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Numerical Heat Transfer, Part B:Fundamentals: An International Journalof Computation and MethodologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/unhb20

MLPG/SUPG Method for Convection-Dominated ProblemsXue-Hong Wu a b , Yan-Jun Dai b & Wen-Quan Tao ba School of Electromechanical Science and Engineering, ZhengzhouUniversity of Light Industry, Zhengzhou, Henan, People's Republic ofChinab Key Laboratory of Thermo-Fluid Science and Engineering, Ministryof Education, Xi'an Jiaotong University, Xi'an, Shannxi, People'sRepublic of ChinaVersion of record first published: 11 Jan 2012.

To cite this article: Xue-Hong Wu , Yan-Jun Dai & Wen-Quan Tao (2012): MLPG/SUPG Method forConvection-Dominated Problems, Numerical Heat Transfer, Part B: Fundamentals: An InternationalJournal of Computation and Methodology, 61:1, 36-51

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MLPG/SUPG METHOD FOR CONVECTION-DOMINATED PROBLEMS

Xue-Hong Wu1,2, Yan-Jun Dai2, and Wen-Quan Tao21School of Electromechanical Science and Engineering, Zhengzhou Universityof Light Industry, Zhengzhou, Henan, People’s Republic of China2Key Laboratory of Thermo-Fluid Science and Engineering,Ministry of Education, Xi’an Jiaotong University, Xi’an, Shannxi,People’s Republic of China

It is well known that convection-diffusion equations suffer from the most difficult problems

to gain a stable and accurate solution. Sometimes, the convection terms may cause oscillat-

ory behavior of solutions. In this article, the streamline upwind Petrov-Galerkin (SUPG)

scheme is applied to eliminate overshoots and undershoots produced by the convection term

in the meshless local Petrov-Galerkin (MLPG) method. The accuracy and stability of the

present method are discussed using two test cases. The results of the present method are

compared with results of other upwind schemes and the finite volume method (FVM) using

a high-order upwind scheme. Our analysis shows that the present method, with its simple

implementation, can give excellent results for convection-dominated problems.

1. INTRODUCTION

To avoid oscillatory behavior and numerical diffusion encountered inconvection-dominant flows, some upwind techniques have been widely applied inthe finite-element method, finite-difference method, finite-volume method, etc. [1, 2].The key of upwind techniques to overcome this problem is to effectively capture theupstream information. Mesh-based methods have been employed as very powerfulnumerical tools for solving fluid flow problems. However, these methods suffer fromsome difficulties of mesh distortion, large deformation, locking, and others. Due tothese reasons, meshless methods, as alternative numerical techniques to eliminate theabove drawbacks based on mesh methods, have attracted much attention andachieved remarkable application. Recently, a number of meshless methods have beendeveloped by different authors. In the following a brief view is presented. Meshlessmethods may be usually categorized in two groups: strong form based and weak

Received 25 May 2011; accepted 12 September 2011.

This work is supported by the National Natural Science Foundation of China (51136004,

21006099), Innovation Scientists and Technicians Troop Construction Projects of Zhengzhou City

(10CXTD151).

Address correspondence to Wen-Quan Tao, Key Laboratory of Thermo-Fluid Engineering, School

of Enengy and Power Engineering of MOE, Xi’an Jiaotong University, 28 Xian Ning Road, Xian,

Shannxi 710049, People’s Republic of China. E-mail: wqtao@mail.xjtu.edu.cn

Numerical Heat Transfer, Part B, 61: 36–51, 2012

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7790 print=1521-0626 online

DOI: 10.1080/10407790.2011.630962

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form based. The strong form methods, such as the smooth particle hydrodynamics(SPH) method proposed by Gingold and Monagha [3], the finite point method(FPM) developed by Onate et al. [4], the radial basis function (RBF) introduced byWendland [5], and the weighted least-square meshless collocation method (WLSM)suggested by Liu et al. [6], build on collocation point schemes and are truly meshlessmethods for which the selection of the collocation points is important. In the weak formmethods the equilibrium equations are satisfied over the global domain. A majority ofmeshless methods belong to this sort, such as the diffuse element method (DEM) [7],the element-free Galerkin (EFG) method [8], the reproducing kernel particle method(RKPM) [9], the hp-clouds method [10], the partition of unity method (PUM) [11],and the point interpolation method (PIM) [12]. They need a background mesh forthe numerical integration and are not really meshless methods. Some investigators pro-posed weak form basedmeshless methods based on the local subdomain; these methodsneed no mesh for either interpolation or numerical integration and are truly meshlessmethods. These methods include the meshless local Petrov-Galerkin (MLPG) method[13], the local boundary integral equation (LBIE)method [14], the finite spheresmethod[15], the local radial point interpolation method (LRPIM) [16], etc.

We gave a brief classification of the meshless methods in the first paragraph.The application of the meshless methods in flow problems is illustrated brieflyin the following. The first work was introduced by Monaghan and Gingold [17] usingthe SPH method. At present, the SPH method had been applied widely in incom-pressible flow and free surface flow problems [18–21]. Onate et al. used the FPMfor flow problem. In their article, the finite increment calculus (FIC) method wasused to handle the nonlinear convection term [4, 22]. Sadat and his co-authorsemployed the DEM method to calculate fluid flow and heat transfer problems[23, 24]. Liu et al. used the reproducing kernel particle method (RKPM) to solveincompressible flow problems. In their study, the streamline upwind Petrov-Galerkinmethod was used to deal with the convection term [25]. Lin and Atluri proposed twokinds of upwind schemes of the meshless local Petrov-Galerkin (MLPG) method tosolve convection-diffusion [26] and incompressible flow problems [27]. Shu et al. pro-posed the RBF-DQ method for incompressible flow [28]. Liu and his collaboratorsdeveloped the mesh-free weak–strong (MWS) form method for 2-D laminarnatural-convection problems [29]. Fries and Matthies developed a coupled methodof the EFG and FEM for incompressible flow [30, 31]. Divo andKassab used the local

NOMENCLATURE

D node distance

M total number of nodes

nj unit normal vector outward to the

boundary

N(x) shape function

Pe Peclet number

�qq heat flux on the boundary

r size of support for the weight

functions

uj velocity

v text function

x spacial coordinates

C, Cu, Ct boundaries

/ temperature

/h(xI) trial functionsb//I fictitious nodal values_UU heat source

k thermal conductivity

X computational domain

MLPG/SUPG METHOD FOR CONVECTION-DOMINATED PROBLEMS 37

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radial basis function method to solve fluid flow and heat transfer problems [32]. Are-fmanesh et al. [33] applied the MLPG method to compute nonisothermal fluid flowproblems with the vorticity-stream function method. Mohanmmadi [34] developed anew upwind scheme based on the meshless finite-volume method to calculate incom-pressible flow problems with a vorticity-stream function formulation. In this meshlessmethod, the Heaviside step function was used as test functions. Zhang et al. [35]applied the EFG method for incompressible flow. Li et al. [36] applied the RBFmethod to solve initial-boundary-value problems for parabolic partial differentialequations with variable coefficients. Wu et al. [37] used the MLPG method for incom-pressible flow problems.

From the above brief review of the applications of the meshless methods incomputational fluid dynamics, we can see that previous researchers have focusedmainly on using the SPH, FPM, EFG, DEM, RKPM, RBF, MLPG method, etc.However, the SPH and FPM are based on strong form methods which are very sensi-tive to the choice of collocation points. The RKPM, DEM, and EFG methods need abackground mesh for the numerical integration. The MLPG method is a truly mesh-less method and does not need a background mesh for numerical integration; itoffers a lot of flexibility to deal with problems of different boundary conditions.Remarkable successes of the MLPG method have been reported in computationalmechanics [38, 39] and heat transfer [37, 40–45]. At the same time, Lin and Atluri con-struct two kinds of upwind schemes to deal with the convection term for convection-diffusion and incompressible flow problems. Their results show that the secondupwind scheme is better, but when this method is used to handle incompressible fluidflow problems for high Reynolds number, it suffers from divergence difficulty.

The present article employs the MLPG method to solve convection-dominatedproblems. The streamline upwind Petrov-Galerkin scheme is developed and appliedin the MLPG method to overcome oscillations produced by convection term. Twocases are applied to validate the stability and accuracy of the present method.

2. UPWIND SCHEMES

Because the MLPG method is based on a local weak form over a local subdo-main, there is a very convenient way to construct upwind scheme. The pioneeringwork on upwind schemes in the MLPG method suggested by Lin and Atluri [26,27], and Mohammadi [34] developed the upwind scheme based on the meshless finitevolume MLPG method. Due to its simple implementation, it can be very conve-niently extended in different meshless methods as an upwind scheme. In this article,the SUPG method is extended in the MLPG method. In the following, the upwindscheme (US2) suggested by Lin and Atluri is described first and the SUPG methodwhich is adopted in this article is then introduced.

2.1. Upwind Scheme Suggested by Lin and Atluri

Lin and Atluri proposed two upwind schemes (US2), which shifts the subdomainopposite to the streamline direction, as shown in Figures 1 and 2. In Figure 2, cri is theshifting distance of the local subdomain, ri is the size of the support domain of theweight function, and ni is the unit vector of the streamline direction at xi.

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2.2. Streamline Upwind Petrov-Galerkin (SUPG) Method

The SUPG method was proposed first by Brooks and Hughes [2, 46], which hasbeen applied widely in the finite-element method for fluid mechanics. In the SUPGmethod, the test function is modified by the streamline upwind method, which isdefined as

vnew ¼ vþ s‘advv ð1Þ

where ‘adv ¼ ujqðÞ=qxj is the convective part of the whole convection-diffusion equa-tion, v is the test function, and s is the stabilization parameter determined by

s ¼ D

2 uk k cothPe

2

� �� 2

Pe

� �ð2Þ

where D is the node distance.

Figure 2. Shift local subdomain.

Figure 1. Upwind scheme.

MLPG/SUPG METHOD FOR CONVECTION-DOMINATED PROBLEMS 39

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Examples of the test functions of the Galerkin method and SUPG method areshown in Figure 3. The magnitude of the modified test function is changed and themaximum value is more than 1 at node A. In the Galerkin method, the value of nodeA equals to 1.

3. GOVERNING EQUATIONS AND LOCAL WEAK FORM

The dimensionless, nonconservative, two-dimensional convection-diffusion equa-tions and boundary conditions in the Cartesian coordinate system are as follows:

ujq/qxj

¼ 1

Pe

q2/qx2j

þ _UU ðj ¼ 1; 2Þ in X ð3Þ

where the Peclet number is defined as

Pe ¼ urefLref

að4Þ

The Dirichlet boundary condition is

/ ¼ �// on Cu ð5Þ

The Neumann boundary condition is

�kq/qxj

nj ¼ �qq on Ct ð6Þ

where / represents the temperature; �// is the given temperature, a is the thermaldiffusive coefficient. k is the thermal conductivity, _UU is the source term, nj is the out-ward unit vector to C, �qq is the given heat flux, uj is the velocity, and Ct and Cu aresubsets of C satisfying Ct\Cu¼; and Ct[Cu¼C.

To satisfy Eq. (3) in a local subdomain XX, the weighted integral form ofEq. (3), is given asZ

Xx

ujq/qxj

� 1

Pe

q2/qx2j

� _UU

!ðvþ sLadvvÞ dXx ¼ 0 ð7Þ

Figure 3. Comparison of test functions of streamline Galerkin and SUPG for node A.

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In the computation, however, we neglect this second-order term s 1Pe

q2/qx2j

ukqvqxk

� �because of the cost of the computations required to calculate the second derivativesof the shape functions. The neglect of these second-derivative stabilization terms is acommon practice in finite-element computations and RKPM and has only a smalleffect on accuracy for most problems [47, 48]. To reduce the order of required differ-entiability on /, we can integrate Eq. (7) by parts. By using Gauss’s theorem, we canobtain the following local weak formulation equation:

ZXx

ujq/qxj

vþ 1

Pe

q/qxj

qvqxj

� _UUv

� �dXx �

ZC

1

Pe

q/qxj

njv dC

þ sZXx

ujq/qxj

ukqvqxk

� _UUukqvqxk

� �dXx ¼ 0

ð8Þ

Substituting Eq. (6) into Eq. (8), we can obtain following equation:

ZXx

ujq/qxj

vþ 1

Pe

q/qxj

qvqxj

� �dX�

ZCI

1

Pe

q/qxj

njv dC�ZCu

1

Pe

q/qxj

njv dCZCt

1

kPe�qqvdC�

ZXx

_UUv dXþ sZXx

ujq/qxj

ukqvqxk

� _UUukqvqxk

� �dX ¼ 0

ð9Þ

where CI is the part of the subdomain boundary included in the global domain.To obtain the discretized equation of each subdomain, the unknown function

can be approximated using the moving least-square (MLS) method:

/hðxÞ ¼ NTðxÞbUU ¼XNI¼1

NI ðxÞ//I ð10Þ

where bUU represents a fictitious nodal value, but not the value of the unknown func-tion. The characteristics of MLS have been discussed widely in the literature [13, 49]and will not be restated here. Substituting Eq. (8) into Eq. (7) for all the nodes, wecan obtain the following discretized system of linear equations:

XMJ¼1

ZXx

ujqNJ bUUJ

qxjvI þ

1

Pe

qNJ bUUJ

qxj

qvIqxj

!dXx �

XMJ¼1

ZCI

1

Pe

qNJ bUUJ

qxjnjvI dC

�XMJ¼1

ZCu

1

Pe

qNJ bUUJ

qxjnjvI dCþ s

ZXx

ujqNJ bUUJ

qxjuk

qvIqxk

dX

¼ZCt

1

kPe�qqvI dCþ

ZXx

_UUvI þ s _UUukqvIqxk

� �dX

ð11Þ

where M is the total number of nodes in the entire domain X. Equation (11) can berewritten as

K � bUU ¼ F ð12Þ

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K and F are the global stiffness matrix and the global vector, respectively,defined as

KIJ ¼ZXx

ujqNJ

qxjvI þ

1

Pe

qNJ

qxj

qvIqxj

� �dX�

ZCu

1

Pe

qNJ

qxjnjvI dC

�ZCI

1

Pe

qNJ

qxjnjvI dCþ s

ZXx

ujqNJ

qxjuk

qvIqxk

dX

ð13Þ

FI ¼ZCt

1

kPe�qqvI dCþ

ZXx

_UUvI þ s _UUukqvIqxk

� �dX ð14Þ

4. NUMERICAL EXAMPLES

In this section, the MLPG method is applied to solve convection-diffusionproblems. In order to illustrate the accuracy and efficiency, the numerical resultsobtained by the present method are compared with results provided by the finite-volume method (FVM) solutions using the QUICK scheme, the upwind scheme pro-posed by Lin and Atluri (MLPG-US2), and the MLPG method without the upwindscheme (MLPG). In all numerical calculations, a quadratic spline weight function isadopted in the meshless computations and the transformation method is applied todeal with the essential boundary conditions.

4.1. Smith-Hutton Problem

Smith and Hutton [50] proposed a convection-diffusion problem. The problemstatement is depicted in Figure 4. Given velocity field and boundary conditions are asfollows:

u ¼ 2yð1� x2Þ v ¼ �2xð1� y2Þ ð15Þ

/inðxÞ ¼ 1þ tan h½að1þ 2xÞ� on � 1 � x � 0; y ¼ 0 ð16Þ

q/qy

¼ �qq ¼ 0 on 0 � x � 1; y ¼ 0 ð17Þ

/ ¼ 0 on other boundaries ð18Þ

where a is the sharp transition coefficient. The inlet temperature distribution is astepwise profile, and the bigger the value of a, the sharper the inlet temperaturedistribution. In the present article, a¼ 100 is selected, representing a very sharp tran-sition. In this computation, the outlet profile is studied.

The numerically predicted outlet temperature profile differs from the exact orreference solution. An error is computed by

Error ¼ /comp � /ref ð19Þ

where /comp is the predicted value and /ref is the reference value or exact solution.

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A uniform 41� 41 grid system is employed for the MPLG method, and 42� 42mesh is applied for the finite-volume method. Figures 5 and 6 show results of outletprofile for Pe¼ 500 and 106 respectively. The reference value is obtained by using aULTRA-3=5=7 upwind scheme on a very fine (160� 80) grid [51]. In the case of

Figure 4. Velocity field and inlet profile (a¼ 100).

Figure 5. Outlet profile at Pe¼ 500.

MLPG/SUPG METHOD FOR CONVECTION-DOMINATED PROBLEMS 43

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Pe¼ 500 (Figure 5), all methods have excellent agreement with the reference values,although the solution from the QUICK scheme is a bit diffusive in thesharp-transition region. At the largest Peclet number (Pe¼ 106), the solutions fromall upwind schemes exhibit some overshoots in the sharp-transition region, but thesolution of the MLPG method is still in good agreement with the exact solution.Figures 7 and 8 show the errors of the outlet profile. We can see that the errors ofthe QUICK scheme and the MLPG-US2 method are more significant in thesharp-transition region than that of the MLPG-SUPG method.

4.2. Brezzi Problem

The Brezzi problem was suggested by Brezzi et al. [52]. The problem statementand the boundary conditions are depicted in Figure 9; the velocity field ðu ¼ �y;v ¼ xÞ is shown in Figure 10.

The solution of this problem shows boundary layers on the upper and left-upper sides and a more complex structure in the vicinity of the reentrant corner, withan interaction between a boundary layer and a corner singularity. We have used the

Figure 6. Outlet profile at Pe¼ 106.

Figure 7. Error of outlet profile at Pe¼ 500.

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Figure 9. Brezzi problem: problem statement.

Figure 8. Error of outlet profile at Pe¼ 106.

Figure 10. Velocity field.

MLPG/SUPG METHOD FOR CONVECTION-DOMINATED PROBLEMS 45

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mesh or node shown in Figure 11. It should be noted that for comparison purposes,the node distribution in the MLPG method is exactly the same as that of the FVM.In the Peclet number definition, the mean velocity of the cross section is taken asthe characteristic flow velocity and the length of the y-direction is taken as the

Figure 11. Mesh or node (2,704 elements, 1,433 nodes).

Figure 12. Comparison of results of all methods at Pe¼ 10.

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characteristic length. In this example, numerical test is performed with Pe¼ 10,1,000, and, 106; The size of the local subdomain of each internal node is taken as1.2 times of the mean node distances (D), which is defined as

�DD ¼ffiffiffiffiA

pffiffiffiffiffiN

p� 1

ð20Þ

where A is the area of the computational domain, and N is the total numbers ofnodes.

Figures 12–14 show the results for Pe¼ 10, 1,000, and 106, respectively. Fromthese results, we can see that all methods can give very coincident results for the lowPeclet number. At Pe¼ 1,000, the MLPG-SUPG and MLPG-US2 methods produceovershoot and undershoot, but the QUICK scheme produces only overshoot. Due tono upwind scheme being used, the MLPG method cannot give a reasonable solution.The maximum of overshoot for the QUICK scheme is 1.13; the maximum values ofovershoot and undershoot for the MLPG-US2 method are 1.13 and �0.02, respect-ively; the maximum values of overshoot and undershoot for the MLPG-SUPGmethod are 1.14 and �0.02, respectively. At Pe¼ 106, the QUICK cannot give areasonable solution. The MLPG-US2 method and the MLPG-SUPG method can

Figure 13. Comparison of results of all methods at Pe¼ 1000.

MLPG/SUPG METHOD FOR CONVECTION-DOMINATED PROBLEMS 47

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give fairly reasonable results. The maximums of overshoot and undershoot for theMLPG-SU2 method are 1.43 and �0.06, respectively; the maximums of overshootand undershoot for the MLPG-SUPG method are 1.34 and �0.08, respectively.These results show that the MLPG-SUPG method is superior not only to theQUICK scheme but also to the MLPG-US2 method.

5. CONCLUSIONS

In this article, the MLPG method, which is a truly meshless method, isextended to solve convection-diffusion equations via the streamline upwind Petrov-Galerkin (SUPG) method. The accuracy and stability of the MLPG-SUPG methodare validated by two test problems and compared with other existing upwind schemeand high order upwind schemes of the FVM. Unlike the FVM, the MLPG methodrequires only a set of nodes for interpolation, so nonuniformly distributed nodes arealso applied in the second case. The results show that our proposed upwind methodhas better accuracy and stability and is superior to the MLPG-US2 method and theQUICK scheme of FVM. Thus the present method sheds light on effectively hand-ling convection-dominated flow problems.

Figure 14. Comparison of results of all methods at Pe¼ 106.

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