + h2

i

2! r+i − ε(q−

i + qci + q+

i ) + (hi−1q−i ai−1 − hiq

+i ai+1)

+(

h2i−1

2! q−i bi−1 + h2

i

2! q+i bi+1

),

T3,i = −h3i−1

3! r−i + h3

i

3! r+i + ε(q−

i hi−1 − q+i hi)

−(

h2i−1

2! q−i ai−1 + h2

i

2! q+i ai+1

)−

(h3

i−1

3! q−i bi−1 − h3

i

3! q+i bi+1

),

T4,i = h4i−1

4! r−i + h4

i

4! r+i − ε

(h2

i−1

2! q−i + h2

i

2! q+i

)

+(

h3i−1

3! q−i ai−1 − h3

i

3! q+i ai+1

)+

(h4

i−1

4! q−i bi−1 + h4

i

4! q+i bi+1

).

It can be easily seen that

T0,i = T1,i = T2,i = T3,i = 0, T4,i = −3ε

(h3

i + h3i−1

hi + hi−1

) [1

4! − 1

2!6]

.

Thus, we have

τi,u = −3ε

(h3

i + h3i−1

hi + hi−1

) [1

4! − 1

2!6]

u(iv)(xi) + O(N−3). (32)

For i = N/2, . . . , N − 1, we can proceed in a similar manner to show that

τi,u = −ε

(hi − hi−1

3

)u(iii)(xi) + 2ε

4!

(h3

i + h3i−1

hi + hi−1

)u(iv)(xi) + O(N−3). (33)

The truncation error at the boundary point x0 is given by

τ0,u = rc0u(x0) + r+

0 u(x1) − qc0f0 − q+

0 f1. (34)

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Again using Equation (1), and the Taylor series expansion for u(x1), the truncation error at x0 canbe given as

τ0,u = −3εh20

(1

4! − 1

2!6)

u(iv)0 (x0) + O(N−3). (35)

Using the bounds of the solution obtained in Section 2, one can prove the following proposition.

PROPOSITION 1 Let u(x) and ui be, respectively, the solutions of (1) and (2) and (24) and (25).Then, the local truncation error satisfies the following bounds:

|τi,u| ≤ CN−2σ 20 ln2 N for 0 ≤ i <

N

2,

|τi,u| ≤ C(N−2ε + N−aσ0) forN

2≤ i ≤ N − 1 and h(2) ≥ √

ε,

|τi,u| ≤ C(N−1ε + N−aσ0) forN

2≤ i ≤ N − 1 and h(2) <

√ε.

5. Numerical experiments

To show the accuracy of the present method, here we have experimented with it for some examples.The results are presented in the form of tables with maximum point-wise errors and rates ofconvergence. The numerical examples are carried out for the values of ε = 10−0, 10−1, . . . , 10−12,whereas in the tables only selected values of ε are presented. The numerical results are comparedwith the classical upwind finite difference scheme on the Shishkin mesh [14].

5.1 Classical upwind finite difference scheme

Here, we present the classical upwind finite difference scheme for the SPP (1) and (2). Thesecond-order and first-order derivatives in the differential equation are, respectively, replaced bythe central difference and the forward difference schemes. The resulting system of equations aregiven in the following compact form:

r−i ui−1 + rc

i ui + r+i ui+1 = q−

i fi−1 + qci fi + q+

i fi+1, i = 1, . . . , N − 1,

rc0u0 + r+

0 u1 = qc0f0 + q+

0 f1,

uN = β, (36)

where

r−i = 2ε

hi−1(hi + hi−1),

rci = −2ε

hi−1(hi + hi−1)− 2ε

hi(hi + hi−1)− ai

hi

− bi,

r+i = 2ε

hi(hi + hi−1)+ ai

hi

,

q−i = 0, qc

i = 1, q+i = 0,

rc0 = b0 + a0

h0, r+

0 = −a0

h0,

qc0 = −1, q+

0 = 0.

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270 R.K. Bawa and S. Natesan

Example 5.1 Consider the following convection–diffusion BVP without the zeroth-order term:

εu′′(x) + (1 + 2x)2u′(x) = −(x3 + exp(−x)), x ∈ (0, 1),

−u′(0) = 1, u(1) = 0.

To calculate the maximum point-wise error and rate of convergence, we use the double-meshprinciple. We calculate the numerical solution UN on �N and the numerical solution UN on themesh �N where the transition parameter is now given by

σ = min

{1

2, σ0ε ln

(N

2

)}.

Define the double-mesh differences to be

ENε = max

xi∈�N

ε

|UN(xj ) − U 2N(xj )| and EN = maxε

ENε ,

where UN(xj ) and U 2N(xj ), respectively, denote the numerical solutions obtained using N and2N mesh intervals. Further, we calculate the order of convergence by

q = log2

(EN

ε

E2Nε

).

The numerical results for the present example are shown in Table 1. The maximum point-wiseerrors are plotted in Figure 1a. Numerical results obtained by the classical upwind scheme (36) areshown in Table 2. From the results given in Tables 1 and 2, one can easily identify the performanceof the proposed method over the simple upwind scheme.

Example 5.2 Consider the following nonself-adjoint two-point boundary-value problem:

εu′′(x) + (1 + x(1 − x))u′(x) − (1 + x4.5)u(x) = −(x3 + exp(−x)), x ∈ (0, 1),

u(0) − u′(0) = 1, u(1) = 0.

Table 1. Maximum point-wise errors ENε , rate of convergence p and ε uniform errors EN for Example 5.1 by the

proposed method.

Number of mesh points N

ε 16 32 64 128 256 512 1024 2048

10−0 2.4092e−3 1.1589e−3 5.6619e−4 2.7960e−4 1.3891e−4 6.9227e−5 3.4557e−5 1.7264e−51.0558 1.0334 1.0179 1.0093 1.0047 1.0024 1.0012

10−2 1.1338e−3 1.2896e−4 3.8017e−5 3.0810e−5 1.7304e−5 8.3840e−6 3.8425e−6 1.7205e−63.1362 1.7622 0.3032 0.8323 1.0454 1.1256 1.1593

10−4 2.6484e−3 6.8858e−4 1.7263e−4 4.2598e−5 1.0332e−5 2.4250e−6 5.2827e−7 9.3387e−81.9434 1.9959 2.0188 2.0437 2.0910 2.1986 2.5000

10−6 2.6687e−3 6.9728e−4 1.7630e−4 4.4196e−5 1.1054e−5 2.7623e−6 6.8981e−7 1.7206e−71.9363 1.9837 1.9961 1.9994 2.0006 2.0016 2.0033

10−8 2.6689e−3 6.9737e−4 1.7634e−4 4.4212e−5 1.1061e−5 2.7657e−6 6.9145e−7 1.7286e−71.9363 1.9836 1.9959 1.9990 1.9997 1.9999 2.0000

10−10 2.6689e−3 6.9737e−4 1.7634e−4 4.4212e−5 1.1061e−5 2.7657e−6 6.9146e−7 1.7288e−71.9363 1.9836 1.9959 1.9990 1.9997 1.9999 1.9999

10−12 2.6689e−3 6.9737e−4 1.7634e−4 4.4212e−5 1.1061e−5 2.7657e−6 6.9146e−7 1.7288e−71.9363 1.9836 1.9959 1.9990 1.9997 1.9999 1.9999

EN 2.6689e−3 1.1589e−3 5.6619e−4 2.7960e−4 1.3891e−4 6.9227e−5 3.4557e−5 1.7264e−5puni 1.2035 1.0334 1.0179 1.0093 1.0047 1.0024 1.0012

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International Journal of Computer Mathematics 271

Figure 1. Loglog plots of N Vs. maximum error.

Table 2. Maximum point-wise errors ENε , rate of convergence p and ε uniform errors EN for Example 5.1 by the

classical upwind scheme (36).

Number of mesh points N

ε 16 32 64 128 256 512 1024 2048

10−0 2.7051e−2 1.2883e−2 6.2827e−3 3.1019e−3 1.5411e−3 7.6812e−4 3.8345e−4 1.9157e−41.0702 1.0360 1.0182 1.0092 1.0046 1.0023 1.0012

10−2 6.4084e−2 2.9018e−2 1.3675e−2 6.5974e−3 3.2234e−3 1.5857e−003 7.8290e−4 3.8727e−41.1430 1.0854 1.0516 1.0333 1.0235 1.0182 1.0155

10−4 6.7088e−2 3.0687e−2 1.4606e−2 7.1169e−3 3.5116e−3 1.7440e−3 8.6901e−4 4.3373e−41.1284 1.0711 1.0373 1.0191 1.0097 1.0050 1.0026

10−6 6.7119e−2 3.0704e−2 1.4616e−2 7.1224e−3 3.5147e−3 1.7457e−3 8.6996e−4 4.3425e−41.1283 1.0709 1.0371 1.0190 1.0096 1.0048 1.0024

10−8 6.7120e−2 3.0705e−2 1.4616e−2 7.1225e−3 3.5148e−3 1.7458e−3 8.6997e−4 4.3426e−41.1283 1.0709 1.0371 1.0190 1.0096 1.0048 1.0024

10−10 6.7120e−2 3.0705e−2 1.4616e−2 7.1225e−3 3.5148e−3 1.7458e−3 8.6997e−4 4.3426e−41.1283 1.0709 1.0371 1.0190 1.0096 1.0048 1.0024

10−12 6.7120e−2 3.0705e−2 1.4616e−2 7.1225e−3 3.5148e−3 1.7458e−3 8.6997e−4 4.3426e−41.1283 1.0709 1.0371 1.0190 1.0096 1.0048 1.0024

EN 6.7120e−2 3.0705e−2 1.4616e−2 7.1225e−3 3.5148e−3 1.7458e−3 8.6997e−4 4.3426e−4puni 1.1283 1.0709 1.0371 1.0190 1.0096 1.0048 1.0024

Table 3. Maximum point-wise errors ENε , rate of convergence p and ε uniform errors EN for Example 5.2 by the

proposed method.

Number of mesh points N

ε 16 32 64 128 256 512 1024 2048

10−0 1.4805e−3 6.8925e−4 3.3153e−4 1.6245e−4 8.0389e−5 3.9986e−5 1.9940e−5 9.9571e−61.1030 1.0559 1.0292 1.0149 1.0075 1.0038 1.0019

10−2 2.4966e−3 8.5495e−4 3.4327e−4 1.6148e−4 8.2964e−5 4.3712e−5 2.2909e−5 1.1880e−51.5461 1.3165 1.0880 0.9608 0.9245 0.9321 0.9473

10−4 1.8892e−3 4.7451e−4 1.2005e−4 3.0720e−5 8.0384e−6 2.1896e−6 6.3792e−7 2.4392e−71.9933 1.9828 1.9664 1.9342 1.8762 1.7792 1.3870

10−6 1.8825e−3 4.7018e−4 1.1758e−4 2.9396e−5 7.3530e−6 1.8400e−6 4.6090e−7 1.1568e−72.0013 1.9996 1.9999 1.9992 1.9986 1.9972 1.9944

10−8 1.8825e−3 4.7018e−4 1.1758e−4 2.9396e−5 7.3530e−6 1.8400e−6 4.6090e−7 1.1568e−72.0014 1.9998 2.0002 1.9999 2.0000 2.0000 1.9999

10−10 1.8824e−3 4.7014e−4 1.1755e−4 2.9383e−5 7.3461e−6 1.8365e−6 4.5914e−7 1.1479e−72.0014 1.9998 2.0002 1.9999 2.0000 2.0000 1.9999

10−12 1.8824e−3 4.7014e−4 1.1755e−4 2.9383e−5 7.3461e−6 1.8365e−6 4.5914e−7 1.1479e−72.0014 1.9998 2.0002 1.9999 2.0000 2.0000 1.9999

EN 4.3279e−3 1.9041e−3 8.3045e−4 3.4956e−4 1.6764e−4 8.3942e−5 4.2003e−5 2.1010e−5puni 1.1845 1.1972 1.2484 1.0602 0.9979 0.9989 0.9994

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272 R.K. Bawa and S. Natesan

Table 4. Maximum point-wise errors ENε , rate of convergence p and ε uniform errors EN for Example 5.2 by the

classical upwind scheme (36).

Number of mesh points N

ε 16 32 64 128 256 512 1024 2048

100 2.1475e−4 8.0127e−005 4.2287e−5 2.2350e−5 1.1519e−5 5.8501e−6 2.9482e−6 1.4799e−61.4223 0.9221 0.9200 0.9562 0.9775 0.9886 0.9943

10−2 1.6099e−2 8.7873e−3 4.5229e−3 2.2743e−3 1.1312e−3 5.5952e−4 2.7599e−4 1.3594e−40.8735 0.9582 0.9918 1.0076 1.0156 1.0196 1.0217

10−4 1.8238e−2 1.0231e−2 5.4342e−3 2.7979e−3 1.4198e−3 7.1510e−4 3.5883e−4 1.7973e−40.8340 0.9129 0.9577 0.9787 0.9894 0.9948 0.9975

10−6 1.8260e−2 1.0247e−2 5.4442e−3 2.8037e−3 1.4230e−3 7.1685e−4 3.5977e−4 1.8022e−40.8336 0.9124 0.9574 0.9784 0.9892 0.9946 0.9973

10−8 1.8261e−2 1.0247e−2 5.4443e−3 2.8038e−3 1.4230e−3 7.1687e−4 3.5978e−4 1.8023e−40.8336 0.9124 0.9574 0.9784 0.9892 0.9946 0.9973

10−10 1.8261e−2 1.0247e−2 5.4443e−3 2.8038e−3 1.4230e−3 7.1687e−4 3.5978e−4 1.8023e−40.8336 0.9124 0.9574 0.9784 0.9892 0.9946 0.9973

10−12 1.8261e−2 1.0247e−2 5.4443e−3 2.8038e−3 1.4230e−3 7.1687e−4 3.5978e−4 1.8023e−40.8336 0.9124 0.9574 0.9784 0.9892 0.9946 0.9973

EN 1.8261e−2 1.0247e−2 5.4443e−3 2.8038e−3 1.4230e−3 7.1687e−4 3.5978e−4 1.8023e−4puni 0.8336 0.9124 0.9574 0.9784 0.9892 0.9946 0.9973

The maximum errors and rates of convergence of this example are given in Table 3. Figure 1bcorresponds to this example. Numerical results obtained by the classical upwind scheme (36) areshown in Table 4. The real need of the cubic spline and midpoint schemes are easily visible forthis example (mainly because of the presence of the zeroth-order term) from the results presentedin Tables 3 and 4.

6. Conclusions

In this article, an hybrid numerical scheme is proposed for convection-dominated two-pointboundary-value problems. The proposed scheme consists of the cubic spline and midpointschemes. We applied this scheme on the layer resolving Shishkin mesh. In the boundary layerregion, where the mesh is fine, the cubic spline scheme is used. In the outer region, i.e. in thecoarse mesh region, the midpoint scheme is used. This has been done mainly to retain the dis-crete stability. Truncation errors are derived. The hybrid scheme produces better results than theclassical upwind finite difference scheme. Nonlinear problems can also be solved by this schemeafter linearization (for example, one can see [8]).

Acknowledgements

This work was supported by the Department of Science and Technology, Government of India, under research grantSR/S4/MS:318/06.

References

[1] H.-G. Roos, M. Stynes, and L.Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer,Berlin, 1996.

[2] R.E. O’Malley, Singular Perturbation Methods for Ordinary Differential Equations, Springer, New York, 1991.[3] E.P. Doolan, J.J.H. Miller, and W.H.A. Schildres, Uniform Numerical Methods for Problems with Initial and

Boundary Layers, Boole Press, Dublin, 1980.[4] J.J.H. Miller, E. O’Riordan, and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World

Scientific, Singapore, 1996.

Dow

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International Journal of Computer Mathematics 273

[5] S. Natesan and N. Ramanujam, ‘Shooting Method’for the solution of singularly perturbed two-point boundary-valueproblems having less severe boundary layers, Appl. Math. Comput. 133 (2002), pp. 623–641.

[6] S. Natesan, J.Vigo-Aguiar, and N. Ramanujam, A Numerical algorithm for singular perturbation problems exhibitingweak boundary layers, Comput. Math. Appl. 45 (2003), pp. 469–479.

[7] J. Vigo-Aguiar and S. Natesan, A parallel boundary value technique for singularly perturbed two-point boundaryvalue problems, J. Supercomput. 27 (2004), pp. 195–206.

[8] S. Natesan and N. Ramanujam, Booster method for singularly perturbed one-dimensional convection–diffusionNeumann problems, J. Optim. Theory Appl. 99 (1999), pp. 53–72.

[9] J. Vigo-Aguiar and S. Natesan, An efficient numerical method for singular perturbation problems, J. Comp. Appl.Math. 192 (2006), pp. 132–141.

[10] M.K. Kadalbajoo and R.K. Bawa, Variable mesh difference scheme for singularly perturbed boundary valueproblems, J. Optim. Theory Appl. 90 (1996), pp. 405–416.

[11] S. Natesan, J. Jayakumar, and J. Vigo-Aguiar, Parameter uniform numerical method for singularly perturbed turningpoint problems exhibiting boundary layers, J. Comput. Appl. Math. 158 (2003), pp. 121–134.

[12] M. Stynes and H.-G. Roos, The midpoint upwind scheme, Appl. Numer. Math. 23 (1997), pp. 361–374.[13] S. Natesan and R.K. Bawa, Second-order parameter-uniform error estimate for convection-dominated two-point

boundary-value problems, Working Paper, 2008.[14] P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan, and G.I. Shishkin, Parameter-uniform numerical methods

for a class of singularly perturbed problems with a Neumann boundary condition, Lect. Notes Comput. Sci. 1988(2001), pp. 292–303.

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