Australian Journal of Basic and Applied Sciences

AESI Journals Australian Journal of Basic and Applied Sciences ISS:1991-8178 Journal home page: www.ajbasweb.com Optimal Radial Basis Function () for Dual Reciprocity Boundary Element Method (DRBEM) applied to Coupled Burgers Equations with Increasing Reynolds umber 1 Sayan Kaennakham, 2 Krittidej Chanthawara and 3 Wattana Toutip 1 Suranaree University of Technology, School of Mathematics, Institute of Science, akhon Ratchasima, 30000, Thailand. 2,3 Khon Kaen University, Department of Mathematics, Faculty of Science, Khon Kaen, 40002, Thailand. A R T I C L E I F O Article history: Received 2 February 2014 Received in revised form 8 April 2014 Accepted 28 April 2014 Available online 25 May 2014 Keywords: The Dual Reciprocity, Boundary Element Method, Radial Basis Function, Burgers Equations, high Reynolds number A B S T R A C T It is well-known that high Reynolds number flows remain challenges in numerical studies. Burgers equations are ones of those considered classical test cases for testing any new numerical method. Growing as an alternative numerical techniques, Dual Reciprocity Boundary Element Method (DRBRM) has been attracting a huge amount of interest from researchers nowadays. This study has two main objectives; to investigate the performance of the Dual Reciprocity Boundary Element Method (DRBEM) when applied to couple Burgers equations with increasing Reynolds number (Re), and to search for an optimal Radial Basis Function () for such case. Taking into consideration also the CPU time requirement for each at each Re, it is found that with appropriate parameter c, the multiquadric type has led to the most satisfactory solution when Re is increased. Moreover, all solutions obtained are in reasonably good agreement with the exact ones implying that DRBEM is able to fairly handle this type of nonlinear and couple equations when the Reynolds number. 2014 AESI Publisher All rights reserved. To Cite This Article: Sayan Kaennakham, Krittidej Chanthawara and Wattana Toutip., Optimal Radial Basis Function () for Dual Reciprocity Boundary Element Method (DRBEM) applied to Coupled Burgers Equations with Increasing Reynolds umber. Aust. J. Basic & Appl. Sci., 8(7): 462-476, 2014 ITRODUCTIO A Radial Basis Function () is a one variable and continuous function defined for r 0 that has been radicalized by composition with the Euclidean norm on R d and can contain a free parameter or the shape parameter, normally denoted by ε. If we are given a set of centers, x 1 c, x 2 c, x 3 c,, x c, a radial basis function then takes the form; b i x = c j =1 α j f j ( x x i 2, ε) (1) where i = 1, 2, 3,, and α j are coefficients to be determined by interpolation. For this task, it is crucial for the final solution that an appropriate interpolation function f j is adopted and there are a number of investigations carried out on this important task. In applications related to Boundary Element Method(BEM), in particular, as pointed out by Golberg and Chen (1994) that the theory of s provides a mathematical basis for the choice of trial function in the application of BEM, clearly indicating that the choice of interpolating functions is the key for the accuracy of the method. If setting r = x 2 = x 2 1 + x 2 2 + x 2 3 + + x 2 d, an early work of Partridge and Brebbia (1989) concluded that a simple linear form denoted as f = 1 + r is usually sufficient. This simple choice seems to be also widely adopted by many authors; Zhu and Liu (2002), Gumgum (2010), Davies et al. (2001),while later the computational work of Golberg (1995) and Zhu et al. (1998) indicated that a combination of the thin plate splines (TPS), f = r 2 log r and some augmented linear terms 1, x, y in R 2 (or 1, x, y, z in R 3 ) could be more reliable. The comparatively higher accuracy, however, comes with the price as it has been demonstrated by Madych and elson (1990), see also Powell (1994) and Wu and Schaback (1992) that this type of is non-differentiable in R 3 and its convergence is very slow. Alongside with many attempts in this optimal searching task, Golberg et al. (1996) have shown that the so-called Multiquadric (MQ) interpolation, denoted as the form f = 1 + (εr) 2, with ε = 1 c is highly effective when applied to the DRBEM. In spite of MQ's excellent performance, its accuracy is greatly affected by the choice of the shape parameter c. It is well-known that large values of parameter ε can result in igher error in solution and decreasing ε can improve the accuracy significantly (Franke, 1982). However, as pointed out by Schaback Corresponding Author: Sayan Kaennakham, Suranaree University of Technology, School of Mathematics, Institute of Science, 30000 Thailand. Ph: (0066-4422-4383), E-mail: sayan_kk@g.sut.ac.th.

463 Sayan Kaennakham et al, 2014 (1995), the direct way of computing this type of interpolatant suffers from severe mathematically illconditioning. This then makes the search for optimal choice of ε a very challenging topic (Fornberg and Wright, 2004), particularly for simulations of flows at high Reynolds number, and remains as one of the main objectives of this paper. Amongst all the well known numerical strategies adopted to tackle problems in nowadays science and engineering worlds, appearing as an alternative numerical method over the last two decades, the boundary element method (BEM) has become an important tool for solving a wide range of applied science and engineering that involve linear as well as certain types of nonlinear partial differential (PDE). The main feature of BEM is that the PDE problem at hand is converted to an equivalent boundary integral Equation (BIE) using Green s theorem and a fundamental solution (Brebbia and Butterfield, (1978). As a result, the discretization process takes place only on the domain boundary rather than the entire domain, as usually required in other classical numerical schemes i.e. Finite Element Method(FEM), Finite Different Method(FDM) or Finite Volume Method(FVM), leading to a much smaller system of Equation and effectively, a considerable reduction of data required in the computing process while satisfactory results accuracy can still be expected. Despite its attractive aspect mentioned, nevertheless, there are some difficulties in extending the method to non-homogeneous, non-linear and time dependent problems. The main drawback encountered in these cases is the need to discretize the domain into a series of internal cells to deal with the terms taken to the boundary by application of the fundamental solution. This additional discretization then inevitably destroys some of the attraction of the method. To remedy this undesirable drawback, one of the attempts that is successful for the task is the so-called Dual Reciprocity Boundary Element Method (DRBEM) which was first proposed by ardini and Brebbia (1983) in 1982. Ever since, the idea has extensively been developed by many researchers such as the perturbation DRBEM (Liu, 2001), the separation of variables DRBEM (Brunton and Pullan, 1996), and the Laplace transform DRBEM (Zhu and Liu, 1998, Zhu et al., 1994). In the process of DRBEM, the solution is divided into two parts: complementary solutions of its homogeneous form and the particular solutions of the inhomogeneous counterpart. Since the particular solutions are not always available especially in complex problems, the inhomogeneous term of the PDE is approximated by a series of simple functions and transformed to the boundary integrals employing particular solutions of considered problem. The most widely used approximating functions in DRBEM are radial basis functions (s) for which particular solutions can be easily determined (Zhu and Liu, (2002). One of the classical non-linear parabolic equation system widely acknowledged as the Burgers Equation, named after the great Physicist Johannes Martinus Burgers (1895-1981), expressed in two dimensions as follows (Bahadir, 2003); u t v t u u + u + v = 1 x y Re v v + u + v = 1 x y Re 2 u + 2 u (2) x 2 y 2 2 v + 2 v (3) x 2 y 2 Where Re being the Reynolds number, the ratio of inertial forces to viscous forces, u x, y, t, v(x, y, t) are the velocity components to be determined. For large values of the Reynolds number (Re), one of the major difficulties is due to in-viscid boundary layers produced by the steepening effect of the nonlinear advection term in Burgers equation. This difficulty is also encountered in an inviscid avier Stokes equation for a convection dominated flow, and in fact, Burgers equation is one of the principle equations used to test new numerical methods (Zhang et al., 1997). For a domain Ω = x, y : a x, y b with its boundary Γ, the above system is subject to the following initial conditions: u x, y, 0 = β 1 x, y, x, y, 0 = β 2 x, y, x, y Ω, x, y Ω, And the boundary conditions: u x, y, t = γ 1 x, y, t, x, y Γ, t > 0, v x, y, t = γ 2 x, y, t, x, y Γ, t > 0, Here β 1, β 2, γ 1 and γ 2 are known functions. The equation retain the nonlinear of the governing equation in a number of applications; flow through a shock wave traveling in viscous fluid, the phenomena of turbulence, sedimentation of two kinds of particles in fluid suspensions under the effect of gravity (see Burgers, 1948, ee and Duan, 1998) and the references therein). With using the Hopf-Cole transformation, an analytic solution of the system is provided in 1983 by

464 Sayan Kaennakham et al, 2014 Fletcher, (1983). Ever since, the coupled two-dimensional Burgers Equation have been attracting and challenging a number of numerical investigations. Some of the recent ones include the application of a fully implicit finite-difference by Bahadir, (2003), that by using Adomain-Pade technique by Dehghan et al., (2007), that by adopting variational iteration method by Soliman (2009), that by applying a meshfree technique by Ali et al., (2009), and also the work of Zhu et al., (2010) (see also the references therein). The main objectives of this study are, therefore, twofold; firstly, to study the overall performance of DRBEM for Burgers equations when Reynolds number increases, and secondly, to search for an optimal Radial Basis Function () judged by determining both aspects including their overall accuracy and CPU time required. In Section 2, all mathematical foundation concerned with DRBEM is provided followed by its computational implementation toward Burgers equations in Section 3. Section 4 then gives a numerical experiment with all results and general discussion before the main findings of this work are listed in Section 5. Mathematical Background: The Dual Reciprocity Boundary Element Method (DRBEM): The way of formulating DRBEM is to consider the Poisson Equation; 2 u = b (4) Where b = b x, y. The solution to above eqn(4) can be expressed as the sum of the solution of homogenous and a particular solution as; u = u + u (5) Where u h is the solution of the homogeneous equation and u is a particular solution of the Poisson equation such that 2 u = b. We approximate b as a linear combination of interpolation functions, as elaborated in Section 2.2, for each of which we can find a particular solution at points which are situated in the domain and on its boundary. If there are boundary nodes and L internal nodes, there will be + L interpolationfunctions,f j,and consequently + L particular solutions, u j. The approximation of b over domain Ω is written, for an i t node, in the following form. b i x, y +L j =1 α j f ij (x, y) (6) Then we have a matrix equation for the unknown coefficients α j : j = 1,2,3,, + L as follows; b = Fα (7) This can easily be expanded as shown below. b 1 (x, y) b 2 (x, y) b 3 (x, y) b +L (x, y) = f 11 (x, y) f 21 (x, y) f (+L)1 (x, y) f 12 (x, y) f 22 (x, y) f (+L)2 (x, y) f 1(+L) (x, y) f 2(+L) (x, y) f (+L)(+L) (x, y) α 1 α 2 α 3 α +L The particular solution, u j,(see Section 2.2) and the interpolation function, f j, are linked through the relation. 2 u j = f j (8) Substituting eqn(8) into eqn(6) leads; +L b = j =1 α( 2 u j ) (9) Therefore, from eqn(4) and with the expression of b in eqn(9), we then obtain; 2 +L u = j =1 α( 2 u j ) (10) Multiplying both sides with the fundamental solution,u, and integrating over the domain yields;

465 Sayan Kaennakham et al, 2014 Ω ( 2 u)u dω +L = j =1 α ( 2 u j )u dω Ω (11) Integrating by parts the Laplacian terms produces the following integral equation for each source note i t (Partridge et al., 1992). c i u i + q udγ u qdγ Γ Γ +L = j =1 α j (c i u ij + q u j dγ u q j dγ) (12) Γ Γ The term q j in eqn (12) is defined as q j = u j, where n is the unit outward normal to Γ, and can be written n as follows; q j = u j x x + u j y n y n (13) Since eqn(12) contains no domain integrals, the next step is to write eqn(12) in discretised form, with summations over the boundary elements replacing the integrals. This gives, for a source node i t, the expression. c i u i + k=1 q udγ k=1 u qdγ = j =1 α j c i u ij + k=1 q u j dγ k=1 u q j dγ (14) Γ k Γ k +L Γ k Γ k After introducing the interpolation function and integrating over each boundary elements, the above eqn(14) can be re-written in terms of nodal values as; c i u i + +L k=1 H ik u k k=1 G ik q k = j =1 α j c i u ij + k=1 H ik u kj k=1 G ik q kj (15) Where the definition of the terms H ik and G ik can be found in Toutip, (2001). The index k is used for the boundary nodes which are the field points. After application to all boundary nodes, using a collocation technique, eqn(15) can be compactly expressed in matrix form as follows; Hu Gq = +L j =1 α j (Hu j Gq j ) (16) If each of the vector u j and q j is considered to be one column of the matrices U j and Q j respectively, then eqn(16) may be written without the summation to produce; Hu Gq = (HU GQ)α (17) By substituting α = F 1 b from eqn(7), into eqn(17) making the right hand side of eqn(17) a known vector. Therefore, it can be rewritten as; Hu Gq = d (18) Where d = (HU GQ)F 1 b. Applying boundary condition(s) to eqn(18), then it can be seen as the simple form as follow; Ax = y (19) Where x contains unknown boundary values of u s and q s. After eqn(19) is solved using standard techniques such as Gaussian elimination, the values at any internal node can be calculated from eqn(15), i.e. c i = 1 as expressed in eqn(20) where each one involving a separate multiplication of known vectors and matrices. u i = +L k=1 H ik u k + k=1 G ik q k = j =1 α j c i u ij + k=1 H ik u kj k=1 G ik q kj (20) Radial Basis Functions () Chosen and Their Particular Solution: Five widely adopted radial basis functions were chosen in this study. They are known as; linear type, thinplate spline type, Multiquadric type, compactly supported type, and quadratic type. Their 2D and 3D profiles are illustrated in Figure 1. In order to get their particular solution, several researchers have already nicely documented and are listed in Table 1 below.

466 Sayan Kaennakham et al, 2014 Table 1: Radial Basis Functions () chosen and their corresponding particular solution (Chen et al., 1999, Li et al., 2002, and Gumgum, 2010). Abbreviation Radial Basis Function Type Corresponding Particular Solution 1 Linear (LR): r 3 1 + r 9 + r2 4 2 Thin-Plate Spline (TPS): r 4 ln r r 2 ln r 16 r4 32 3 Multiquadric (MQ): c 3 ln c r r 2 + c 2 3 2 + c 2 + c 2 + 1 9 r2 + 4c 2 r 2 + c 2 4r 7 49c 5 5r6 12c 4 + 4r5 5c 3 5r4 8c 2 + r2, r c 4 4 Compactly Supported (CS) 1 r 4 4 r 529c 2 + 1, 0 r c 5880 + c2 14 ln r, r > c c c c 0, r > c 5 Quadratic (QC): r 2 1 + r + r 2 4 + r3 9 + r4 16 Multiquadric 8 7 Compactly Supported 1 Thin-Plate Spline Linear Quadratic 6 5 1 0.8 0.8 4 0.6 0.6 3 0.4 2 0.2 0.4 1 0-2 -1 0 1 2-1 -2 0 1 0 y -1-1 -0.5 x 0 0.5 1 0.2 0 Fig. 1: Chosen Radial Basis Functions and their two and three dimensional images. Computer Implementation of the Drbem For 2d-Coupled Burgers Equasions: Computing matrices construction: We first substitute eqn(17) into eqn(16) to get the equation system matrix which expressed as; Hu Gq = (HU GQ)F 1 b (21) Setting S = (HU GQ)F 1 (22) Then eqn(21) becomes Hu Gq = Sb (23) Consider the following system of two-dimensional Burgers Equation u t v t u u + u + v = 1 x y Re ( 2 u) v v + u + v = 1 x y Re ( 2 v) (24) (25) Subject to the initial conditions: u x, y, 0 = β 1 x, y, v x, y, 0 = β 2 (x, y, ) (26) (27)

467 Sayan Kaennakham et al, 2014 where x, y ε Ω And the boundary conditions: u x, y, t = γ 1 x, y, t, v x, y, t = γ 2 (x, y, t) (28) (29) where x, y ε Ω, t > 0 and D is a domain of the problem, D is its boundary; u(x, y, t) and v(x, y, t) are the velocity components to be determined, β 1, β 2, γ 1 and γ 2 are known functions and Re is the Reynolds number, which described in Zhu (2010). The analytic solution of eqn(24) and eqn(25) was given by Fletcher using the Hopf-Cole transformation (Fletcher, 1983). The numerical solutions of this equation system have been studied by several authors. Dehgham et al. (2007)developed two algorithms based on cubic spline function technique. Soliman (2009) has discussed the comparison of a number of different numerical approaches. As described in ee and Duan (1998), u u u u u, u, v and v are approximated similarly. x y x y Then we obtain u u F = U x x F 1 u v u F = V y y F 1 u u v F = U x x F 1 v v u F = V y y F 1 v (30) (31) (32) (33) From eqn (21) and eqn(23) we obtain Hu Gq = (HU GQ)F 1 b 1 (34) and Hv Gz = (HV GZ)F 1 b 2 (35) where q = u v, z = and n n b 1 = Re( u u u + u + v ) t x y b 2 = Re( v v v + u + v ) t x y (36) (37) In the case, if we consider U, V to be generated by using the same redial basis function, then we get; U = V (38) which yield Q = Z (39) Setting A = (HU GQ)F 1 (40) Then eqn(34) and eqn(35) becomes Hu Gq = Sb 1 (41)

468 Sayan Kaennakham et al, 2014 and Hv Gz = Sb 2 (42) For the time derivatives, we use the forward difference method to approximate time derivative u and v. The forward difference is expressed as; t t u = ut+1 u t t v = vt+1 v t t (43) (44) where u = u v and v = t t Substituting eqn(30) to eqn(33), eqn(36) and eqn(37) in eqn(41) and eqn(42), then (Hu Gq) = ARe(u + U F x F 1 u + V F y F 1 u) (45) (Hv Gz) = ARe(v + U F x F 1 v + V F y F 1 v) (46) Setting C = U F x F 1 + V F x F 1 (47) Then eqn(45) and eqn(46), respectively, becomes; Hu Gq = ARe(u + Cu) (48) and Hv Gz = ARe(v + Cv) (49) Substituting eqn(43) and eqn(44) in above eqns(48-49). The following expressions are obtained. (Re)AC + SRe t H ut+1 + Gq t+1 = ReS t ut (50) (Re)AC + SRe t H vt+1 + Gz t+1 = ReS t vt (51) ote that the elements of matrices H, G and A depend only on geometrical data. Thus, they can all be computed once and stored. umerical Algorithm: In the computing step, we apply the initial conditions to the right hand side of eqn(50) and eqn(51), and boundary conditions to the left hand side. In each time step, the iteration is utilized to get the suitable solution as described below; 1. Constructing matrix C in eqn(47) by setting it to zero for the first iterative loop for both U and V. 2. Setting all the values appearing in eqn(50) and eqn(51), and then applying the initial conditions to the right hand side and boundary condition to the left hand side, reducing to the form Ax = y. 3. Using the iterative scheme briefly expressed, running with k = 1, 2, as A (k) x (k) = y k, we get the solutions of U and V. With a ad-hoc value of ε, the iteration process stops when; max i x i (k+1) x i (k) < ε (52) 4. The solutions obtained from step 3 will be used as the initial conditions in next time-step and the computing process continues until the time level given is reached.

469 Sayan Kaennakham et al, 2014 umerical Experiment: The example being numerically carried out is the well-known form of Burgers equations where its exact solutions for validation are provided by using a Hopf-Cole transformation nicely done by Fletcher (1983) and expressed as follows; u x, y, t = 3 4 1 4 v x, y, t = 3 4 + 1 4 Re 1 + exp 4x + 4y t 32 Re 1 + exp 4x + 4y t 32 1 1 (53) (54) Where both the initial and boundary conditions to be imposed to the equation system are generated directly from the above exact form over the domain Ω = x, y : 0 x 1, 0 y 1, Fig. 2. With its rich of challenging features, the equations have been received interests from several researchers and been treated with variety of powerful numerical techniques. evertheless, it can be clearly seen that with the reason mentioned in Section 1, these studies are concerned with only the case of relatively low Reynolds numbers, as shown in Table 2. This work, on the other hand, focuses on how well DRBEM could perform when higher Reynolds numbers are tackled. Table 2: Related researches on coupled Burgers Equations with different Reynolds numbers. ames/year Method Highest Re Studied Chanthawara et al. (2014) DRBEM 80 Aminikhah (2012) Hybrid of Laplace transform method and homotopy 1 perturbation method (LTHPM) Siraj-ul-Islam et al. (2012) Local radial basis functions collocation method 1,000 (LCM) Zhao et al. (2011) The local discontinuous Galerkin(LDG) finite element 20 method Srivastava et al. (2011) Crank-icolson finite-difference method 500 Zhu et al. (2010) The discrete domain decomposition method(adm) 100 Biazar and Aminikhah (2009) Variational iteration method 10 Soliman (2009) Variational iteration method 100 Ali et al. (2009) The mesh free collocation method based on MQ 200 Kaennakham (2004) DRBEM 100 Fig. 2: umerical Domain and i-th computational internal nodes. In order to have the whole feature of s performances to judge which one is optimal, a large number of simulations were required. The criteria adopted to determine which should be crowned the winner are; the overall accuracy and CPU time needed to run each case. Fig. 3 and Fig. 4 illustrates comparisons of computed U-value and V-value obtained from DRBEM with 80 boundary nodes and 13 internal nodes against the exact ones, with time step t = 0.0001 and t = 0.1 at Re = 200, 400 and Re = 600, 800 respectively. It can be seen that at Re lower than 300, overall solutions numerically solved by DRBRM at all selected points remain close to each other and also close to the exact ones. For Reynolds number above 400, nevertheless, it is noticeable that 1 st -, 4 th -, 7 th -, 10 th -, and 13 rd point reveal slightly differences in both U- and V-velocity calculated values from the exact. This is no coincidence as clearly displayed by Fig. 6 and Fig. 7Fig. 7, these points are located exactly where the both profiles increase sharply. Apart from these points, other points at other locations where relatively smoother profiles are founds have predicted solutions remained close to the analytic ones. All calculated solutions for Re = 1,000 with time step t = 0.0001 and t = 0.1 are compared in digits against the analytics in Table 3 and Table 4.

U-value at Re = 800 V-value at Re = 800 U-value at Re = 600 V-value at Re = 600 U-value U-value at Re = at 400 Re = 400 V-value V-value at Re = at 400 Re = 400 U-value U-value at Re = at 200 Re = 200 V-value V-value at Re = at 200 Re = 200 470 Sayan Kaennakham et al, 2014 In the case of 4, Multiquadric, in particular, our previous study Chanthawara et al. (2014), has discovered that the best parameter, C, lies between 1.1 and 1.5. It is, nevertheless, for simulating Burgers equations with Reynolds number below 100 which is obviously not the case here. Observing behavior of most suitable C value for each Re from Fig. 5, it is found that optimal C value decrease rapidly during100 Re 350. Beyond this point, optimal C value remains nearly constant, at approximately 0.05 for 400 Re 650 before beginning to continue to decrease gradually and reaches 0.03 at Re = 1,000 Exact Solution Exact Solution (C= (C= i-th Internal ode Exact Solution Exact Solution (C= (C= i-th Internal ode i-th Internal ode i-th Internal ode Exact Solution Exact Solution (C= (C= i-th Internal ode Exact Solution Exact Solution (C= (C= i-th Internal ode Fig. 3: Comparisons of computed U-value and V-value obtained from DRBEM with 80 boundary nodes and 13 i-th Internal ode i-th Internal ode internal nodes against the exact ones, at Re = 200 and 400, with time step t = 0.0001 and t = 0.1. Exact Solution (C= Exact Solution (C= i-th Internal ode i-th Internal ode Exact Solution (C= Exact Solution (C= i-th Internal ode i-th Internal ode Fig. 4: Comparisons of computed U-value and V-value obtained from DRBEM with 80 boundary nodes and 13 internal nodes against the exact ones, at Re = 600 and 800, with time step t = 0.0001 and t = 0.1. In the case of 4, Multiquadric, in particular, our previous study Chanthawara et al. (2014), has discovered that the best parameter, C, lies between 1.1 and 1.5. It is, nevertheless, for simulating Burgers

MQ : Best C-value 471 Sayan Kaennakham et al, 2014 equations with Reynolds number below 100 which is obviously not the case here. Observing behavior of most suitable C value for each Re from Fig. 5, it is found that optimal C value decrease rapidly during100 Re 350. Beyond this point, optimal C value remains nearly constant, at approximately 0.05 for 400 Re 650 before beginning to continue to decrease gradually and reaches 0.03 at Re = 1,000. Table 3: umerical solutions of U-value and V-value obtained from DRBEM with 80 boundary nodes and 13 internal nodes, compared against the exact ones, at Re = 1000, with time step t = 0.0001 and t = 0.1. i-th ode Exact Sol. 1 2 3 4(.03) 5 1 (0.1,0.1) 0.51052193 0.59917266 0.62957349 0.59969551 0.63599644 0.64507757 2 (0.5,0.1) 0.50000000 0.49237926 0.50284521 0.49161027 0.47380002 0.50696804 3 (0.9,0.1) 0.50000000 0.50135488 0.50184921 0.50076389 0.48898800 0.50036983 4 (0.3,0.3) 0.51052193 0.61719995 0.62574658 0.61718138 0.58309217 0.62963531 5 (0.7,0.3) 0.50000000 0.49283875 0.49835652 0.49278298 0.43501623 0.49847695 6 (0.1,0.5) 0.75000000 0.75782662 0.74930116 0.75928725 0.74378806 0.74864571 7 (0.5,0.5) 0.51052193 0.61600506 0.61162080 0.61611656 0.56321118 0.60598567 8 (0.9,0.5) 0.50000000 0.49784315 0.48825392 0.49852565 0.45021784 0.48483543 9 (0.3,0.7) 0.75000000 0.75125230 0.74891875 0.75139474 0.73911872 0.74810550 10 (0.7,0.7) 0.51052193 0.61437019 0.62129960 0.61441573 0.57498292 0.63232717 11 (0.1,0.9) 0.75000000 0.74958203 0.74900516 0.74926099 0.74237450 0.75060594 12 (0.5,0.9) 0.75000000 0.73690362 0.74942268 0.73426018 0.75169213 0.75025181 13 (0.9,0.9) 0.51052193 0.57169037 0.53702044 0.56999302 0.53312358 0.54901515 Table 4: umerical solutions of U-value and V-value obtained from DRBEM with 80 boundary nodes and 13 internal nodes, compared against the exact ones, at Re = 1000, with time step t = 0.0001 and t = 0.1. i-th ode Exact Sol. 1 2 3 4(.03) 5 1 (0.1,0.1) 0.98947806 0.90954507 0.87388408 0.91072271 0.83379588 0.84773759 2 (0.5,0.1) 1.00000000 1.01278161 0.99929148 1.01488002 1.00058607 1.00756279 3 (0.9,0.1) 1.00000000 1.00159273 0.99976180 1.00279264 0.99990834 0.99744408 4 (0.3,0.3) 0.98947806 0.87937287 0.88073139 0.87855524 0.88194082 0.88572733 5 (0.7,0.3) 1.00000000 1.00655831 1.00429211 1.00649387 1.04433813 1.01375114 6 (0.1,0.5) 0.75000000 0.74618390 0.75252971 0.74576865 0.73458043 0.76273016 7 (0.5,0.5) 0.98947806 0.88343190 0.88925849 0.88323163 0.91692724 0.90296469 8 (0.9,0.5) 1.00000000 0.99839281 1.00994020 0.99672832 1.05458462 1.00200762 9 (0.3,0.7) 0.75000000 0.74949692 0.75201338 0.74954832 0.74715508 0.75922561 10 (0.7,0.7) 0.98947806 0.89029296 0.87530740 0.89126672 0.92961286 0.84785250 11 (0.1,0.9) 0.75000000 0.75038793 0.75071798 0.75072007 0.75465741 0.74984592 12 (0.5,0.9) 0.75000000 0.75724268 0.74716235 0.75832416 0.76004286 0.72809930 13 (0.9,0.9) 0.98947806 0.91809696 0.95178365 0.91766848 0.98969331 0.95946481 Reynolds umber (Re) Fig. 5: Decrease of optimal C values for Multiquadric with increasing Reynolds number (Re). In terms of Root Mean Square (RMS) error of velocity U and V obtained from each, Fig. 8 displays the overall curves. As expected, every chosen has an exponent increase of error with the increase of Reynolds number. evertheless, from roughly Re = 500 onwards, the differences become clearer. From both graphs, while 5 seems to yield the highest error at each Re, 4 with its best parameter C, remains the lowest in terms of the RMS error. Considering the computational time required to run each simulation, shown in Fig. 9, all s seem to take approximately the same amount of computational effort.

472 Sayan Kaennakham et al, 2014 Fig. 6: U-Profiles obtained from each Radial Basis Function () for Re = 1000, time step t = 0.0001 and t = 0.1: b) 1, c)2, d)3, e)4(with C=0.03), and f)5 compared with a) Exact solution.

473 Sayan Kaennakham et al, 2014 Fig. 7: V-Profiles obtained from each Radial Basis Function () for Re = 1000, time step t = 0.0001 and t = 0.1: b) 1, c)2, d)3, e)4(with C=0.03), and f)5 compared with a) Exact solution.

Computatinal Time RMS Error for U RMS Error for V 474 Sayan Kaennakham et al, 2014 Reynolds umber Fig. 8: Growth of Root Mean Square (RMS) error of computed U and V with increasing Reynolds number. 66 65.5 65 64.5 64 63.5 63 62.5 62 61.5 1 2 3 4 5 61 100 200 300 400 500 600 700 800 900 1000 Reynolds umber Fig. 9: CPU time required to run each simulation at each Reynolds umber,intel(r) Core(TM) i3 M370 with 2GB of RAM. Conclusions: This study aims to explore answers to two challenging questions; how well DRBRM could perform in numerically solving Burgers' equations at high Reynolds number, and which provides best solution in terms of accuracy and CPU time requirement when Reynolds number increases. To shed more light into these questions, a large number of simulations were carried out and important pieces of data were gathered and present. The Reynolds number involved ranks from 80 up to 1,000 as highest and crucial findings discovered from the experiments are as follows; 1. In general, DRBRM performed reasonably well with numerically predicted solutions obtained from the method remained in good agreement with the exact solutions. 2. As expected, when increasing Reynolds number, all resulted in increasing level of RMS error while there is no noticeable difference in CPU time needed for each case study. 3. Taking into consideration both the overall accuracy and CPU time requirement, the Multiquadric radial basis function ( 4) type that provided the best solution. For Multiquadric, nevertheless, while it seems to lead to optimal solutions for all Reynolds numbers, determining the optimal parameter C is not an easy task and deserves more investigations. REFERECES Reynolds umber Ali, A., Siraj-ul-lslam and S. Haq, 2009. A Computational Meshfree Technique for the umerical Solution of the Two-Dimensional Coupled Burgers Equations. International Journal for Computational Methods in Engineering Science and Mechanics, 10(5): 406-422. Aminikhah, H., 2012. A ew Efficient Method for Solving Two-Dimensional Burgers Equation. International Scholarly Research etwork Computational Mathematics, 2012: 1-8. Bahadir, A.R., 2003. A Fully Implicit Finite Difference Scheme for Two Dimensional Burgers equations. Applied Mathematics and Computation, 137: 131-137. Biazar, J. and H. Aminikhah, 2009. Exact and umerical Solutions for on-linear Burgers' Equation by VIM. Mathematical and Computer Modelling, 49(7): 1394-1400.

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