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Higher-order time accurate numericalmethods for singularly perturbedparabolic partial differential equationsRajdeep Deb a & Srinivasan Natesan ba Department of Chemical Engineering, Indian Institute ofTechnology Guwahati, Guwahati, Indiab Department of Mathematics, Indian Institute of TechnologyGuwahati, Guwahati, Assam, IndiaVersion of record first published: 17 Jun 2009.

To cite this article: Rajdeep Deb & Srinivasan Natesan (2009): Higher-order time accuratenumerical methods for singularly perturbed parabolic partial differential equations, InternationalJournal of Computer Mathematics, 86:7, 1204-1214

To link to this article: http://dx.doi.org/10.1080/00207160701798764

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International Journal of Computer MathematicsVol. 86, No. 7, July 2009, 1204–1214

Higher-order time accurate numerical methods for singularlyperturbed parabolic partial differential equations

Rajdeep Deba and Srinivasan Natesanb*

aDepartment of Chemical Engineering, Indian Institute of Technology Guwahati, Guwahati, India;bDepartment of Mathematics, Indian Institute of Technology Guwahati, Guwahati, Assam, India

(Received 26 February 2007; revised version received 26 May 2007; second revision received 01 August 2007;accepted 01 November 2007 )

This article presents two numerical methods for singularly perturbed time-dependent reaction-diffusioninitial–boundary-value problems. The spatial derivative is replaced by a hybrid scheme, which is a com-bination of the cubic spline and the classical central difference scheme in both the methods. In the firstmethod, the time derivative is replaced by the Crank–Nicolson scheme, whereas in the second methodthe time derivative is replaced by the extended-trapezoidal scheme. These schemes are applied on thelayer resolving piecewise-uniform Shishkin mesh. Some numerical examples are carried out to show theaccuracy and efficiency of these methods.

Keywords: singular perturbed parabolic problem; cubic spline; piecewise-uniform Shishkin mesh;Crank–Nicolson scheme; extended-trapezoidal scheme

2000 AMS Subject Classification: 65M06; 65M12

CCS Category: G1.8

1. Introduction

In this article, we consider the following singularly perturbed initial–boundary-value problem(IBVP):

∂u

∂t− ε

∂2u

∂x2+ b(x, t)u(x, t) = f (x, t), (x, t) ∈ � = (0, 1) × (0, T ],

u(x, 0) = s(x), on Sx = {(x, 0) : 0 ≤ x ≤ 1},u(0, t) = a0(t), on S0 = {(0, t) : 0 ≤ t ≤ T },u(1, t) = a1(t), on S1 = {(1, t) : 0 ≤ t ≤ T },

(1)

where 0 < ε � 1 is a small parameter, and the functions b, f are sufficiently smooth functionssuch that b(x, t) ≥ β > 0 on �. Under suitable continuity and compatibility conditions on the

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

ISSN 0020-7160 print/ISSN 1029-0265 online© 2009 Taylor & FrancisDOI: 10.1080/00207160701798764http://www.informaworld.com

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data, a unique solution u(x, t) of Equation (1) exists. Parabolic boundary layers occur in thesesolutions when ε → 0. Classical numerical methods such as finite difference methods and finiteelement methods fail to yield satisfactory numerical results on uniform mesh, because of the pres-ence of boundary layers. To overcome these difficulties one has to use specially fitted nonuniformmesh. More details on special meshes for singularly perturbed parabolic problems can be seen inthe books of Farrell et al. [2], Miller et al. [7], and Roos et al. [11].

There are several articles dealt with the numerical methods for time-dependent convection-diffusion and reaction-diffusion problems. To cite a few: Stynes and O’Riordan [12] presented auniformly convergent finite element method for these types of problems using exponential basisfunctions. In [6], finite element method of exponentially fitted lumped schemes were given. Farrellet al. [3–5] proposed numerical methods for IBVPs of the form (1). Recently, Natesan and Deb[10] developed a robust numerical scheme for the parabolic reaction-diffusion IBVP (1), andobtained ε-uniform error estimates of order O(N−2

x ln2 Nx + N−1t ).

In this article, we propose two numerical schemes for the parabolic IBVP (1), which are ofhigher-order convergent in the time variable. More precisely, first we derive the schemes for thespatial derivative on variable meshes, and then we apply it on a piecewise-uniform Shishkin mesh.In both the methods the spatial derivatives are replaced by a hybrid scheme, in which the cubicspline scheme (for the boundary layer regions) and the classical finite difference scheme (forthe regular region) are combined suitably. In the first difference scheme (denoted as differencescheme-I), the time derivative is replaced by a second-order O(N−2

t ) accurate Crank–Nicolsonscheme, whereas in the second difference scheme (denoted as difference scheme-II) the timederivative is replaced by a third-order O(N−3

t ) accurate extended-trapezoidal scheme used byChawla and Al-Zanaidi [1]. These schemes provide higher-order accuracy for the time part, andthe hybrid scheme for the spatial derivative takes care of the boundary layer effect. By this waythe proposed schemes resolve the boundary layer with higher-order approximation for the timecomponent.

Section 2 presents the two numerical schemes for the IBVP (1). Some numerical experimentsare carried out in Section 3. Basically, the numerical examples are tested to show the accu-racies for both the spatial and temporal components separately, and also we have calculatedthe normalized flux by these schemes, which will be very useful to calculate the errors in theH 1-energy norm.

2. The numerical schemes

In this section, first, we present the piecewise-uniform Shishkin mesh for the spatial discretizationof the domain. And then, we derive the difference schemes on the Shishkin mesh. In both theseschemes, we discretize the spatial derivative by a hybrid difference scheme, which is a combinationof the cubic spline scheme and the classical central difference scheme. In the first method, the timederivative is replaced by the Crank–Nicolson scheme, whereas the extended-trapezoidal schemeis used in the second method for the time derivative.

2.1 Discretization of the domain

Consider the domain � = (0, 1) × (0, T ]. First, we present the Shishkin mesh for the spatial part:let D = [0, 1] be the spatial domain, which is divided into three subintervals as D = [0, σ ) ∪[σ, 1 − σ ] ∪ (1 − σ, 1] for some σ such that 0 < σ ≤ 1/4. On the subintervals [0, σ ), and (1 −σ, 1] a uniform mesh with Nx/4 mesh-intervals are placed, where [σ, 1 − σ ] has a uniform meshwith Nx/2 mesh-intervals. It is obvious that the mesh is uniform when σ = 1/4, and it is fitted to

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1206 R. Deb and S. Natesan

the problem by choosing σ be the following function of Nx, ε

σ = min

{1

4, σ0

√ε ln Nx

},

where σ0 ≥ 2 is a constant.Here, we use the multi-index notation N = (Nx, Nt), where Nt denotes the number of mesh

elements in the t-direction. We shall introduce the meshes in the spatial and temporal variables,respectively, as ωNx : 0 = x0 < x1 < . . . < xNx

= 1, and ωNt : 0 = t0 < t1 < . . . < tNt= T . Let

the meshes in � be the tensor product of the one-dimensional meshes ωNx and ωNt , and denote

it by �Nx,Nt

ε . Further, let hi = xi+1 − xi be the mesh diameter in the spatial dimension, and τj =tj+1 − tj , with �i = (hi−1 + hi)/2, h = 4σ/Nx, H = 2(1 − 2σ)/Nx , and k = maxj=1,...,Nt

τj .The domain is discretized in the x-direction with Shishkin mesh and uniform mesh in thet-direction.

2.2 Difference scheme-I

Difference scheme-I consists of the Crank–Nicolson scheme for the time derivative, and the hybriddifference scheme for the spatial derivatives. Consider the Crank–Nicolson scheme(

uj+1i − u

j

i

k

)= 1

2

(g

j

i + gj+1i

), i = 1, . . . , Nx − 1, j = 0, . . . , Nt − 1, (2)

where

gj

i = εδ2hbdu

j

i − bj

i uj

i + fj

i ,

gj+1i = εδ2

hbduj+1i − b

j+1i u

j+1i + f

j+1i .

Let Mj

i = δ2hbdu

j

i , where δ2hbd is such that for 0 < i < N/4, 3N/4 < i < N :

δ2hbd =

((D−/hi−1) − (2�i/hihi−1) + (D+/hi)

)(((hi−1D−)/6) + (2�i/3) + ((hiD+)/6))

,

and for N/4 ≤ i ≤ 3N/4:

δ2hbd =

(D−

hi−1− 2�i

hihi−1+ D+

hi

),

here D+uj

i = uj

i+1, D−u

j

i = uj

i−1. Then, we have for 0 < i < N/4, 3N/4 < i < N :

hi−1

6M

j

i−1 +(

hi−1 + hi

3

)M

j

i + hi

6M

j

i+1 = uj

i−1

hi−1− 2�iu

j

i

hi−1hi

+ uj

i+1

hi

,

and for N/4 ≤ i ≤ 3N/4:

Mj

i = uj

i−1

�ihi−1− 2u

j

i

hi−1hi

+ uj

i+1

�ihi

.

The above system of equations can be represented in the following compact form

r−i−1M

j+1i−1 + rc

i Mj+1i + r+

i+1Mj+1i+1 = q−

i−1uj+1i−1 + qc

i uj+1i + q+

i+1uj+1i+1 , (3)

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International Journal of Computer Mathematics 1207

where for 0 < i < N/4, 3N/4 < i < N ,

r−i−1 = hi−1

6, rc

i = 2�i

3, r+

i+1 = hi

6,

q−i−1 = 1

hi−1, qc

i = −2�i

hi−1hi

, q+i+1 = 1

hi

,

and for N/4 ≤ i ≤ 3N/4,

r−i−1 = 0, rc

i = 1, r+i+1 = 0,

q−i−1 = 1

hi−1�i

, qci = −2

hi−1hi

, q+i+1 = 1

hi�i

.

The system (3) can be written in the matrix form as

AMj+1 = BUj+1 + Cj+1, j = 0, . . . , Nt − 1,

where

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

rc1 r+

2

r−1 rc

2 r+3

· · ·· · ·

· · ·r−Nx−2 rc

Nx−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦Nx−1×Nx−1

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

qc1 q+

2

q−1 qc

2 q+3

· · ·· · ·

· · ·q−

Nx−2 qcNx−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦Nx−1×Nx−1

Mj+1 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

Mj+11

Mj+12

...

Mj+1Nx−2

Mj+1Nx−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦Nx−1×1

Uj+1 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

uj+11

uj+12

...

uj+1Nx−2

uj+1Nx−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦Nx−1×1

Cj+1 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

q−0 u

j+10 − r−

0 Mj+10

0

...

0

q+Nx

uj+1Nx

− r+Nx

Mj+1Nx

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦Nx−1×1

.

In a similar way, we obtain

AMj = BUj + Cj ,

and, so

gj = εMj − Bj

1 Uj + F j ,

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1208 R. Deb and S. Natesan

where

Bj

1 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

bj

1

bj

2

. . .

bj

Nx−2

bj

Nx−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦Nx−1×Nx−1

F j =

⎡⎢⎢⎢⎢⎢⎢⎣f

j

1

fj

2

...

fj

Nx−1

⎤⎥⎥⎥⎥⎥⎥⎦Nx−1×1

Thus,

gj = ε(A−1BUj + A−1Cj

) − Bj

1 Uj + F j . (4)

Similarly, we have

gj+1 = ε(A−1BUj+1 + A−1Cj+1

) − Bj+11 Uj+1 + F j+1. (5)

Using the values of Equations (4) and (5) in Equation (2), we have the following relation

Uj+1 − Uj =(

k

2

) [ε(A−1BUj + A−1Cj

) − Bj

1 Uj + F j]+

+(

k

2

) [ε(A−1BUj+1 + A−1Cj+1

) − Bj+11 Uj+1 + F j+1

].

(6)

The difference scheme (6) can be written in the compact form as

Pj+11 Uj+1 = Q

j

1Uj + R

j+11 , j = 0, . . . , Nt − 1, (7)

where

Pj+11 = I − kε

2A−1B + k

2B

j+11 ,

Qj

1 = I + kε

2A−1B − k

2B

j

1 ,

Rj+11 = k

2

(F j + F j+1 + εA−1Cj + εA−1Cj+1

).

One can see from the truncation errors using Taylor’s expansion that the Crank–Nicolsonscheme is of O(N−2

t ) accurate for the time derivative. In [9], the hybrid scheme has been shownof O(N−2

x ln2 Nx) for the spatial derivatives. The error estimates of this difference will be carriedout in our forthcoming article [10].

2.3 Difference scheme-II

Difference scheme-II uses the extended-trapezoidal formula for the time derivative, and the hybridscheme for the spatial derivative.

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International Journal of Computer Mathematics 1209

Here, we use the extended-trapezoidal formula to discretize the time derivative, as

uj+1i − u

j

i

k=

(5g

j

i + 8gj+1i − g

j+2i

12

), i = 1, . . . , Nx − 1, j = 0, . . . , Nt − 1, (8)

where

gj

i = εMj

i − bj

i uj

i + fj

i ,

gj+1i = εM

j+1i − b

j+1i u

j+1i + f

j+1i ,

gj+2i = εM

j+2i − b

j+2i u

j+2i + f

j+2i ,

uj+2i = 5u

j

i − 4uj+1i + 2k

(g

j

i + 2gj+1i

).

In vector form uj+2i can be written as

U j+2 = 5Uj − 4Uj+1 + 2k[(

εMj − Bj

1 Uj + F j)

+ 2(εMj+1 − B

j+11 Uj+1 + F j+1

)].

After rearranging the terms, and using Equations (4) and (5), we have

U j+2 =(DjUj + Ej+1Uj+1 + F

j+11

),

where

Dj =(

5I + 2kεA−1B − 2kBj

1

),

Ej+1 =(−4I + 4kεA−1B − 4kB

j+11

),

Fj+11 = 2k

(F j + 2F j+1 + εA−1Cj + 2εA−1Cj+1

).

Now, substituting all these values in Equation (8), and rearranging it, we obtain

(Pj+12 − R

j+22 Ej+1)Uj+1 = (Q

j

2 + Rj+22 Dj)Uj + (R

j+22 F

j+11 + F

j+22 ), j = 0, . . . , Nt − 1

(9)where

Pj+12 = 1

3

(3I − 2kεA−1B + 2kB

j+11

), R

j+22 = k

12

(−εA−1B + B

j+21

),

Qj

2 = 1

12

(12I + 5kεA−1B − 5kB

j

1

), F

j+11 = 2k

(F j + 2F j+1 + εA−1Cj + 2εA−1Cj+1

),

Fj+22 = k

12

(5F j + 8F j+1 − F j+2 + 5εA−1Cj + 8εA−1Cj+1 − εA−1Cj+2

).

The truncation error of order O(N−3t ) and unconditional stability for the extended-trapezoidal

formula for parabolic reaction-diffusion problems on uniform mesh are derived in [1].We postponethe error analysis of this scheme to our future article [10].

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1210 R. Deb and S. Natesan

3. Numerical results

In this section, we implement the proposed schemes to two test problems studied by variousresearchers. The numerical results have been given in terms of maximum point-wise errors, andrate of convergence. The maximum point-wise error and the ε-uniform errors are obtained to showseparately the spatial effect and the temporal effect, mainly because the difference scheme-I isof O(N−2

t ) accurate and Difference scheme-II is of O(N−3t ) accurate. Further, we have obtained

approximations for the normalized flux, which will play a crucial role in the H 1-energy normerror estimates of finite element analysis.

Example 3.1 [8] Consider the following parabolic IBVP:

ut (x, t) − εuxx(x, t) + u(x, t) = f (x, t), (x, t) ∈ (0, 1) × (0, 1]. (10)

The right-hand side source term, initial and boundary conditions have been calculated from theexact solution

u(x, t) =(

t + x2

2ε

)erfc

(x

2√

εt

)−

√t

πεxe−x2/4εt .

The exact solution is used to calculate the maximum nodal error. More precisely, we determinethe maximum error and ε-uniform error by

ENx,Nt

ε = max�

Nx ,Ntε

|u(xi, tj ) − uj

i |, and ENx,Nt = maxε

ENx,Nt

ε ,

where u(x, t) denotes the exact solution, and uj

i stands for the numerical solution obtained by using

Nx, Nt mesh-intervals in the domain �Nx,Nt

ε . In addition, the rate of convergence is calculated by

p = log2

(ENx

ε

E2Nxε

), and p

uni= log2

(ENx

E2Nx

).

Also, we have calculated the normalized flux

Fε(x, t) = √ε

∂u(x, t)

∂x,

and its numerical approximation

FNx

ε (x, t) = √εD+

x u(x, t).

The maximum error and ε-uniform error in the normalized fluxes have been calculated as

QNx

ε = max0≤t≤T

|Fε(0, t) − FNx

ε (0, t)|, and QNx = maxε

QNx

ε .

The rates of convergence for the normalized flux are determined by

q = log2

(QNx

ε

Q2Nxε

), and q

uni= log2

(QNx

Q2Nx

).

The numerical experiments are carried out to show two important points: (1) the spatial effecton the error of the solution and the normalized flux; and (2) the temporal effect in the solutionand the normalized flux.

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International Journal of Computer Mathematics 1211

Table 1. ε-Uniform error ENx,Nt and rate of convergence puni for Example 3.1 by difference scheme-I(spatial effect).

Nx 16 32 64 128 256 512ENx,Nt 4.8208e-2 1.8812e-2 7.4719e-3 2.5722e-3 8.4513e-4 2.7074e-4puni 1.3577 1.3321 1.5385 1.6058 1.6423

Table 2. ε-Uniform error QNx and rate of convergence quni for Example 3.2 by difference scheme-I(spatial effect).

Nx 16 32 64 128 256 512QNx 6.4800e-1 4.8285e-1 3.2692e-1 2.0564e-1 1.2282e-1 7.0844e-2quni 0.4244 0.5626 0.6688 0.7436 0.7938

Table 3. ε-Uniform error ENx,Nt and rate of convergence puni for Example 3.1 by differencescheme-II (spatial effect).

Nx 16 32 64 128 256 512ENx,Nt 5.3210e-2 2.1149e-2 7.7576e-3 2.6409e-3 8.5688e-4 2.7257e-4puni 1.3311 1.4469 1.5546 1.6239 1.6525

Table 4. ε-Uniform error QNx and rate of convergence quni for Example 3.1 by differencescheme-II (spatial effect).

Nx 16 32 64 128 256 512QNx 6.4559e-1 4.8105e-1 3.2594e-1 2.0518e-1 1.2264e-1 7.0780e-2quni 0.4244 0.5616 0.6677 0.7425 0.7930

The ε-uniform error of the solution and the normalized flux, and the rate of convergence forthe difference scheme-I, which show the spatial effect are presented in Tables 1 and 2. The samecan be shown for the difference scheme-II in Tables 3 and 4.

The ε-uniform error of the solution and the normalized flux showing the temporal effect fordifference schemes I and II are, respectively given in Tables 5–8. From the results of Table 5, onecan notice that the difference scheme-I provides O(N−2

t ) accuracy for the temporal variable. Thishigher order accuracy can also be reflected in the normalized flux results shown in Table 6.

Table 7 highlights the O(N−3t ) accurate results for time variable in the solution by the difference

scheme-II, and Table 8 shows the same for the corresponding normalized flux. These results reveal

Table 5. ε-Uniform error ENx,Nt and rate of convergence puni for Example 3.1 bydifference scheme-I (temporal effect).

Nt 8 16 32 64 128ENx,Nt 6.1977e-3 1.6928e-3 4.4301e-4 1.1244e-4 2.8503e-5puni 1.8511 1.9040 1.9369 1.9682

Table 6. ε-Uniform error QNx and rate of convergence quni for Example 3.1 bydifference scheme-I (temporal effect).

Nt 8 16 32 64 128QNx 3.1731e-1 1.7280e-1 9.0350e-2 4.5924e-2 2.3223e-2quni 0.8768 0.9355 0.9763 0.9837

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1212 R. Deb and S. Natesan

Table 7. ε-Uniform error ENx,Nt andrate of convergence puni for Example 3.1by difference scheme-II (temporal effect).

Nt 8 16ENx,Nt 8.9952e-4 1.1378e-4puni 2.9829

Table 8. ε-Uniform error QNx and rateof convergence quni for Example 3.1 bydifference scheme-II (temporal effect).

Nt 8 16QNx 1.2410e-1 4.5813e-2quni 1.4377

our claim of higher-order accuracy for the temporal part in these newly proposed differenceschemes.

Also, one can observe that the proposed method produces ε-uniform numerical results, this isthe outcome of the proposed hybrid scheme on the Shishkin mesh. Thus, we have resolved theboundary layers as well as obtained higher-order time accurate results by these schemes.

Further, we consider the following variable coefficient parabolic PDE to examine the differenceschemes.

Example 3.2 Consider the following parabolic IBVP:

ut (x, t) − εuxx(x, t) + √x + 1u(x, t) = 1, (x, t) ∈ (0, 1) × (0, 1]

u(0, t) = 0, u(1, t) = 0, for t ∈ (0, 1]u(x, 0) = 0, for x ∈ [0, 1] (11)

The exact solution of the IBVP (3.2) is not available, therefore, we determine the accuracy ofthe numerical solution by comparing it with the numerical solution computed on a fine mesh. Wedetermine the maximum point-wise error, and ε-uniform error respectively, by

ENx,Nt

ε = max�

Nx ,Ntε

||uNx,Nt − u4Nx,4Nt ||, and ENx,Nt = maxε

ENx,Nt

ε ,

where uNx,Nt denotes numerical solution obtained by using Nx and Nt mesh-intervals, respectively,

on the x- and t-axes, i.e., on the discretized domain �Nx,Nt

ε . Then bisect twice each interval ωNx andωNt and take their tensor product to obtain a mesh �4Nx,4Nt

ε that has 16 times as many subintervals

as �Nx,Nt

ε (by this way, one can obtain the same transition points as in �Nx,Nt

ε ). Let u4Nx,4Nt denotesthe numerical solution computed on this mesh. In addition, the rate of convergence is calculated by

p = log2

(ENx

ε

E2Nxε

), and p

uni= log2

(ENx

E2Nx

).

The maximum point-wise error and ε-uniform error in the normalized flux have been calculated as

QNx

ε = max�

Nx ,Ntε

||F Nx

ε (xi, tj ) − F 4Nx

ε (xi, tj )||, and QNx = maxε

QNx

ε ,

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International Journal of Computer Mathematics 1213

Table 9. ε-Uniform error ENx,Nt and rate of convergence puni for Example 3.2 by difference scheme-I(spatial effect).

Nx 16 32 64 128 256 512ENx,Nt 7.1535e-3 2.8045e-3 1.0236e-3 3.4895e-4 1.1398e-4 3.6035e-5puni 1.3509 1.4540 1.5526 1.6142 1.6613

Table 10. ε-Uniform error QNx and rate of convergence quni for Example 3.2 by difference scheme-I(spatial effect).

Nx 16 32 64 128 256 512QNx 1.7713e-1 1.2658e-1 8.3911e-2 5.3076e-2 3.0833e-2 1.7737e-2quni 0.4848 0.5931 0.6608 0.7836 0.7977

Table 11. ε-Uniform error ENx,Nt and rate of convergence puni for Example 3.2 bydifference scheme-II (spatial effect).

Nx 16 32 64 128 256ENx,Nt 8.5433e-2 4.4195e-2 1.8666e-2 7.3712e-3 2.7092e-3puni 0.9509 1.2435 1.3405 1.4440

Table 12. ε-Uniform error QNx and rate of convergence quni for Example 3.2 bydifference scheme-II (spatial effect).

Nx 16 32 64 128 256QNx 1.0829e-1 6.8780e-2 4.5416e-2 3.1807e-2 2.0389e-2quni 0.6548 0.5988 0.5139 0.6415

where

F Nx

ε (xi, t) = √εD+

x u(x, t), for i = 1, . . . , Nx − 1, and F Nx

ε (xNx, t) = √

εD−x u(xNx

, t).

The numerical results obtained by Difference scheme-I are given in Tables 9 and 10 for thesolution and the normalized flux, respectively. Tables 11 and 12 show the same for the differencescheme-II.

Although we have not supplied the numerical results supporting the temporal effect of thesedifference schemes (mainly, to reduce the number of tables), we have experimented and observedthe same higher-order accuracy as like in the constant coefficient problem given in Example 3.1.Further the independency of ε can be observed from the numerical results.

4. Conclusions

This paper presents two numerical schemes for singularly perturbed parabolic reaction-diffusionIBVPs of the type (1). In both the schemes the spatial derivatives are replaced by a hybrid scheme,which combines the cubic spline and classical finite difference scheme in an appropriate manner.The time derivative is replaced by the O(N−2

t ) accurate Crank–Nicolson scheme, and the O(N−3t )

accurate extended-trapezoidal scheme for the first and second difference schemes, respectively.The numerical experiments reveal that the proposed schemes are capable of resolving the boundarylayer and producing higher-order time accurate approximations.

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1214 R. Deb and S. Natesan

Acknowledgement

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

References

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