Applying the differential equation maximum principle with cubic spline method to determine the error...

International Communications in Heat and Mass Transfer 37 (2010) 147–155

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International Communications in Heat and Mass Transfer

j ourna l homepage: www.e lsev ie r.com/ locate / ichmt

Applying the differential equation maximum principle with cubic spline method todetermine the error bounds of forced convection problems☆

Chi-Chang Wang ⁎Department of Technology Management, Hsing Kuo University of Management, No. 89, Yuying St., Tainan, Taiwan 709, People's Republic of China

☆ Communicated by: Dr. W.J. Minkowycz⁎ Corresponding author.

E-mail address: ccwang123@mail.hku.edu.tw.

0735-1933/$ – see front matter © 2009 Elsevier Ltd. Aldoi:10.1016/j.icheatmasstransfer.2008.11.017

a b s t r a c t

Available online 13 October 2009

Keywords:Maximum principleError analysisResidual correctionForced convectionCubic spline

This article uses the concept of differential equation maximum principle as well as the technique of virtualtime to establish the solution's monotonic relation with residual of forced convection. To obtain error boundsof approximate solution, the article first uses cubic spline approximation to discretize the differentialequation, then applies “residual correction method” newly put forth by it to convert the once complexinequation constraint mathematical programming problem into a simple problem of equation iteration.What is more, not only the obtained upper and lower approximate solutions of the differential equation cancorrectly analyze error range, but also it is found that the new method helps increase the accuracy of meanapproximate solutions.

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© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In the analysis of heat transfer and fluid flow, various kinds ofdifferential equations and systems of coupling differential equationsare commonly used. However, to date, only a limited number ofsimple or special boundary value problems can obtain exact solutionsof differential equation, while others can only get approximatesolutions of the differential equation(s) through approximate meth-od. A review of research methods in the past will show us that manymethods, such as finite volume method (Capm et al. [1] and Pan andChang [2]), finite element method (Bruch and Zyvoloski [3]),boundary element method (Ramirez et al. [4]), hybrid numericalmethod (Chu and Chang [5]) and decomposition method (Chiu andChen [6]), have been applied to many heat transfer and convectionproblems successfully. The previous researches show the errorbetween approximate solution and exact solution usually reduceswith the increase of grid points or numbers of approximate function,but it has to sacrifice more computer internal memory and morecalculation time. More importantly, it is unable to fully determine theaccuracy of approximate solutions. For this reason, another theorybased on maximum principle of differential equation problem(Protten [7]) was put forth to estimate the error range betweenapproximate solution and exact solution (Finlayson and Scriven [8]).Nevertheless, this method has to obtain the optimal solution ofinequation, resulting in rather complex and time-consuming calcu-lation. As far as I know, only Chang and Lee [9] and Wang [10] did

correlative researches by genetic algorithms method and residualcorrection method, recently.

This article tries to propose a new residual correction method toovercome traditional problems in solving mathematical programmingunder inequation constraint condition. Its idea is to first use splineapproximation to discretize differential equation, then use iterationtechnique of residual correction and the concept of virtual time (ifneeded) to covert the once complex inequation constraint problemsinto simple problems of equation iteration. For numerical testing, asimple differential equationwill be utilized to elaborate on how to carryouterror analysis on exact solutions atfirst, followedby further applyingand disseminating it to heat transfer problems of forced convection.

2. Problem formulation for lower and upper solutions

2.1. The maximum principle of the differential equation

In order to obtain the upper and lower solutions of a differentialequation, in most cases, it has to rely on maximum principle of thedifferential equation to establish a solution's monotonic relation withresidual of the differential equation. According to this principle, anycontinuously differentiable function that satisfies the inequality f″(x)>0 on an interval must reach the maximum value of the function f(x) atone end point. More generally, if a function satisfies a differentiableinequality on a domain and reaches its maximum value on the domainboundary, we can say that this differentiable equation satisfies themaximum principle and has monotonicity. To describe the contents ofthis theory, let's consider a simple nonlinear boundary value problem inthe following form:

RðxÞ = u ″ + Hðx;u;u ′Þ; x ∈ ða; bÞ ð1aÞ

Nomenclature

Ea(x) actual errorEmax(x) maximum possible errorf variableNi maximum number of grid pointsPr Prandtl numberR residualx axialu dimensionless velocity

Greek symbolsθ dimensionless temperatureη dimensionless axialτ virtual time

Superscriptsm iteration timesn virtual time index− mean approximate solution~ approximate function∩, ∪ upper and lower approximate solutions^ value of previous iteration

Subscriptsi serial number of calculation grid points

148 C.-C. Wang / International Communications in Heat and Mass Transfer 37 (2010) 147–155

with boundary conditions:

Rðx = aÞ = u ′ðaÞ cos θ−uðaÞ sin θ + γa ð1bÞ

Rðx = bÞ = u ′ðbÞ cosϕ−uðaÞ sinϕ + γb ð1cÞ

where 0≤θ≤π/2, 0≤ϕ≤π/2 and they are not zero at the same time.Function R(x) is called residual or residual function of the differentialequation when x∈(a, b) or on boundaries. On assumption thatfunction u(x) is the exact solution that makes the residuals ofEqs. (1a)–(1c) all satisfy with zero, and the approximate functionsu(x) and ŭ(x) have definition within x∈(a, b) and continuous

derivatives till the second order, if H,∂H∂u ,

∂H∂u′

are continuous and

∂H∂u ≤ 0 ð2Þ

then, when

Rðx; ⌣u; ⌣u ′; ⌣u ″Þ ≥ Rðx;u;u′;u″Þ = 0 ≥ Rðx; ⌢u; ⌢u ′; ⌢u ″Þ; x ∈ ða; bÞ ð3aÞ

Rbðx = a; ⌣u; ⌣u ′Þ ≥ Rðx = a;u;u ′Þ = 0 ≥ Rbðx = a; ⌢u; ⌢u ′Þ ð3bÞ

Rbðx = b; ⌣u; ⌣u ′Þ ≥ Rðx = b;u;u ′Þ = 0 ≥ Rbðx = b; ⌢u; ⌢u ′Þ ð3cÞ

hold, the following relations will also hold (Protten [7])

⌣uðxÞ ≤ uðxÞ ≤ ⌢uðxÞ ð4Þ

in which approximate solutions ŭ(x) and u(x) are called the lower andupper solutions of the exact solution, while any differential equationwith such relation is consideredmonotonic in solutions. In addition tothe basic descriptions given above, for looser discrimination methodsfor monotonicity, please refer to discussions made by Wang and Hu[11].

Although the maximum principle for differential equations can beadopted to build a commonmonotonic relation like Expression (4), twodifficult issues still exist when this principle is applied practically; one isthat approximate functions must be supplemented with the character-istics of continuity and differentiability, and the other is how to quicklyfind the approximate solutions that satisfy inequalities with constraintconditions, as shown in Expressions (3a, 3b, and 3c). Discussions in suchrespect will be given in next two sections respectively.

2.2. Residual correction method for boundary value problem

The traditional method to obtain approximate solutions of adifferential equation is nothingbut discretizing the differential equationinto algebraic equation or using trial function as its approximatesolution. However, such methods can't fully satisfy the differentialequation, so residual of a differential equation was proposed. For theinfluence of residual on approximate solutions, in this article, if theequation is monotonic and the residual of approximate function ũ(x)satisfies constraint conditions

Rðx; u; u′; u″Þ ≥ 0; a ≤ x ≤ b ð5aÞ

Rbðx = a; u; u′Þ ≥ 0 ð5bÞ

Rbðx = b; u; u′Þ ≥ 0 ð5cÞ

then, an optimal objective function (approximate value) can be found

⌣uðxÞ = Maxð uðxÞÞ ð6Þ

and it can be guaranteed that approximate function ŭ(x) is the optimalapproximate solution smaller than or equal to the exact solution. On theother hand, if the residual of function u(x) satisfies constraint conditions

Rðx; u; u′; u″Þ ≤ 0; a ≤ x ≤ b ð7aÞ

Rbðx = a; u; u′Þ ≤ 0 ð7bÞ

Rbðx = b; u; u′Þ ≤ 0 ð7cÞ

then, a corresponding optimal objective function can be found, too,

⌢uðxÞ = Minð uðxÞÞ ð8Þ

making the approximate function u(x) as the optimal approximatesolution that is greater than or equal to the exact solution. Although theacquisition of upper and lower approximate solutions are good foraccuracy and credibility analysis of a solution, owing to the tremendouscomplexity and difficulty in extracting the optimal solution of mathe-matical programming problems under such constraint conditions, thetheory of such kind has stopped at theoretic studies for quite some timeand failed to solve complex problems in practical application. For thisreason, the article puts forth the concept of residual correctionmethod bywhich the once inequation mathematical programming problems aresimplified into equation problems similar to traditional differentialequations. The concept is to use iteration technique to discretizeExpressions (5a, 5b, and 5c) or (7a, 7b, and 7c) and rewrite inequationsinto the following equations:

Rmi = u″m + 1

i + Hðxi;um + 1i ;u′m + 1

i Þ; i = 1;2;3; :::;Ni−1 ð9aÞ

Rm0 = u′m + 1

i cos θ−um + 10 sin θ + γa ð9bÞ

RmNi

= u′m + 1Ni

cosϕ−um + 1Ni

sinϕ + γb ð9cÞ

where the superscriptm is iteration times, subscript i=0, 1, 2, ...,Ni is theserial number of discretized calculation grid points, and Ri

m are residual

149C.-C. Wang / International Communications in Heat and Mass Transfer 37 (2010) 147–155

correction values on calculation grid points used to correct the residualvalues on calculation gridpoints andensure that the residual values of theneighboring subinterval of i grid point (xi−1≤x≤xi+1) are all positive ornegative. The complete process for residual correction is as follows:

1. Assume the residual on each grid point Ri0=0;2. Use Eqs. (9a)–(9c) to obtain new values uim+1, u′im+1 and u″i

m+1;3. Estimate Ri

m+1 required for next iteration according to the residualvalue of the neighboring subinterval of i grid point (xi−1≤x≤xi+1).Its residual correction relations are as follows:(i) When extracting lower approximate solution,

Rm + 1i = Rm

i −MinðRðxÞÞ; xi−1 ≤ x ≤ xi + 1 ð10aÞ

Rm + 10 = Rm

0 −Min ðRðxÞ;RbðaÞÞ; a ≤ x ≤ x1 ð10bÞ

Rm + 1Ni

= RmNi−Min ðRðxÞ;RbðbÞÞ; xNi−1 ≤ x ≤ b: ð10cÞ

(ii) When extracting upper approximate solution,

Rm + 1i = Rm

i −MaxðRðxÞÞ; xi−1 ≤ x ≤ xi + 1 ð11aÞ

Rm + 10 = Rm

0 −Max ðRðxÞ;RbðaÞÞ; a ≤ x ≤ x1 ð11bÞ

Rm + 1Ni

= RmNi−MaxðRðxÞ;RbðbÞÞ; xNi−1 ≤ x ≤ b ð11cÞ

where R(x) is the residual function obtained by putting the functionand its derivatives obtained from the previous step into Eq. (1a);4. Advance to next m value and repeat Steps 2 and 3, till all residual

values on the intervals are smaller or larger than zero.

Moreover, the correction procedures as indicated above onlyinvolve residual correction. For new solutions to be sought in Step 2,any numerical method for solving equations may apply.

2.3. Cubic spline approximation

To solve residual correction Eqs. (10a)–(11c), it must obtain theresiduals of neighboring subregions. However, the functions andderivatives obtained from traditional difference are only those on gridpoints. They are not continuous on non-calculation grid points, sothey are not applicable to this method. For this reason, this articleadopts the concept of spline approximation to discretize equations,making the functions and derivatives of various orders continuous inany area. Talking about the origin of spline, it was often found in pastresearches on numerical methods that if using a single polynomial toapproximate a function, the result is usually not so good. To tackle thisproblem, traditional finite difference method divides function regioninto a number of subregions, then approximate derivatives at the endsof each subregion, and the calculation result was markedly improved.Whereas the shortcoming of this effort is that the functions andderivatives are discontinuous at the boundaries of subregions. Itundermines value accuracy. Therefore, how to solve these disconti-nuity points and make them, including derivatives of various orders“continuous and smooth” at the boundaries of subregions has becomethe research direction of spline function. The earliest application ofcubic spline in numerical approximation is to the extract boundaryvalue of two points. Later on, numerous scholastic researches wereconducted that further proved that cubic spline fixation method hasthe following advantages (Rubin and Graves [12] and Wang andKahawita [13]):

1. Governing equation can be discretized into an algebraic expressioncontaining only function, or first derivative, or second derivativeand its coefficient matrix is a tri-diagonal matrix, so it can be solvedquickly by Thomas Algorithm.

2. It is more accurate than traditional finite difference method.When grid points are evenly collocated, the obtained first andsecond derivatives have the accuracy of the fourth and thirdderivatives respectively. Even if the grid points are unevenlycollocated, they still have the accuracy of the third and secondderivatives respectively.

3. The obtained function and its first and second derivatives are allcontinuous in every interval.

4. The method applies to uneven grid points and it won't increasecalculation procedures and complexity.

Based on the above advantages, many scholars carried outresearches relevant with spline recently. Generally speaking, the firstcalculation step is to rewrite the differential equation into the followingnormal formula (Wang and Kahawita [13])

ui−Giu′i−Siu″

i = Fi; i = 0;1;2; :::;Ni ð12Þ

where Fi, Gi and Si are known coefficients. To get solutions of suchexpression, the cubic spline functional relation is used to convert theformula into an algebraic equation that only involves ui or firstderivative mi or second derivative Mi. For example, Eq. (12) istransformed into equation sets only containing second derivatives inthe following form, as indicated below

AiMi−1 + BiMi + CiMi + 1 = Di; i = 0;1;2; :::;Ni ð13aÞ

Ai =hi6

+Gi + 2Gi−1

6Δi− Si−1

hiΔið13bÞ

Bi =hi + hi + 1

3−Gi + 1 + 2Gi

6Δi + 1+

2Gi + Gi−1

6Δi+ Si

1Δi + 1hi + 1

+1

Δihi

� �ð13cÞ

Ci =hi + 1

6−2Gi + ! + Gi

6Δi + 1− Si + 1

hi + 1Δi + 1ð13dÞ

Di =Fi + 1−Fi

Δi + 1hi + 1− Fi−Fi−1

Δihið13eÞ

Δi = 1−Gi−Gi−1

hi;Δi + 1 = 1−Gi + 1−Gi

hi + 1; hi = xi−xi−1: ð13fÞ

These formulae are recurrence relations, and can be expressed inmatrix form as

B0 C0 ⋅ ⋅ ⋅ ⋅ ⋅A1 B1 C1 ⋅ ⋅ ⋅ ⋅⋅⋅

⋅⋅

⋅⋅

⋅⋅

⋅⋅

⋅⋅

⋅⋅

⋅ ⋅ ⋅ ⋅ ANi−1 BNi−1 CNi−1⋅ ⋅ ⋅ ⋅ ⋅ ANi

BNi

26666664

37777775

M0M1⋅⋅

MNi−1MNi

26666664

37777775=

D0D1⋅⋅

DNi−1DNi

26666664

37777775

ð14Þ

where B0, C0, D0, ANi, BNi

and DNican be acquired based on boundary

conditions. Since Formula (14) consists of Ni+1 equations and takesthe form of a tri-diagonal matrix, then Thomas Algorithm can be usedto calculate the second-order differential value of the differentialequation swiftly. This paper seeks to use Formulae (13a, 13b, 13c, 13d,13e, and 13f) to obtain the second derivativeMi

n+1 of a second partialdifferential equation; then trace back to extract the first differentialderivativemi

n+1 and the functional value uin+1 using the basic relation

150 C.-C. Wang / International Communications in Heat and Mass Transfer 37 (2010) 147–155

of cubic spline function. In addition, the value of the function at anypoint can be expressed as follow:

uiðxÞ = Mi−1ðxi−xÞ3

6hi+ Mi

ðx−xi−1Þ36hi

+ ui−1−Mi−1h

2i

6

!xi−xhi

+ ui−Mih

2i

6

!x−xi−1

hi; x ∈ ½xi−1; xi�:

ð15Þ

Generally when the cubic spline approach is used for solutions,each time a different boundary is encountered, the cubic splinerelationmust be employed individually to get the formulae relating toB0, C0, D0, ANi

, BNiand DNi

, making this method rather inconvenient forsolutions. On this ground, this paper relies on mixed boundaryconditions to deduce general formulae of B0, C0, D0, ANi

, BNiand DNi

.Let's begin by considering mixed boundary conditions

α0uðx0Þ + β0u′ðx0Þ = γ0 ð16aÞ

αNiuðxNi

Þ + βNiu′ðxNi

Þ = γNið16bÞ

where α0, β0, γ0 and αNi, βNi

, γNiare constants.

Based on the theory of cubic spline, Formula (16a) can bediscretized into

B0M0 + C0M0 = D0 ð17aÞ

B0 = 3G0G1h1α0−2G0h21α0−6G1S0α0 + 6h1S0α0 + 3G1h1β0

−2h21β0−6S0β0

ð17bÞ

C0 = ðG0α0 + β0Þð3G1h1−h21−6S1Þ ð17cÞ

D0 = 6ðF0ðG1α0−h1α0 + β0Þ−F1ðG0α0 + β0Þ + γ0ðG0−G1 + h1ÞÞ:ð17dÞ

Similarly at the boundary of the other end, the following expressioncan be generated:

ANiMNi−1 + BNi

MNi= DNi

ð18aÞ

ANi= −ðGNi

αNi+ βNi

Þð3GNi−1hNi+ h2Ni

−6SNi−1Þ ð18bÞ

BNi= −3GNi

GNi−1hNiαNi

−2GNih2Ni

αNi−6GNi−1SNi

αNi

−6hNiSNi

αNi−3GNi−1hNi

βNi−2h2Ni

βNi−6SNi

βNi

ð18cÞ

DNi= 6ðFNi

ðGNi−1αNi+ hNi

αNi+ βNi

Þ−FNi−1ðGNiαNi

+ βNiÞ

+ γNiðGNi

−GNi−1−hNiÞÞ:

ð18dÞ

3. Result and discussion

In this section, two examples are given to explain how to useresidual correction method. While in order to research how big theerror of approximate solutions is, we first define mean approximatesolution u(x), maximum possible error Emax(x) and mean approxi-mate error Ea(x) as

―uðxÞ =⌢uðxÞ + ⌢uðxÞ

2ð19Þ

EmaxðxÞ =⌢uðxÞ− ⌣uðxÞ

Minð j⌢uðxÞ j ; j⌣uðxÞ jÞ ð20Þ

EaðxÞ =j―uðxÞ−uðxÞ j

juðxÞ j ð21Þ

where u(x), ŭ(x) and u(x) are the exact solution, lower and upperapproximate solutions respectively. u(x) is the average of the lowerand upper approximate solutions.

Example 1. Inorder toexplicitlydescribe the stepsof residual correction,we first consider the following simple equation in which the exact solu-tion exists.

u″−u = − sin½πðx−0:3Þ� ð22aÞ

with boundary condition

uð0Þ = uð1Þ = 0: ð22bÞ

As spline approximation can fully satisfy the boundary condition,there is no need to consider boundary residual. What needs to beconsidered is only the internal residual equation as described below:

Rðx;u;u′;u″Þ = u″ðxÞ−uðxÞ + sin½πðx−0:3Þ�; 0 ≤ x ≤ 1: ð23Þ

Apparently, the above equation accords with Eq. (2), therefore, itssolutions are monotonic. In other words, when approximate functionũ(x) satisfies constraint condition

Rðx; u; u′; u″Þ = u″ðxÞ− uðxÞ + sin½πðx−0:3Þ� ≥ 0; 0 ≤ x ≤ 1 ð24Þ

and objective function is

⌣uðxÞ = MaxðuðxÞÞ ð25Þ

the lower value of optimal approximate solution can be extracted. Onthe other hand, when approximate function ũ(x) satisfies constraintcondition

Rðx; u; u′; u″Þ = u″ðxÞ− uðxÞ + sin½πðx−0:3Þ� ≤ 0; 0 ≤ x ≤ 1 ð26Þ

and objective function is

⌢uðxÞ = MinðuðxÞÞ ð27Þ

then, the upper value of optimal approximate solution is u(x).Use the concept of residual correction to convert the above

inequation mathematical programming problem into equation iter-ative operation as per the following steps:

1. Discretize Eq. (23) on each calculation grid point by cubic splinefixation method and rewrite it into the following equation:

u″m + 1i −um + 1

i + sin½πðxi−0:3Þ� = Rmi ð28Þ

where Rim is the corrected residual value of calculation grid point i. It is

used to correct the residual value on a calculation grid point and ensurethat the residual values within neighboring subinterval (xi−1≤x≤xi+1)are all positive or negative;2. Set m=0 and assume residual correction value Ri

m=0;3. Calculate Eq. (28) by cubic spline method to obtain the approx-

imate solution um+1 and its derivatives u′m+1 and u″m+1;4. Put the approximate solution obtained from step 3 into residual

Expression (23) to obtain residualR(x), and calculate out new residualcorrection according to Eqs. (10a, 10b, and 10c) or Eqs. (11a, 11b,and 11c);

5. Progress onem value, repeat steps 3 and 4 till all residual values onthe intervals are smaller or larger than zero.

In order to observe whether the result from residual correctionmethod is correct, Figs. 1 and 2 illustrate the residual distribution beforeand after residual correction respectively. It can be seen from Fig. 1 that

Fig. 1. Distribution of uncorrected residuals when the number of grid points is 6 and 11respectively.

Fig. 3. Distribution curves of upper and lower approximate solutions when the numberof grid points is 6 and 11 respectively.

151C.-C. Wang / International Communications in Heat and Mass Transfer 37 (2010) 147–155

although the uncorrected residuals of the differential equation of theapproximate solutions are zero on all calculation grid points, they are notzero on non-calculation grid points as the approximate solutions fail tosatisfy all the inner regions within neighboring subintervals. Furtherobservation reveals whether the residuals in regions where residuals arenot zero are greater than zero or smaller than zero is affected by thefeatures of the equation and the approximate solution function, thus isunpredictable. In this example, the residuals of the differential equationare all smaller than or equal to zero when x-coordinate value is smallerthan 0.3 approximately. In other regions, the residuals are greater thanzero. The outcome is obviously affected by the last term of differentialEq. (22a). Due to such unpredictable residual distribution, the traditionalapproximate solutions are always unable to determine their relationwithexact solution. In order to find out the monotonicity of approximatesolution with residual of differential equation, Fig. 2 draws a distributioncurve of residuals that are the correction of the residuals described inFig. 1. It can be seen in Fig. 2 that when extracting a lower approximate

Fig. 2. Distribution curves of corrected residuals of upper and lower approximatesolutions when the number of grid points is 6 and 11 respectively.

solution, the residual of Eq. (23) must satisfy Expression (24). Therefore,as shown in Fig. 2, residuals on calculation grid points where x≤0.3 areadded during residual correction to make the residuals within neighbor-ing subintervals all greater thanor equal to zero. On the other hand,whenextracting upper approximate solution, it can be seen apparently as theresiduals in the region where about x≤0.3 satisfy Expression (26), thereis no need to correct any residual. On the contrary, correction is made onthe residuals on gridpoints of other regions tomake themslightly smallerthan zero and satisfy the condition that the residuals in neighboringsubregions are all smaller than or equal to zero.

Since the residuals in Fig. 2 satisfy Expression (24) or (26), Fig. 3—the approximate solution curve where the residuals in Fig. 2 aredrawn in can ensure its relation with exact solution: the top twocurves in Fig. 3 are the upper approximate solutions of exact solutions,while the other two curves below them are the lower approximatesolutions of exact solutions. In other words, the curves of the upperand lower approximate solutions limit the values of the exact solutionwithin these two curves. In order to more accurately validatecorrectness and error range, Table 1 lists lower, average and uppersolutions and relevant error of different grid points when x=0.5. Itcan be seen that with the increase of grid points, the lowerapproximate values are always smaller than their respective uppervalue and the twos gradually approximate each other, so themaximum error extracted from Eq. (20) reduces with the increaseof grid points, just as expected. Notably, the maximum possible errorhere refers to the possible maximum error between exact solutionsand the approximate value within the range between the upperapproximate solution and lower approximate solution. In practicalapplication, the mean approximate solutions extracted from Eq. (19)

Table 1Approximate solutions and errors at x=0.5 under different numbers of calculation gridpoints.

Ni ũ(0.5) Emax Ea

Lower solutionsŭ

Average valuesu

Upper solutions u

5 0.05192437 0.05357006 0.05521576 0.06338 0.009356810 0.05360204 0.05397624 0.05435045 0.01396 0.001845520 0.05395839 0.05405091 0.05414344 0.00342 0.000464750 0.05405723 0.05407201 0.05408678 0.00054 0.0000746100 0.05407134 0.05407503 0.05407873 0.00013 0.0000186

152 C.-C. Wang / International Communications in Heat and Mass Transfer 37 (2010) 147–155

can be used as the final and better numerical approximate solutions.In this example, the exact solution is

uðxÞ = −11=5e−x

2ð−1 + eÞð1 + π2Þ

� ð−1 + ð−1Þ3=5Þð−e + e2xÞ+ ð−1 + eÞexðð−1 + ð−1Þ3=5ÞcosðπxÞ + ð−i + ð−1Þ1=10ÞsinðπxÞÞ

!

ð29Þ

While the exact value at x=0.3 is

uð0:5Þ = −13=10−ð−17=10Þ2 + 2π2 ≈ 0:0540760483: ð30Þ

To put it into Eq. (21), it can be found that the error of meanapproximate solution in the last column of Table 1 is much smallerthan the maximum possible error, indicating the upper and lowersolutions obtained by using residual correction method in this articlecan be used to analyze the maximum possible error of a solution.Moreover, as the residuals of these two approximate solutions arealways distributed on two sides of zero in a basically symmetricmanner as shown in Fig. 2, the errors between the upper/lowerapproximate solutions extracted from Fig. 3 and the exact solution aresymmetric to considerable degree, and the values of mean approx-imate solutions extracted from Eq. (19) are rather close to their exactsolutions even if there are only very a few grid points. Therefore, it canbe said that the residual correction also indirectly increases theaccuracy of mean approximate solutions.

Example 2. Considering the problem of boundary layer forcedconvection when Newtonian fluid passes through a wedge, flowfield and temperature field can be denoted as (Bejan [14])

2f ‴ + ðm + 1Þff ″ + 2mð1−f ′2Þ = 0 ð31aÞ

θ″ +Pr2ðm + 1Þf θ′ = 0 ð31bÞ

with boundary conditions

On plate surface ðη = 0Þ : f = f 0 = θ = 0 ð31cÞ

Far away from the surface ðη→∞Þ : f 0 = θ = 1 ð31dÞ

where f(η) and θ(η) are dimensionless flow field and temperatureparameter on η coordinate respectively; m=β /(2π−β) is related tothe wedge angle β and Pr is Prandtl number.

3.1. Proof of monotonicity

In their researches, Wang and Hu [11] have proposed a method todecide monotonicity of individual differential equations under looseconditions. However, as Eqs. (31a) and (31b) forma systemof nonlinearequations. The monotonicity of their solutions with residuals probablydoesn't exist at all, or even though it does exist, the analysis on it isconsiderably difficult. As far as I know, there isn't any monotonicityanalysis of such kind so far. The proof of monotonicity described belowintroduces a new concept that is to use the concept of virtual time toconvert nonlinear into uncoupling linear equations, enabling theextension of the concept of maximum principle to once non-monotonicequation or system of equations. By using the concept of virtual time,Eqs. (31a) and (31b) are rewritten into new differential equations

f ′−u = u−u ð32aÞ

2u″ + ðm + 1Þ fu′ + 2mð1−u2Þ−u−uΔτ

= −ðm + 1Þðf− f Þu′−u−uΔτð32bÞ

θ″ +Pr2ðm + 1Þ fθ′− θ− θ

Δτ= −Pr

2ðm + 1Þðf− f Þθ′− θ− θ

Δτð32cÞ

where functionu is thefirst differential valueof f, andEqs. (32a) and (32b)are the order-reduced expressions of Eq. (31a). Δτ denotes the positiveincrement of virtual time. Superscript ^ denotes the given or obtainedfunction value in previous time that may be greater or smaller than theexact solution respectively when solving for upper or lower approximatevalue.

No doubt, in most cases, it is impossible to obtain an exact solutionthat satisfies Eqs. (32a)–(32c), so the relation between approximatefunctions and exact solutions are assumed as follows:

f = f−δf ð33aÞ

u = u−δu ð33bÞ

θ = θ−δθ ð33cÞ

where f , ũ and θ are the approximate solutions of exact solutions u, fand θ respectively, and δ refers to the difference between their exactsolutions and approximate solutions. Put Eqs. (33a, 33b, and 33c) intoEqs. (32a, 32b, and 32c) to obtain the new residual expression

Rf ðηÞ = f′−u = δf ′ + ðu−uÞ = 0 ð34aÞ

R uðηÞ = 2 u″ + ðm + 1Þ f u0 + 2mð1− u2Þ− u−uΔτ

= 2δu″ + ðm + 1Þ f δu′

− 4mu +1Δτ

� �δu− ðm + 1Þðf− f Þu′ +

u−uΔτ

� �

ð34bÞ

R θ ðηÞ = θ″ +Pr2ðm + 1Þ f θ′− θ− θ

Δτ= δθ″ +

Pr2ðm + 1Þ f δθ′− 1

Δτδθ

− Pr2ðm + 1Þðf− f Þθ′ + θ−θ

Δτ

!ð34cÞ

Whereas cubic spline approximation can satisfy Eq. (34a) in allcalculation intervals, the residual here can be set as zero. Frommaximum principle, we can know the residual equations on the rightof the second equal marks of Eqs. (34b) and (34c) in accord with

∂H∂ðδuÞ = − 4mu +

1Δτ

� �≤ 0 and

∂H∂ðδθÞ = − 1

Δτ≤ 0 ð35Þ

Therefore, by using virtual time concept, the monotonicity ofEqs. (34a)–(34c) is successfully proved. In practical application, as it isknown that

u′ðηÞ ≥ 0 ð36Þ

θ0ðηÞ ≥ 0 ð37Þ

then, when Eqs. (34a)–(34c) are in accord with

Rf ðηÞ = 0 ð38aÞ

RuðηÞ ≥ 0 ð38bÞ

RθðηÞ ≥ 0 ð38cÞ

Fig. 4. Distribution of uncorrected dimensionless velocity and temperature residualswhen m=0 and the number of grid points is 21.

Fig. 5. Distribution of corrected residuals of upper and lower approximate solutions ofdimensionless velocity and temperature whenm=0 and the number of grid points is 21.

153C.-C. Wang / International Communications in Heat and Mass Transfer 37 (2010) 147–155

and

f ≥ f ; u ≥ u; θ ≥ θ ð38dÞ

we can conclude that the value in the brace of the last termof Eqs. (34b)and (34c) are positive. Based on Eqs. (3a, 3b, and 3c) and (4), we canknow δf, δu and δθ have lower approximate solutions that are smallerthan or equal to zero (because their exact solutions shall be zero).Therefore, fromEqs. (33a, 33b, and 33c),we can know that the obtainedapproximate solutions ũ(η), f (η) and θ(η) are the lower approximatesolutions of their respective exact solutions when Eqs. (38a)–(38d) aresatisfied. Of course, it can be simply proved that when Eqs. (38a)–(38d)are adverse, the obtained approximate solutions are the upperapproximate solutions of their respective exact solutions.

3.2. Procedures for numerical calculation

Discretize the differential terms in Eqs. (32a)–(32c) by cubic splinemethod and rewrite them according to normal Formula (12) into

f ′n + 1i = un

i ð39aÞ

−1Δτ

−2muni

� �un + 1i + ðm + 1Þf nu′n + 1

i + 2u″n + 1i = − un

i

Δτ−2m + Ru jmi

ð39bÞ

− 1Δτ

θn + 1i +

Pr2ðm + 1Þf ni θ′n + 1

i + θ″n + 1i = − θni

Δτ+ Rθ jmi ð39cÞ

where n is the virtual time index. In Eq. (39b), the order reductionmethod is used to obtain the values of uin+1, u′i n+1 and u″i

n+1 oncalculation grid points at first. After that, by referring to cubic splinetheory, any function g(η) at any point of a subinterval can beexpressed with the functions and derivatives of its two ends as

gðηÞ = g″i−1ðηi−ηÞ36Δηi

+ g″iðη−ηi−1Þ3

6Δηi+ ηi−1−

g″i−1Δη2i

6

!ηi−ηΔηi

+ gi−g″iΔη

2i

6

!η−ηi−1

Δηi

ð40Þ

where Δηi=ηi−ηi−1 is the space between grid points. Therefore, inEqs. (39b) and (39c), the values of function f on grid points can beaccurately obtained by using

fi = ∫ηi

0udη = ∑

i

k=1ðuk−1 + ukÞ

Δηk

2−Δη3

k

24ðu″

k−1 + u″kÞ

!: ð41Þ

In this way, the values of coefficients Fi, Gi and Si in Eqs. (39b)and (39c) can all be determined by the solutions obtained in previousvirtual time. The whole procedure to obtain solutions is the same asthe former example except that the initial conditions (n=0) shall,when extracting lower approximate values (the given values shall besmaller than exact solutions), be

f ðηÞ = uðηÞ = θðηÞ = 0 ð42Þ

and when extracting upper approximate values (the given valuesshall be greater than exact solutions), be

f ðηÞ = x; uðηÞ = 1 and θðηÞ = 1 ð43Þ

so as to ensure that the approximate values obtained from eachiteration shall either be smaller than or greater than the exact values.

3.3. Numerical results

Let's begin by considering a special case where fluids flow througha flat plate at m=0 (i.e. Blasius's equation). By using the above steps,the uncorrected residual distribution as shown in Fig. 4 can becorrected into the distribution of residuals that are all greater orsmaller than zero as shown in Fig. 5, while the obtained upper andlower approximate solutions of dimensionless velocity and temper-ature are shown in Fig. 6. In Fig. 4, it can be seen that the residualdistribution seems more complex than the previous example's.However the residual correction method put forth in this article canstill accurately correct their residuals into those shown in Fig. 5. Inaddition, in Fig. 5, most of the dimensionless temperature residualvalues are greater than those of velocity equation, so it can be seen inFig. 6 that at the same grid point, the upper and lower curves ofvelocity are more approximate than the upper and lower curves of

Fig. 6. Distribution curves of upper and lower approximate solutions of dimensionlessvelocityand temperaturewhenm=0andthenumberofgridpoints is 11and21 respectively.

Fig. 7. Distribution curves of upper and lower approximate solutions of dimensionlessvelocity when Δη=1.

154 C.-C. Wang / International Communications in Heat and Mass Transfer 37 (2010) 147–155

temperature. As the ultimate purpose of this problem is mostly for theresearch of fluid's shear stress and thermal conductivity on the wall,Tables 2 and 3 list the differential values of dimensionless velocity andtemperature on different grid points on the wall (η=0). From thetables, we can see with the increase of grid points, that the lower andupper differential values on the wall get closer and as expected, themaximum possible error reduces, too. For instance: when the numberof grid points is 200, it is confirmed that

0:33205400 ≤ u′ð0Þ ≤ 0:33206088 ð44aÞ

0:57667056 ≤ θ′ð0Þ ≤ 0:57670425: ð44bÞ

This result is rather close to traditional velocity differential value(Bejan [14])

u′ð0Þ ≈ 0:33206 ð45Þ

Table 2Approximate values and errors of dimensionless velocity u′(0) under different numbersof calculation grid points.

Ni ũ′(0) Emax Ea

Lower solutionsŭ′

Average solutionsu ′

Upper solutionsu ′

10 0.32255938 0.33289023 0.34322109 0.064056 0.00250820 0.33000348 0.33212953 0.33425558 0.012885 0.00021750 0.33173462 0.33206801 0.33240140 0.00201 3.18E-05100 0.33197678 0.33206003 0.33214329 0.000502 7.8E-06200 0.33205400 0.33205744 0.33206088 2.07E-05 –

Table 3Approximate values and errors of dimensionless temperature θ′(0) under differentnumbers of calculation grid points.

Ni θ′(0) Emax Ea

Lower solutionsθ′

Average solutionsθ ′

Upper solutionsθ′

10 0.53084676 0.59271398 0.65458120 0.233089 0.02778720 0.56738873 0.57747369 0.58755864 0.035549 0.0013650 0.57529989 0.57676661 0.57823333 0.005099 0.000133100 0.57634522 0.57670824 0.57707126 0.00126 3.22E-05200 0.57667506 0.57668966 0.57670425 5.06E-05 –

and has a greater error with temperature differential value (Bejan[14])

θ′ð0Þ ≈ 0:332Pr1=3 ≈ 0:56781 ð46Þ

because Eq. (46) is just an approximate formula. However, no matterwhether their exact solutions are knownor not, the article can obtain anerror analysis of approximate solutions by usingmonotonicity analysis.

To further verify applicability of thismethod, Fig. 7 shows distributionof upper and lower approximate solutions of the velocity curve atdifferent wedge angles. According to this figure, when the distancebetween grid points isfixed, i.e.Δη=1, the upper and lower approximatesolutions obtained with this method will expectedly outline the areawhere analytical solutions exist. It is worth noticing that the size of thisarea reflects the magnitude of the error. It seems that the magnitude ofthe error varieswith changes in thickness of boundary layers and value ofm.Moreover, Fig. 8 shows the errordistributionoutlinedby theupper and

Fig. 8. Distribution curves of upper and lower approximate solutions of dimensionlesstemperature when m=1 and Δη=1/2.

155C.-C. Wang / International Communications in Heat and Mass Transfer 37 (2010) 147–155

lower approximate solutions of a temperature approximation curveunder different Pr numbers. It can be seen clearly from thisfigure that theerror of approximate solutions tends to become bigger as the Pr numberincreases, when the distance between grid points is fixed. As the errordistribution curves in Figs. 7 and 8 show, however, no matter howparameters of differential equations change, themethod proposed in thispaper will make it easy to indicate the magnitude of the error betweenapproximate solutions and analytical solutions, thus making it unneces-sary to confirm correctness of solutions through grid point testing.

4. Conclusions

Byestablishing amonotonic relation for forced convection problems,we can find that the residual correction method put forth by this articlecan accurately correct the residuals of approximate solutions of adifferential equation and satisfy the condition that residuals are allgreater than or smaller than zero.Moreover, the process to obtain upperand lower approximate solutions by this method doesn't need extrainternal memory and the iteration times are rather short. Comparingwith traditional calculation, it only spends somemore time in iteration.The numerical findings also indicate that the method has the feature ofcorrecting residuals symmetrically. Even if there are a limitednumber ofgrid points, the mean approximate solutions obtained are still ratherclose to the exact solutions. In addition, it is very easy to obtain themaximumpossible error with the help of upper and lower approximatesolutions obtained by this method. In this way, it can avoid blindincrease of grid points for increasing numerical accuracy and credibility.

Acknowledgement

Thanks to the subsidy of the Outlay NSC96-2221-E-432-002 givenby the National Science Council, Republic of China, for helping usfinish this special research successfully.

References

[1] A. Camp, K. Tebeest, U. Lacoa, C.J. Morales, Application of a finite volume basedmethod of lines to turbulent forced convection in circular tubes, Numerical HeatTransfer; Part A: Applications 30 (1996) 503–517.

[2] D. Pan, C.H. Chang, Upwind finite-volume method for natural and forcedconvection, Numerical Heat Transfer Part B: Fundamentals 25 (1994) 177–191.

[3] J.C. Bruch, G. Zyvoloski, Transient two-dimensional heat conduction problemssolved by the finite element method, International Journal for Numerical Methodsin Engineering 9 (1974) 481–494.

[4] C. Ramirez, G. Ramiro, J.R. Barbosa, The boundary element method applied toforced convection heat problems, International Communications in Heat andMassTransfer 35 (2008) 1–11.

[5] S.S. Chu, W.J. Chang, Hybrid numerical method for transient analysis of two-dimensional pin fins with variable heat transfer coefficients, InternationalCommunications in Heat and Mass 29 (2002) 367–376.

[6] C.H. Chiu, C.K. Chen, A decomposition method for solving the convectivelongitudinal fins with variable heat transfer, International Journal of Heat andMass Transfer 45 (2002) 2067–2075.

[7] M.H. Protter, Maximum Principles in Differential Equations, Prentice-Hall, Engle-wood Cliffs, NJ, 1967.

[8] B.A. Finlayson, L.E. Scriven, Upper and lower bounds for solutions to the transportequations, AICHE Journal 12 (1966) 1151–1157.

[9] C.L. Chang, Z.Y. Lee, Applying the double side method to solution nonlinearpendulum problem, Applied Mathematics and Computation 149 (2004) 613–624.

[10] C.C.Wang,Use residual correctionmethod to calculate theupper and lower solutionsof initial value problem, Applied Mathematics and Computation 181 (2006) 29–39.

[11] C.C.Wang,H.P. Hu, Study onmonotonicity of boundary value problems of differentialequations, Applied Mathematics and Computation 202 (2008) 383–394.

[12] S.G. Rubin, R.A. Graves, Viscous flow solution with a cubic spline approximation,Computers and Fluids 1 (1975) 1–36.

[13] P. Wang, R. Kahawita, A two-dimensional numerical model of estuarine circulationusing cubic spline, Canadian Journal of Civil Engineering 10 (1983) 116–124.

[14] A. Bejan, Convection Heat Transfer, 2e,Wiley-Interscience, New York, 1995, pp. 46–52.