radial basis function collocation method for the numerical solution of the two-dimensional transient...

Applied Mathematical Modelling 36 (2012) 1148–1160

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Radial basis function collocation method for the numerical solution of thetwo-dimensional transient nonlinear coupled Burgers’ equations

Siraj-ul-Islam a,c,⇑, Bozidar Šarler a, Robert Vertnik b, Gregor Kosec a

a Laboratory for Multiphase Processes, University of Nova Gorica, Vipavska 13, SI-5000 Nova Gorica, Sloveniab Technical Development, Štore Steel d.o.o., Zelezarska 3, SI-3220 Štore, Sloveniac University of Engineering and Technology, Peshawar, Pakistan

a r t i c l e i n f o

Article history:Received 23 December 2010Received in revised form 8 July 2011Accepted 12 July 2011Available online 29 July 2011

Keywords:Local collocationMeshless methodsMultiquadric radial basis functionsAdaptive upwindBurgers’ equationsDiffusion

0307-904X/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.apm.2011.07.050

⇑ Corresponding author at: University of EngineeE-mail addresses: [email protected], siraj-ul-isl

a b s t r a c t

This paper examines the numerical solution of the transient nonlinear coupled Burgers’equations by a Local Radial Basis Functions Collocation Method (LRBFCM) for large valuesof Reynolds number (Re) up to 103. The LRBFCM belongs to a class of truly meshless meth-ods which do not need any underlying mesh but works on a set of uniform or randomnodes without any a priori node to node connectivity. The time discretization is performedin an explicit way and collocation with the multiquadric radial basis functions (RBFs) areused to interpolate diffusion–convection variable and its spatial derivatives on decom-posed domains. Five nodded domains of influence are used in the local support. Adaptiveupwind technique [1,54] is used for stabilization of the method for large Re in the caseof mixed boundary conditions. Accuracy of the method is assessed as a function of timeand space discretizations. The method is tested on two benchmark problems having Dirich-let and mixed boundary conditions. The numerical solution obtained from the LRBFCM fordifferent value of Re is compared with analytical solution as well as other numerical meth-ods [15,47,49]. It is shown that the proposed method is efficient, accurate and stable forflow with reasonably high Reynolds numbers.

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1. Introduction

The use of radial basis functions (RBFs) in the numerical solution of partial differential equations (PDEs) has gained pop-ularity in the engineering and science community as it is meshless and can readily be extended to multi-dimensional prob-lems. In recent years RBFs have been extensively used in different context [1–6] and emerged as a potential alternative in thefield of numerical solution of PDEs.

The radial basis functions interpolation was introduced in [7] to approximate two-dimensional geographical surfacesbased on scattered data. In [8] the author derived a meshless method based on multiquadrics RBFs for the numerical solutionof PDEs. This idea was extended later on by Golberg et al. [9]. The existence, uniqueness, and convergence of the RBFsapproximation was discussed by Franke and Schaback [10], Madych and Nelson [11], and Micchelli [12]. The importanceof shape parameter c in the MQ method was elaborated by Tarwater [13] and Huang et al. [14]. Solvability of the systemof equations obtained by this method for distinct interpolation points is addressed in [12]. The authors [15–22] have veryrecently used the Global Radial Basis Collocation Method (GRBFCM) to obtain meshless numerical solution of the nonlinearcoupled PDEs.

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ring and Technology, Peshawar, Pakistan. Tel.: +92 091 [email protected] ( Siraj-ul-Islam).

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Siraj-ul-Islam et al. / Applied Mathematical Modelling 36 (2012) 1148–1160 1149

Contrary to the mesh based methods like the FEM, FVM and FDM, meshless methods use a set of uniform or randompoints which are not necessarily interconnected in the form of a mesh. Due to this advantageous feature, meshless methodshave got increased acceptance since mesh generation in multi-dimensional problems is a non-trivial task. The benefits ofmeshless approximation by the RBFs is somehow over shadowed by the dense and ill-conditioned matrix, especially inthe large scale simulations. The non-singularity of the RBF’s interpolation matrix depends on the shape parameter andthe size of the domain. In the GRBFCM the collocation matrix is constructed by taking into consideration the whole domain.This limits the applicability of the GRBFCM to solve large scale problems. Many remedies like point collocation, local sym-metric weak form and local boundary-integral-equation formulation, domain decomposition by Mai-Duy and Tran-Cong[23], multi-grid approach and compactly supported RBFs by Chen et al. [24] have been suggested in the literature to circum-vent this problem. These approaches result in a substantial complication of the original simple method on one hand with avery limited advantages on the other hand. Various localized meshless methods such as [25–29] have been successfully usedin many practical problems for localization of the domain size while maintaining simplicity of the RBF approach. TheLRBFCM was first introduced for diffusion problems in [29]. The results in the paper show accuracy and efficiency. Manyauthors have applied the LRBFCM to more complex problems such as convection–diffusion problems with phase-change[30], continuous casting [31], solid–solid phase transformations [32], Navier–Stokes equations [33], Darcy flow [34] and tur-bulent flow [35], etc. The main idea of the LRBFCM is the collocation on overlapping sub-domains of the whole domain. Theoverlapping sub-domains drastically reduce the collocation matrix size at the expense of solving many small matrices withthe dimension of the number of nodes included in the domain of influence for each node instead of a large collocation matrix.Circular and rectangular domains are most commonly used in the literature which can either be overlapping or non-overlapping. Detailed discussions on meshless methods and its application to many complex PDEs, industrial and large-scaleproblems can be found in [2,36,34,37,38,32,5,39–41,35,31,42,43] and the references therein.

Mathematical models of basic flow equations describing unsteady transport problems consist of system of nonlinearparabolic and hyperbolic PDEs [2]. The coupled Burgers’ equations [44] form an important class of such PDEs. This classis related to a large number of physical problems such as the phenomena of turbulence and supersonic flow, flow of ashock wave traveling in a viscous fluid, sedimentation of two kinds of particles in fluid suspensions under the effect ofgravity, acoustic transmission, traffic and aerofoil flow theory, as well as a prerequisite to the Navier–Stokes equations(see [15,2,44–49] and the references therein for details). Exact solution for the special cases of the two dimensional cou-pled Burgers’ equations is given by Fletcher [50] using Hopf–Cole transformation. Various numerical methods have beenintroduced for the numerical solution of the two dimensional coupled Burgers’ equations. A comparison of linear, qua-dratic and cubic finite element methods and three, five and seven points finite difference methods for the numerical solu-tion of the two dimensional coupled Burgers’ equations is given in [51]. Numerical solution of the problem using fourth-order accurate two-point compact scheme and fourth-order accurate Dufort–Frankel scheme is the subject of the paper[52]. The author [2] has proposed MLPG for the steady state Burgers’ equation. A fully implicit finite-difference schemefor the numerical solution of the problem has been used in [53]. Recently, different numerical methods like diffuseapproximate method [46], method of fundamental solutions [47], element-free characteristic method [49], Chebyshevpseudospectral method [48] and global radial basis collocation method (GRBFCM) [15] are focused on the numerical solu-tion of the two dimensional coupled Burgers’ equations.

The structure of the rest of the present paper is organized as follows. In Section 2, we introduce the governing equationsconsidered in this paper. In Section 3, we discuss the numerical method. In Section 4, the adaptive upwind technique is de-scribed. Section 5 is devoted to the numerical tests on benchmark problems with Dirichlet and Neumann’s boundary con-ditions with and without analytical solution. At the end, we draw conclusions of the work.

2. Governing equations

Consider the dimensionless form of two dimensional Burgers’ equation defined on two dimensional domain X withboundary C

@ux

@tþ ux

@ux

@xþ uy

@ux

@y¼ 1

Re@2ux

@x2 þ@2ux

@y2

" #;

@uy

@tþ ux

@uy

@xþ uy

@uy

@y¼ 1

Re@2uy

@x2 þ@2uy

@y2

" #;

ð1Þ

where Re ¼ qjujLl ;l is the viscosity and L is the characteristic length.

Initial conditions for the system of PDEs are

uðp; t0Þ ¼ u0ðpÞ; p 2 X [ C; ð2Þ

where tr is the notation used for transposition, u = [ux, uy]tr,p = [x,y]tr and t0 is the current time. The boundary C is dividedinto necessarily connected Dirichlet and Neumann parts CD, CN with Dirichlet and Neumann boundary conditions respec-tively. At point p with normal nC, these boundary conditions are defined as

Fig. 1.m = 5.

1150 Siraj-ul-Islam et al. / Applied Mathematical Modelling 36 (2012) 1148–1160

uðp; tÞ ¼ f1ðp; tÞ for p 2 CD

@uðp; tÞ@nC

¼ f2ðp; tÞ for p 2 CN :ð3Þ

Eq. (3) can be written in a more compact form as

Bu ¼ fðp; tÞ; ð4Þ

where B ¼ 1; @@nC

h iand f(p,t) = [f1(p, t), f2(p, t)]tr are the known functions of p and t.

3. Numerical method

Let Dt be the time step size, and t = t(n) = t0 + nDt be the time discretization. The time derivative in Eq. (1) is approximatedby Euler formula in the following manner

@uðp; tÞ@t

� uðp; tÞ � uðp; t0ÞDt

: ð5Þ

In order to reduce the size we use a local meshless scheme. For each xj 2X [ C, j = 1,2, . . . ,N we choose m nearest neighbor-ing points contained in the domain of influence jX ¼ fjxkgm

k¼1, where k denotes the local indexing for each collocation point xj

being center of jX instead of the whole domain. The schematic diagram of 11 � 11 uniform node distribution, and localdomains of influence containing five points, i.e. m = 5, at the interior, near the boundary and corner points are shown inFig. 1.

To approximate the solution u(p, t) over X [ C,we consider collocation on the domain of influencejX ¼ fjpkg

mk¼1; j ¼ 1; . . . ;N instead of the whole domain X [ C. The variable u = [ux,uy]tr can be approximated on each domain

of influence in the following form,

uxðjp; tÞ ¼Xm

k¼1

/ðkjp� jpkkÞjak; jp 2 jX; ð6Þ

uyðjp; tÞ ¼Xm

k¼1

/ðkjp� jpkkÞjbk; jp 2 jX; ð7Þ

where jak and jbk are the collocation coefficients. The RBF / in the above expressions is Hardy’s multiquadrics (MQ) which isdefined as

/ðkjp� jpkkÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikjp� jpkk

2 þ c2q

ð8Þ

where c is the shape parameter. It follows that for j = 1,2, . . . ,N,

The 11 � 11 uniform node arrangement and the schematics of the local domains of influence in the interior, near boundary and corner points using


ruxðjp; tÞ ¼Xm

k¼1

r/ðkjx� jxkkÞjak;

ruyðjp; tÞ ¼Xm

k¼1

r/ðkjx� jxkkÞjbk;

ð9Þ

@

@nuxðjp; tÞ ¼

Xm

k¼1

@

@n/ðkjp� jpkkÞjak; n ¼ x; y

@

@nuyðjp; tÞ ¼

Xm

k¼1

@

@n/ðkjp� jpkkÞjbk:

ð10Þ

The coefficients jak and jbk are determined by collocation in the following form:

csX

Xm

k¼1

/ðkjps � jpkkÞjak þ csC

Xm

k¼1

B/ðkjps � jpkkÞjak ¼ csX uxðjps; t0Þ� �

þ csCfðjps; t0Þ; ð11Þ

csX

Xm

k¼1

/ðkjps � jpkkÞjbk þ csC

Xm

k¼1

B/ðkjps � jpkkÞjbk ¼ csX uyðjps; t0Þ� �

þ csCfðjps; t0Þ; ð12Þ

where s = 1,2, . . . ,m. The symbols csX and cs

C in the above equations are domain and boundary indicators which are defined as:

csX ¼

1; ps 2 X;

0; ps R X;

�cs

C ¼1; ps 2 C;

0; ps R C;

�ð13Þ

where s = 1,2, . . . ,m. The above linear system can be written in matrix notation as

jUja ¼ jb; ð14ÞjUjb ¼ jc; ð15Þ

where ja = [ja1, ja2, . . . , jam]tr, jb = [jb1, jb2, . . . , jbm]tr, jb = [jb1, jb2, . . . , jbm]tr, jc = [jc1, jc2, . . . , jcm]tr. The matrices jU, jb and jc aredefined as

jbs ¼ csXðuxðjps; t0ÞÞ þ cs

Cf1ðjps; t0Þ;jcs ¼ cs

Xðuyðjps; t0ÞÞ þ csCf2ðjps; t0Þ;

jUsk ¼ csX/ðkjps � jxkkÞ þ cs

CBi/ðkjps � jpkkÞ; s; k ¼ 1;2; . . . ;m;

where jU ¼ ½jUsk� 2 Rm�m: We determine the coefficients ja, jb by inverting jU

ja ¼ jU�1

jb; ð16Þ

jb ¼ jU�1

jc; ð17Þ

which implies that

jas ¼Xm

k¼1jU�1sk jbk; s ¼ 1;2; . . . ;m; ð18Þ

jbs ¼Xm

k¼1jU�1sk jck; s ¼ 1;2; . . . ;m; ð19Þ

where jU�1sk denotes the matrix element of the matrix jU

�1. The solution u = [ux,uy] can be approximated at the interior pointpj in the following form:

uxðpj; tÞ ¼ uxðpj; t0Þ � Dtuxðpj; t0ÞXm

s¼1

@

@x/ðkpj � jpskÞ

Xm

k¼1jU�1sk jbk � Dtuyðpj; t0Þ

Xm

s¼1

@

@y/ðkpj � jpskÞ

Xm

k¼1jU�1sk jbk

þ DtXm

s¼1

r/ðkpj � jpskÞXm

k¼1jU�1sk jbk: ð20Þ

uyðpj; tÞ ¼ uyðpj; t0Þ � Dtuxðpj; t0ÞXm

s¼1

@

@x/ðkpj � jpskÞ

Xm

k¼1jU�1sk jck � Dtuyðpj; t0Þ

Xm

s¼1

@

@y/ðkpj � jxskÞ

Xm

k¼1jU�1sk jck

þ DtXm

s¼1

r/ðkpj � jpskÞXm

k¼1jU�1sk jbk: ð21Þ


For boundary point pj, from Eq. (6) we have

uxðpj; tÞ ¼Xm

s¼1

/ðkpj � jpskÞXm

k¼1jU�1sk jbk; ð22Þ

uyðpj; tÞ ¼Xm

s¼1

/ðkpj � jpskÞXm

k¼1jU�1sk jck: ð23Þ

This completes the formulation of the LBRFCM.

4. Stability and adaptive upwind technique

An explicit time discretization is conditionally stable. Like the finite difference methods, the stability criterion in the caseof LRBFCM approximately follow almost similar pattern for the diffusion dominated problems which is defined by the Fou-rier number Fo

Fo ¼ lDt

qðhÞ2; ð24Þ

where h is the distance between two consecutive nodes. The condition Fo 6 12 leads to the following restriction on the time

step used in the explicit time stepping procedure

Dt 6ðhÞ2q

2l: ð25Þ

Similarly the stability criterion for the convection dominated problems is defined by the Courant number

Cr ¼jvjDt

h: ð26Þ

The condition Cr 6 1 leads to the following restriction on the time step used in the explicit time stepping procedure

Dt 6hjvj : ð27Þ

For convection–diffusion problem Dt must satisfy both criteria (25) and (27) to ensure stability of the numerical method.For high values of the Reynolds number Re > 100 the flow becomes convection dominated and as a result a sharp front

appears in the solution which causes instabilities in terms of spurious oscillations in the case of uniform grid [49] for theproblem with Dirichlet–Neumann boundary conditions. Even a very fine grid cannot get rid of the oscillatory behaviorcaused by a sharp gradient. The spurious oscillations for large Reynold’s number has been reported previously in the caseof FEM, FDM and element free Gelarkin method [49]. LRBFCM exhibits similar behavior for large Reynold’s number in thecase of Neumann boundary conditions on one pair of opposite edges and Dirichlet boundary conditions on another pairof boundaries of the square domain. To stabilize the solution in the case of convection dominated flow, the upwind techniquebased on the local Reynolds number (Ref) [1,54] is used. This technique has been successfully used in the numerical mod-eling of the heat transfer in the continuous casting and turbulent flows [35]. In this technique, the expansion coefficients arefirst calculated by LRBFCM. The convection terms are then calculated at the point pfD, shifted by the central offset distanceDpf in the opposite direction of the current velocity as shown Fig. 2. The position of the shifted point is defined as

pfD ¼ pf � Dpf; f ¼ x; y: ð28Þ

Fig. 2. Schematic diagram of upwind technique.


The central offset distance is calculated by
lo
g 10(L

∞)

Dpf ¼ signðufÞdupd2

; f ¼ x; y; ð29Þ

where dup is the upwind function shown in Fig. 2 and d is the distance between the central point and the neighboring pointsin the opposite direction. The upwind function is defined as

dup ¼ coth Refj j � 1Refj j ; f ¼ x; y; ð30Þ

where

Ref ¼ux pþx � p�x� �

l;

l being kinematic viscosity.

5. Numerical results

Throughout this section, we investigate the performance of the LRBFCM that can be implemented both on evenly or ran-domly distributed nodes. MATLAB 9 platform is used for implementation of the method. Two kinds of errors, maximumabsolute error and root mean squared error

L1 ¼max juðpj; tÞ � uðpj; tÞj; j ¼ 1;2; . . . ;N;

Lrms ¼1N

XN

j¼1

juðpj; tÞ � uðpj; tÞj2

" #1=2

are considered in this paper. u(pj, t) and u(pj, t) in the above equations represent exact and numerical solutions of the givenpartial differential equation, respectively. In our case we take N = Nx � Ny, where Nx is the number of nodes on x-axis and Ny

is the number of nodes on y-axis. A scaling technique similar to the one introduced in [29] is used to alleviate the difficulty ofchoosing different values of shape parameter in MQ RBFs. The scaling parameter jr0 is the maximum nodal distances in thesub-domains

jr0 ¼maxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikjpk � pjk

2q

j; k ¼ 1; . . . ;N: ð31Þ

The parameter c in all RBFs and corresponding derivatives are replaced by cjr0. Hence, a large shape parameter of the MQ RBFcan be used in the numerical implementation. Scaling of the shape parameter is performed to make MQ RBFs approximationinsensitive to various dimensions of the domain. Thus, the LRBFCM is less sensitive with respect to the shape parameter un-like the GRBFCM. The number of nodes in each sub-domain is chosen as m = 5. For numerical validation, the following exam-ples are considered. The method is applied on two benchmark problems [15,53,52,47] (Ex. 1), [49] (Ex. 2) with Dirichlet andmixed boundary conditions. In the case of Ex. 1 we found L1 and Lrms and compared the numerical method with the onesreported earlier. In Ex. 2 graphical results are compared with [49] as we do not have the analytical solution of the problem.Numerical results related to Ex. 1 are presented in Figs. 3–6 and Tables 1–4. Numerical results related to Ex. 2 are presentedin Figs. 7–13 and Table 5.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

t

961 nodes2610 nodes10200 nodes40400 nodes

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−8

−7

−6

−5

−4

−3

−2

−1

0

t

log 1

0(L

∞)

961 nodes2610 nodes10200 nodes40400 nodes

Fig. 3. L1 error norms on the left and Re = 100 on the right, Re = 1000.

0

0.2

0.4

0.6

0.8

1 00.2

0.40.6

0.81

0.5

0.55

0.6

0.65

0.7

0.75

yx

ux

0

0.5

1

00.2

0.40.6

0.81

0.75

0.8

0.85

0.9

0.95

1

xy

uy

Fig. 4. Numerical solutions ux on the left and uy on the right, Re = 1000, N = 40,401.

Fig. 5. Labs error norms on the left ux on the right uy, Re = 1000, N = 40,401.

10 15 20 25 30 35 40 45 50 55 600

1000

2000

3000

4000

5000

6000

7000

8000

N

CPU

tim

e

CPU time of LRBFCM CPU time of GRBFCM

Fig. 6. Ex. 1 CPU time comparison of GRBFCM [15] and the present method, t = 0.1, Dt = 0.0001, R = 100.


Table 1Numerical results for ux at t = 2.0, Re = 100, Dt = 0.0001, N = 441.

(x,y) (0.1,0.1) (0.3,0.3) (0.5,0.5) (0.3,0.7) (0.1,0.9) (0.5,0.9)

Exact 0.500482 0.500482 0.500482 0.555675 0.744256 0.555675LRBFCM 0.500470 0.500441 0.500414 0.554805 0.744197 0.554489[15] 0.50035 0.50042 0.50046 0.55609 0.74409 0.55604[47] 0.50012 0.50042 0.50041 0.55587 0.74416 0.55637[53] 0.49983 0.49977 0.49973 0.55429 0.74340 0.55413

Table 2Numerical results for uy at t = 2.0, Re = 100, Dt = 0.0001, N = 441.

(x,y) (0.1,0.1) (0.3,0.3) (0.5,0.5) (0.3,0.7) (0.1,0.9) (0.5,0.9)

Exact 0.999518 0.999518 0.999518 0.944325 0.755744 0.944325LRBFCM 0.999530 0.999559 0.999586 0.945195 0.755803 0.945511[15] 0.99936 0.99951 0.99958 0.94387 0.75592 0.94392[47] 0.99946 0.99938 0.99941 0.94387 0.75558 0.94345[53] 0.99826 0.99861 0.99821 0.94409 0.75500 0.94441

Table 3Error Norms for ux and uy at t = 2.0, Re = 100.

ux uy

Dt N L1 Lrms L1 Lrms

1.000 � 10�4 961 3.566 � 10�1 4.264 � 10�2 4.685 � 10�1 4.263 � 10�2

5.000 � 10�5 2601 5.322 � 10�2 7.673 � 10�3 5.568 � 10�1 4.657 � 10�1

1.000 � 10�5 10,201 1.325 � 10�2 0.383 � 10�3 1.552 � 10�2 1.383 � 10�3

1.000 � 10�6 40,401 3.222 � 10�3 2. 971 � 10�3 3.068 � 10�3 2.971 � 10�4

Table 4Error norms for ux and uy at t = 2.0, Re = 100, 2601 nodes.

ux uy

Dt L1 Lrms L1 Lrms

1.000 � 10�2 5.394 � 10�4 2.113 � 10�4 8.647 � 10�4 2.113 � 10�4

1.000 � 10�3 3.006 � 10�4 9.106 � 10�5 3.308 � 10�4 9.105 � 10�5

1.000 � 10�4 2.891 � 10�4 8.097 � 10�5 2.833 � 10�4 8.096 � 10�5

1.000 � 10�5 2.881 � 10�4 8.000 � 10�5 2.739 � 10�4 8.000 � 10�5

1.000 � 10�6 2.881 � 10�4 7.991 � 10�5 2.784 � 10�4 7.991 � 10�5

Fig. 7. Solution with N = 1681, t = 1.0, Dt = 10�4, x = 0.5, Re = 100, on the left ux and on the right uy.


Fig. 8. Solution with N = 6561, t = 1.0, Dt = 10�4, x = 0.5, Re = 100, on the left ux and on the right uy.

Fig. 9. Solution with N = 25,921, t = 1.0,Dt = 10�4, x = 0.5, Re = 100, on the left ux and on the right uy.

Fig. 10. Solution with N = 1681, t = 0.4,Dt = 10�4, x = 0.5, Re = 1000, on the left ux and on the right uy.


Example 1. Consider the governing equations (1). The computational domain is xa 6 x 6 xb, ya 6 y 6 yb, t P t0 = 0,xa = ya = t0 = 0 and xb = yb = 1. Initial condition and boundary conditions can be derived form the analytical solution whichis given by Refs. [15,53,52]

uxðx; y; tÞ ¼34� 1

4 1þ exp ð�4xþ 4y� tÞ Re32

� �� ;uyðx; y; tÞ ¼

34þ 1

4 1þ exp ð�4xþ 4y� tÞ Re32

� �� : ð32Þ

Fig. 11. Solution with N = 6561, t = 0.4,Mt = 10�4, x = 0.5, Re = 1000, on the left ux and on the right uy.

Fig. 12. Solution with N = 25,921, t = 0.4,Dt = 10�4, x = 0.5, Re = 1000, on the left ux and on the right uy.

Fig. 13. LRBFCM with upwind, N = 1681, 6561, 25,921, t = 1.0, Dt = 10�4, x = 0.5, Re = 1000, on the left ux and on the right uy.


For larger values of Re, convection dominates the flow which causes formation of sharp gradients in the solutions asshown in Fig. 4. The results produced through LRBFCM based on uniform node arrangement from N = 961 to N = 40,401and Re = 100, 1000 are shown in Figs. 3–5 and Tables 1–4 respectively. Logarithmic plots of L1 of ux versus different numberof nodes = 961, 2601, 10,201, 40,401 and Dt = 10�4, 5.0 � 10�5, 10�5, 10�6 respectively are shown in Fig. 3 for Re = 100, 1000.These figures show a steady decrease in the error with increase in number of collocation points. Plots of absolute errors cor-responding N = 40,401, Dt = 10�6 and R = 1000 are shown in Fig. 5.

In Tables 1 and 2 performance of LRBFCM is shown against the other methods, [15,53,47] at selected points. In Tables 3and 4 the method is tested in terms error measures L1, Lrms for various Dt’s and different number of collocation points. Theseresults show accuracy and stable performance of the LRBFCM for large number of nodal density and large values of Re (whichcause steeper gradients).

Table 5Solution for ux and uy at Dt = 10�4, t = 1.0x = 0.5, N = 25,921.

Re = 100 (LRBFCM without upwind) Re = 1000 (LRBFCM with upwind)

y ux uy y ux uy

0.000 0.3751 0.0000 0.000 0.3761 0.00000.100 0.3712 0.0730 0.100 0.3758 0.07420.200 0.3698 0.1457 0.200 0.3747 0.14820.300 0.3670 0.2173 0.300 0.3730 0.22180.400 0.3448 0.2700 0.400 0.3703 0.29470.410 0.3364 0.2690 0.450 0.3687 0.33080.420 0.3250 0.2651 0.455 0.3685 0.33440.430 0.3090 0.2566 0.460 0.3682 0.33800.440 0.2878 0.2428 0.465 0.3679 0.34140.450 0.2598 0.2224 0.470 0.3673 0.34460.460 0.2234 0.1934 0.475 0.3660 0.34700.470 0.1786 0.1561 0.480 0.3616 0.34640.480 0.1249 0.1099 0.485 0.3477 0.33630.490 0.0644 0.0569 0.490 0.3068 0.29920.495 0.0325 0.0287 0.495 0.2070 0.20320.500 0.0000 0.0000 0.500 0.0000 0.0000


Compared with performance via method of fundamental solution [47], fully implicit finite-difference method [53],GRBFCM [15], the results obtained from the LRBFCM are fairly accurate and stable. As shown in Fig. 4, formation of sharpfront (when Re = 1000) is captured in a stable and accurate fashion by LRBFCM. In the previous studies [15,53,52,47] thesame model was tested up to Re = 100. Performance of these numerical methods are tightly related to a small value of Re. Inthe case GRBFCM [15] the permissible range of the values of Re is roughly 0 < Re 6 100, otherwise it becomes morechallenging for the method to produce stable results due ill conditioning of the large collocation matrix. For detail view ofcomparative performance of the global and local radial basis based collocation methods, the readers are referred to a recentpapers by Refs. [28,43].

The LRBFCM works for larger amount of nodes, large values of Re and larger range of the shape parameter, but theGRBFCM [15] perform more accurate with smaller number of nodes, small values of Re subject to proper choice of the shapeparameter c.

To show computational efficiency of the LRBFCM, we have done CPU time comparison with GRBFCM [15] and the resultsare shown in Fig. 6. From these results it is obvious LRBFCM is more cost effective than GRBFCM and that LRBFCM outperforms GRBFCM in this respect as well. The CPU time is calculated by Dell Core I 5 machine.

Example 2. Consider the non-dimensional form of a two-dimensional equations (1). The computational domain is a squaredefined by xa 6 x 6 xb, ya 6 y 6 yb, t P t0 = 0, xa = ya = t0 = 0 and xb = yb = 1. Initial condition and boundary conditions aregiven by Zhang et al. [49]:

uxðx; y;0Þ ¼ sinðpxÞ cosðpyÞ;uyðx; y;0Þ ¼ cosðpxÞ sinðpyÞ;uxðxa; y; tÞ ¼ uxðxb; y; tÞ ¼ 0:0;uyðx; ya; tÞ ¼ uyðx; yb; tÞ ¼ 0:0;uxðx; ya; tÞ

@n¼ uxðx; yb; tÞ

@n¼ 0;

uyðxa; y; tÞ@n

¼ uyðxb; y; tÞ@n

¼ 0:

ð33Þ

Due to non-availability of the analytical solution of the above problem, we compare our results with those reported in [49].Like the test problem 1, a sharp front appears in the solution for larger values of Re in this case as well. This phenomenon isshown in Figs. 10–13. For smaller values of Re the solution is smooth enough that it does not cause any instability near thefront hence no upwind technique is needed as shown in Figs. 7–9. Due to Neumanns boundary condition instabilities interms of over shooting and under shooting appear near the sharp gradient for the uniform node arrangement without up-wind technique for Re = 1000. These spurious oscillation has also been reported in [49] for t P 0.6 despite of using element-free characteristic Galerkin method to obtain a smooth solution near the sharp gradient.

In our case the oscillatory behavior is shown in Figs. 10–12. To avoid these wiggles, we have used adaptive unwindtechnique [1,54,35] to stabilize the solution near the sharp front. Improvement of LRBFCM coupled with the adaptive upwindtechnique versus LRBFCM without adaptive upwind is shown in Fig. 10–12 at x = 0.5 and 0 6 y 6 1. In the case of uniformnodes without upwind, the solution blows up beyond t > 0.4 and Re = 1000. If adaptive upwind is used, then the numericalsolution can be found up to t = 1.0 for N = 1681, 6561, 25,921 as shown in Fig. 13.


Improvement in terms of smoothing and controlling the unrealistic oscillations is achieved by LRBFCM coupled withadaptive upwind technique. In Table 5 values of the velocities ux and uy are shown at x = 0.5 and for various specified valuesof y.

6. Conclusions

In this paper we presented a truly meshless local approach LRBFCM based on the MQ RBFs for the numerical solution oftwo dimensional coupled Burgers’ equation. Based on the findings investigated in the previous sections, we summarize themain features of the LRBFCM as follows:

(i) LRBFCM employs local configuration of the nodes falling in the domain of influence.(ii) LRBFCM approach provides a very efficient and accurate method to solve Burgers’ equation for wide range of Re (10–

1000).(iii) LRBFCM is fully explicit, thus it is very simple to implement and parallelize.(iv) Computational cost of the local version is considerably smaller than the global one as it requires only solution of a

small system of the same size as the number of nodes in the domain of influence.(v) LRBFCM coupled with adaptive upwind technique is used to remove the wiggles and over come instabilities for large

values of Re in the case of mixed boundary conditions for both ux and uy.

Acknowledgments

The authors are grateful to the financial support provided from the 6th EU framework project INSPIRE and to the Slove-nian Grant Agency through Grant No. P2.-0379 Programme Group Modeling of Material and Process.

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