a weighted finite difference method involving nine-point ... · nine-point formula for...

International Mathematical Forum, 5, 2010, no. 68, 3379 - 3398

A Weighted Finite Difference Method Involving

Nine-Point Formula for Two-Dimensional

Convection-Diffusion Equation

A. S. J. Al-Saif and Muna O. Al-Humedi

Department of Mathematics, College of Education University of Basrah, Iraq

[email protected]

Abstract: New issues of finite difference method are proposed in this paper. Successfully, we have generalized and extended the works in [7] and [1], respectively. By a new version" nine-point finite difference formula" more accurate results are obtained than those in the literature. Mathematics Subject Classifications: 65M06, 65M12 Keywords: Accuracy, Convection-diffusion equation, Weighted-nine-point finite difference formula. 1. Introduction

Accurate solution of the two-dimensional convection-diffusion equation is desirable.

This requires the development of numerical schemes that can remain accurate even for large amount of computational time, several effective schemes have been proposed to achieve this goal [1,2,6,11,12]. In the field of finite difference methods (FDMs), there are many articles that were carried out to numerically solve the convection-diffusion equation in both one- and two-dimensional forms successfully [1-12].

During the last two decades, some new techniques involving weighted discretization and modified equivalent partial differential equation have been introduced to develop FDMs. For example, the explicit, and implicit weighted finite difference method with three-point formula[9], and five-point formula[3,5] are used to solve the one- dimensional convection-diffusion equation, for solving the two-dimensional convection-diffusion equation three-point semi-implicit and implicit formula FDMs[6,11,12,8], and five-point explicit and semi-implicit formula FDM[1] have been used successfully . The motives to present this work are: there are perhaps only few (if there are any) papers that deal with nine-point weighted FDM for solving the two-dimensional convection-diffusion equation, in addition to extending our previous work[7],and to generalizing [1].

mailto:[email protected]

3380 A. S. J. Al-Saif and M. O. Al-Humedi In this paper, we extended and developed the weighted FDM. The development

reported here is nine-point weighted FDM. When compared with [1,6,11,12,8],present numerical results are in agreement with the existing results, and show that our method is efficient for giving accurate solution for two-dimensional convection-diffusion equation and have reasonable stability. 2. The mathematical formulas

2.1 Governing equation

The two-dimensional convection-diffusion equation for a transport scalar function is ),,( tyxT

0),,(),,(),,(),,(),,(2

2

2

2

=∂

∂−

∂∂

−∂

∂+

∂∂

+∂

∂

444444 3444444 2144444 344444 21termsDiffusion

yx

termsConvection

ytyxT

xtyxT

ytyxTv

xtyxTu

ttyxT αα (1)

where x , y are the distances, is the time, are the constants speeds of convection in t 0, >vux and y direction, 0, >yx αα are the diffusivity in x and y direction.

Equation (1) can be solved in the interior of the region ax ≤≤0 , , of ( space, where a , are constants, subject to appropriate initial and boundary conditions defined as;

by ≤≤0 0≥t),, tyx b

}byaxyxhyxT ≤≤≤≤= 00),()0,,( (2)

⎪⎪⎭

⎪⎪⎬

⎫

>≤≤=>≤≤=>≤≤=>≤≤=

00),(),,(00),(),0,(00),(),,(00),(),,0(

0

0

taxtxgtbxTtaxtxgtxTtbytyftyaTtbytyftyT

b

a (3)

where are known data. ba ggffh ,,,, 00

2.2 The modified equivalent partial differential equation Consider the finite difference equation (FDE)

0][,

2

2

,2

2

,,., =

∂∂

−∂∂

−∂∂

+∂∂

+∂∂

=Δ

n

kjy

n

kjx

n

kj

n

kj

n

kj

nkj yxy

vx

ut

L ταταττττ (4) which is consistent with partial differential equation (1), where is a finite difference operator and is the approximation values at a set of grid points

, where , .The distance between the

points on lines which parallel to x -axis , y -axis and t are

respectively, where Τ is an optimal time and are positive integer numbers. The equivalent partial differential equation (EPDE) to equation (1), can be obtained by converting it

ΔL),,( tnyjxiT ΔΔΔ= τ

),,( tnyjxi ΔΔΔ 1)1(1 1 −= Mi 1)1(1 2 −= Mj 1)1(1 −= Nn

Ntand

Mby

Max Τ

=Δ=Δ=Δˆ

,21

ˆ NMM ,, 21

Weighted finite difference method 3381 to finite difference equation and then expanding the resulting equation by Taylor's series around the grid point

∑∑∑∞

= = =−−−− =

∂∂∂∂

+∂∂

−∂∂

−∂∂

+∂∂

+∂∂

2 0 0,,2

2

2

2

0p

p

q

q

rqprqr

P

qprqryx yxtF

yxyv

xu

tττατατττ (5)

Any (FDE) which is consistent with convection-diffusion equation(1) after expanding each term of it in Taylor series about some fixed point in the solution domain, has a modified equivalent partial differential equation(MPDE) of the form

∑∑∞

= =−− =

∂∂∂

+∂∂

−∂∂

−∂∂

+∂∂

+∂∂

2 0,2

2

2

2

0p

p

qqpq

P

qpqyx yxC

yxyv

xu

tττατατττ (6)

where, the term under the summation signs form is the truncation error. A (FDEs) for solving (1) is said to have accuracy of thr order if for0, =−qpqC pq )1(0= is valid, for all ,where at least rp ,....,2= 01,1 ≠−++ qrrC 1,...,1,0 += rq . Using the weighted difference method is useful to develop the FDMs to obtain high accurate numerical solution [1-10]. 3. Accurate (FDEs) by weighted-differences

Using weighted differencing in order to construct higher-order schemes weights

are used to eliminate from the MEPDE as many as possible of the terms containing the

derivatives ,qpp

p

yx −∂∂∂ τ L,3,2,1)1(1 =−= ppq to develop FDMs to higher order of accuracy

than conventional methods. For the stencil of Figure (1a), the first-order derivatives of ),,( tyxτ can be

approximated by using three-point formulas (3pt.) at as follows thnkj ),,(

)(1,,

,

xOyy

nkj

nkj

n

kj

Δ+Δ

−=

∂∂ −τττ

)( xO Δ+,1,

, xx

nkj

nkj

n

kj Δ

−=

∂∂ −τττ

)(,

1,

,tO

t

nkj

nkjn

kjtΔ+

Δ

−+

=∂∂ τττ

These approximations are called forward scheme (FS) and backward scheme (BS). For the stencil of figure (1b), the first -order and the second-order derivatives of ),,( tyxτ can

be approximated by using nine-point formulas (9pt.) at as follows thnkj ),,(

3382 A. S. J. Al-Saif and M. O. Al-Humedi

j

1+kk

)(b

3+k2+k

1+kk

1−k2−k

3−k

3−j 2−j 1−j j 1+j 2+j 3+j

leveltimen )1( +

4−k

4+k

4−j 4+j

leveltimen)(),,( nkj

1−j 1+j1−k

)(a

Figure (1). The stencil in plane for : (a) three-point (b) nine-point. ),( yx

)()(10080

366866524652468636 8,4,3,2,1,1,2,3,4

,

xOxx

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

n

kj

Δ+Δ

−−−+−++≅

∂∂ ++++−−−− τττττττττ

)()(10080

366866524652468636 84,3,2,1,1,2,3,4,

,

yOyy

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

n

kj

Δ+Δ

−−−+−++≅

∂∂ ++++−−−− τττττττττ

)()(20160

481372260964935026096137248 82

,4,3,2,1,,1,2,3,4

,2

2

xOxx

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

n

kj

Δ+Δ

−−−+−+−−−=

∂∂ ++++−−−− ττττττττττ

)()(20160

481372260964935026096137248 82

4,3,2,1,,1,2,3,4,

,2

2

yOyy

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

n

kj

Δ+Δ

−−−+−+−−−=

∂∂ ++++−−−− ττττττττττ

These approximations are called central scheme (CS). We can approximate all the derivatives in equation (4) with weighted or non-weighted discretization as the following formulas;

Weighted finite difference method 3383 3.1 The forward time- central space scheme with nine-point formulas (FTCS 9pt.) Using a first-order forward difference approximation equation for the time derivative and eighth-order central difference approximation for the first and second spatial derivatives to approximate equation (4), leads to a new formula as,

nkj

yy

nkj

yynkjyy

nkjyy

nkjyy

nkjyy

nkj

yynkj

yynkj

xx

nkj

xxnkjxx

nkjxx

nkjyx

nkjxx

nkjxx

nkj

xxnkj

xxnkj

SC

SCSCSCSC

SCSCSCSC

SCSCSCSS

SCSCSCSC

4,

3,2,1,1,

2,3,4,,4

,3,2,1,

,1,2,3,41

,

)2016010080

(

)420280

()(5040343)2(

25201631)2(

25201631

)(5040343)

420280()

2016010080()

2016010080(

)420280

()(5040343)2(

25201631))(

201649351(

)2(25201631)(

5040343)

420280()

2016010080(

+

+++−

−−−+

+++

−−−−+

−+

−+−+−−++

+−+−+−−+

−+−+−−+−+

+++−+−+−=

τ

ττττ

ττττ

ττττ

τττττ

( 7 )

where 22 )(,

)(,,

yS

xS

yC

xC y

yx

xyx Δ

tttvtu Δ=

Δ=

Δ=

Δ=

ΔΔΔ αα are the Courant numbers and diffusion

umbers respectively. n The MEPDE of finite difference equation (7) is equivalent to equation (6), and it has the coefficients of the leading (first-order) error terms given by

2,,

2 2,01,10,2yx CtuvCC

yCvxCu Δ=Δ==

his implies that the FTCS 9pt. method introduces numerical diffusion in both the

Δ

x -and y - established by using the Von Neumann method,

where the Von Neumann amplification factor of equation (7) is given by

Tdirections. The stability of equation (7) can be

]1,1[,)(12 −∈=− χχfG

where 7710) χb += (8)

6

65

54

43

32

2( χχχχχχχχ bbbbbbbbf +++++++

h

88

,( ymBxmBhereBBB yyxxyx Δ=Δ=== ππ ,,in whic )cos(B=χ and ( ) Fourier components)and the coefficients

24247

222

23236

221

225

22220

10524.910175.3353.0

10628.310209.1074.7086.1126.32

10702.610175.3826.2413.10811.4

045.0025.0)603.2(248.9)624.4(

CSS

CSbCSSb

CSbCSSb

CSbCSSb

−−

−−

−−

×+×−−=

×−×=−−=

×−×=−+−=

−=+−=

ymxm ,

25268

223 10008.110519.2872.2076.0300.5 CSbCSSb −− ×−×=+−−=

23324b


ere and in the next stability analysis, we setting

H and SSS yx == . CCC yx ==To satisfy the stability criterion of equation (7), we require ,for 11 ≤≤− χ .Thus, If 0)( ≤χf

0=χ , then for 0)0( ≤f ,we have

]20649685435826[0 CS< m 104361211

2)2913(

1260−≤ .

. and if 198.10 ≤< C 1−=χ ,then for 0)1( ≤−fThis is applied to ,yields

]22)630(

5491581416962457613024[2347552

0 S ≤< 630 C+m .

In this section, we use the first-order forward and backward approximation of the first time and spatial derivatives, while the second spatial derivatives are approximated by the eighth-order central difference. Thus, a new formula can be obtained as,

This condition is applied to C > 0. 3.2 The Upwind difference scheme with nine-point formulas (UDS 9pt.)

nkjy

nkjyy

nkj

nkjy

nkj

nkj

ynkj

nkj

y

nkjx

nkjyxyx

nkjxx

nkj

nkjx

nkj

nkj

xnkj

nkj

xnkj

SSC

SSS

SSSCCSC

SSS

1,1,

2,2,3,3,4,4,

,1,,1

,2,2,3,3,4,41

,

12601631)

12601631(

)(5040343)(

420)(

20160

12601631))(

201649351()

12601631(

)(5040343)(

420)(

20160

+−

+−+−+−

+−

+−+−+−+

+++

+−+−+−

++−+−+++

+−+−+−=

ττ

ττττττ

τττ

τττττττ

(9)

of this FDE has the coefficients of the leading (first- order) error with coefficients The MEPDEgiven by

2

)1(2,0,1,1,

2)1(

0,2yCyv

CtuvCxCxuC

−Δ−=Δ=

−Δ−=

mplification factor with

We noted that the coefficients 2,01,10,2 ,, CCC are not equal to zero , therefore the upwind method introduced numerical diffusion in both the x- and y-directions. The application of Von Neumann stability analysis shows that the equation (9) has the

G )(χfa having the same form in equation (8) with the coefficients:

Weighted finite difference method 3385

SbSC

SbSCSb

SbCCSCSb

CSbCSCSb

)149.010175.3(3

10519.2)095.276

10209.1)086.1498.11(126.32

175.384)321.39413.10(810.4

10349.6025.04)248.9575.18()624.4(

32

268

247

22

26

221

2325

2220

+×−

×=+

×=−+=

=−+−+−=

×−=+−+=

−

−

−

−

The stability criterion

086.1498.11(126.32

175.384)321.39413.10(810.4

10349.6025.04)248.9575.18()624.4(

32

268

247

22

26

221

2325

2220

+×−

×=+

×=−+=

=−+−+−=

×−=+−+=

−

−

−

−

The stability criterion

Sb 0.0(300.5 23 −−= 23 −−=

b 35.04 −=4 −= SCS

]2441

277633191363517033942276)582611702[(2)2913(

600 CCCS +−+−≤< m

≤<

12

C for 0 9.0 when , and for 0 < C <0.1 when0)0( ≤f 0)1( ≤−f is

]22)630(

6754598486300 S ≤< 1152052088169624576)1302444948[(2347552

CCC +−+− m

e i

Note: these schemes (FTCS 9pt.) and (UDS 9pt.) are unconditionally stabl f 1=χ for

rmulas (WFDS 9pt.) e, when discretizing equation (4) at the grid point ,( , we used

all S,C > 0 . 3.3 Weighted finite difference scheme with nine-point fo To get this schem , thnkj )weights 1,0 ≤≤ γφ in the approximation

CSBS yx ∂

CSBS ×−+×≈∂

×−+×≈ )1(,)1( γγ∂∂ τφφτ

with the forward time difference approximation for time derivative and eight-order

central-space approximation for 2

2

x∂

∂ τ and

2

2

y∂

∂ τ the resulting (WFDE 9pt.) was

(10)

nkj

yynkj

yynkjyy

nkjyy

nkjyyy

nkjyy

nkj

yynkj

yynkj

xx

nkj

xxnkjxx

nkjxx

nkjyxyx

nkjxxx

nkjxx

nkj

xxnkj

xxnkj

SCSCSC

SCSCCSC

SCSCSC

SCSCSC

SSCCSCC

SCSCSC

4,3,2,

1,1,2,

3,4,,4

,3,2,1

,,1

,2,3,41

,

)2016010080

)1(()

420280)1(

())1((5040343

)2)1((25201631)2)1((

25201631())1((

5040343

)420280

)1(()

2016010080)1(

()2016010080

)1((

)420280

)1(())1((

5040343)2)1((

25201631

))(201649351())2)1((

25201631(

))1((5040343)

420280)1(()

2016010080)1((

+++

+−−

−−+

+++

−

−−−+

−−

+−−

+−−+

−−−+−+++−−

+−

−+−

−−−

+

−−

+−−+−−−

+−−−++−++

+−−+−

−+−

−=

τγ

τγ

τγ

τγτγγτγ

τγ

τγ

τφ

τφ

τφτφ

τγφτφφ

τφτφ φ

τ τ

3386 A. S. J. Al-Saif and M. O. Al-Humedi The finite diequation (6).

fference equation has MEPDE with coefficients, which are the same as those of Therefore, equation (10) has the first-order truncation error with coefficients

2

)(2,0,1,1,

)(0,2

yCyvCtuvCxCxu

C2

−Δ−=Δ=

−Δ−=

γφ

diffusion that numerical introduced by the first and last of these equations may be CandC

The eliminated by setting x y== γφ , but there rem involving ains a first-order error term

yx∂∂∂ τ . The V

2on Neumann amplification factor with G )(χf has the same form in equation (8)

with the coefficients

223236

2322325

2223324

22223

22222

22222221

2222222220

)519.2

)2(10675.110175.3

))(2(10175.3)2(011.0)(10175.3025.0

))(2(057.0)2(10381.2))(075.010175.3(353.0

))(2(539.0)2(718.0))(048.1076.0(017.53

))(2(603.26)2(769.1)086.1)(749.5(126.32

)(2)(2)()2(542.0)2(707.0))(660.19413.10(810.4

)(2))(2(603.2)2()302.1()248.9)(287.9()624.4(

γφ

γφγφγφγφ

γφγφγφγφ

γφγφγφγφ

γφγφγφγφ

γφγφγφγφγφγφ

γφγφγφγφγφ

×−

−−×−×=

+−−×+−−−+×−=

+−−+−−×+++×−−=

+−−+−−++−−−=

+−−−−−−−++=

+−+++−−−−−−+−+−=

+++−−+−−+−++=

−−

−−

−−

−−

CSb

CCCSSb

CCSCSb

CCSCSb

CCSCSb

CCCCSCSb

CCCSCSb

y the stability criterion of equation (10), we require

226268

7

2(1010519.2

)2(10070.910209.1

γφ

γφ

−−×=

−−×−×=−− CSb

CS 22524b

To satisf for 11 ≤≤− χ0)( ≤χf . Thus if x = 0, then for , we have 0)0( ≤f

]4441

3320615173441

47006280260232536942276)258[(212600 CS +−≤<

441331170226

)2913(CCC +− m

o 20 ≤<C . If 1−=xThis is applied t , then for 0)1( ≤−f we have

m)4494813024[(2347552

6300 2CS −≤<

]630)630()630()630(

this condition is applied to 77.00 ≤<

1284580456422212125181416902169624576 23

22

2 CCC −+++13191938 54C

C . Note: thes schem s (WFDS 9pt.) are unconditionally stable if 1e e =χ for all S,C > 0 .

ow this equation reduces to some well-known finite difference equation when it contains alues of

So, we can say that equation (10) is a generalization to the pervious schemes. We show now hv φ and γ are chosen. If substituting 0== γφ ,in equation (10),we


et (FTCS 9pt.)equation (7).If we put

in equation (10) UDS 9pt. is obtained, g 1== γφequation (9). If C== γφ , we get weighted Difference scheme of nine points.

To obtain FDMs. of greater accuracy, more grid points must be used in order to introduce erro

four extra grid points

3.4 Modified weighted finite difference scheme (MWFDS. 9pt)

weights so that more terms in the truncation r of the MEPDE may be eliminated. Adding ),1,1(,),1,1(,),1,1(,),1,1( nkjnkjnkjnkj ++−++−−−

s involving

on the stencil

of Figure (1b) and the error termyx∂∂

∂ τ2 in the MEPDE of (10) may be eliminated by

using the approximation

)(4

11,11,11,11,1

,

2n

kjn

kjn

kjkj

n

kj yxyx −+−−+++− −+ΔΔ

≈∂∂

∂ τττ

(11)

22

n +− ττ

ch has a truncation error of )(,){( yxO ΔΔ . The method of obtaining a more accurate error term

ional case by [5]. Consider the finite difference equation (10). Written in the form

(12)

(13)

, aandaaaaaa −−−− are e corresponding coefficients in (10). Using the results of MEPDE in (10) with

whi }FDE through the replacement of s in the MEPDE has been described for the one-dimens

0][ , =Δn

kjL τwhere

in which ,,,,,,,,,, aaaaaaaaaa −−−−

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

nkj

aaaaaa

aaaaaa

aaaaaL

4,4,03,3,02,2,01,1,01,1,02,2,0

3,3,04,4,0,0,0,40,4,30,3,20,2

,10,1,10,1,20,2,30,3,40,41

,, ][

++++−−−−

−−−−+++

+−−−−−−−−+

Δ

−−−−−−

−−−−−

−−−−−−≡

ττττττ

ττττττ

τττττττ

4,03,02,01,01,02,03,04,00,00,40,30,20,10,10,20,30,4 ,,,,φ=xCth and

γ=yC the alternative differential form may be written as;

n

kjyx

nkj yx

tuvyxt

tL 22, ][][ L+∂∂

Δ+∂

−∂

−∂

Δ≡Δ αατ (14y

vx

u,

222 ∂∂∂∂∂

+∂∂

+∂ ττττττ

)

term

yx

tuv∂∂

∂Δ

τ22)(Subtracting the error from (14) is equivalent to subtracting from (15) the

1,11,11n

kjn

kj −+−− −+ ττ

earrangement gives the second-order FDE

term ( ,11,1

nkj

nkjyxCC +++− +− ττ 4/)

The largest error terms in the brackets on the right-hand side of (14) are now }2)(,2){( yxO ΔΔ . R


−=

++−=

+ =4

4,

4

4

1,

p

nkpj

q

nkj ττ (15)

qp except and

he MEPDE of equation (15) contain first-order error terms and has the following oefficients of the second-order error terms:

∑ , qqpa ∑ In which a ,4)1(40, −=== qpfor qp aa ,00, , 4/),(),(),(),( 1,11,11,11,1 yxyxyxyxyx CCCCaCCaCCaCCa ===−= −−−−

T s noc

yyyCyCyCvyC αΔ−−Δ

= )2)((6

2)(3,0 C yyCxyCx=3,1 Δ Δ α

)(2

2)(2,1 yCuy

yxxCC Δ−Δ−= α

)(20,4 xxC

ux

xCxxC Δ+

Δ=

αα

2

xyCyyxCx

yxyCxCuxC αααα 2)(

2

2)(2)(2

2)(2,2

Δ+

Δ+

Δ=

2)(2

2)(1,2 xCvx

xyyCC Δ−Δ−= α xyCxyCxC αΔΔ=1,3

)(24,0 v

C ΔΔ

= yy

yy yCCy+

αα xxxCxCxCuxC αΔ−−

Δ= )2)((

6

2)(0,3

Setting SSS yx == , CCC yx == and carrying out a numerical stability analysis yields the

f these sc

NWSWSE . ay be an implicit

e function value at the adjacent points to it when

stability of second order weighted difference scheme which is identical to the stability of WFDS 9pt. equation (13). 4. Adjacent boundary domain When applying the difference schemes which depend on formulas of the nine-point to solve the two-dimensional transport equation (1), we find that there are difficulties in computing the values of the numerical approximation for the transport variable T(x, y, t) at the adjacent points of boundaries Figure (2). We cannot use the formulas o hemes directly to compute these values, so the method of computing the approximate values of the transport variable in equation (1) has been done on two stages, the first stage is done by using the formulas of schemes explained previously (for the bounded area by ENWS ,,, ), whereas the second stage is by computing the approximate values for the transport variable at the adjacent points of the boundaries in ,,,,( NSWSEN ),,, It is suggested that there mscheme resulting as an idea that the value of the function at any point is equal to the average of th yx Δ=Δ .This scheme will be illustrated in

e following algorithms. th


ts of boundary

rithms for computation of adjacent boundary points

For region of boundary layers

jdoWElayersofgridatfunctionvaluescomputeTo

MkdonnleveltimetheFor

kjkj

nkjM

nkjM

nkjM

nkj

nkj

nkj

2/)

2:21,2,3

&4.,,.........5,4

......2,1,0;1

1,1,1,1

1),2(

1),1(

1,

1,2

1,1

1,

2

111

+−+

++−

++−

+−

++

++

+

+

−=−=

−=

−==+

τ

ττττττ

For region of boundary layers at the corners

enddo

jdoNWSWlayersofgridatfunctionvaluescomputeTo

kdonnleveltimethe

y

Figure (2) Stencil of Adjacent poin Algo

enddo

Mjorjforenddo

nn (:11 11 ++ =−== ττ nkj

enddo

Mjorjfor

For

nnn

nkjM

nkjM

nkjM

nkj

nkj

nkj 2:2

1,2,3&

1,1,3......2,1,0;1

111

1),2(

1),1(

1,

1,2

1,1

1, 111

+kjkjkj 2/)(:11 ,1,1,1 −

++

+

++−

++−

+−

++

++

+ −=−=

−=

−==+

ττττττ

:WandE

:NWandSW

+=−== τττ

S

N

W E1+ntimelvelforfunctionvalues

NE

SESW

Known

NW

31−M 21−M 11 −M 1M

M

M

2M

1

22 −

12 −M

32 −

1

2

2

3

3

00 x

3390 if and M. O. Al-Humedi A similar procedure can be used for other regions SEandNESN ,,, . These algorithms were applied to compute the adjacent points of the boundaries in the ni

A. S. J. Al-Sa

ne-point (weighted and non- eighted) finite difference schemes .These schemes can be regarded as general ones that can be

pplied to treat the same problems that occur when the five-point finite difference schemes are to treat other schemes.

e method, which depend on the formulas of the nine-

oint (with weighted and without weighted) to solve unsteady two-dimensional convection –em, in order to demonstrate the validity and effectiveness of this method. sented for two problems.

waused . This procedure may be a useful tool

5. Discussion of numerical results

In this section, we test our differencpdiffusion probl

esults are preR Problem 1. The first test for our pr roblem in the unit square [ ]opose method is the diffusion p [ ]1,01,0 × , which is obtained by setting 1== yx αα and 0== vu in the equation (1). The exact solution of

this problem is given by; )cos()sin(),,(22 yxetyxu t πππ−= . This problem is given in[12].The

initial and boundar itions(2-3) are directly taken from this solution. Computations were carried out at different times over the problem domain [0,1]. The results are documented in Table 1 and 2. 2L norm error of our method is documented at 125.0

y cond

=t in table 1 for various mesh size, and at 25.0=t in table 2 and figure 3 for various time step size. We conclude that the accuracy of present method is increased with increasing number of gird points in space and levels in time respectively. The present simulation exhibited more accurate results comparison to the rival Tain and Ge method [12]. Figure 4 show that the FD9pt. method has a good solution agreement with the exact solution. All comparison shows that the current method offers better results than the other methods.

0.0 0.2 0.4 0.6 0.8 1.0

0.00E+000

5.00E-013

1.00E-012

1.50E-012

2.00E-012

2.50E-012

3.00E-012

L 2 nor

m e

rror

xFigure(3)L2norm errors at diagonal of a square region [0,1]x[0,1] for t=1.25

003125.0=Δt 0025.0=Δt 00625.0=Δt


(a) Exact solution with grids (b) Numerical solution with grids ith um rical solution with grids

Ta r a

Figure(4) Comparison between exact solution and approximation solution at t=0.125, contours plots of; 1111× 1111× (c) Exact solution w grids (d) N e4141× 4141×

ble-1:- 2L norm erro t 125.0=t with 2)( xt Δ=Δ Grid Tain&Ge [12] FD9pt. method 2L norm error 2L norm error 11x11 8.55134E-05 3.066732E-08 21x21 5.19160E-06 1.142410E-09

41 3.17475E-07 2.777872E-10 41x Table-2:- 2L norm error at 1.0=Δ=Δ yx , 25.0=t

tΔ Tain&Ge [12] FD9pt. method 2L norm error 2L norm error 0.0025 2.64692E-05 4.017735E-12 0.00625 5.40813E-06 1.032878E-12

7.18040E-07 2.515195E-13 0.003125 Problem 2.

c solutio n [1,12,8] of the problem is applied to this numerical test given by An analyti n i

),,( tyxT = 0,))5.0()5.0(exp(1≥

−−−

−−− tvtyutx

)14(

2

)14(

2

14 +++ tytxt αα

0.0050

-0.014

0.024

-0.033

0.043

-0.052

0.062

-0.071

0.081

(d)

0.0050

-0.014

0.024

-0.033

0.043

-0.052

0.062

-0.071

0.081

(c)

0.0050

-0.014

0.024

-0.033

0.043

-0.052

0.062

-0.071

(a)

0.0050

-0.014

0.024

-0.033

0.043

-0.052

0.062

-0.071

(b)


small and in the MWFDS 9pt.scheme there is no rical diffusio

The initial and boundary conditions (2-3) are directly taken from this solution. It is noticed that when comparing the altitude of Gauss pulse, it has the largest numerical approximation to T(x, y, t) with Gauss pulse height Figures (5 a-f), explained also by contour drawings of the Figures (6 a-f). The WFD 9pt. and MWFDS 9pt. produces an approximate value identical to the analytic value and better than the schemes FTCS 9pt. and Upwind 9pt. The main reason for this can be attributed to the fact that the effect of the numerical diffusion in WFDS 9pt. scheme is nume n. Table (3) explains the average error and the maximum error for the nine-point finite difference schemes at time 25.1=t , 8.0== vu , 2.0== yx CC , 1.0== yx SS and 00625.0=Δt . Figures (5,6 ), and Table (3) show that the weighted modified difference schemes for the nine-point (MWFDS 9pt.) are better than other schemes because of their second-order accuracy, and

igure(5) The surface plots of (a) exact solution ,(b) (MWFD 9pt.), (c)(WFD 9pt.

their having absolute error less than the other schemes since the value of the numerical solution is approximate to that of the analytic one Figures (5 a,b)

F C== γφ ), (d) (FTCS 9pt.), (e) (Upwind 9pt.) and (f) (WFD 9pt. 5.0== γφ ) at in plane [1,2] 25.1=t

1 .0

1 .5

2 .0

0 .0 0

0 .0 2

0 .0 4

0 .0 6

0 .0 8

0 .1 0

0 .1 2

0 .1 4

0 .1 6

0 .1 8

1 .0

1 .5

2 .0

(a )

Tra

nspo

rt va

riabl

e (T

)

Y

X

1 .0

1 .5

2 .0

0 .0 0

0 .0 2

0 .0 4

0 .0 6

0 .0 8

0 .1 0

0 .1 2

0 .1 4

0 .1 6

0 .1 8

1 .0

1 .5

2 .0

(b )

Tran

spor

t var

iabl

e (τ

)

Y

X

1 .0

1 .5

2 .0

0 .0 0

0 .0 2

0 .0 4

0 .0 6

0 .0 8

0 .1 0

0 .1 2

0 .1 4

0 .1 6

0 .1 8

1 .0

1 .5

2 .0

(c )

Tran

spor

t var

icbl

e (τ

)

Y

X

1 .0

1 .5

2 .0

0 .0 00 .0 2

0 .0 4

0 .0 6

0 .0 8

0 .1 0

0 .1 2

0 .1 4

0 .1 6

0 .1 8

0 .2 0

0 .2 2

1 .0

1 .5

2 .0

(d )

Tran

spor

t bar

iabl

e (τ

)

Y

X

1 .00 .0 0

0 .0 2

0 .0 4

0 .0 6

0 .0 8

0 .1 0

0 .1 2

0 .1 4

0 .1 6

0 .1 8

1 .5

2 .0 1 .0

1 .5

2 .0

(e )

Tran

spor

t var

iabl

e (τ

)

2 .0

0 .0 2

0 .0 4

0 .0 6

0 .0 8

0 .1 0

0 .1 2

0 .1 4

0 .1 6

0 .1 8

Y

X

1 .0

1 .5

0 .0 0

1 .0

1 .5

( f )

Tran

spor

t var

iabl

e (τ

)

2 .0

Y

X


igure(6) The contours plots of (a) exact solution ,(b) (MWFD 9pt.), (c)(WFD 9pt.

0.0180.0360.0540.0720.0900.110.130.14

(a)

0.0180.0360.0540.0720.0900.110.130.14

(b)

0.0180.0360.0540.0720.0900.110.130.14

(c)

0.0220.0440.0660.0880.110.130.150.18

(d)

0.011

0.0220.0330.0440.0550.0660.0770.088

(e)

0.0140.0280.0420.0560.0700.0840.0980.110.13

(f)

C== γφF ),(d) (FTCS 9pt.), (e) (Upwind 9pt.) and (f) (WFD 9pt. 5.0== γφ ) at in plane [1,2]

have first-

25.1=t and this is clear from contour drawings as Figures (6 a and b), Furthermore , when compared to the other schemes found in this table with MWFDS 9pt., it is noticed that all these schemes

order error accuracy . Generally, it is clear that one kind of nine-point schemes, is identical in identifying that the absolute error value and the WFDS are the most accurate when

C== γφ and they have less error than the other schemes in terms of average absolute error a ximum absolute error.Table (4) explains the e y of these schem for solving the two-dimensional transport equation (1) when taking different values for Reynolds number (

nd ma fficienc es

SCR = ) following its difference in dimension steps yx ΔΔ , and time step tΔ , noticing that

the accuracy of these schemes on the behavior of the numerical solution becomes better gradually through values changeability of these parameters.


ge absolu

Figures (7 a, b) show the drawing of avera te errors curves against the time for every new FDS suggested in this study. From Figure (7 a), we notice that the error average of WFDS 9pt. for all the time-level with the weight C== γφ is better than the other schemes which are

he nine-point formula. Figure (7 b) explains the drawing of average absolute error for the MWFDS 9pt. From this Figure, we can at the MWFDS 9pt. with the weight has

C==

based on th notice t

γφ less error average for all the time-levels than the other schemes. Generally, it is bserved that FDS based on the weights C== γφ o has less errors average than the other chemes.

T

s

able-3:- Error measurements at 25.1=t with 00625.0=Δt 8.0== vu , and 2.0== yx CC The Method Average |error| Maximum |error| Type of formula weight values Upwind 9pt. 9.787177E-05 4.420168E-03 orderfirst − ----- FTCS 9pt. 5.826575E-05 1.418220E-03 orderfirst − ----- WFDS 9pt 4.925594E-05 1.067123E-03 orderfirst − φ =γ =0.5 2.794456E-05 1.607237E-04 orderfirst − φ =γ =C MWFDS 9pt. 9.054745E-06 2.557348E-05 orderSecond − φ =γ =C

T

able-4:- Error measurements at 25.1=t with different yxandtcR Δ=ΔΔΔ= s=,

The Method Average |error| Maximum |error| tΔ Δ R Upwind 9pt. 9.787177E-05 4.420168E-03 0.00625 0.025 2 2.828835E-04 8.997762E-03 0.0125 0.05 4 7.431901E-04 1.225278E-02 0.025 0.1 8

3.586615E-03 1.669250E-01 0.025 0.1 8

5.630792E-04 3.575699E-03 0.025 0.1 8

2.819292E-04 1.335344E-03 0.025 0.1 8 1.392667E-03 1.966503E-02 0.0375 0.15 12

1.661878E-03 2.684112E-02 0.0375 0.15 12 FTCS 9pt. 5.826575E-05 1.418220E-03 0.00625 0.025 2 3.283737E-04 1.442082E-02 0.0125 0.05 4 3.586615E-03 1.669250E-01 0.0375 0.15 12 WFDS 9pt. 2.794456E-05 1.607237E-04 0.00625 0.025 2 1.128239E-04 5.783730E-04 0.0125 0.05 4 1.378246E-03 1.946828E-02 0.0375 0.15 12 MWFDS 9pt. 9.054745E-06 2.557348E-05 0.00625 0.025 2 3.419029E-05 1.195652E-04 0.0125 0.05 4


umber of w ghts given tables (5) andTable (5) showspoints formula

6. Comparisons with the results of other researchers Confidence in the present results is gained by the comparison of the results obtained using the present numerical methods with those previously published in the literature. These comparisons are given in the study of error measurements (average absolute error and maximum absolute error), type of formula and n ei in (6).

a comparison of the explicit finite difference schemes of three , five and nine kind at time 25.1=t , when 0125.0=Δt , 8.0== vu , 4.0== CC and yx

01.0== yx αα . This table reveals that the results of MWFD 9pt. and WFDS 9pt. are better in accuracy than the other resea es . Table (6) illustrates a comparison of explicit FDS and implicit for a formula including three, five and nine points when 00625.0=Δt , 2.0

rcher's schem

== yCxC , noticing that the MWFD 9pt. is better than the others in accuracy. The new suggested schemes in this study are better in the

easurement errors than that of the traditional and suggested schemes by other researchers. ased on these results, our conclusions are made in the next section.

e (7)Comparison between average absolute error for (a) WFD 9pt. (b) MWFD 9pt.

mB

Figur

0.00 0.25 0.50 0.75 1.00 1.250.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.00 0.25 0.50 0.75 1.00 1.250.00000

0.00002

0.00004

0.00006

0.00008

0.00010

0.00012

0.00014(a)

Time level

(b)

φ=γ=1.0 φ=γ=1.0 φ=γ=0.5

φ=γ=0,0 φ=γ=c

Aver

rage

|erro

r|

Time level

φ=γ=0.5 φ=γ=0.0 φ=γ=C

Avar

eage

|err

or|


T

able-5:- Errors at 25.1=t with 0125.0=Δt , 8.0== vu , and 4.0== y x CCThe Method Average |error| hts Maximum |error| No. of weig FTCS 3pt. 3.94E-03 1.12E-01 -----

WFDS 9pt. 3.17E-04 5.47E-03 2 MWFDS 9pt. 7.31E-05 3.53E-04 2

FTCS 5pt. [1] 2.22E-03 8.00E-02 ----- WDF 5pt. [1] 3.88E-04 3.64E-03 4 Noye [6] 3.33E-04 6.03E-03 2

Table-6:- Errors at 25.1=t with 00625.0=Δt 8.0== vu , and = 2.0= yx CC

The Method * Average |error| Maximum |error| No. of weights Upwind 3pt. 2.65E-03 6.63E-02 ----- Upwind 5pt.[1] 2.62E-03 6.69E-02 ----- ADI [11] 9.22E-06 5.93E-05 ----- EC-ADI [12] 9.66E-06 6.19E-05 -----

-

WFDS 9pt. 2.79E-05 1.61E-04 2 2.56E-05 2

Dehghan&Mohebbi[8] 9.48E-06 2.47E-04 ---- Noye [6] 1.43E-05 4.84E-04 8

MWFDS 9pt. 9.05E-06 * The m thods in [6,11,12,8] are implicit type.

:

e

7. Conclusions

From the numerical results, we can conclude the following

The WFDS have been used successfully with γφ and to introduce explicit new schemes with high accuracy and the results were better at error measurements. It is preferable to choose ideal values for the weights to give us numerical results with high accuracy to solve the two-dimensional transition equation compared with the other schemes explained in Figures(7) and Tables (5)and (6).The numerical accuracy depends on the number of the selected points. The tables and Figures show that increasing the number of the grid points gives more accurate results. We notice that’s the MPDE method has been successfully used with weighted to develop several new explicit finite difference method for solving transport equation, and the use of the modified equivalent equation permits a proper determination of the accuracy order of the finite difference method. We have developed an improved finite difference schemes with weighted by the MEPDE, since it is allowed to apply two equations of

nite equation which have the same order to give high order and more accurate schemes. fiNumerical treatment for the adjacent boundary is useful and successful to handle difficulties


d on nine-point formula with weights

which appear through using new methods for solving two-dimensional transport equation. Moreover, it is applicable to obtain excellent results in the accuracy and the stability. The new explicit FDS base ),( γφ have produced better results compared to other results of other researchers. Further study is apply WFDS to solve Navier-tokes equations.

cknowledgment:

s

A The authors would like to thank prof. Majeed H.Jasim (Ph. D in English language) for his valuable comments through review of the grammar of manuscript linguistically.

g five-point formulas solving two-dimensional transport equation, M.Sc thesis in Mathematics,

aif, Improved of modified finite difference schemes for solving a eady-state two-dimensional heat transfer problem, J. Basrah Research,

aif, A fourth order accurate weighted difference method for time-pendent one-dimensional transport equation, Basrah J. Science, A,

] A.S.J. Al-Saif and A.J.Mohmmad, Finite difference methods involving five-point

B. J. Noye, Computational Techniques for Differential Equation, (North-Holland

] B. J. Noye, Finite difference methods for solving two-dimensional advection-

] H.O.Al-Humedi and A.S.J. Al-Saif, Finite difference scheme involving nine-

compact boundary value method for e solution of unsteady convection–diffusion problems,mathematics and

ansport equation, Anziam J.,42(2000), 1179-1198.

References

[1] A.J. Al-Rahmani, New finite difference schemes involvinforMathematics Depart ment,Basrah University,IRAQ,1999. [2] A. S. J. Al-Sst16(1998),72-85. [3] A. S. J. Al-Sde21(2003),7-15. [4formula for convection-diffusion equation, Basrah J. Science,A, 16(1998),87-94. [5]and Mathematics studies; 83) Amsterdam, the Natherlands 1984. [6diffusion equation, Int. J. Meths. In Fluid, 9(1989),75-98. [7point formula for transfer equation, Mustenseria J. Science, A,13(2002),59-75. [8] M. Dehghan and A.Mohebbi, High-orderthcomputer in simulation, 79(2008),683-699. [9] R. J. May, An Implicit finite difference approximation to one-dimensional tr


ighted ifference schemes for solving one-dimensional heat transfer , M.Sc thesis in

d Computer imulation, Department of Computer Science University of

.B.Ge, A fourth-order compact ADI method for solving two-imensional Unsteady convection-diffusion problems, J. Comput. Appl. Math., 98 (2007),268-286.

eceived: July, 2010

[10] S. A., Hedaad, Improved five-point formula by using two-level wedMathematics, Mathematics Department, Basrah University, IRAQ,1998. [11] S. Karaa, A high-order ADI method for solving unsteady convection-diffusion problems, Laboratory for high Performance Scientific Computing anSKentucky,773AndersonHall, Lexington,KY 40506-0046, (2003), USA. [12] Z. F. Tain and Yd1

R

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