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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 596218 10 pageshttpdxdoiorg1011552013596218

Research ArticleComparison Study on the Performances of Finite VolumeMethod and Finite Difference Method

Renwei Liu1 Dongjie Wang1 Xinyu Zhang1 Wang Li12 and Bo Yu1

1 National Engineering Laboratory for Pipeline Safety and Beijing Key Laboratory of Urban Oil and Gas Distribution TechnologyChina University of Petroleum Beijing 102249 China

2 PetroChina Southwest Pipeline Company Chengdu 610041 China

Correspondence should be addressed to Bo Yu yuboboxvip163com

Received 26 March 2013 Revised 9 June 2013 Accepted 9 June 2013

Academic Editor Shuyu Sun

Copyright copy 2013 Renwei Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Vorticity-stream function method and MAC algorithm are adopted to systemically compare the finite volume method (FVM)and finite difference method (FDM) in this paper Two typical problemsmdashlid-driven flow and natural convection flow in a squarecavitymdashare taken as examples to compare and analyze the calculation performances of FVM and FDMwith variant mesh densitiesdiscrete forms and treatments of boundary condition It is indicated that FVM is superior to FDM from the perspective of accuracystability of convection term robustness and calculation efficiency Particularly when the mesh is coarse and taken as 20 times 20 theresults of FDM suffer severe oscillation and even lose physical meaning

1 Introduction

In the numerical solution of flow and heat transfer problemsthe concepts of conservative and nonconservative equationswere firstly proposed in [1 2] in the 1980s It is noteworthythat from the perspective of differential unit the conservativeand nonconservative equations are equivalent which areboth the mathematical expression of physical conservationlaw Nevertheless the numerical calculation is implementedon the calculation unit of finite size for which the twoforms of equations are of different characteristics Practicalcalculation process shows that the influence of differencesbetween conservative and nonconservative forms on accu-racy stability and efficiency of numerical calculation issignificant

Conservative governing equation and the correspondingdiscrete form show some advantages for example in the con-trol volume with finite size only the conservative equationcan guarantee that the conservation law of the problem stud-ied is satisfied [3ndash5] The result obtained by the conservativeequation is of higher accuracy generally In the calculation offlow problem involving shock wave the obtained flow field isusually smooth and stable employing the conservation formof governing equation while using the non-conservation

equation might lead to unsatisfactory spatial oscillations inthe upstream and downstream regions of the shock wave[6ndash8] Moreover when the conservation equation is usedto a body-fitted coordinate system the conservativeness ofgoverning equation can still be satisfied [9 10] More relatedresearches and applications are presented in the literature[11ndash15] Hence the conservation property of discretizedequations is desirable in engineering calculation Based onthis it is of great significance to investigate the influencefrom conservation property on the calculation performancesand provide guidance for the better implementation of itThe present paper compares the calculation performances byanalyzing the finite volumemethodwhich is conservative andfinite difference method which is nonconservative

Some researchers made comparisons of the twomethodsPatankar [16] and West and Fuller [17] pointed out thatthe results obtained in the situation that grids numbersdecrease and mass conservation is rigorously controlled loseits physical meaning Leonard [18] made a comparison ofthe accuracy of truncation error of the convection term inFVM and FDM By theoretical derivation he indicated thatthe truncation error of FVM is smaller than that of FDMin second-order central difference and second-order upwinddifference Botte et al [19] made comparison of the accuracy

2 Journal of Applied Mathematics

mass conservation and computation time It was validated byexamples that FDM cannot ensure mass conservation whileFVM can Meanwhile the influence of boundary conditionson the accuracy of algorithm was taken into consideration

Previous researchers mainly focused on the comparisonof accuracy There are few systemic studies on the stabilityrobustness and computation efficiencyThis paper comparedthe calculation performances of FVM and FDM based onprevious research Comprehensive comparison of accuracystability of convection term robustness and calculationefficiency was performed The results obtained may enrichthe study on the calculation performances of conservativeand nonconservativemethods which are expected to providesome reference for the practical calculation

2 Physical Problem

Tomake the study possess general significance it is necessaryto analyze the calculation performances of the two methodsbased on variant problems Lid-driven flow and naturalconvection flow in a square cavity which are both typicalproblems in computational fluid mechanics and numericalheat transfer are solved in this paper to make a systemicanalysis on the subject

As Figure 1(a) shows in the lid-driven flow the lid moveshorizontally at a constant velocity 119906top while the other threeboundaries keep still The Reynolds number Re (Reynoldsnumber) adopted in this paper is 1000

Figure 1(b) shows the schematic of natural convectionflow in a square cavity The left and right boundaries are thefirst boundary conditions between which the temperatureon the left boundary is higher The other two boundaries areadiabatic ones In calculation Pr (Prandtl number) is takenas 071 and Ra (Rayleigh number) is taken as 106

3 Numerical Method

Considering that the calculation approach is influential tothe calculation in order to compare the difference of cal-culation performances between FVM and FDM this paperadopts both primitive variable method and nonprimitivevariable method to solve the governing equations For prim-itive variable method MAC algorithm is adopted and fornonprimitive variable method the generally used vorticity-stream function algorithm is adopted The conservationproperty generally refers to the convection term and thusonly the difference of convective term between conservativeand nonconservative forms is compared MAC method ischaracterized by the direct solving of velocity and pressureThe key point of this method is to obtain a nondivergentvelocity field in every time step or iteration step That is thevelocity field of every time layer or iteration layer shouldsatisfy the continuity equation and pressure and the velocityis decoupled by solving Poisson equation which is relevant topressure

31 Governing Equation In primitive variable method pres-sure and velocity are directly used as variables to solve the

Table 1 Parameters of equations for MAC of lid-driven flow

Name of variables Φ Γ120601

119878120601

119883 direction 1198801

Reminus120597119875

120597119883

119884 direction 1198811

Reminus120597119875

120597119884

Table 2 Parameters of equations for MAC of natural convectionflow


119878120601

119883 direction 119880 1 minus120597119875

120597119883

119884 direction 119881 1 RaΘPr

minus120597119875

120597119884

Temperature Θ1

Pr0

equations As to the incompressible fluid the dimensionlessunsteady-state conservative and nonconservative equationscan be written as respectively

120597Φ

120597120591+120597 (119880Φ)

120597119883+120597 (119881Φ)

120597119884= Γ120601(1205972Φ

1205971198832+1205972Φ

1205971198842) + 119878120601 (1)

120597Φ

120597120591+ 119880

120597Φ

120597119883+ 119881

120597Φ

120597119884= Γ120601(1205972Φ

1205971198832+1205972Φ

1205971198842) + 119878120601 (2)

where the values of Φ Γ120601 and 119878

120601for lid-driven flow and

natural convection flow are shown in Table 1 and Table 2respectively From (1) and (2) we can see that the convec-tion term of conservative equation is given in the form ofdivergence while that in nonconservative equation is notwhere the nondimensional excessive temperature can bedescribed as Θ = (119879 minus 119879

119897)(119879ℎminus 119879119897) and U V are defined as

nondimensional velocities in separate directions (X directionor Y direction) Governing equations of conservative andnonconservative equations of primitive variable method aregiven before In the calculation process MAC algorithmis adopted to obtain the result in steady state by unsteadycalculation over enough time The result of steady state isanalyzed as follows from the angle of primitive variablemethod Taking the vorticity-stream function method asexample the governing equations of conservative form andnonconservative form can be written as respectively

119886120601(120597 (119880Φ)

120597119883+120597 (119881Φ)

120597119884) = Γ120601(1205972Φ

1205971198832+1205972Φ

1205971198842) + 119878120601 (3)

119886120601(119880

120597Φ

120597119883+ 119881

120597Φ

120597119884) = Γ120601(1205972Φ

1205971198832+1205972Φ

1205971198842) + 119878120601 (4)

As to the vorticity-stream function method only the lid-driven flow is studied in this paper Values ofΦ 119886

120601 Γ120601 and 119878

120601

are shown in Table 3 where ldquo120596rdquo stands for vorticity and ldquo120595rdquostands for stream function

Journal of Applied Mathematics 3

l

l

y

x

119984top

(a) Lid-Driven Flow

l

l

y

x

gTh Tl

Adiabatic

Adiabatic

(b) Natural Convection Flow

Figure 1 Schematic of different physical problem in a square cavity

Table 3 Parameters of equations for vorticity-stream functionalgorithm of lid-driven flow

Name of variables Φ 119886120601

Γ120601

119878120601

Vorticity equation 120596 1 1

Re0

Stream function equation 120595 0 1 minus120596

where ldquo120596rdquo stands for vorticity and ldquo120595rdquo stands for stream function

32 Discretization of Equations Uniform mesh is adopted inthis paper In order to compare the calculation performancesof FVM and FDM with variant discretized schemes centraldifference and second-order upwind scheme are adopted Asto FVM taking e interface on X direction as example theexpressions can be written as

Φ+

119890= Φminus

119890=Φ119875+ Φ119864

2 (5)

Φ+

119890= 15Φ

119875minus 05Φ

119882 119880

119890ge 0 (6a)

Φminus

119890= 15Φ

119864minus 05Φ

119864119864 119880

119890lt 0 (6b)

where ldquoerdquo stands for the east interface between two nodes ldquoErdquostands for east node and ldquoWrdquo means west node and ldquo119864119864rdquo canbe defined as the east node of ldquoErdquo node

As to FDM taking node119875 as example the expressions canbe written as

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875

=120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875

=Φ119864minus 2Φ119875+ Φ119882

2Δ119883 (7)

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875

=3Φ119875minus 4Φ119882+ Φ119882119882

2Δ119883 119880

119875ge 0 (8a)

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875

=minus3Φ119875+ 4Φ119864minus Φ119864119864

2Δ119883 119880

119875lt 0 (8b)

xy

ΔY (i 1)

(i 2)

(i 3)

Figure 2 Schematic of the treatment for boundary conditions ofvorticity-stream function method

33 Treatment of the Boundary Condition of Vorticity Sincethe influence from boundary condition of vorticity on thesolution of equations is significant in this section the calcula-tion performances of FVM and FDMwith variant treatmentsof the boundary condition for vorticity are studied

Three boundary conditions of vorticity are comparedand studied In the boundary as shown in Figure 2 theirexpressions Thom equations [20] Woods equation [21] andJenson equation [22] as well as corresponding accuracy areshown in (9)ndash(11)

Thom equation

1205961198941=2 (1205951198942minus 1205951198941)

1205751198842

+ 119874 (Δ119884) (9)

Woods equation

1205961198941=3 (1205951198942minus 1205951198941)

Δ1198842

minus1

21205961198942+ 119874 (Δ119884

2) (10)

Jenson equation

1205961198941=minus71205951198941+ 81205951198942minus 1205951198943

2Δ1198842

+ 119874 (Δ1198842) (11)


34 Deferred Correction When MAC is adopted momen-tum and energy equations are both solved in explicit schemewhile in vorticity-stream function method the convectionterm is treated in implicit scheme by deferred correctionwhich can guarantee the main-diagonal domination of thesolvingmatrix and thus guarantee the calculation stability Asto FVM taking Φ

119890as example the expressions of deferred

correction are as follows

when 119880119890ge 0 Φ

119890= Φ119875+ (Φ+

119890minus Φ119875)lowast

when 119880119890lt 0 Φ

119890= Φ119864+ (Φminus


(12)

where ldquolowastrdquo in these equations stands for the former calculationstep

As to FDM similarly to the treatment of Φ119890in FVM the

deferred correction for node 119875 can be written as the sum offirst-order upwind gradient and correction gradient which isshown as follows

when 119880119875ge 0 119880

119875

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875

= 119880119875(Φ119875minus Φ119882

Δ119883)

+ [119880119875(120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875

minusΦ119875minus Φ119882

Δ119883)]

lowast

(13)

when 119880119875lt 0 119880

119875

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875

= 119880119875(Φ119864minus Φ119875

Δ119883)

+ [119880119875(120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875


Δ119883)]

lowast

(14)

The first terms in the right side of (12)ndash(14) are all first-orderupwind term The second terms are incorporated into thesource term as the form of correction term The discretizedequation is solved by G-S solver with underrelaxation ofwhich the discrete expression is shown as (15) and thediscretized equations of FVM and FDM are shown in (16a)(16b) (17a) and (17b) respectively

119886119875Φ119875= 119886119864Φ119864+ 119886119882Φ119882+ 119886119873Φ119873+ 119886119878Φ119878+ 119887 (15)

119886119864=max (minus119880

119890 0)

Δ119883+

1

ReΔ1198832

119886119882=max (119880

119908 0)

Δ119883+

1

ReΔ1198832

119886119873=max (minus119881

119899 0)

Δ119884+

1

ReΔ1198842

119886119878=max (119881

119904 0)

Δ119884+

1

ReΔ1198842

119886119875=

max (119880119890 0)

Δ119883+max (minus119880

119908 0)

Δ119883+max (119881

119899 0)

Δ119884

+max (minus119881

119904 0)

Δ119884+

2

ReΔ1198832+

2

ReΔ1198842

(16a)

119887 = minusmax (119880119890 0) [

Φ+

119890minus Φ119875

Δ119883] +max (minus119880

119890 0) [

Φminus

119890minus Φ119864

Δ119883]

+max (119880119908 0)[

Φ+

119908minus Φ119882

Δ119883]minusmax (minus119880

119908 0)[

Φminus

119908minus Φ119875

Δ119883]

minusmax (119881119899 0) [

Φ+

119899minus Φ119875

Δ119884] +max (minus119881

119899 0) [

Φminus

119899minus Φ119873

Δ119884]

+max (119881119904 0) [

Φ+

119904minus Φ119878

Δ119884] minusmax (minus119881

119904 0) [

Φminus

119904minus Φ119875

Δ119884]

(16b)

119886119864=max (minus119880

119875 0)

Δ119883+

1

ReΔ1198832

119886119882=max (119880

119875 0)

Δ119883+

1

ReΔ1198832

119886119873=max (minus119881

119875 0)

Δ119884+

1

ReΔ1198842

119886119878=max (119881

119875 0)

Δ119884+

1

ReΔ1198842

119886119875=

max (minus119880119875 0)

Δ119883+max (119880

119875 0)

Δ119883+max (minus119881

119875 0)

Δ119884

+max (119881

119875 0)

Δ119884+

2

ReΔ1198832+

2

ReΔ1198842

(17a)

119887 = minusmax (119880119875 0) [

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875


Δ119883]

+max (minus119880119875 0) [

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875


Δ119883]

minusmax (119881119875 0) [

120597Φ

120597119884

10038161003816100381610038161003816100381610038161003816

+

119875


Δ119884]

+max (minus119881119875 0) [

120597Φ

120597119884

10038161003816100381610038161003816100381610038161003816

minus

119875


Δ119883]

(17b)

where 119886119873 119886119875 119886119864 119886119882 and 119886

119878are coefficients in discretized

equations and 119887 is the source term in discretized equations

4 Results Comparison and Analysis

In this section the accuracy stability of convection termrobustness and calculation efficiency of variant algorithmsschemes and treatments of boundary condition which havebeen discussed previously are systemically compared andanalyzed based on the calculation results of lid-driven flowand natural convection flow in a square cavity

41 Accuracy Firstly the accuracy of FVM and FDM isevaluated by comparing benchmark solution with the results


Table 4 Truncation error of FVM and FDMgoverning equations for vorticity-stream function algorithm of lid driven flow in a square cavity

Convection item Diffusion item

FVM (format)Φ119890

Δ119883minusΦ119908

Δ119883

1

Δ119883

120597Φ

120597119909

10038161003816100381610038161003816100381610038161003816119890

minus1

Δ119883

120597Φ

120597119909

10038161003816100381610038161003816100381610038161003816119908

FVM (TE) minusΔ1198832

8Φ101584010158401015840

119875minusΔ1198834

128Φ(5)

119875minus sdot sdot sdot minus

Δ1198832

24Φ(4)

119875minus 13

Δ1198834

5760Φ(6)

119875minus sdot sdot sdot

FDM (format) 120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816119875

1205972Φ

1205971198832

100381610038161003816100381610038161003816100381610038161003816119875

FDM (TE) minusΔ1198832

6Φ101584010158401015840

119875minusΔ1198834

120Φ(5)


Δ1198832

12Φ(4)

119875minusΔ1198834

360Φ(6)


obtained from variant mesh density discrete schemes andboundary conditions Through precision analysis FVM andFDM are respectively adopted with the Taylor expansioncommenced at midpoint (P node) and their respectivetruncation error (TE) is just as shown Table 4 [19] Thetruncation errors of FVM are obviously less than the FDMrsquos

411 Influence of Mesh Density In [4] it is indicated thatthe conservative property can affect the accuracy while thenonconservative FDM loses its conservative property in thesituation when the grid is too small Tomake further analysisthe calculation results obtained by FVM and FDM withvariant mesh densities are compared The results of thevelocity along the horizontal central axisV and velocity alongthe vertical central axis U obtained by FDM and FVM withcentral difference scheme of MAC algorithm are comparedand illustrated in Figure 3

From Figure 3 we can see that when the mesh densityis taken as 80 times 80 the results obtained by FVM and FDMboth agree well with the benchmark solutionWhen themeshdensity is reduced to 40 times 40 the results obtained by FVMare closer to the benchmark solution and if the mesh densityis further reduced to 20 times 20 the results obtained by FDMdiffer from the benchmark solution greatly in which FVMshows its advantage in accuracyThis phenomenon is in goodagreement with the theoretical analysis in [18] that is theconservative property of FVM better ensures the accuracyof calculation Therefore whether the conservative propertycan be realized is of much significance to the accuracy of theobtained result

412 Influence of Boundary Condition In the followingsection FVM and FDM with variant boundary conditionsare compared from the perspective of accuracy by vorticity-stream function method The results of the velocity alongthe horizontal central axis V and velocity along the verticalcentral axis U obtained by FDM and FVM with centraldifference scheme are compared and illustrated in Figure 4and the influence of boundary condition of vorticity is alsogiven From Figure 4 we can see that no matter whichtreatment of boundary condition is adopted the accuracy ofFVM is higher than that of FDM in vorticity-stream functionmethod

413 Influence of Discrete Scheme The selection of discreteforms is also influential to the accuracy Thus the resultsobtained with variant discrete forms are studied Taking thevorticity-stream function with boundary condition treatedby first-order Thom equation as example the results of thevelocity along the horizontal central axis V and velocityalong the vertical central axis U obtained by FDM and FVMwith variant discrete schemes are compared and illustratedin Figure 5 from which we can see that whether second-order central difference or second-order upwind scheme isadopted the accuracy of FVM is higher than that of FDM thecomparison between (a) and (b) indicates that second-orderupwind difference possesses higher accuracy than second-order central difference

414 Influence of Physical Problem Finally two methods invariant physical problems are compared from the perspectiveof accuracy based on the convection flow in a square cavityTaking the result obtained by MAC algorithm as exampleFigure 6 shows the comparison of results of the velocity alongthe horizontal central axis V and velocity along the verticalcentral axisU obtained by FDMandFVMTemperature fieldsobtained by FVM and FDM are compared with the grid-independent solution (obtained with grid of 256 times 256) asFigure 7 shows

From the comparison in Figure 6 and Figure 7 we cansee that based on the results obtained by MAC of naturalconvection flow in a square cavity the accuracy of FVM isgenerally superior to that of FDM Thus it is indicated bycomparison that no matter which mesh density scheme andtreatment of boundary condition is adopted FVM showsbetter accuracy than FDM

42 Stability of Convection Term Stability of mathematicalmeaning can only ensure that the oscillation of solution iscontrolled within a range but cannot eliminate the absenceof oscillation The amplification of oscillation may result indivergence Stability of convection term originates from var-ied difference schemes for convective itemand is independentof the introducing of rounding error in the calculation Thisstability is the key point to obtain a solution with physicalmeaning It is concluded that all the unstable schemes willresult in the oscillation of solution Stability condition of


0

04

08

12

0

03

06

U V

X(Y)

0 02 04 06 08 1minus04

minus06

minus03

(a) 80 times 80

X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(b) 40 times 40

FDMFVM (U) benchmark solution

(V) benchmark solution

X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(c) 20 times 20

Figure 3 Velocity on the central axis of lid-driven flow obtained by MAC with variant mesh densities



X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(a) Thom



X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(b) Woods

Figure 4 Velocity along the central axis obtained by second-order central difference scheme with two treatments of boundary conditionwhen the mesh density is 40 times 40




X(Y)

0 02 04 06 08 1

0

04

08

12U

minus04

0

03

06

V

minus06

minus03

(a) Second-order central difference scheme



X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(b) Second-order upwind difference scheme

Figure 5 Velocity along the central axis obtained by vorticity-stream function method for lid-driven flow with different schemes when themesh density is 40 times 40

FDMFVM (U) grid-independent

(V) grid-independent

X(Y)

0 02 04 06 08 1

0

60

120

U

minus120

minus60

0

200

400

V

minus400

minus200

(a) 40 times 40



X(Y)

0 02 04 06 08 1

0

60

120

U

minus120

minus60

0

200

400

V

minus400

minus200

(b) 80 times 80

Figure 6 Velocity along the central axis of convection flow in a square cavity with variant gird densities

40 times 40

Grid-independent

(a) 40 times 40

80 times 80

Grid-independent

(b) 80 times 80

Figure 7 Comparison of temperature contours of natural convection flow in a square cavity with variant mesh densities (FVM in the leftcolumn and FDM in the right column)


(a) 40 times 40 (b) 20 times 20

Figure 8 Comparison of stream lines in lid-driven flow with variant mesh densities (FVM in the left column and FDM in the right column)

(a) Thom (b) Woods

(c) Jenson

Figure 9 Stream functions of lid-driven flow under variant boundary treatments with the grid of 20 times 20 (FVM in the left column and FDMin the right column)

the convection item (Mesh Peclet number) is affected bythe following factors such as mesh density and boundarycondition Taking lid-driven flow as example the stabilityof convection term of FVM and FDM with variant meshdensities treatments of boundary condition is compared

421 Influence of Mesh Density Figure 8 shows the compar-ison of stream lines between FVM and FDM with variantmesh densities With the grid of 40 times 40 the two methodsboth can capture the structure of primary vortex and sec-ondary vortex as Figure 8(a) shows It is seen in Figure 8(b)that when the mesh of 20 times 20 is adopted concussionphenomenon appears which ismore serious in FDM and thelocation of the vortex center obviously moves upward

422 Influence of Boundary Condition Similarly to the accu-racy analysis the results obtained by variant treatments for

vorticity of wall boundary are compared as Figure 9 showsThree treatments Thom Woods and Jenson are analyzedFrom Figure 9 we can see that only when Jenson equationis adopted the results obtained by FVM and FDM arerelatively close while when Thom and Woods are adoptedFVM shows better results than FDM The results obtainedby FDM suffer severe oscillation and even lack physicalmeaning It is clear that based on the lid-driven flow stabilityof convection term of FVM is superior to that of FDMThe influence from the same boundary error (truncationerror and rounding error) on difference solving methodsis illustrated in Figure 9 With the treatment of boundarycondition with the same truncation error or rounding errorthe stability of FVM is better than that of FDM especiallyin the situation with boundary treatment of (a) and (b)with relatively low precision truncation error FDM is veryunstable and very easy to slip into oscillation


120572

02 04 06 08 1

Nite

r

80 times 80

40 times 4020 times 20

FVM FDM

5

4

3

2

1

0

times104


02 04 06 08 1120572

Nite

r

5

4

3

2

1

0

80 times 80


FVM FDM

times104

(b) Second-order upwind scheme

Figure 10 Comparison of the robustness of lid-driven flow with variant schemes

43 Robustness Robustness is a very critical index to evaluatethe numerical approaches In this section the robustness ofFVM and FDM is compared based on the lid-driven flowproblem which is solved by vorticity-stream function Gridsof 20 times 20 40 times 40 and 80 times 80 are adopted respectively inthe calculation Relaxation factor is tentatively taken up from02 to the maximum which can ensure the convergence ofcalculation Iterations are correspondingly recorded as well

It is shown in Figure 10 when the whether the cen-tral difference scheme or second-order upwind scheme isadopted with the refining of calculation grids the maximumrelaxation factor (120572) increases in both FVM and FDM therobustness is enhanced and the iterations (Niter) corre-spondingly decrease as well

As Figure 10(a) shows in the situation that central differ-ence scheme is adopted with the grid of 20 times 20 and 40 times 40the maximum relaxation factor of FVM is bigger than thatof FVM while with the grid of 80 times 80 the relaxation factorcan be taken as 10 in bothmethods Similarly as Figure 10(b)shows generally speaking when second-order upwind dif-ference scheme is adopted the maximum relaxation factor isbigger than that of FDM Based on the result of lid-drivenflow the conservative FVM is superior to the nonconservativeFDM from the perspective of numerical robustness

In the numerical simulation of complex flow field cal-culation efficiency is another critical index on which thispapermakes analysis FromFigure 10we can see that with theincreasing of relaxation factor the iterations of two methodsboth decrease generally What needs to be notified is that theiterations of FVM are smaller than those of FDM

44 Efficiency In the numerical simulation of complex flowfield calculation efficiency is another critical index on whichthis paper makes analysis

As is shown in Table 5 although the calculation timerequired by every iteration step in FVM and FDM is almostthe same with the increasing of mesh numbers calculationtime required by FVM slightly decreases Thus total calcula-tion time of FVM is much smaller than that of FDMThat isFVM is much more efficient that FDM Thus it is validatedthat the efficiency of FVM is better than that of FDM in thesame calculation condition

5 Conclusions

Based on vorticity-stream function and MAC algorithmlid-driven flow and natural convection in a square cavityare calculated The accuracy stability of convective termrobustness of FVM and FDM with variant mesh densitiesdiscrete forms and treatments of boundary condition arecompared The following conclusions are drawn

(1) No matter which algorithm discrete form or treat-ment of boundary condition is adopted the resultsobtained by conservative FVM are closer to bench-mark solution than those obtained by FDM Theaccuracy of FVM is higher than FDM especiallywhen the mesh density is relatively small

(2) The results of lid-driven flow indicate that the resultsobtained by FDM show more serious oscillation andthe results obtained even lose physicalmeaningThusthe stability of convection term of FVM is superiorthan that of FDM

(3) As to the lid-driven flow it is indicated that whenvorticity-stream function method is adopted therelaxation factor of conservative method is biggerthan that of the nonconservative one while theiterations are less which shows that the robustness


Table 5 Computation iterations and time of FVM and FDM in second-order central difference scheme for vorticity-stream functionalgorithm of lid driven flow

Mesh 120572FVM FDM

Iterations Time(s) Time of each iteration Iterations Time(s) Time of each iteration20 02 4950 738 0015 2100 307 001520 01 10500 1544 0015 4300 616 001440 06 3250 1519 0047 4350 1993 004640 05 4450 2035 0046 5800 2629 004540 04 6300 2963 0047 7950 3693 004680 08 4950 8223 0166 5050 8584 017080 06 8500 14134 0166 9000 15395 017180 04 17250 29694 0172 17750 32420 0183

and calculation efficiency of conservative method arebetter than those of the nonconservative method

(4) In the lid-driven flow with vorticity-stream functionmethod the calculation time required by every itera-tion step of FVM is slightly smaller than that of FDMTogether with the smaller total iteration steps FVMis more efficient than FDM

Acknowledgment

The study is supported by the National Science Foundation ofChina (nos 51134006 51176204 and 51276198)

References

[1] J D Anderson Jr Computational Fluid Dynamics The BasicsWith Applications McGraw-Hill 1995

[2] P J Roache Computational Fluid Dynamics Hermosa 1976[3] W Q Tao Heat Transfer Science Press 2001[4] W Q Tao Advances in Computational Heat Transfer Science

Press 2000[5] Y Jaluria and K E Torrance Computational Heat Transfer

Hemisphere Publishing Corporation 1986[6] P D Lax ldquoWeak solutions of nonlinear hyperbolic equations

and their numerical computationrdquo Communications on Pureand Applied Mathematics vol 7 pp 159ndash193 1954

[7] P Lax and B Wendroff ldquoSystems of conservation lawsrdquo Com-munications on Pure and Applied Mathematics vol 13 no 2 pp217ndash237 1960

[8] A Jameson ldquoTransonic potential flow calculations using con-servation formrdquo in Proceedings of the Computational fluiddynamics conference pp 148ndash161 Hartford Conn USA 1975

[9] M Vinokur ldquoConservation equations of gasdynamics in curvi-linear coordinate systemsrdquo Journal of Computational Physicsvol 14 pp 105ndash125 1974

[10] T J Bridges ldquoConservation laws in curvilinear coordinatesa short proof of Vinokurrsquos theorem using differential formsrdquoApplied Mathematics and Computation vol 202 no 2 pp 882ndash885 2008

[11] R J DiPerna ldquoDecay of solutions of hyperbolic systems ofconservation laws with a convex extensionrdquoArchive for RationalMechanics and Analysis vol 64 no 1 pp 1ndash46 1977

[12] A Bressan ldquoHyperbolic systems of conservation lawsrdquo RevistaMatematica Complutense vol 12 no 1 pp 135ndash198 1999

[13] R K S Hankin ldquoThe Euler equations for multiphase com-pressible flow in conservation form-Simulation of shock-bubbleinteractionsrdquo Journal of Computational Physics vol 172 no 2pp 808ndash826 2001

[14] C Lai R A Baltzer andRW Schaffranek ldquoConservation-formequations of unsteady open-channel flowrdquo Journal of HydraulicResearch vol 40 no 5 pp 567ndash578 2002

[15] L M Gao K T Li B Liu and J Su ldquoGeneral integralconservation form of Navier-Stokes equations and numericalapplicationrdquo Chinese Journal of Computational Physics vol 25no 2 pp 172ndash178 2008

[16] S Patankar Numerical Heat Transfer and Fluid Flow Hemi-sphere Publishing 1980

[17] A West and T Fuller ldquoInfluence of rib spacing in proton-exchange membrane electrode assembliesrdquo Journal of AppliedElectrochemistry vol 26 no 6 pp 557ndash565 1996

[18] B P Leonard ldquoComparison of truncation error of finite-difference and finite-volume formulations of convection termsrdquoApplied Mathematical Modelling vol 18 no 1 pp 46ndash50 1994

[19] G G Botte J A Ritter and R E White ldquoComparison of finitedifference and control volume methods for solving differentialequationsrdquoComputers and Chemical Engineering vol 24 no 12pp 2633ndash2654 2000

[20] AThom ldquoThe flow past cylinders at low speedsrdquo Proceedings ofthe Royal Society A vol 141 pp 651ndash659 1993

[21] L C Woods ldquoA note on the numerical solution of fourth orderdifferential equationsrdquo vol 5 pp 176ndash182 1954

[22] V G Jenson ldquoViscous flow round a sphere at low Reynoldsnumbers (Re iexcl 40)rdquo Proceedings of the Royal Society A vol 249pp 346ndash366 1959

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


mass conservation and computation time It was validated byexamples that FDM cannot ensure mass conservation whileFVM can Meanwhile the influence of boundary conditionson the accuracy of algorithm was taken into consideration

Previous researchers mainly focused on the comparisonof accuracy There are few systemic studies on the stabilityrobustness and computation efficiencyThis paper comparedthe calculation performances of FVM and FDM based onprevious research Comprehensive comparison of accuracystability of convection term robustness and calculationefficiency was performed The results obtained may enrichthe study on the calculation performances of conservativeand nonconservativemethods which are expected to providesome reference for the practical calculation

2 Physical Problem

Tomake the study possess general significance it is necessaryto analyze the calculation performances of the two methodsbased on variant problems Lid-driven flow and naturalconvection flow in a square cavity which are both typicalproblems in computational fluid mechanics and numericalheat transfer are solved in this paper to make a systemicanalysis on the subject

As Figure 1(a) shows in the lid-driven flow the lid moveshorizontally at a constant velocity 119906top while the other threeboundaries keep still The Reynolds number Re (Reynoldsnumber) adopted in this paper is 1000

Figure 1(b) shows the schematic of natural convectionflow in a square cavity The left and right boundaries are thefirst boundary conditions between which the temperatureon the left boundary is higher The other two boundaries areadiabatic ones In calculation Pr (Prandtl number) is takenas 071 and Ra (Rayleigh number) is taken as 106

3 Numerical Method

Considering that the calculation approach is influential tothe calculation in order to compare the difference of cal-culation performances between FVM and FDM this paperadopts both primitive variable method and nonprimitivevariable method to solve the governing equations For prim-itive variable method MAC algorithm is adopted and fornonprimitive variable method the generally used vorticity-stream function algorithm is adopted The conservationproperty generally refers to the convection term and thusonly the difference of convective term between conservativeand nonconservative forms is compared MAC method ischaracterized by the direct solving of velocity and pressureThe key point of this method is to obtain a nondivergentvelocity field in every time step or iteration step That is thevelocity field of every time layer or iteration layer shouldsatisfy the continuity equation and pressure and the velocityis decoupled by solving Poisson equation which is relevant topressure

31 Governing Equation In primitive variable method pres-sure and velocity are directly used as variables to solve the

Table 1 Parameters of equations for MAC of lid-driven flow


119878120601

119883 direction 1198801

Reminus120597119875

120597119883

119884 direction 1198811

Reminus120597119875

120597119884

Table 2 Parameters of equations for MAC of natural convectionflow


119878120601

119883 direction 119880 1 minus120597119875

120597119883

119884 direction 119881 1 RaΘPr

minus120597119875

120597119884

Temperature Θ1

Pr0

equations As to the incompressible fluid the dimensionlessunsteady-state conservative and nonconservative equationscan be written as respectively

120597Φ

120597120591+120597 (119880Φ)

120597119883+120597 (119881Φ)

120597119884= Γ120601(1205972Φ

1205971198832+1205972Φ

1205971198842) + 119878120601 (1)

120597Φ

120597120591+ 119880

120597Φ

120597119883+ 119881

120597Φ

120597119884= Γ120601(1205972Φ

1205971198832+1205972Φ

1205971198842) + 119878120601 (2)

where the values of Φ Γ120601 and 119878

120601for lid-driven flow and

natural convection flow are shown in Table 1 and Table 2respectively From (1) and (2) we can see that the convec-tion term of conservative equation is given in the form ofdivergence while that in nonconservative equation is notwhere the nondimensional excessive temperature can bedescribed as Θ = (119879 minus 119879

119897)(119879ℎminus 119879119897) and U V are defined as

nondimensional velocities in separate directions (X directionor Y direction) Governing equations of conservative andnonconservative equations of primitive variable method aregiven before In the calculation process MAC algorithmis adopted to obtain the result in steady state by unsteadycalculation over enough time The result of steady state isanalyzed as follows from the angle of primitive variablemethod Taking the vorticity-stream function method asexample the governing equations of conservative form andnonconservative form can be written as respectively

119886120601(120597 (119880Φ)

120597119883+120597 (119881Φ)

120597119884) = Γ120601(1205972Φ

1205971198832+1205972Φ

1205971198842) + 119878120601 (3)

119886120601(119880

120597Φ

120597119883+ 119881

120597Φ

120597119884) = Γ120601(1205972Φ

1205971198832+1205972Φ

1205971198842) + 119878120601 (4)

As to the vorticity-stream function method only the lid-driven flow is studied in this paper Values ofΦ 119886

120601 Γ120601 and 119878

120601

are shown in Table 3 where ldquo120596rdquo stands for vorticity and ldquo120595rdquostands for stream function


l

l

y

x

119984top

(a) Lid-Driven Flow

l

l

y

x

gTh Tl

Adiabatic

Adiabatic





Γ120601

119878120601


Re0




Φ+

119890= Φminus

119890=Φ119875+ Φ119864

2 (5)

Φ+

119890= 15Φ

119875minus 05Φ

119882 119880

119890ge 0 (6a)

Φminus

119890= 15Φ

119864minus 05Φ

119864119864 119880

119890lt 0 (6b)



120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875

=120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875

=Φ119864minus 2Φ119875+ Φ119882

2Δ119883 (7)

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875

=3Φ119875minus 4Φ119882+ Φ119882119882

2Δ119883 119880

119875ge 0 (8a)

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875

=minus3Φ119875+ 4Φ119864minus Φ119864119864

2Δ119883 119880

119875lt 0 (8b)

xy

ΔY (i 1)

(i 2)

(i 3)




Thom equation

1205961198941=2 (1205951198942minus 1205951198941)

1205751198842

+ 119874 (Δ119884) (9)

Woods equation

1205961198941=3 (1205951198942minus 1205951198941)

Δ1198842

minus1

21205961198942+ 119874 (Δ119884

2) (10)

Jenson equation

1205961198941=minus71205951198941+ 81205951198942minus 1205951198943

2Δ1198842

+ 119874 (Δ1198842) (11)





when 119880119890ge 0 Φ

119890= Φ119875+ (Φ+


when 119880119890lt 0 Φ

119890= Φ119864+ (Φminus


(12)




when 119880119875ge 0 119880

119875

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875

= 119880119875(Φ119875minus Φ119882

Δ119883)

+ [119880119875(120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875


Δ119883)]

lowast

(13)

when 119880119875lt 0 119880

119875

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875

= 119880119875(Φ119864minus Φ119875

Δ119883)

+ [119880119875(120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875


Δ119883)]

lowast

(14)


119886119875Φ119875= 119886119864Φ119864+ 119886119882Φ119882+ 119886119873Φ119873+ 119886119878Φ119878+ 119887 (15)

119886119864=max (minus119880

119890 0)

Δ119883+

1

ReΔ1198832

119886119882=max (119880

119908 0)

Δ119883+

1

ReΔ1198832

119886119873=max (minus119881

119899 0)

Δ119884+

1

ReΔ1198842

119886119878=max (119881

119904 0)

Δ119884+

1

ReΔ1198842

119886119875=

max (119880119890 0)

Δ119883+max (minus119880

119908 0)

Δ119883+max (119881

119899 0)

Δ119884

+max (minus119881

119904 0)

Δ119884+

2

ReΔ1198832+

2

ReΔ1198842

(16a)

119887 = minusmax (119880119890 0) [

Φ+

119890minus Φ119875

Δ119883] +max (minus119880

119890 0) [

Φminus

119890minus Φ119864

Δ119883]

+max (119880119908 0)[

Φ+

119908minus Φ119882


119908 0)[

Φminus

119908minus Φ119875

Δ119883]

minusmax (119881119899 0) [

Φ+

119899minus Φ119875

Δ119884] +max (minus119881

119899 0) [

Φminus

119899minus Φ119873

Δ119884]

+max (119881119904 0) [

Φ+

119904minus Φ119878


119904 0) [

Φminus

119904minus Φ119875

Δ119884]

(16b)

119886119864=max (minus119880

119875 0)

Δ119883+

1

ReΔ1198832

119886119882=max (119880

119875 0)

Δ119883+

1

ReΔ1198832

119886119873=max (minus119881

119875 0)

Δ119884+

1

ReΔ1198842

119886119878=max (119881

119875 0)

Δ119884+

1

ReΔ1198842

119886119875=

max (minus119880119875 0)

Δ119883+max (119880

119875 0)

Δ119883+max (minus119881

119875 0)

Δ119884

+max (119881

119875 0)

Δ119884+

2

ReΔ1198832+

2

ReΔ1198842

(17a)

119887 = minusmax (119880119875 0) [

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875


Δ119883]

+max (minus119880119875 0) [

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875


Δ119883]

minusmax (119881119875 0) [

120597Φ

120597119884

10038161003816100381610038161003816100381610038161003816

+

119875


Δ119884]

+max (minus119881119875 0) [

120597Φ

120597119884

10038161003816100381610038161003816100381610038161003816

minus

119875


Δ119883]

(17b)

where 119886119873 119886119875 119886119864 119886119882 and 119886










Δ119883minusΦ119908

Δ119883

1

Δ119883

120597Φ

120597119909

10038161003816100381610038161003816100381610038161003816119890

minus1

Δ119883

120597Φ

120597119909

10038161003816100381610038161003816100381610038161003816119908


8Φ101584010158401015840

119875minusΔ1198834

128Φ(5)


Δ1198832

24Φ(4)

119875minus 13

Δ1198834

5760Φ(6)



120597119883

10038161003816100381610038161003816100381610038161003816119875

1205972Φ

1205971198832

100381610038161003816100381610038161003816100381610038161003816119875


6Φ101584010158401015840

119875minusΔ1198834

120Φ(5)


Δ1198832

12Φ(4)

119875minusΔ1198834

360Φ(6)











0

04

08

12

0

03

06

U V

X(Y)

0 02 04 06 08 1minus04

minus06

minus03

(a) 80 times 80

X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(b) 40 times 40



X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(c) 20 times 20




X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(a) Thom



X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(b) Woods





X(Y)

0 02 04 06 08 1

0

04

08

12U

minus04

0

03

06

V

minus06

minus03




X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03





X(Y)

0 02 04 06 08 1

0

60

120

U

minus120

minus60

0

200

400

V

minus400

minus200

(a) 40 times 40



X(Y)

0 02 04 06 08 1

0

60

120

U

minus120

minus60

0

200

400

V

minus400

minus200

(b) 80 times 80


40 times 40

Grid-independent

(a) 40 times 40

80 times 80

Grid-independent

(b) 80 times 80



(a) 40 times 40 (b) 20 times 20


(a) Thom (b) Woods

(c) Jenson







120572

02 04 06 08 1

Nite

r

80 times 80


FVM FDM

5

4

3

2

1

0

times104


02 04 06 08 1120572

Nite

r

5

4

3

2

1

0

80 times 80


FVM FDM

times104









5 Conclusions







Mesh 120572FVM FDM




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










l

l

y

x

119984top

(a) Lid-Driven Flow

l

l

y

x

gTh Tl

Adiabatic

Adiabatic





Γ120601

119878120601


Re0




Φ+

119890= Φminus

119890=Φ119875+ Φ119864

2 (5)

Φ+

119890= 15Φ

119875minus 05Φ

119882 119880

119890ge 0 (6a)

Φminus

119890= 15Φ

119864minus 05Φ

119864119864 119880

119890lt 0 (6b)



120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875

=120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875

=Φ119864minus 2Φ119875+ Φ119882

2Δ119883 (7)

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875

=3Φ119875minus 4Φ119882+ Φ119882119882

2Δ119883 119880

119875ge 0 (8a)

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875

=minus3Φ119875+ 4Φ119864minus Φ119864119864

2Δ119883 119880

119875lt 0 (8b)

xy

ΔY (i 1)

(i 2)

(i 3)




Thom equation

1205961198941=2 (1205951198942minus 1205951198941)

1205751198842

+ 119874 (Δ119884) (9)

Woods equation

1205961198941=3 (1205951198942minus 1205951198941)

Δ1198842

minus1

21205961198942+ 119874 (Δ119884

2) (10)

Jenson equation

1205961198941=minus71205951198941+ 81205951198942minus 1205951198943

2Δ1198842

+ 119874 (Δ1198842) (11)





when 119880119890ge 0 Φ

119890= Φ119875+ (Φ+


when 119880119890lt 0 Φ

119890= Φ119864+ (Φminus


(12)




when 119880119875ge 0 119880

119875

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875

= 119880119875(Φ119875minus Φ119882

Δ119883)

+ [119880119875(120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875


Δ119883)]

lowast

(13)

when 119880119875lt 0 119880

119875

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875

= 119880119875(Φ119864minus Φ119875

Δ119883)

+ [119880119875(120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875


Δ119883)]

lowast

(14)


119886119875Φ119875= 119886119864Φ119864+ 119886119882Φ119882+ 119886119873Φ119873+ 119886119878Φ119878+ 119887 (15)

119886119864=max (minus119880

119890 0)

Δ119883+

1

ReΔ1198832

119886119882=max (119880

119908 0)

Δ119883+

1

ReΔ1198832

119886119873=max (minus119881

119899 0)

Δ119884+

1

ReΔ1198842

119886119878=max (119881

119904 0)

Δ119884+

1

ReΔ1198842

119886119875=

max (119880119890 0)

Δ119883+max (minus119880

119908 0)

Δ119883+max (119881

119899 0)

Δ119884

+max (minus119881

119904 0)

Δ119884+

2

ReΔ1198832+

2

ReΔ1198842

(16a)

119887 = minusmax (119880119890 0) [

Φ+

119890minus Φ119875

Δ119883] +max (minus119880

119890 0) [

Φminus

119890minus Φ119864

Δ119883]

+max (119880119908 0)[

Φ+

119908minus Φ119882


119908 0)[

Φminus

119908minus Φ119875

Δ119883]

minusmax (119881119899 0) [

Φ+

119899minus Φ119875

Δ119884] +max (minus119881

119899 0) [

Φminus

119899minus Φ119873

Δ119884]

+max (119881119904 0) [

Φ+

119904minus Φ119878


119904 0) [

Φminus

119904minus Φ119875

Δ119884]

(16b)

119886119864=max (minus119880

119875 0)

Δ119883+

1

ReΔ1198832

119886119882=max (119880

119875 0)

Δ119883+

1

ReΔ1198832

119886119873=max (minus119881

119875 0)

Δ119884+

1

ReΔ1198842

119886119878=max (119881

119875 0)

Δ119884+

1

ReΔ1198842

119886119875=

max (minus119880119875 0)

Δ119883+max (119880

119875 0)

Δ119883+max (minus119881

119875 0)

Δ119884

+max (119881

119875 0)

Δ119884+

2

ReΔ1198832+

2

ReΔ1198842

(17a)

119887 = minusmax (119880119875 0) [

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875


Δ119883]

+max (minus119880119875 0) [

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875


Δ119883]

minusmax (119881119875 0) [

120597Φ

120597119884

10038161003816100381610038161003816100381610038161003816

+

119875


Δ119884]

+max (minus119881119875 0) [

120597Φ

120597119884

10038161003816100381610038161003816100381610038161003816

minus

119875


Δ119883]

(17b)

where 119886119873 119886119875 119886119864 119886119882 and 119886










Δ119883minusΦ119908

Δ119883

1

Δ119883

120597Φ

120597119909

10038161003816100381610038161003816100381610038161003816119890

minus1

Δ119883

120597Φ

120597119909

10038161003816100381610038161003816100381610038161003816119908


8Φ101584010158401015840

119875minusΔ1198834

128Φ(5)


Δ1198832

24Φ(4)

119875minus 13

Δ1198834

5760Φ(6)



120597119883

10038161003816100381610038161003816100381610038161003816119875

1205972Φ

1205971198832

100381610038161003816100381610038161003816100381610038161003816119875


6Φ101584010158401015840

119875minusΔ1198834

120Φ(5)


Δ1198832

12Φ(4)

119875minusΔ1198834

360Φ(6)











0

04

08

12

0

03

06

U V

X(Y)

0 02 04 06 08 1minus04

minus06

minus03

(a) 80 times 80

X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(b) 40 times 40



X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(c) 20 times 20




X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(a) Thom



X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(b) Woods





X(Y)

0 02 04 06 08 1

0

04

08

12U

minus04

0

03

06

V

minus06

minus03




X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03





X(Y)

0 02 04 06 08 1

0

60

120

U

minus120

minus60

0

200

400

V

minus400

minus200

(a) 40 times 40



X(Y)

0 02 04 06 08 1

0

60

120

U

minus120

minus60

0

200

400

V

minus400

minus200

(b) 80 times 80


40 times 40

Grid-independent

(a) 40 times 40

80 times 80

Grid-independent

(b) 80 times 80



(a) 40 times 40 (b) 20 times 20


(a) Thom (b) Woods

(c) Jenson







120572

02 04 06 08 1

Nite

r

80 times 80


FVM FDM

5

4

3

2

1

0

times104


02 04 06 08 1120572

Nite

r

5

4

3

2

1

0

80 times 80


FVM FDM

times104









5 Conclusions







Mesh 120572FVM FDM




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













when 119880119890ge 0 Φ

119890= Φ119875+ (Φ+


when 119880119890lt 0 Φ

119890= Φ119864+ (Φminus


(12)




when 119880119875ge 0 119880

119875

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875

= 119880119875(Φ119875minus Φ119882

Δ119883)

+ [119880119875(120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875


Δ119883)]

lowast

(13)

when 119880119875lt 0 119880

119875

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875

= 119880119875(Φ119864minus Φ119875

Δ119883)

+ [119880119875(120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875


Δ119883)]

lowast

(14)


119886119875Φ119875= 119886119864Φ119864+ 119886119882Φ119882+ 119886119873Φ119873+ 119886119878Φ119878+ 119887 (15)

119886119864=max (minus119880

119890 0)

Δ119883+

1

ReΔ1198832

119886119882=max (119880

119908 0)

Δ119883+

1

ReΔ1198832

119886119873=max (minus119881

119899 0)

Δ119884+

1

ReΔ1198842

119886119878=max (119881

119904 0)

Δ119884+

1

ReΔ1198842

119886119875=

max (119880119890 0)

Δ119883+max (minus119880

119908 0)

Δ119883+max (119881

119899 0)

Δ119884

+max (minus119881

119904 0)

Δ119884+

2

ReΔ1198832+

2

ReΔ1198842

(16a)

119887 = minusmax (119880119890 0) [

Φ+

119890minus Φ119875

Δ119883] +max (minus119880

119890 0) [

Φminus

119890minus Φ119864

Δ119883]

+max (119880119908 0)[

Φ+

119908minus Φ119882


119908 0)[

Φminus

119908minus Φ119875

Δ119883]

minusmax (119881119899 0) [

Φ+

119899minus Φ119875

Δ119884] +max (minus119881

119899 0) [

Φminus

119899minus Φ119873

Δ119884]

+max (119881119904 0) [

Φ+

119904minus Φ119878


119904 0) [

Φminus

119904minus Φ119875

Δ119884]

(16b)

119886119864=max (minus119880

119875 0)

Δ119883+

1

ReΔ1198832

119886119882=max (119880

119875 0)

Δ119883+

1

ReΔ1198832

119886119873=max (minus119881

119875 0)

Δ119884+

1

ReΔ1198842

119886119878=max (119881

119875 0)

Δ119884+

1

ReΔ1198842

119886119875=

max (minus119880119875 0)

Δ119883+max (119880

119875 0)

Δ119883+max (minus119881

119875 0)

Δ119884

+max (119881

119875 0)

Δ119884+

2

ReΔ1198832+

2

ReΔ1198842

(17a)

119887 = minusmax (119880119875 0) [

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

+

119875


Δ119883]

+max (minus119880119875 0) [

120597Φ

120597119883

10038161003816100381610038161003816100381610038161003816

minus

119875


Δ119883]

minusmax (119881119875 0) [

120597Φ

120597119884

10038161003816100381610038161003816100381610038161003816

+

119875


Δ119884]

+max (minus119881119875 0) [

120597Φ

120597119884

10038161003816100381610038161003816100381610038161003816

minus

119875


Δ119883]

(17b)

where 119886119873 119886119875 119886119864 119886119882 and 119886










Δ119883minusΦ119908

Δ119883

1

Δ119883

120597Φ

120597119909

10038161003816100381610038161003816100381610038161003816119890

minus1

Δ119883

120597Φ

120597119909

10038161003816100381610038161003816100381610038161003816119908


8Φ101584010158401015840

119875minusΔ1198834

128Φ(5)


Δ1198832

24Φ(4)

119875minus 13

Δ1198834

5760Φ(6)



120597119883

10038161003816100381610038161003816100381610038161003816119875

1205972Φ

1205971198832

100381610038161003816100381610038161003816100381610038161003816119875


6Φ101584010158401015840

119875minusΔ1198834

120Φ(5)


Δ1198832

12Φ(4)

119875minusΔ1198834

360Φ(6)











0

04

08

12

0

03

06

U V

X(Y)

0 02 04 06 08 1minus04

minus06

minus03

(a) 80 times 80

X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(b) 40 times 40



X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(c) 20 times 20




X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(a) Thom



X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(b) Woods





X(Y)

0 02 04 06 08 1

0

04

08

12U

minus04

0

03

06

V

minus06

minus03




X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03





X(Y)

0 02 04 06 08 1

0

60

120

U

minus120

minus60

0

200

400

V

minus400

minus200

(a) 40 times 40



X(Y)

0 02 04 06 08 1

0

60

120

U

minus120

minus60

0

200

400

V

minus400

minus200

(b) 80 times 80


40 times 40

Grid-independent

(a) 40 times 40

80 times 80

Grid-independent

(b) 80 times 80



(a) 40 times 40 (b) 20 times 20


(a) Thom (b) Woods

(c) Jenson







120572

02 04 06 08 1

Nite

r

80 times 80


FVM FDM

5

4

3

2

1

0

times104


02 04 06 08 1120572

Nite

r

5

4

3

2

1

0

80 times 80


FVM FDM

times104









5 Conclusions







Mesh 120572FVM FDM




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Δ119883minusΦ119908

Δ119883

1

Δ119883

120597Φ

120597119909

10038161003816100381610038161003816100381610038161003816119890

minus1

Δ119883

120597Φ

120597119909

10038161003816100381610038161003816100381610038161003816119908


8Φ101584010158401015840

119875minusΔ1198834

128Φ(5)


Δ1198832

24Φ(4)

119875minus 13

Δ1198834

5760Φ(6)



120597119883

10038161003816100381610038161003816100381610038161003816119875

1205972Φ

1205971198832

100381610038161003816100381610038161003816100381610038161003816119875


6Φ101584010158401015840

119875minusΔ1198834

120Φ(5)


Δ1198832

12Φ(4)

119875minusΔ1198834

360Φ(6)











0

04

08

12

0

03

06

U V

X(Y)

0 02 04 06 08 1minus04

minus06

minus03

(a) 80 times 80

X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(b) 40 times 40



X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(c) 20 times 20




X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(a) Thom



X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(b) Woods





X(Y)

0 02 04 06 08 1

0

04

08

12U

minus04

0

03

06

V

minus06

minus03




X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03





X(Y)

0 02 04 06 08 1

0

60

120

U

minus120

minus60

0

200

400

V

minus400

minus200

(a) 40 times 40



X(Y)

0 02 04 06 08 1

0

60

120

U

minus120

minus60

0

200

400

V

minus400

minus200

(b) 80 times 80


40 times 40

Grid-independent

(a) 40 times 40

80 times 80

Grid-independent

(b) 80 times 80



(a) 40 times 40 (b) 20 times 20


(a) Thom (b) Woods

(c) Jenson







120572

02 04 06 08 1

Nite

r

80 times 80


FVM FDM

5

4

3

2

1

0

times104


02 04 06 08 1120572

Nite

r

5

4

3

2

1

0

80 times 80


FVM FDM

times104









5 Conclusions







Mesh 120572FVM FDM




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

04

08

12

0

03

06

U V

X(Y)

0 02 04 06 08 1minus04

minus06

minus03

(a) 80 times 80

X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(b) 40 times 40



X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(c) 20 times 20




X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(a) Thom



X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03

(b) Woods





X(Y)

0 02 04 06 08 1

0

04

08

12U

minus04

0

03

06

V

minus06

minus03




X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03





X(Y)

0 02 04 06 08 1

0

60

120

U

minus120

minus60

0

200

400

V

minus400

minus200

(a) 40 times 40



X(Y)

0 02 04 06 08 1

0

60

120

U

minus120

minus60

0

200

400

V

minus400

minus200

(b) 80 times 80


40 times 40

Grid-independent

(a) 40 times 40

80 times 80

Grid-independent

(b) 80 times 80



(a) 40 times 40 (b) 20 times 20


(a) Thom (b) Woods

(c) Jenson







120572

02 04 06 08 1

Nite

r

80 times 80


FVM FDM

5

4

3

2

1

0

times104


02 04 06 08 1120572

Nite

r

5

4

3

2

1

0

80 times 80


FVM FDM

times104









5 Conclusions







Mesh 120572FVM FDM




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












X(Y)

0 02 04 06 08 1

0

04

08

12U

minus04

0

03

06

V

minus06

minus03




X(Y)

0 02 04 06 08 1

0

04

08

12

U

minus04

0

03

06

V

minus06

minus03





X(Y)

0 02 04 06 08 1

0

60

120

U

minus120

minus60

0

200

400

V

minus400

minus200

(a) 40 times 40



X(Y)

0 02 04 06 08 1

0

60

120

U

minus120

minus60

0

200

400

V

minus400

minus200

(b) 80 times 80


40 times 40

Grid-independent

(a) 40 times 40

80 times 80

Grid-independent

(b) 80 times 80



(a) 40 times 40 (b) 20 times 20


(a) Thom (b) Woods

(c) Jenson







120572

02 04 06 08 1

Nite

r

80 times 80


FVM FDM

5

4

3

2

1

0

times104


02 04 06 08 1120572

Nite

r

5

4

3

2

1

0

80 times 80


FVM FDM

times104









5 Conclusions







Mesh 120572FVM FDM




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) 40 times 40 (b) 20 times 20


(a) Thom (b) Woods

(c) Jenson







120572

02 04 06 08 1

Nite

r

80 times 80


FVM FDM

5

4

3

2

1

0

times104


02 04 06 08 1120572

Nite

r

5

4

3

2

1

0

80 times 80


FVM FDM

times104









5 Conclusions







Mesh 120572FVM FDM




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120572

02 04 06 08 1

Nite

r

80 times 80


FVM FDM

5

4

3

2

1

0

times104


02 04 06 08 1120572

Nite

r

5

4

3

2

1

0

80 times 80


FVM FDM

times104









5 Conclusions







Mesh 120572FVM FDM




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Mesh 120572FVM FDM




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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