research article numerical solution of advection-diffusion...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 672936 7 pageshttpdxdoiorg1011552013672936

Research ArticleNumerical Solution of Advection-Diffusion Equation Usinga Sixth-Order Compact Finite Difference Method

Gurhan Gurarslan1 Halil Karahan1 Devrim Alkaya1 Murat Sari2 and Mutlu Yasar1

1 Department of Civil Engineering Faculty of Engineering Pamukkale University 20070 Denizli Turkey2Department of Mathematics Faculty of Art and Science Pamukkale University 20070 Denizli Turkey

Correspondence should be addressed to Gurhan Gurarslan gurarslanpauedutr

Received 15 February 2013 Accepted 25 March 2013

Academic Editor Guohe Huang

Copyright copy 2013 Gurhan Gurarslan et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This study aims to produce numerical solutions of one-dimensional advection-diffusion equation using a sixth-order compactdifference scheme in space and a fourth-order Runge-Kutta scheme in time The suggested scheme here has been seen to be veryaccurate and a relatively flexible solution approach in solving the contaminant transport equation for Pe le 5 For the solution of thepresent equation the combined technique has been used instead of conventional solution techniques The accuracy and validity ofthe numerical model are verified through the presented results and the literature The computed results showed that the use of thecurrent method in the simulation is very applicable for the solution of the advection-diffusion equation The present technique isseen to be a very reliable alternative to existing techniques for these kinds of applications

1 Introduction

Problems of environmental pollution can always be reducedto the solution of a mathematical model of advection dif-fusion The unknown quantity in these cases is the con-centration 119862 a scalar physical quantity which representsthe mass of a pollutant or the salinity or temperature ofthe water [1] Advection-diffusion equation (ADE) illustratesmany quantities such as mass heat energy velocity andvorticity [2] The ADE has been used as a model equation inmany engineering problems such as dispersion of tracers inporous media [3] pollutant transport in rivers and streams[4] the dispersion of dissolved material in estuaries andcoastal seas [5 6] contaminant dispersion in shallow lakes[7] long-range transport of pollutants in the atmosphere [8]thermal pollution in river systems [9] and flow in porousmedia [10] The advection-diffusion transport equation inone-dimensional case without source terms is as follows

120597119862

120597119905+ 119880

120597119862

120597119909minus 119863

1205972119862

1205971199092= 0 0 le 119909 le 119871 0 lt 119905 le 119879 (1)

with initial condition

119862 (119909 119905 = 0) = 1198620(119909) (2)

and boundary conditions

119862 (119909 = 0 119905) = 1198910(119905)

119862 (119909 = 119871 119905) = 119891119871(119905) or minus 119863

120597119862 (119871 119905)

120597119909= 119892 (119905)

(3)

where 119905 is time 119909 is space coordinate 119863 is diffusion coef-ficient 119862(119909 119905) is concentration 119880(119909 119905) is velocity of waterflow and 119871 is length of the channel respectively Here 119862

0 1198910

119891119871 and 119892 are prescribed functions whilst 119862 is the unknown

function Notice that 119863 gt 0 and 119880 gt 0 are considered tobe positive constants quantifying the diffusion and advectionprocesses respectively

It is known that the use of the standard finite differenceand finite element method is not effective and often leads tounreal results For that reason several alternative methodsare proposed in the literature for solving the ADE with highaccuracy [11] These include method of characteristic with

2 Mathematical Problems in Engineering

Galerkinmethod (MOCG) [11] finite difference method [12ndash14] high-order finite element techniques [15] high-orderfinite difference methods [16ndash24] Green-element method[25] cubic B-spline [26] cubic B-spline differential quadra-turemethod (CBSDQM) [27] method of characteristics inte-grated with splines (MOCS) [28ndash30] Galerkin method withcubic B-splines (CBSG) [31] Taylor-Collocation (TC) andTaylor-Galerkin (TG) methods [32] B-spline finite elementmethod [33] Least squares finite element method (FEMLSFand FEMQSF) [34] Lattice Boltzmann method [35] Taylor-Galerkin B-spline finite element method [36] and meshlessmethod [37 38]

Utility of higher-order numerical methods in solvingmany problems accurately is required Lately a noticeableinterest in the development and application of CD methodsfor solving the Navier-Stokes [39ndash41] and other partialdifferential equations [42ndash46] has been renovated Narrowerstencils are required in the CD schemes and by a comparisonto classical difference schemes they have less truncationerror In the current paper accurate solutions of the ADEare obtained by using a sixth-order compact difference (CD6)[47] a fourth-order Runge-Kutta (RK4) schemes in space andtime respectively

2 The Compact Finite Difference Method

CDmethods are very popular in the fluid dynamics commu-nity because of their high accuracy and advantages associatedwith stencils [48] These methods are efficient for higheraccuracies without any increase in a stencil while traditionalhigh-order finite difference methods use larger stencil sizesthat make boundary treatment hard It can also be noted thatthe CD schemes have been demonstrated to be more preciseand computationally economic Use of smaller stencil sizesin the CD methods is useful when dealing with nonperiodicboundary conditions

A uniform one-dimensional mesh is considered consist-ing of119873points119909

1 1199092 119909

119894minus1 119909119894 119909119894+1 119909

119873with themesh

size ℎ = 119909119894+1minus 119909119894 The first-order derivatives of the unknown

function can be given at interior nodes as follows [47]

1205721198621015840

119894minus1+ 1198621015840

119894+ 1205721198621015840

119894+1= 119887119862119894+2minus 119862119894minus2

4ℎ+ 119886119862119894+1minus 119862119894minus1

2ℎ

(4)

leading to an 120572-family of fourth-order tridiagonal schemeswith

119886 =2

3(120572 + 2) 119887 =

1

3(4120572 minus 1) (5)

A sixth-order tridiagonal scheme is obtained by 120572 = 13

1198621015840

119894minus1+ 31198621015840

119894+ 1198621015840

119894+1=1

12ℎ(119862119894+2+ 28119862

119894+1minus 28119862

119894minus1+ 119862119894minus2)

(6)

Approximation formulae for the derivatives of nonperiodicproblems can be derived with the consideration of one-sided schemes for the boundary nodes Interested readers arereferred to the work of Lele [47] for details of the derivationsfor the first- and second-order derivatives

The third-order formula at boundary point 1 is as follows

1198621015840

1+ 21198621015840

2=1

ℎ(minus5

21198621+ 21198622+1

21198623) (7)

The fourth-order formula at boundary points 2 and119873 minus 1 isas follows

1198621015840

119894minus1+ 41198621015840

119894+ 1198621015840

119894+1=3

ℎ(119862119894+1minus 119862119894minus1) (8)

The third-order formula at boundary point119873 is as follows

21198621015840

119873minus1+ 1198621015840

119873=1

ℎ(5

2119862119873minus 2119862119873minus1

minus1

2119862119873minus2) (9)

Use of the above formulae leads to followingmatrix equation

B1198621015840 = A119862 (10)

where 119862 = (1198621 119862

119899)119879 Consideration of the first-order

operator twice will give us the second-order derivative termsthat is

B11986210158401015840 = A1198621015840 (11)

The RK4 scheme is considered to obtain the temporalintegration in the present study Utility of the CD6 techniqueto (1) gives rise to the following differential equation in time

119889119862119894

119889119905= 119871119862119894 (12)

where 119871 indicates a spatial linear differential operator Thespatial and temporal terms are approximated by the CD6and the RK4 schemes respectivelyThe semidiscrete equation(12) is solved using the RK4 scheme through the followingoperations

119862(1)= 119862119899+1

2Δ119905119871 (119862

119899)

119862(2)= 119862119899+1

2Δ119905119871 (119862

(1))

119862(3)= 119862119899+ Δ119905119871 (119862

(2))

119862119899+1=119862119899+1

6Δ119905 [119871 (119862

119899)+2119871 (119862

(1))+2119871 (119862

(2))+119871 (119862

(3))]

(13)

To obtain the approximate solution of (1) with the boundaryand initial conditions using the CD6-RK4 the domain [0 119871]is first discretized such that 0 = 119909

1lt 1199092lt sdot sdot sdot lt 119909

119873= 119871

where119873 is the number of grid points

3 Numerical Illustrations

Let us consider the advection-diffusion equation with theinitial and boundary conditions The numerical results arecompared with the exact solutions The differences betweenthe computed solutions and the exact solutions are shownin Tables 1ndash5 Three examples for which the exact solutions

Mathematical Problems in Engineering 3

Table 1 Peak concentration values at 119905 = 9600 s for various Cr numbers (Δ119905 = 50 s)

Cr 025 050 075 100MOCS [28] 9677 9756 9805 10000MOCG [11] 9816 9836 9934 10000CBSG [31] 9986 9986 9993 9986FEMLSF [34] 9647 9864 9918 9943FEMQSF [34] 9926 9932 9949 9961TC [32] 9940 9984 9993 9986TG [32] 9989 9991 9996 9991CD6 9999 10000 10000 10000Exact 10000 10000 10000 10000

Table 2 Error norms for various Cr values at 119905 = 9600 s

Cr ℎCSDQM [27] FEMLSF [34] FEMQSF [34] CD61198712

119871infin

1198712

119871infin

1198712

119871infin

1198712

119871infin

01250 200 34734 1156 32874 1350 12555 0518 08511 0429302500 100 2685 0136 10596 0494 7951 0376 00218 0010005000 50 0170 0008 7984 0380 7908 0373 00024 0000810000 25 0023 0001 7881 0377 7908 0379 00029 00007

Table 3 Comparison between numerical solutions and the exact solution

119909MOCS MOCG CBSG FEMLSF FEMQSF TC TG CD6 Exact[28] [11] [31] [34] [34] [32] [32] Δ119905 = 10 s Δ119905 = 1 s

0 1000 1000 1000 1000 1000 1000 1000 1000 1000 100018 1000 1000 1000 1000 1000 1000 1000 1000 1000 100019 1000 0999 1000 1000 1000 0999 0999 0999 0999 099920 1000 0998 0999 0999 1000 0999 0998 0998 0998 099821 1000 0996 0996 0997 0999 0999 0996 0996 0996 099622 1000 0990 0991 0993 0996 0998 0991 0992 0991 099123 1000 0978 0981 0985 0989 0994 0980 0982 0982 098224 1000 0957 0961 0970 0974 0987 0960 0965 0964 096425 1000 0922 0927 0943 0946 0972 0926 0936 0935 093426 0996 0870 0874 0902 0900 0945 0874 0891 0889 088927 1013 0799 0800 0842 0832 0902 0800 0827 0824 082328 1047 0708 0706 0763 0743 0838 0705 0743 0739 073829 0897 0602 0596 0666 0638 0755 0595 0641 0637 063630 0457 0488 0479 0556 0524 0653 0479 0528 0523 052331 0067 0375 0366 0442 0411 0541 0366 0413 0408 040832 minus0036 0272 0265 0332 0306 0427 0264 0306 0301 030133 minus0010 0185 0181 0235 0218 0320 0181 0212 0208 020834 0002 0118 0118 0156 0147 0227 0117 0138 0135 013535 0000 0070 0072 0096 0095 0152 0072 0084 0082 008236 0000 0038 0042 0055 0058 0096 0041 0048 0047 004637 0000 0020 0023 0030 0034 0057 0023 0025 0025 002438 0000 0009 0012 0015 0019 0032 0012 0012 0012 001239 0000 0004 0006 0007 0010 0017 0006 0006 0005 000540 0000 0002 0003 0003 0005 0008 0002 0002 0002 000241 0000 0001 0001 0001 0003 0004 0001 0001 0001 000142 0000 0000 0001 0000 0001 0001 0000 0000 0000 0000


Table 4 A comparison of the peak errors of different solutiontechniques for Pe = 4 with ℎ = 0025

Cr Δ119905 CN [12] FD3 [17] CD60016 00005 154119864 minus 03 913119864 minus 04 564119864 minus 09

0032 0001 151119864 minus 03 931119864 minus 04 564119864 minus 09






080 0025 NA NA 179119864 minus 06

Table 5 A comparison of analytical and CD6 solutions for variousvalues of 119909 with ℎ = 0025 Δ119905 = 0005 Cr = 016 and Pe = 4

119909 Exact CD6 Absolute error350 00000000 00000000 378119864 minus 13

400 00000159 00000159 102119864 minus 09

450 00201770 00201769 194119864 minus 08

500 02182179 02182179 110119864 minus 08

550 00201770 00201770 251119864 minus 08

600 00000159 00000160 945119864 minus 10

650 00000000 00000000 440119864 minus 13

are known are used to test the method described for solvingthe advection-diffusion equationThe technique is applied tosolve the ADE with 119862

0(119909) 119891

0(119905) 119891119871(119905) and 119892(119905) prescribed

To test the performance of the proposed method 1198712and 119871

infin

error norms are used as follows

119871infin= max119894

10038161003816100381610038161003816119888exact119894

minus 119888numerical119894

10038161003816100381610038161003816

1198712= radic

119873

sum

119894=1

1003816100381610038161003816119888exact119894


1003816100381610038161003816

2

(14)

Example 1 Here pure advection equation is considered in aninfinitely long channel of constant cross-section and bottomslope and velocity is taken to be 119880 = 05ms Concentrationis accepted to be the Gaussian distribution of 120588 = 264m andinitial peak location is 119909

0= 2000mThe initial distribution is

transported downstream in a long channel without change inshape by the time 119905 = 9600 s Exact solution of this problemis as follows [11]

119862 (119909 119905) = 10 exp[minus(119909 minus 119909

0minus 119880119905)

2

21205882

] (15)

At the boundaries the following conditions are used

119862 (0 119905) = 0

minus119863(120597119888

120597119909) (9000 119905) = 0

(16)

Initial conditions can be taken from exact solution Theinitial Gaussian pulse at 119905 = 0 the concentration distri-bution obtained using the CD6 solution and concentration

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

1

2

3

4

5

6

7

8

9

10

11

InitialCD6Exact

119909

119862

Figure 1 Comparison of the exact solution and the numericalsolution obtained with CD6 scheme for ℎ = 25 and Δ119905 = 50 s

distribution obtained using exact solution at 119905 = 9600 sare compared in Figure 1 Both quantitative and qualitativeagreements between the exact and the CD6 solutions areexcellent (see Tables 1 and 2 Figure 1)

As seen from Table 1 the CD6 method has given theclosest result to the exact peak concentration value Note thatsince an explicit time integration scheme RK4 is used in thisstudy the CD6 scheme cannot produce any result for Cr gt 1To show the accuracy of the obtained results 119871

2and 119871

infin

error norms have been calculated using the CD6 scheme andexhibited in Table 2 As seen in the corresponding table theCD6 solution is better than its rivals

Example 2 Flow velocity and diffusion coefficient are takento be 119880 = 001ms and 119863 = 0002m2s in this experimentLet the length of the channel be 119871 = 100m and be dividedinto 100 uniform elements The Pe number is accepted to be5 The Cr numbers are selected as 001 01 and 06 for thepresent work Exact solution of the current problem is [11]

119862 (119909 119905) =1

2erfc(119909 minus 119880119905

radic4119863119905

) +1

2exp (119880119909

119863) erfc(119909 + 119880119905

radic4119863119905

)

(17)


119862 (0 119905) = 1

minus119863(120597119888

120597119909) (119871 119905) = 0

(18)

Initial conditions can be taken from exact solution Com-parison between numerical solutions and the exact solutionis given in Table 3 In the calculation of the exact resultsgiven by Szymkiewicz [11] there has erroneously been a


0 10 20 30 40 50 60 70 80 90 1000

02

04

06

08

1

CD6Exact

119909

119862

Figure 2 Comparison of the exact solution and the numericalsolution obtained with CD6 scheme for ℎ = 1m and Δ119905 = 10 s

mistake Therefore the exact results have been recalculatedin MATLAB As seen in Table 3 the solutions produced byother researchers [11 28 31 32 34] for Cr = 06 do notconverge enoughThis case proves that the selected time step(Δ119905 = 60 s) is greater than it needs to be Note also thatthe CD6 scheme gives stable results for Cr = 06 and thecomputed results are nearly as good as in FEMQSF [34] Sincethis problem cannot be solved accurate enough for Δ119905 = 60 sthe calculations have been repeated for the cases Δ119905 = 10 s(Cr = 01) and Δ119905 = 1 s (Cr = 001) and the correspondingresults have been given in Table 3 The results produced bythe CD6 scheme for Δ119905 = 1 s are the same as with the exactsolution while the results of theCD6 scheme forΔ119905 = 10 s areseen to be acceptable level Comparison of the exact solutionand the numerical solution obtained with CD6 scheme forℎ = 1m and Δ119905 = 10 s is shown in Figure 2 As can be seenin Figure 2 there is an excellent agreement between CD6 andexact solutions

Example 3 Consider the quantities 119880 = 08ms and119863 = 0005m2s in (1) The following exact solution for thisexample can be found in [49]

119862 (119909 119905) =1

radic4119905 + 1

exp[minus(119909 minus 1 minus 119880119905)2

119863 (4119905 + 1)] (19)


119862 (0 119905) =1

radic4119905 + 1

exp[minus(minus1 minus 119880119905)2

119863(4119905 + 1)]

119862 (9 119905) =1

radic4119905 + 1

exp[minus (8 minus 119880119905)2

119863(4119905 + 1)]

(20)

Initial conditions can be taken from exact solution Thedistribution of the Gaussian pulse at 119905 = 5 s is computedusing the exact solution and comparedwith the concentration

35 4 45 5 55 6 650

005

01

015

02

CD6Exact

119909

119862

Figure 3 Comparison of analytical and CD6 solutions for transportof one-dimensional Gaussian pulse

distribution obtained using the CD6 solution as shown inFigure 3 As can be seen in Table 4 the CD6 results inExample 3 are far more accurate comparison to Crank-Nicolson (CN) scheme [12] and third-order finite difference(FD3) scheme [17] In Table 5 a comparison of analyticaland CD6 solutions is carried out for various values of 119909 withℎ = 0025 Δ119905 = 0005 Cr = 016 and Pe = 4

4 Conclusions

This paper deals with the advection-diffusion equation usingthe CD6 scheme in space and the RK4 in timeThe combinedmethod worked very well to give very reliable and accuratesolutions to these processes The CD6 scheme provides anefficient and alternative way for modeling the advection-diffusion processes The performance of the method for theconsidered problems was tested by computing 119871

2and 119871

infin

error norms The method gives convergent approximationsfor the advection-diffusion problems for Pe le 5 Notethat numerical solution cannot obtained while Pe gt 5 andCr gt 1 To overcome these disadvantages upwind compactschemes and implicit time integration need to be used Forfurther research special attention can be paid on the useof compact difference schemes in computational hydraulicproblems such as sediment transport in stream and lakescontaminant transport in groundwater and flood routing inriver and modeling of shallow water waves

References

[1] C G Koutitas Elements of Computational Hydraulics PentechPress London UK 1983

[2] B J Noye Numerical Solution of Partial Differential EquationsLecture Notes 1990


[3] Q N Fattah and J A Hoopes ldquoDispersion in anisotropichomogeneous porousmediardquo Journal of Hydraulic Engineeringvol 111 no 5 pp 810ndash827 1985

[4] P C Chatwin and C M Allen ldquoMathematical models ofdispersion in rivers and estuariesrdquo Annual Review of FluidMechanics vol 17 pp 119ndash149 1985

[5] F M Holly and J M Usseglio-Polatera ldquoDispersion simulationin two-dimensional tidal flowrdquo Journal ofHydraulic Engineeringvol 110 no 7 pp 905ndash926 1984

[6] J R Salmon J A Liggett andRHGallagher ldquoDispersion anal-ysis in homogeneous lakesrdquo International Journal for NumericalMethods in Engineering vol 15 no 11 pp 1627ndash1642 1980

[7] H Karahan ldquoAn iterative method for the solution of dispersionequation in shallow waterrdquo in Water Pollution VI ModellingMeasuring and Prediction C A Brebbia Ed pp 445ndash453Wessex Institute of Technology Southampton UK 2001

[8] Z Zlatev R Berkowicz and L P Prahm ldquoImplementationof a variable stepsize variable formula method in the time-integration part of a code for treatment of long-range transportof air pollutantsrdquo Journal of Computational Physics vol 55 no2 pp 278ndash301 1984

[9] M H Chaudhry D E Cass and J E Edinger ldquoModelingof unsteady-flow water temperaturesrdquo Journal of HydraulicEngineering vol 109 no 5 pp 657ndash669 1983

[10] N Kumar ldquoUnsteady flow against dispersion in finite porousmediardquo Journal of Hydrology vol 63 no 3-4 pp 345ndash358 1983

[11] R Szymkiewicz ldquoSolution of the advection-diffusion equationusing the spline function and finite elementsrdquo Communicationsin Numerical Methods in Engineering vol 9 no 3 pp 197ndash2061993

[12] H Karahan ldquoImplicit finite difference techniques for theadvection-diffusion equation using spreadsheetsrdquo Advances inEngineering Software vol 37 no 9 pp 601ndash608 2006

[13] H Karahan ldquoUnconditional stable explicit finite differencetechnique for the advection-diffusion equation using spread-sheetsrdquo Advances in Engineering Software vol 38 no 2 pp 80ndash86 2007

[14] H Karahan ldquoSolution of weighted finite difference techniqueswith the advection-diffusion equation using spreadsheetsrdquoComputer Applications in Engineering Education vol 16 no 2pp 147ndash156 2008

[15] H S Price J C Cavendish and R S Varga ldquoNumericalmethods of higher-order accuracy for diffusion-convectionequationsrdquo Society of Petroleum Engineers vol 8 no 3 pp 293ndash303 1968

[16] K W Morton ldquoStability of finite difference approximationsto a diffusion-convection equationrdquo International Journal forNumerical Methods in Engineering vol 15 no 5 pp 677ndash6831980

[17] H Karahan ldquoA third-order upwind scheme for the advection-diffusion equation using spreadsheetsrdquoAdvances in EngineeringSoftware vol 38 no 10 pp 688ndash697 2007

[18] Y Chen and R A Falconer ldquoAdvection-diffusion modellingusing the modified QUICK schemerdquo International Journal forNumerical Methods in Fluids vol 15 no 10 pp 1171ndash1196 1992

[19] M Dehghan ldquoWeighted finite difference techniques for theone-dimensional advection-diffusion equationrdquo Applied Math-ematics and Computation vol 147 no 2 pp 307ndash319 2004

[20] C Man and C W Tsai ldquoA higher-order predictor-correctorscheme for two-dimensional advection-diffusion equationrdquoInternational Journal for Numerical Methods in Fluids vol 56no 4 pp 401ndash418 2008

[21] M Dehghan and A Mohebbi ldquoHigh-order compact boundaryvalue method for the solution of unsteady convection-diffusionproblemsrdquo Mathematics and Computers in Simulation vol 79no 3 pp 683ndash699 2008

[22] G Gurarslan and H Karahan ldquoNumerical solution ofadvection-diffusion equation using a high-order MacCormackschemerdquo in Proceedings of the 6th National Hydrology CongressDenizli Turkey September 2011

[23] M Sari GGurarslan andA Zeytinoglu ldquoHigh-order finite dif-ference schemes for solving the advection-diffusion equationrdquoMathematical and Computational Applications vol 15 no 3 pp449ndash460 2010

[24] A Mohebbi andM Dehghan ldquoHigh-order compact solution ofthe one-dimensional heat and advection-diffusion equationsrdquoApplied Mathematical Modelling vol 34 no 10 pp 3071ndash30842010

[25] A E Taigbenu and O O Onyejekwe ldquoTransient 1D transportequation simulated by a mixed green element formulationrdquoInternational Journal for Numerical Methods in Fluids vol 25no 4 pp 437ndash454 1997

[26] R C Mittal and R K Jain ldquoNumerical solution of convection-diffusion equation using cubic B-splines collocation methodswith Neumannrsquos boundary conditionsrdquo International Journal ofAppliedMathematics and Computation vol 4 no 2 pp 115ndash1272012

[27] A Korkmaz and I Dag ldquoCubic B-spline differential quadraturemethods for the advection-diffusion equationrdquo InternationalJournal of Numerical Methods for Heat amp Fluid Flow vol 22no 8 pp 1021ndash1036 2012

[28] F M Holly and A Preissmann ldquoAccurate calculation of trans-port in two dimensionsrdquo Journal of Hydraulic Division vol 103no 11 pp 1259ndash1277 1977

[29] T L Tsai J C Yang and L H Huang ldquoCharacteristics methodusing cubic-spline interpolation for advection-diffusion equa-tionrdquo Journal of Hydraulic Engineering vol 130 no 6 pp 580ndash585 2004

[30] T L Tsai S W Chiang and J C Yang ldquoExamination ofcharacteristics method with cubic interpolation for advection-diffusion equationrdquo Computers and Fluids vol 35 no 10 pp1217ndash1227 2006

[31] L R T Gardner and I Dag ldquoA numerical solution of theadvection-diffusion equation using B-spline finite elementrdquo inProceedings International AMSE Conference lsquoSystems AnalysisControl amp Designlsquo pp 109ndash116 Lyon France July 1994

[32] I Dag A Canivar and A Sahin ldquoTaylor-Galerkin method foradvection-diffusion equationrdquo Kybernetes vol 40 no 5-6 pp762ndash777 2011

[33] S Dhawan S Kapoor and S Kumar ldquoNumerical method foradvection diffusion equation using FEM and B-splinesrdquo Journalof Computational Science vol 3 pp 429ndash437 2012

[34] I Dag D Irk and M Tombul ldquoLeast-squares finite elementmethod for the advection-diffusion equationrdquo Applied Mathe-matics and Computation vol 173 no 1 pp 554ndash565 2006

[35] B Servan-Camas and F T C Tsai ldquoLattice Boltzmann methodwith two relaxation times for advection-diffusion equationthird order analysis and stability analysisrdquo Advances in WaterResources vol 31 no 8 pp 1113ndash1126 2008

[36] M K Kadalbajoo and P Arora ldquoTaylor-Galerkin B-spline finiteelement method for the one-dimensional advection-diffusionequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 5 pp 1206ndash1223 2010


[37] M Zerroukat K Djidjeli and A Charafi ldquoExplicit and implicitmeshless methods for linear advection-diffusion-type partialdifferential equationsrdquo International Journal for NumericalMethods in Engineering vol 48 no 1 pp 19ndash35 2000

[38] J Li Y Chen and D Pepper ldquoRadial basis function methodfor 1-D and 2-D groundwater contaminant transportmodelingrdquoComputational Mechanics vol 32 no 1-2 pp 10ndash15 2003

[39] Y V S S Sanyasiraju and V Manjula ldquoHigher order semicompact scheme to solve transient incompressible Navier-Stokes equationsrdquo Computational Mechanics vol 35 no 6 pp441ndash448 2005

[40] Z Tian and Y Ge ldquoA fourth-order compact finite differencescheme for the steady stream function-vorticity formulation ofthe Navier-StokesBoussinesq equationsrdquo International Journalfor NumericalMethods in Fluids vol 41 no 5 pp 495ndash518 2003

[41] Z Tian X Liang and P Yu ldquoA higher order compact finitedifference algorithm for solving the incompressible Navier-Stokes equationsrdquo International Journal for Numerical Methodsin Engineering vol 88 no 6 pp 511ndash532 2011

[42] M Sari and G Gurarslan ldquoA sixth-order compact finite differ-ence scheme to the numerical solutions of Burgersrsquo equationrdquoApplied Mathematics and Computation vol 208 no 2 pp 475ndash483 2009

[43] M Sari G Gurarslan and I Dag ldquoA compact finite differencemethod for the solution of the generalized burgers-fisherequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 1 pp 125ndash134 2010

[44] M Sari and G Gurarslan ldquoA sixth-order compact finite differ-ence method for the one-dimensional sine-Gordon equationrdquoInternational Journal for Numerical Methods in BiomedicalEngineering vol 27 no 7 pp 1126ndash1138 2011

[45] G Gurarslan ldquoNumerical modelling of linear and nonlin-ear diffusion equations by compact finite difference methodrdquoAppliedMathematics andComputation vol 216 no 8 pp 2472ndash2478 2010

[46] M Sari ldquoSolution of the porous media equation by a compactfinite difference methodrdquo Mathematical Problems in Engineer-ing vol 2009 Article ID 912541 13 pages 2009

[47] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[48] J C Kalita and A K Dass ldquoHigh-order compact simulation ofdouble-diffusive natural convection in a vertical porous annu-lusrdquo Engineering Applications of Computational Fluid Dynamicsvol 5 no 3 pp 357ndash371 2011

[49] S Sankaranarayanan N J Shankar and H F Cheong ldquoThree-dimensional finite difference model for transport of conserva-tive pollutantsrdquo Ocean Engineering vol 25 no 6 pp 425ndash4421998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Galerkinmethod (MOCG) [11] finite difference method [12ndash14] high-order finite element techniques [15] high-orderfinite difference methods [16ndash24] Green-element method[25] cubic B-spline [26] cubic B-spline differential quadra-turemethod (CBSDQM) [27] method of characteristics inte-grated with splines (MOCS) [28ndash30] Galerkin method withcubic B-splines (CBSG) [31] Taylor-Collocation (TC) andTaylor-Galerkin (TG) methods [32] B-spline finite elementmethod [33] Least squares finite element method (FEMLSFand FEMQSF) [34] Lattice Boltzmann method [35] Taylor-Galerkin B-spline finite element method [36] and meshlessmethod [37 38]

Utility of higher-order numerical methods in solvingmany problems accurately is required Lately a noticeableinterest in the development and application of CD methodsfor solving the Navier-Stokes [39ndash41] and other partialdifferential equations [42ndash46] has been renovated Narrowerstencils are required in the CD schemes and by a comparisonto classical difference schemes they have less truncationerror In the current paper accurate solutions of the ADEare obtained by using a sixth-order compact difference (CD6)[47] a fourth-order Runge-Kutta (RK4) schemes in space andtime respectively

2 The Compact Finite Difference Method

CDmethods are very popular in the fluid dynamics commu-nity because of their high accuracy and advantages associatedwith stencils [48] These methods are efficient for higheraccuracies without any increase in a stencil while traditionalhigh-order finite difference methods use larger stencil sizesthat make boundary treatment hard It can also be noted thatthe CD schemes have been demonstrated to be more preciseand computationally economic Use of smaller stencil sizesin the CD methods is useful when dealing with nonperiodicboundary conditions

A uniform one-dimensional mesh is considered consist-ing of119873points119909

1 1199092 119909

119894minus1 119909119894 119909119894+1 119909

119873with themesh

size ℎ = 119909119894+1minus 119909119894 The first-order derivatives of the unknown

function can be given at interior nodes as follows [47]

1205721198621015840

119894minus1+ 1198621015840

119894+ 1205721198621015840

119894+1= 119887119862119894+2minus 119862119894minus2

4ℎ+ 119886119862119894+1minus 119862119894minus1

2ℎ

(4)

leading to an 120572-family of fourth-order tridiagonal schemeswith

119886 =2

3(120572 + 2) 119887 =

1

3(4120572 minus 1) (5)

A sixth-order tridiagonal scheme is obtained by 120572 = 13

1198621015840

119894minus1+ 31198621015840

119894+ 1198621015840

119894+1=1

12ℎ(119862119894+2+ 28119862

119894+1minus 28119862

119894minus1+ 119862119894minus2)

(6)

Approximation formulae for the derivatives of nonperiodicproblems can be derived with the consideration of one-sided schemes for the boundary nodes Interested readers arereferred to the work of Lele [47] for details of the derivationsfor the first- and second-order derivatives

The third-order formula at boundary point 1 is as follows

1198621015840

1+ 21198621015840

2=1

ℎ(minus5

21198621+ 21198622+1

21198623) (7)

The fourth-order formula at boundary points 2 and119873 minus 1 isas follows

1198621015840

119894minus1+ 41198621015840

119894+ 1198621015840

119894+1=3

ℎ(119862119894+1minus 119862119894minus1) (8)

The third-order formula at boundary point119873 is as follows

21198621015840

119873minus1+ 1198621015840

119873=1

ℎ(5

2119862119873minus 2119862119873minus1

minus1

2119862119873minus2) (9)

Use of the above formulae leads to followingmatrix equation

B1198621015840 = A119862 (10)

where 119862 = (1198621 119862

119899)119879 Consideration of the first-order

operator twice will give us the second-order derivative termsthat is

B11986210158401015840 = A1198621015840 (11)

The RK4 scheme is considered to obtain the temporalintegration in the present study Utility of the CD6 techniqueto (1) gives rise to the following differential equation in time

119889119862119894

119889119905= 119871119862119894 (12)

where 119871 indicates a spatial linear differential operator Thespatial and temporal terms are approximated by the CD6and the RK4 schemes respectivelyThe semidiscrete equation(12) is solved using the RK4 scheme through the followingoperations

119862(1)= 119862119899+1

2Δ119905119871 (119862

119899)

119862(2)= 119862119899+1

2Δ119905119871 (119862

(1))

119862(3)= 119862119899+ Δ119905119871 (119862

(2))

119862119899+1=119862119899+1

6Δ119905 [119871 (119862

119899)+2119871 (119862

(1))+2119871 (119862

(2))+119871 (119862

(3))]

(13)

To obtain the approximate solution of (1) with the boundaryand initial conditions using the CD6-RK4 the domain [0 119871]is first discretized such that 0 = 119909

1lt 1199092lt sdot sdot sdot lt 119909

119873= 119871

where119873 is the number of grid points

3 Numerical Illustrations

Let us consider the advection-diffusion equation with theinitial and boundary conditions The numerical results arecompared with the exact solutions The differences betweenthe computed solutions and the exact solutions are shownin Tables 1ndash5 Three examples for which the exact solutions






119871infin

1198712

119871infin

1198712

119871infin

1198712

119871infin

01250 200 34734 1156 32874 1350 12555 0518 08511 0429302500 100 2685 0136 10596 0494 7951 0376 00218 0010005000 50 0170 0008 7984 0380 7908 0373 00024 0000810000 25 0023 0001 7881 0377 7908 0379 00029 00007



0 1000 1000 1000 1000 1000 1000 1000 1000 1000 100018 1000 1000 1000 1000 1000 1000 1000 1000 1000 100019 1000 0999 1000 1000 1000 0999 0999 0999 0999 099920 1000 0998 0999 0999 1000 0999 0998 0998 0998 099821 1000 0996 0996 0997 0999 0999 0996 0996 0996 099622 1000 0990 0991 0993 0996 0998 0991 0992 0991 099123 1000 0978 0981 0985 0989 0994 0980 0982 0982 098224 1000 0957 0961 0970 0974 0987 0960 0965 0964 096425 1000 0922 0927 0943 0946 0972 0926 0936 0935 093426 0996 0870 0874 0902 0900 0945 0874 0891 0889 088927 1013 0799 0800 0842 0832 0902 0800 0827 0824 082328 1047 0708 0706 0763 0743 0838 0705 0743 0739 073829 0897 0602 0596 0666 0638 0755 0595 0641 0637 063630 0457 0488 0479 0556 0524 0653 0479 0528 0523 052331 0067 0375 0366 0442 0411 0541 0366 0413 0408 040832 minus0036 0272 0265 0332 0306 0427 0264 0306 0301 030133 minus0010 0185 0181 0235 0218 0320 0181 0212 0208 020834 0002 0118 0118 0156 0147 0227 0117 0138 0135 013535 0000 0070 0072 0096 0095 0152 0072 0084 0082 008236 0000 0038 0042 0055 0058 0096 0041 0048 0047 004637 0000 0020 0023 0030 0034 0057 0023 0025 0025 002438 0000 0009 0012 0015 0019 0032 0012 0012 0012 001239 0000 0004 0006 0007 0010 0017 0006 0006 0005 000540 0000 0002 0003 0003 0005 0008 0002 0002 0002 000241 0000 0001 0001 0001 0003 0004 0001 0001 0001 000142 0000 0000 0001 0000 0001 0001 0000 0000 0000 0000










080 0025 NA NA 179119864 minus 06



400 00000159 00000159 102119864 minus 09

450 00201770 00201769 194119864 minus 08

500 02182179 02182179 110119864 minus 08

550 00201770 00201770 251119864 minus 08

600 00000159 00000160 945119864 minus 10

650 00000000 00000000 440119864 minus 13


0(119909) 119891

0(119905) 119891119871(119905) and 119892(119905) prescribed


infin


119871infin= max119894

10038161003816100381610038161003816119888exact119894


10038161003816100381610038161003816

1198712= radic

119873

sum

119894=1

1003816100381610038161003816119888exact119894


1003816100381610038161003816

2

(14)




119862 (119909 119905) = 10 exp[minus(119909 minus 119909

0minus 119880119905)

2

21205882

] (15)


119862 (0 119905) = 0

minus119863(120597119888

120597119909) (9000 119905) = 0

(16)


0 1000 2000 3000 4000 5000 6000 7000 8000 90000

1

2

3

4

5

6

7

8

9

10

11

InitialCD6Exact

119909

119862




2and 119871

infin



119862 (119909 119905) =1

2erfc(119909 minus 119880119905

radic4119863119905

) +1

2exp (119880119909

119863) erfc(119909 + 119880119905

radic4119863119905

)

(17)


119862 (0 119905) = 1

minus119863(120597119888

120597119909) (119871 119905) = 0

(18)



0 10 20 30 40 50 60 70 80 90 1000

02

04

06

08

1

CD6Exact

119909

119862




119862 (119909 119905) =1

radic4119905 + 1


119863 (4119905 + 1)] (19)


119862 (0 119905) =1

radic4119905 + 1


119863(4119905 + 1)]

119862 (9 119905) =1

radic4119905 + 1

exp[minus (8 minus 119880119905)2

119863(4119905 + 1)]

(20)


35 4 45 5 55 6 650

005

01

015

02

CD6Exact

119909

119862



4 Conclusions


2and 119871

infin


References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119871infin

1198712

119871infin

1198712

119871infin

1198712

119871infin

01250 200 34734 1156 32874 1350 12555 0518 08511 0429302500 100 2685 0136 10596 0494 7951 0376 00218 0010005000 50 0170 0008 7984 0380 7908 0373 00024 0000810000 25 0023 0001 7881 0377 7908 0379 00029 00007



0 1000 1000 1000 1000 1000 1000 1000 1000 1000 100018 1000 1000 1000 1000 1000 1000 1000 1000 1000 100019 1000 0999 1000 1000 1000 0999 0999 0999 0999 099920 1000 0998 0999 0999 1000 0999 0998 0998 0998 099821 1000 0996 0996 0997 0999 0999 0996 0996 0996 099622 1000 0990 0991 0993 0996 0998 0991 0992 0991 099123 1000 0978 0981 0985 0989 0994 0980 0982 0982 098224 1000 0957 0961 0970 0974 0987 0960 0965 0964 096425 1000 0922 0927 0943 0946 0972 0926 0936 0935 093426 0996 0870 0874 0902 0900 0945 0874 0891 0889 088927 1013 0799 0800 0842 0832 0902 0800 0827 0824 082328 1047 0708 0706 0763 0743 0838 0705 0743 0739 073829 0897 0602 0596 0666 0638 0755 0595 0641 0637 063630 0457 0488 0479 0556 0524 0653 0479 0528 0523 052331 0067 0375 0366 0442 0411 0541 0366 0413 0408 040832 minus0036 0272 0265 0332 0306 0427 0264 0306 0301 030133 minus0010 0185 0181 0235 0218 0320 0181 0212 0208 020834 0002 0118 0118 0156 0147 0227 0117 0138 0135 013535 0000 0070 0072 0096 0095 0152 0072 0084 0082 008236 0000 0038 0042 0055 0058 0096 0041 0048 0047 004637 0000 0020 0023 0030 0034 0057 0023 0025 0025 002438 0000 0009 0012 0015 0019 0032 0012 0012 0012 001239 0000 0004 0006 0007 0010 0017 0006 0006 0005 000540 0000 0002 0003 0003 0005 0008 0002 0002 0002 000241 0000 0001 0001 0001 0003 0004 0001 0001 0001 000142 0000 0000 0001 0000 0001 0001 0000 0000 0000 0000










080 0025 NA NA 179119864 minus 06



400 00000159 00000159 102119864 minus 09

450 00201770 00201769 194119864 minus 08

500 02182179 02182179 110119864 minus 08

550 00201770 00201770 251119864 minus 08

600 00000159 00000160 945119864 minus 10

650 00000000 00000000 440119864 minus 13


0(119909) 119891

0(119905) 119891119871(119905) and 119892(119905) prescribed


infin


119871infin= max119894

10038161003816100381610038161003816119888exact119894


10038161003816100381610038161003816

1198712= radic

119873

sum

119894=1

1003816100381610038161003816119888exact119894


1003816100381610038161003816

2

(14)




119862 (119909 119905) = 10 exp[minus(119909 minus 119909

0minus 119880119905)

2

21205882

] (15)


119862 (0 119905) = 0

minus119863(120597119888

120597119909) (9000 119905) = 0

(16)


0 1000 2000 3000 4000 5000 6000 7000 8000 90000

1

2

3

4

5

6

7

8

9

10

11

InitialCD6Exact

119909

119862




2and 119871

infin



119862 (119909 119905) =1

2erfc(119909 minus 119880119905

radic4119863119905

) +1

2exp (119880119909

119863) erfc(119909 + 119880119905

radic4119863119905

)

(17)


119862 (0 119905) = 1

minus119863(120597119888

120597119909) (119871 119905) = 0

(18)



0 10 20 30 40 50 60 70 80 90 1000

02

04

06

08

1

CD6Exact

119909

119862




119862 (119909 119905) =1

radic4119905 + 1


119863 (4119905 + 1)] (19)


119862 (0 119905) =1

radic4119905 + 1


119863(4119905 + 1)]

119862 (9 119905) =1

radic4119905 + 1

exp[minus (8 minus 119880119905)2

119863(4119905 + 1)]

(20)


35 4 45 5 55 6 650

005

01

015

02

CD6Exact

119909

119862



4 Conclusions


2and 119871

infin


References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















080 0025 NA NA 179119864 minus 06



400 00000159 00000159 102119864 minus 09

450 00201770 00201769 194119864 minus 08

500 02182179 02182179 110119864 minus 08

550 00201770 00201770 251119864 minus 08

600 00000159 00000160 945119864 minus 10

650 00000000 00000000 440119864 minus 13


0(119909) 119891

0(119905) 119891119871(119905) and 119892(119905) prescribed


infin


119871infin= max119894

10038161003816100381610038161003816119888exact119894


10038161003816100381610038161003816

1198712= radic

119873

sum

119894=1

1003816100381610038161003816119888exact119894


1003816100381610038161003816

2

(14)




119862 (119909 119905) = 10 exp[minus(119909 minus 119909

0minus 119880119905)

2

21205882

] (15)


119862 (0 119905) = 0

minus119863(120597119888

120597119909) (9000 119905) = 0

(16)


0 1000 2000 3000 4000 5000 6000 7000 8000 90000

1

2

3

4

5

6

7

8

9

10

11

InitialCD6Exact

119909

119862




2and 119871

infin



119862 (119909 119905) =1

2erfc(119909 minus 119880119905

radic4119863119905

) +1

2exp (119880119909

119863) erfc(119909 + 119880119905

radic4119863119905

)

(17)


119862 (0 119905) = 1

minus119863(120597119888

120597119909) (119871 119905) = 0

(18)



0 10 20 30 40 50 60 70 80 90 1000

02

04

06

08

1

CD6Exact

119909

119862




119862 (119909 119905) =1

radic4119905 + 1


119863 (4119905 + 1)] (19)


119862 (0 119905) =1

radic4119905 + 1


119863(4119905 + 1)]

119862 (9 119905) =1

radic4119905 + 1

exp[minus (8 minus 119880119905)2

119863(4119905 + 1)]

(20)


35 4 45 5 55 6 650

005

01

015

02

CD6Exact

119909

119862



4 Conclusions


2and 119871

infin


References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 80 90 1000

02

04

06

08

1

CD6Exact

119909

119862




119862 (119909 119905) =1

radic4119905 + 1


119863 (4119905 + 1)] (19)


119862 (0 119905) =1

radic4119905 + 1


119863(4119905 + 1)]

119862 (9 119905) =1

radic4119905 + 1

exp[minus (8 minus 119880119905)2

119863(4119905 + 1)]

(20)


35 4 45 5 55 6 650

005

01

015

02

CD6Exact

119909

119862



4 Conclusions


2and 119871

infin


References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article numerical solution of advection-diffusion...

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