Applied Mathematics and Computation Volume 219 issue 3 2012 [doi 10.1016_j.amc.2012.07.007]...

8/10/2019 Applied Mathematics and Computation Volume 219 issue 3 2012 [doi 10.1016_j.amc.2012.07.007] Chi-Chang Wan

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Hybrid spline difference method for the Burgers equation

Chi-Chang Wang a, , Wu-Jung Liao a, Yin-Sung Hsu ba Ph.D. Program of Mechanical and Aeronautical Engineering, Feng Chia University, No. 100 Wenhwa Rd., Seatwen, Taichung 40724, Taiwan, ROC b Ph.D. Program in Civil and Hydraulic Engineering, Feng Chia University, No. 100 Wenhwa Rd., Seatwen, Taichung 40724, Taiwan, ROC

a r t i c l e i n f o

Keywords:Hybrid spline difference methodBurgers equationParametric splineFinite difference methodHigh accuracy

a b s t r a c t

This study developed a high accuracy hybrid spline difference method, and deduced a dif-ference equation that is very similar to nite difference method from the concept of splinedifference. As validated by nonlinear Burgers equation, the concept of difference made thecomputational process as simple as the nite difference method, and easy to be imple-mented. The free parameters a ! 1=12 and a ! 1=6 were combined in the concept of spline hybrid, in order to increase the accuracy of the rst and second derivatives of spacefrom Oh

2

of nite difference method to the Oh4

. The accuracy was improved, and thenumerical oscillation with the increase in parameter Re was improved greatly. 2012 Elsevier Inc. All rights reserved.

1. Introduction

In order to describe the behavioral model of uid, previous studies have proposed the Burgers equation that is similar tohydrokinetic NavierStokes equation in the early 19th century. The one-dimensional mathematical general equation isshown below (Kemal and Turgut [1]):

@ u@ t

u@ u@ x

v @ 2 u@ x2

1

where u x; t is the velocity for space x and time t , v is the kinematics viscosity parameter related to the parameter Re (=1/ v ).Burgers equation has been extensively used in uid turbulence problem, aerodynamics, heat conduction problem, elasticityproblem, concentration diffusion and as the mathematical model of modern trafc ow dynamics solving. Although thisequation is simple, its solving difculty grows as the Parameter Re increases. Thus, there is a general solution only in specialboundary conditions.

As Burgers equation is in simple form and difcult to be solved, many numerical methods, such as Wavelet-TaylorGalerkin method, hybrid method, cubic B-splines, homotopy perturbation method, semi-implicit nite difference schemesand reproducing kernel function [27] use this equation as the test object of numerical values. Reviewing these numericalmethods, it is found that different solving modes have different advantages and characteristics, such as rapid convergence,high accuracy, efcient processing of transient term, and easy processing of nonlinear equation. In addition, the smallviscosity parameter solving of this equation often results in numerical oscillation. The upwind techniques (Volkwein [8] )are usually used to avoid instability of values, but the accuracy of values is sometimes reduced slightly.

Although different numerical methods have their own advantages, the nite difference method is still the most exten-sively used method at present due to its simple concept and easy practice. However, the function discontinuity and inaccu-rate of this method are the shortcoming. Therefore, the spline function with smooth continuity and higher numerical

0096-3003/$ - see front matter 2012 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2012.07.007

Corresponding author.E-mail address: [email protected] (C.-C. Wang).

Applied Mathematics and Computation 219 (2012) 10311039

Contents lists available at SciVerse ScienceDirect

Applied Mathematics and Computation

j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a te / a m c
http://dx.doi.org/10.1016/j.amc.2012.07.007mailto:[email protected]://dx.doi.org/10.1016/j.amc.2012.07.007http://www.sciencedirect.com/science/journal/00963003http://www.elsevier.com/locate/amchttp://www.elsevier.com/locate/amchttp://www.sciencedirect.com/science/journal/00963003http://dx.doi.org/10.1016/j.amc.2012.07.007mailto:[email protected]://dx.doi.org/10.1016/j.amc.2012.07.007


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accuracy than nite difference method has been proposed to reinforce the nite difference method. For the earlier cubic

spline (Wang and Kahawita [9]), if the grid points are uniform, the rst derivative and second derivative have fourth-orderand second-order accuracy respectively, so the spline value solving has been concerned and applied. In recent years, therehave been many studies developed from spline theory, such as high-accuracy cubic spline collocation method [10] , quarticspline method [11] , residual correction method [12] and parameter spline method [13,14] etc.

The spline method is characterized by high accuracy. The parameter spline has been well evaluated in some special heatproblems such as hyperbolic heat problem [13,14] ; however, the calculation procedure of this method is too complicatedand how to determine the optimum parameter is not yet solved. Therefore, based on the spline difference concept proposedby Wang and Lee [13,15] , this paper uses Burgers equation for discussion, besides expressing the function and its derivativeas adjacent splines in the concept of spline difference similar to the discrete mode of nite difference, to construct a simplespline difference calculation procedure. This study develops the skill of hybrid spline, so as to make the rst-order and sec-ond-order numerical derivative accuracies reach Oh

4

at the same time.

2. Hybrid spline difference method

2.1. Origin of parametric spline

In studies of numerical methods, it is often found that the effect of using a single polynomial to approximate to a functionis unsatisfactory. In order to make improvement, the region of function can be divided into multiple subregions, and eachsubregion is expressed by simple fundamental functional equations such as polynomial. However, how to solve thecontinuity of the functions at the subregion junctures and their derivatives to make them at these subregion juncturesand multi-order derivatives continuously smooth is the direction of research on spline function. The traditional cubicspline function was mostly assumed to be a simple cubic polynomial in early stages (Wang and Kahawita [9]), and itscurvature must be linear relationship after second differentiation, its expression is:

s00i x s00i 1 x xi 1

s00i s00i x

xi x ; x 2 xi 1 ; xi 2

Nomenclature

h interval dened as xi xi 1N t number of grid points of timeN i number of grid points of spaceOh truncation error of spatial term p unknown coefcientR residual value of grid point ( xi)Re parameter, 1 =v S base function of splinet timeu unknown function on x 1 ; xN 1 x axial z location of transition layer

Greek symbolsa parameter, a x cscx 1 =x 2b parameter, b 1 x cot x =x 2e constantc; x free parameter

s undetermined parameterm kinematics viscosity parameter

Superscript n serial number of calculation grid point of time

Subscript i serial number of calculation grid point of space

1032 C.-C. Wang et al./ Applied Mathematics and Computation 219 (2012) 10311039


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where si 1 x and si x are the cubic spline approximation curves in the interval xi 2 ; xi 1 and xi 1 ; xi respectively, and s00i 1and s00i are the second order derivatives on the calculation grid point i-1 and i respectively. In recent years, in order to improvethe accuracy of values, some studies [13,14] have added an undetermined parameter in traditional spline, and assumed therelationship of second differentiation to be

u00i x; c su i x; c u00i xi 1 ; c su i xi 1 ; c xi x

h i u00i xi; c su i xi; c

x xi 1hi

; x 2 xi 1 ; xi 3

wheres

is the undetermined parameter,c

P 0 is a free parameter; x 2

xi 1 ;

xi

and xi are discrete grid points in computation

space x0 ; xN ; the spacing h is dened as xi xi 1 , and u i x; c is the unknown function dened in x 1 ; xN 1 . Make Eq. (3)integral, and the integration constant is determined by the endpoint relationships si xi 1 u i 1 and si xi u i to obtainu i x; c zui zui 1 h

2

i g z u00i g z u

00i 1 .x 2 4

where x h ffiffiffi cp ; z x xi=h; z 1 z ; g z z sinx z =sin x and u00i u00i xi are second differential values of end-points. The subscript i in Eq. (4) is replaced by i + 1, the u i1 x; c can be obtained, and then the following fundamental rela-tion of parameter spline function can be deduced from the functions u i x; cand u i1 x; cmeeting the continuity condition of the rst and second derivatives at the intersection point.u0i

u i 1 u i 12 h

ahu00i 1 u00i 1

2 5 :1

au0i 1 2 bu0i au0i 1 a b

h ui 1 u i 1 5 :2

au00i 1 2 bu00i au00i 1 1

h2 ui 1 2 u i u i 1 5 :3

where a x cscx 1 =x 2 ; b 1 x cotx =x 2 .

3. Introduction of concept of spline difference

In the past, the differential equation solution often uses spline for solution in relational expression (5) . however, the func-tions and derivatives must be obtained by solving recurrence relation (5) . The process is complicated, and it is different fromtraditional nite difference. Therefore, in order to simplify traditional spline calculation, the approximate function of differ-ential equation is assumed to be composed of multiple different parameter splines S x xi=h; c, the form is

u x; c XN 1

i 1 piS

x x i

h ;c 6

where S is the fundamental function of spline, the unknown coefcient p i is the value of spline at grid point i. The aboveequation is substituted in relationship (5) for solving, fortunately, the discrete relationship of spline at the grid point canbe obtained

u i a pi 1 2 b pi a pi 1 7

u0i pi 1 pi 1

2 h 8

u00i pi 1 2 pi pi 1

h2 9

Interestingly, in the above equation, in addition to the functions consisting of adjacent parameter splines, and the discretemode of the rst and second differentials of function u is similar to traditional nite difference method. Therefore, the dif-ferential equation can be dissociated by Eq. (7) into equation set expressed as pi 1 ; pi; pi1 , the discrete mode, calculationprocedure and differential solving mode are very similar to nite difference, and the complicated calculation of traditionalspline is avoided completely. Therefore, this solving concept is called parameter spline difference method in this paper.

4. Truncation error analysis

If the function u x is the correct solution of differential equation, its Taylor series at xi can be expressed as

u xi h X1

k0

hk

k!dku xi

dx e Dhu xi 10

where Diu x ui x. The above equation is substituted in Eqs. (5.2) and (5.3) , and e Dh is developed to obtain

C.-C. Wang et al. / Applied Mathematics and Computation 219 (2012) 10311039 1033
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u0i a b

heDh e Dh

aeDh 2 b ae Dh u xi

a b720 D1 120 h2 D3 6 h4 D5 720 a bD0 360 ah2 D2 30 ah4 D4

u xi 9 :1

u00i 1

h2eDh 2 e Dh

aeDh 2 b ae Dh u xi

360 D2 30 h2D4 h

4D6

720 a bD0 360 ah2 D2 30 ah4 D4 u xi 9 :2

The truncation errors of the rst and second derivatives of approximate function can be expressed as

u0i u0 xi e 1 u0 xi e 1

6 e a h2

u3

xi Oh4

10 :1

u00i u00 xi e 1 u00 xi e

1

12 ea h2 u4 xi Oh

4 10 :2

where e 1=2a 2b. According to the above equation, there is better accuracy when a b 1=2, and1. If a 1=12 and a 1=6, the accuracy of the rst and second derivatives of approximate function is Oh

2

, the nite dif-ference method using fa ; bg ! f0; 1=2g is the most famous one among this type of numerical methods;2. If a 1=6, the accuracy of the rst and second derivatives of approximate function is Oh4

and Oh2

respectively, thistype of numerical method is the common cubic spline method;3. If a 1=12, the accuracy of the rst and second derivatives of approximate function is Oh

2

and Oh4

respectively, as theaccuracy of the second derivative is similar to Oh4

of quartic spline method [11] , it is very applicable to equations with-out rst derivative term.Although different parameters have different characteristics and numerical accuracy, the parameters fa ; bgare difcult to bedetermined in practical equation solving. Therefore, the concept of hybrid spline will be adopted in the next section, so as tomake the rst and second derivatives of approximate function reach the accuracy of Oh

4

at the same time.

5. Concept of hybrid spline

The direction of parameter selection can be known from the above simple derivation, but it seems that the truncationerror of h2 is eliminated only in special cases. Furthermore, in practical physical problems, the coefcients of the rst andsecond differential terms of differential equation are usually variables. Therefore, how to determine the values of parametersis always a trouble. This paper attempts to propose a concept of hybrid spline out of the original parameter spline thought tosolve this problem.

Carefully, observe Eqs. (10.1) and (10.2) that if a

1=12, the accuracy of the rst and second derivatives of approximate

function is Oh2

and Oh4

respectively. In order to improve the accuracy, if the e 16e a h2u3 xion the right of equal sign inEq. (10.1) is moved to the left of the equal sign as the correction term, the accuracy of the rst derivative of approximate

function will increase to the same accuracy Oh4

of the second derivative. Therefore, the discrete relationships of spaceand time with the concept of hybrid spline can be redened asuni

pni 1 10 pni p

ni 1

12 11 :1

u x ni

pni 1 pni 1

2 h D u0 ni 11 :2

u xx ni

pni 1 2 pni p

ni 1

h2 11 :3

u t ni

un 1i uniD t

pn 1i 1 10 pn 1i pn 1i 1 =12 uni

D t 11 :4

where i and n are serial numbers of calculation grid point of space and time, respectively. D u0 ni is the discrete expression of

correction term e 16e a h2u xxx t n ; xi , the boundary and internal grid point can be dissociated as:

D u0 ni h2

12u xxx t n ; xi

3 u xx ni 4 u xx ni 1 u xx

ni 2 h

24 ; i 0u xx ni 1 u xx

ni 1 h

24 ; i 1 ; 2 ; . . . ; N 1 at t t n3 u xx ni 4 u xx

ni 1 u xx

ni 2 h

24 ; i N

8>>>>>>>:11 :5

Based on the concept of hybrid parameter, the advantages of a ; b ! 1=12 ; 5=12 and a ; b ! 1=6; 1=3 can be possessedat the same time, so the accuracy of the rst differential term and the second differential term can be increased to Oh4

atthe same time. The only defect is that the D u0

n

i must be obtained from last result, so the calculation shall be iterated to con-

vergence. However, as the correction term is very small, the convergence is very fast. In addition, the iterative procedure is



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originally unavoidable in nonlinear problems, so the calculation of D u0 ni can be processed simultaneously in the course of

nonlinear iteration.

6. Numerical results and discussion

The computing modes and accuracies of nite difference method and different spline methods are compared in Table 1 .As seen, thespline difference method proposed in this paper simplies the calculation procedureof traditional spline greatly,

and the concept of difference is very close to the nite difference method. Furthermore, the values of optimum parametersa ; bdo not need to be determined if the concept of hybrid spline is added, the overall spatial accuracy of numerical solutionwill be Oh4

.According to Table 1 , the hybrid spline relationship of Eq. (11) dissociates the Burgers equation of Eq. (1) in the way sim-ilar to the discrete mode of nite difference method [16]

a pn 1i 1 2 b pn 1i a pn

1i 1 u

ni

D t uni

pn 1i 1 pn 1i 1

2 h D u0 ni ! u x ni a pn 1i 1 2 b pn 1i a pn 1i 1 uni u x ni

1

Re pn 1i 1 2 p

n 1i p

n 1i 1

h2 ! for 0 6 i 6 N 12 :1 The initial and boundary conditions:

a p0

i 1 2

b p0

i a p0

i 1 f x for 0 6

i6

N 12

:2

a pn 11 2 b pn 10 a pn 11 c 1 for i 0 12 :3 a pn 1N 1 2 b pn

1N a pn

1N 1 c 2 for i N 12 :4

where f x is the function of x; c 1 and c 2 are constants; i and n are serial number of calculation grid point of space and time,respectively. And N is the calculated number of grid points, a and b are randomly selected spline parameters, meeting therelations of 0 6 a 6 1=2 and b 1=2 a . Eq. (12.1) is changed to tridiagonal matrix for solving, the form is shown below:

aD t

uni2 h

a u x ni 1

Reh2 ! pn 1i 1 2 bD t 2 b u x ni 2Reh2 ! pn 1i aD t u

ni

2 h a u x ni

1

Reh2 ! pn 1i 1

uniD t

D u0 ni uni u x

ni u

ni for 0 6 i 6 N 13

(12.3) and (12.4) are substituted in the above equation and pn11

and pn1N 1

are removed, and then the unknown pn1i

can beobtained by Thomas Algorithm [17] . Thus, the function uni on computational grid point and its differential term within sec-ond order can be obtained quickly from Eq. (11) .

Example 1. In order to evaluate whether the hybrid spline difference method is applicable to Burgers equation, this paperconsiders 1-D steady-state Burgers equation having a general solution (Xiu and Karniadakis [18] )

u x Atanh A2 v

x z ex ; 1 < x 6 0 14 :1

The boundary conditions are:

u0 0 ; u 1 A 14 :2

Table 1Computing mode and accuracy of different numerical methods. a

Item Finite difference

a ; b ! 0; 1=2Traditional spline method Proposed method

Cubic spline [9]

a ; b ! 1=6; 1=3Parametric spline [14]

a ; b ! 1=12 ; 5=12Parametric spline different method0 6 a 6 1=2, b 1=2 a

Hybrid splinedifferent method

u i au0i 1 2b u0i au0i1 abh ui1 u i 1 a a pi 1 2b pi a pi1 pi 1 10 pi pi112

u0i u i1 u i 12h ui1 u i 12h ahu00i1 u00i 1 2

pi1 pi 12h pi1 pi 12h D u0i

u00i u i 1 2u iu i1h2

au00i 1 2b u00i au00i1 1h2

ui1 2u i u i 1 a pi 1 2 pi pi1h2 pi 1 2 pi pi1h2First differential

accuracyOh

2 Oh

4 Oh

2 Oh

2 Oh

4 Oh

4

Seconddifferentialaccuracy

Oh2 Oh

2 Oh

4 Oh

2 Oh

4 Oh

4

a A recurrence relation, cannot be expressed directly by adjacent spline function.

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Let A 1; z ex and A @ u=@ xj x z ex are the location of transition layer and the slope of the location of u z ex 0 respectively. Interms of physical signicance, the transition layer is between the laminar and turbulent, z ex is set as 0 in this paper, meaningthat the uid in the x 2 1; 0 interval is turbulent. When x ! 1 , the velocity reaches u 1 resulting in velocity uc-tuation, so this paper uses 99% of free stream velocity u 1 as the x value of boundary conditions of equation.This paper uses nite difference method, cubic spline difference method, parameter spline difference method and the hy-brid spline difference method proposed in this paper to solve the mean error of Parameter Re 10 and Re 3000 respec-tively. The difference among the solution methods is that when the nite difference method, cubic spline differencemethod and parameter spline difference method are used, Eq. (13) neglects D u

0i, and the corresponding parameters are

a ! 0; a ! 1=6 and a ! 1=12 respectively. The hybrid spline difference method proposed in this paper lets a ! 1=12 inEq. (13) , and considers the hybrid correction effect of D u0i. According to Tables 2 and 3 , the descending order of numericalsolution accuracies of the above four methods are: hybrid spline difference method, cubic spline difference method, param-eter spline difference method and nite difference method. In addition, according to Tables 2 and 3 , the hybrid spline dif-ference method not only has high accuracy, but also the best convergence. This is because of the hybrid spline differencemethod has the calculation of Eq. (11.2) . The accuracy of the rst differential term and the second differential term canbe increased to Oh

4

at the same time.

Table 2

If Re 10, mean error analysis of approximate solution in Example 1 x 2 1; 0 .

Grids(N )

Finite difference

a ; b

! 0; 1=2

Cubic spline

a ; b

! 1=6; 1=3

Parameter spline

a ; b

! 1=12 ; 5=12

Proposed method Hybridspline

10 1.196E 2 2.543E 3 5.537E 3 6.894E 420 2.895E 3 6.891E 4 1.327E 3 4.494E 540 7.185E 4 1.772E 4 3.28E 4 2.798E 680 1.792E 4 4.458E 5 8.174E 5 1.308E 7100 1.147E 4 2.856E 5 5.228E 5 3.201E 8

meanerror PN i0

uni uexact xiN .

Table 3

If Re 3000, mean error analysis of approximate solution in Example 1 x 2 3:5E 3; 0 .

Grids

(N )

Finite difference

a ; b ! 0; 1=2

Cubic spline

a ; b ! 1=6; 1=3

Parameter spline

a ; b ! 1=12 ; 5=12

Proposed method Hybrid

spline10 1.263E 2 2.649E 3 5.868E 3 7.965E 420 3.051E 3 7.204E 4 1.401E 3 5.182E 540 7.564E 4 1.864E 4 3.461E 4 3.246E 680 1.887E 4 4.697E 5 8.623E 5 1.589E 7100 1.207E 4 3.011E 5 5.514E 5 4.183E 8

Fig. 1. Numerical solution oscillation of different numerical methods in Example 1 (Re 3000, N = 5 and x 2 3:5E 3; 0 ).

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According to previous studies, the numerical solution of Burgers equation is likely to have oscillation as the Parameter Reincreases. Therefore, if Re 3000, Figs. 1 and 2 show the numerical solution oscillation when number of grid points N = 5 andN = 10 respectively. It is observed that the oscillation amplitude of numerical solution of any numerical method declines asthe number of grid points increases. In addition, according to the oscillation amplitude in Fig. 1, the cubic spline method hasthe minimum oscillation, and then the hybrid spline difference method proposed in this paper. However, in terms of overallaccuracy, as seen in Table 3 and Fig. 2, the hybrid spline difference method is still the best. Therefore, the hybrid spline dif-ference method proposed in this paper not only improves the numerical accuracy, but also reduces the numerical oscillation.

Finally, as for the selection of optimum parameter a , Tables 2 and 3 show the numerical accuracy of several common spe-cial parameters, and Figs. 3 and 4 show the solution error with and without the concept of hybrid spline varying with param-eter a . As shown in Figs. 3 and 4 , the different Parameter Re and number of grid points of parameter spline difference methodwithout hybrid spline affect the value of optimum parameter a . According to Tables 2 and 3 , the numerical accuracy of cubicspline method (i.e. a ! 1=6) is higher than that of parameter spline method (i.e. a ! 1=12). Therefore, improper selection of parameter often makes the result of parameter spline method worse than the expected one. However, how to nd out theoptimum parameter before calculation is always a trouble. As shown in Figs. 3 and 4 , when a ! 1=12 is adopted to correctthe rst differential term of hybrid spline as proposed in this paper, the optimum parameter of numerical solution of hybridspline is nearby a ! 1=12. Therefore, the hybrid spline concept proposed in this paper is an effective method that overcom-ing the problem in selecting the optimal parameter and improving numerical accuracy.

Fig. 2. Numerical solution of different numerical methods in Example 1 (Re 3000, N = 10 and x 2 3:5E 3; 0 ).

Fig. 3. Variation of optimum parameter a of parameter spline difference method with and without concept of hybrid spline in Example 1(N = 100, x 2 1; 0 ).



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Example 2. Next, let us consider another Burgers equation whose analytic solution (Chrisite et al. [19] ) is:

ut ; x a l l aeg

1 eg ; 0 6 x 6 1 ; t > 0 15 :1

Where g Re a x l t c; a ; l and c are constant values, and assumed to be 0.4, 0.6 and 0.125 [4,20] , respectively, in thispaper. Thus, the initial and boundary conditions of this Burgers equation will be:u0 ; x

a l eRea x cl a1 eRe a x c

for 0 6 x 6 1 15 :2

ut ; 0 1 and ut ; 1 0 :2 for t > 0 15 :3

Fig. 5 compares the transient approximate solutions and analytic solutions of nite difference method and the method of this paper when Re

250, N i

50, D t

0:005 and x

2 0; 1 . It is observed that at the time points of t

0:2; t

0:5 and

t 1:0, the hybrid spline difference method proposed in this paper effectively avoids the nite difference method havinglarge numerical oscillation where there is drastic velocity change. Secondly, Table 4 lists the mean error analysis of differentnumerical methods when Re 10 ; N i 50 and t 2 0; 1 . It is observed that the hybrid spline difference method proposed inthis paper not only has better convergence rate, but also results in better numerical accuracy than other methods at differenttime-steps.

Fig. 4. Variation of optimum parameter a inuenced by grids in Example 1 (Re 3000 and x 2 3:5E 3; 0 ).

Fig. 5. Numerical solution oscillation of different numerical methods with time item in Example 2 (Re 250, N i 50, D t 0:005 and x 2 0; 1 ).



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7. Conclusions

According to the validation of numerical values, the spline difference method proposed in this paper can simplify the cal-culation procedure of traditional spline theory. The differential discrete spirit is very similar to nite difference method. Sec-ondly, the problem in determining the optimal spline parameter can be avoided by using the concept of hybrid spline. Thederivative accuracy within second order is obviously increased by two orders and the numerical oscillation is improvedgreatly as compared with the nite difference method. Considering these advantages, the hybrid spline difference methodof this paper is a simple and very potential numerical method, and it may be a feasible alternative the users when applyingnite difference method in the future.

Acknowledgement

Thanks for the subsidy of the Outlay NSC 100-2628-E-035-009-MY2 given by National Science Council, the Republic of China, to help us nish this special research successfully.

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Table 4

If Re 10 ; N i 50 and t 2 0; 1 mean error analysis of approximate solution in Example 2 x 2 0; 1 .

Grids(D t )

Finite difference

a ; b ! 0; 1=2Cubic spline

a ; b ! 1=6; 1=3Parameter spline

a ; b ! 1=12 ; 5=12Proposed method Hybridspline

1E 3 3.642E 5 4.011E 5 3.159E 5 3.094E 55E 4 3.368E 5 2.466E 5 2.015E 5 1.548E 51E 4 3.654E 5 1.229E 5 1.646E 5 3.084E 65E 5 3.723E 5 1.075E 5 1.668E 5 1.535E 6

1E 5 3.783E 5 9.531E 6 1.700E 5 2.963E 7

meanerror PN t n0P

N ii0

uni uexact t n ; xiN t N i

, where N t t D t .


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