The Eulerian Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers equations

Transcription

1 Engineering Analysis with Boundary Elements 32 (2008) The Eulerian Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers equations D.L. Young a,, C.M. Fan a, S.P. Hu a, S.N. Atluri b a Department of Civil Engineering and Hydrotech Research Institute, National Taiwan University, Taipei, Taiwan b Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, USA Received 27 October 2006; accepted 17 August 2007 Available online 24 October 2007 Abstract The Eulerian Lagrangian method of fundamental solutions is proposed to solve the two-dimensional unsteady Burgers equations. Through the Eulerian Lagrangian technique, the quasi-linear Burgers equations can be converted to the characteristic diffusion equations. The method of fundamental solutions is then adopted to solve the diffusion equation through the diffusion fundamental solution; in the meantime the convective term in the Burgers equations is retrieved by the back-tracking scheme along the characteristics. The proposed numerical scheme is free from mesh generation and numerical integration and is a truly meshless method. Twodimensional Burgers equations of one and two unknown variables with and without considering the disturbance of noisy data are analyzed. The numerical results are compared very well with the analytical solutions as well as the results by other numerical schemes. By observing these comparisons, the proposed meshless numerical scheme is convinced to be an accurate, stable and simple method for the solutions of the Burgers equations with irregular domain even using very coarse collocating points. r 2007 Elsevier Ltd. All rights reserved. Keywords: Eulerian Lagrangian method; Method of fundamental solutions; Burgers equations; Diffusion fundamental solution; Meshless method 1. Introduction The Burgers equation was initially studied for the weather problem in 1915 by Bateman [1] and was extended to model turbulence and shock wave by Burgers [2]. Besides, the Burgers equation is a useful model for many interesting physical problems [3], such as shock wave, acoustic transmission, traffic and aerofoil flow theory, turbulence and supersonic flow as well as a prerequisite to the Navier Stokes equations. The problems modeled by the Burgers equation can be considered as an evolutionary process in which a convective phenomenon is in contrast with a diffusive phenomenon. It is possible to obtain the exact solutions of the Burger s equation for simple geometry by the Cole Hopf transformation [4,5]. The known exact solutions of the Burgers equation are tabulated by Benton and Platzman [6] as well as Fletcher [7]. Corresponding author. Fax: address: dlyoung@ntu.edu.tw (D.L. Young). Although there are some analytic solutions available in the literature, the exact solutions for the practical applications are very limited due to the complex geometry and complicated initial and boundary conditions. The numerical methods developed more than three decades seem to serve as a satisfactory alternative to solve the unsteady Burger s equations. Most of the existing numerical methods in previous studies were reported successfully to be able to solve the Burgers equations, such as the finite difference method (FDM) [8 10], the finite element method (FEM) [11,12] and the boundary element method (BEM) [13,14]. For example, for the Burgers equation Bahadir [8] proposed a fully implicit finite difference scheme and Radwan [10] used a fourth-order compact scheme and the fourth-order Du Fort Frankel algorithm. In addition, Froncioni et al. [11] proposed the discontinuous-galerkin space time finite element formulation using the simplextype meshes. In the meantime, Kutluay et al. [12] used the least-squares quadratic B-spline FEM to handle the unsteady Burgers equations. In comparing with FDM and FEM, the BEM appears to be a better alternative to /$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi: /j.enganabound

2 396 D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) simulate the physical problems due to the reduction of one dimension. Kakuda and Tosaka [14] adopted the generalized BEM to treat the Burgers equations while Chino and Tosaka [13] used the dual reciprocity BEM. The numerical methods discussed above can be used to solve the unsteady Burgers equations; however the large amount of efforts should be paid during the numerical implementation. The time-consuming mesh generation of FDM and FEM as well as the complicated singular integrals of BEM always bothered researchers. The drawbacks make these conventional numerical methods very difficult to efficiently deal with the Burgers equations especially for treating the nonlinear, multidimensional flows and irregular domain problems. The developments of the so-called meshless or meshfree methods catch the researchers attentions recently. There t Field Points (x) Source Points (ξ) t (n+1) Δt (n+1-λ) Δt Y (n+1) Δt C A (n) Δt X (n-λ) Δt (n) Δt B x Fig. 1. (a) Schematic diagram for the location of source and field points on the space time domain in 2-D problem. (b) Schematic diagram for the characteristic AB. Fig. 2. Velocity profiles of problem 1 at different time levels (Re ¼ 1, Dt ¼ 0.01, N ¼ 64). (a) t ¼ 0.10; (b) t ¼ 1.00; (c) t ¼ 3.00; (d) t ¼ 9.00.

3 D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) Fig. 3. Error profiles of problem 1 at different time levels (Re ¼ 1, Dt ¼ 0.01, N ¼ 64). (a) t ¼ 0.10; (b) t ¼ 1.00; (c) t ¼ 3.00; (d) t ¼ Fig. 4. Velocity profiles of problem 1 at different time levels (Re ¼ 20, Dt ¼ 0.001, N ¼ 441). (a) t ¼ 0.50; (b) t ¼ 0.75; (c) t ¼ 1.00; (d) t ¼ 1.25.

4 398 D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) are several meshless methods developed in the past decade and some available methods are the multiquadrics (MQ) method [15,16], the meshless local Petrov-Galerkin (MLPG) method [17 20] and the method of fundamental solutions (MFS) [21 29]. Hon and Mao [15] applied the MQ method to the one-dimensional unsteady Burgers equation, while Li et al. [16] used the MQ method to solve two-dimensional problems. Though the MQ method can simply solve the Burgers equations, the choice of a suitable shape parameter which will influence the stability of the numerical scheme is still an open topic. This handicap drastically limits the developments and applications of the MQ method. The MFS, which is similar to the BEM due to the reduction of one dimension, is free from the mesh generation and numerical integration. The MFS was originally proposed by Kupradze and Aleksidze [24] and has been extended to the solution of Poisson s equation by Golberg [21]. Karageorghis and Fairweather [23] adopted the MFS to model the biharmonic equation. On the other hand, Young and Ruan [26] analyzed the electromagnetic waves scattering problems by MFS, and Young et al. [27] used the Stokeslets and MFS to simulate the Stokes flow in a rectangular cavity with cylinders. Under the novel concept of time space unification, Young et al. [28,29] solved the time-dependent diffusion equations by the diffusion fundamental solution and MFS which can avoid the Laplace transform or finite difference method in discretizing the time state. The time-dependent MFS is further applied to the Stokes first and second problems in a semi-infinite domain by Hu et al. [22]. The MFS is successfully applied to solve the linear diffusion equation since the numerical results are assumed to be the linear combination of the time-dependent diffusion fundamental solutions. Due to the existence of the convective term in the Burgers equations, the MFS cannot be used directly for the Burgers equations. The convective term of the unsteady Burgers equation can be dealt with by the Eulerian Lagrangian method (ELM) [25,30]. The ELM combines the computational powers of the Eulerian and Lagrangian approaches, so as to incorporate the merits of a fixed Eulerian coordinate and a moving Lagrangian coordinate. The combination of the ELM and BEM has been successfully applied to the advection diffusion equations [30], while the same problems are also simulated by using the Eulerian Lagrangian method of fundamental solutions (ELMFS) [25]. The use of ELM can be regarded as changing the physical viewpoint of the problem from a fixed to moving path. In this study, the Burgers equations will be converted to the characteristic diffusion equations by ELM, and then Fig. 5. Error profiles of problem 1 at different time levels (Re ¼ 20, Dt ¼ 0.001, N ¼ 441). (a) t ¼ 0.50; (b) t ¼ 0.75; (c) t ¼ 1.00; (d) t ¼ 1.25.

5 D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) Table 1 Numerical solutions of different time levels in some specific points of problem 1 (Re ¼ 20, Dt ¼ 0.001, N ¼ 441) x y t ¼ 0.50 t ¼ 0.75 t ¼ 1.00 t ¼ 1.25 Analytical solution ELMFS ERROR Analytical solution ELMFS ERROR Analytical solution ELMFS ERROR Analytical solution ELMFS ERROR E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 05 the diffusion equations will be solved by the MFS. After the diffusion solutions are found by MFS, the convective term of the Burgers equations can be obtained by the back-tracking scheme through the characteristics [25,30]. This ELMFS technique has been successfully applied to the one-dimensional unsteady Burgers equations by Young [31]. The aim of this study is to demonstrate the capability and simplicity of the ELMFS to solve the unsteady nonlinear two-dimensional Burgers equations. The governing equations and numerical method will be explained in Sections 2 and 3, respectively. The numerical results and conclusions will be provided, respectively, in Sections 4 and 5. There are three case study problems adopted in this article and the numerical results are compared very well with the analytical solutions as well as other numerical solutions. 2. Governing equations The two-dimensional Burgers equations with two variables are similar to the incompressible Navier Stokes equations without considering pressure term and continuity equation. We will consider the following system of the two-dimensional Burgers equations: qu qt þ u qu qx þ v qu qy ¼ 1 Re q 2 u qx 2 þ q2 u qy 2, (1) qv qt þ u qv qx þ v qv qy ¼ 1 q 2 v Re qx 2 þ q2 v qy 2 (2) subject to the initial conditions: uðx; y; t 0 Þ¼f 1 ðx; yþ ðx; yþ 2O, (3) vðx; y; t 0 Þ¼f 2 ðx; yþ ðx; yþ 2O (4) and the boundary conditions: Luðx; y; tþ ¼f 3 ðx; y; tþ ðx; yþ 2qO; (5) Lvðx; y; tþ ¼f 4 ðx; y; tþ ðx; yþ 2qO; (6) where O and qo denote the computational domain and the associated boundary. L is a boundary differential operator. uðx; y; tþ and vðx; y; tþ are the two unknown variables which can be regarded as the velocities in fluid-related problems. f 1 ðx; yþ, f 2 ðx; yþ, f 3 ðx; y; tþ and f 4 ðx; y; tþ are all known functions. Re is the Reynolds number, and t 0 is the initial time. According to the relative weighting of the diffusive and convective terms (Re) in the Burgers equations, the Burgers equations will behave as elliptic, parabolic or hyperbolic type of partial differential equations. 3. Numerical method The two-dimensional Burgers equations, Eqs. (1) and (2), can be transferred to the following two characteristic

6 400 D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) diffusion equations using the ELM: Du Dt ¼ 1 q 2 u Re qx 2 þ q2 u qy 2, (7) Dv Dt ¼ 1 q 2 v Re qx 2 þ q2 v qy 2, (8) where the total or material derivative including the convective term is defined as [25,30] D Dt ¼ q qt þ u q qx þ v q qy. (9) Since the two-dimensional Burgers equations are converted to the characteristic diffusion equations, the MFS is first adopted to solve the diffusion equations [28,29]. In MFS, the diffusion solution can be represented as the linear combination of the diffusion fundamental solutions with different source intensities. The fundamental solution of the linear diffusion equation is governed by qgð~x; t; ~ x; tþ qt ¼ 1 Re r2 Gð~x; t; ~ x; tþþdð~x ~ xþdðt tþ, (10) where Gð~x; t; ~ x; tþ is the fundamental solution of the linear diffusion equation. ~x ¼ðx; yþ and ~ x ¼ðx; ZÞ are the spatial coordinates of the field and source points. t and t are the temporal coordinates of the field and source points. d() is the well-known Dirac delta function. Fig. 6. Velocity profiles of problem 2 at different time levels (Re ¼ 100, Dt ¼ 0.005, N ¼ 441). (a) t ¼ 0.01; (b) t ¼ 0.50; (c) t ¼ 2.00.

7 D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) By using the integral transform theory of Eq. (10), the free-space Green s function or the fundamental solution of the linear diffusion equation can be obtained as Gð~x; t; ~ x; tþ ¼ e j~x ~ xj 2 =½4ð1=ReÞðt tþš d=2 Hðt tþ, (11) 4pð1=ReÞðt tþ where d is the dimension of the problem and is equal to two in this study. H( ) is the Heaviside step function. Based on the time-dependent MFS concept, we can express the diffusion solutions of Eqs. (7) and (8) by the combination of the diffusion fundamental solutions, Eq. (11), as uð~x; tþ ¼ XN j¼1 vð~x; tþ ¼ XN j¼1 a j Gð~x; t; ~ x j ; t j Þ, (12) b j Gð~x; t; ~ x j ; t j Þ, (13) where N is the number of source point. a j and b j are the unknown coefficients which denote the source intensities of the corresponding fundamental solutions. Once the coefficients are obtained, the velocity of any field points in the Fig. 7. Error profiles of problem 2 at different time levels (Re ¼ 100, Dt ¼ 0.005, N ¼ 441). (a) t ¼ 0.01; (b) t ¼ 0.50; (c) t ¼ 2.00.

8 402 D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) time space domain can be acquired by using Eqs. (12) and (13) accordingly. In our numerical experiments, the numbers of field and source points are chosen the same, and both are equal to N so that square matrices are formed. The locations of field and source points are illustrated in Fig. 1(a), and the field and source points are located at the same spatial positions but at different time levels. In Fig. 1(a), the parameter, l, is chosen as a function of the maximum distance of the spatial domain (R) and it can be expressed as lðdtþ ¼mR. By observing the diffusion fundamental solution, it is noted that the temporal difference (t t) between field and source points is proportional to their spatial distance ðj~x ~ xjþ. Hence we will use the empirical formula to determine the temporal location of the source points. In the section of numerical results, it will be elaborated that the proposed formula performs well and provides a useful guide to determine the time level of the source points. m is an adaptive parameter which can be chosen by the trial and error process. By collocating the initial and boundary conditions, two matrices are formed by utilizing Eqs. (3) (6), (12) and (13): ½A u Šfag ¼½f u Š, (14) ½A v Šfbg ¼½f v Š. (15) The components of [A u ]and[a v ] are the representation of the fundamental solutions. ff u g is the combination of f 1 ðx; yþ and f 3 ðx; y; tþ, and ff v g is the combination of f 2 ðx; yþ and f 4 ðx; y; tþ. The unknown coefficients, or the source intensities of the fundamental solutions, can be obtained by inverting the above two matrices, Eqs. (14) and (15). The function values inside the time space box at t ¼ðnþ1ÞDtcan thus be acquired from Eqs. (12) and (13). The results of the Burgers equations with convective term can be retrieved from the numerical diffusion results by back-tracking the particles along the line of characteristics. In the ELM, the convective velocities in the Burgers equations are expressed in terms of the spatial and time increments as follows: u ¼ dx dt ¼ xnþ1 x n, (16) Dt v ¼ dy dt ¼ ynþ1 y n, (17) Dt x n ¼ x nþ1 u Dt, (18) y n ¼ y nþ1 v Dt. (19) In Fig. 1(b), the line AB is the characteristic path on which the transport of the scalar quantity can be traced. If the velocities at point A are required, the spatial location of point B can be traced by Eqs. (18) and (19). When the spatial location of point B is determined, the solutions along characteristics AB will follow the characteristic diffusion operators, Eqs. (7) and (8), according to the material derivative and the diffusion equations. After the diffusion process is calculated by the time-dependent MFS, the velocities at point C can be obtained to represent the velocities at point A. Points B and C are located at the same spatial position but at different time levels (Fig. 1(b)). The velocities at point A are properly replaced by the diffusion results at point C, and then the results of the Burgers equations at t ¼ðn þ 1ÞDt thus can be acquired. This procedure can be repeated until either the terminal time or steady-state solution is achieved. Maximum absolute error 1.0E E E E E u Δt = 0.01 Δt = 0.05 Δt = Time v Δt = 0.01 Δt = 0.05 Δt = Fig. 8. Time history of maximum absolute errors of u and v of problem 2 for different size of time step (Re ¼ 100, N ¼ 441). Maximum absolute error 1.0E E E E E-004 u λ (Δt)=0.5R λ (Δt)=1R λ (Δt)=5R λ (Δt)=10R 1.0E Time 2 v λ (Δt)=0.5R λ (Δt)=1R λ (Δt)=5R λ (Δt)=10R Fig. 9. Time evolution history of maximum absolute errors of u and v of problem 2 for temporal locations of source point (Re ¼ 100, N ¼ 441). 2.5

9 D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) Numerical results To illustrate the high performance of ELMFS described in the previous section, three Burgers problems will be considered. The first one is the two-dimensional Burgers equation in one variable and the numerical results are compared with the analytical solutions. The second and the third ones are the two-dimensional Burgers equations in two variables with regular and irregular domains. The results of the second problem are in good agreement with the analytical solutions and show better performance than the FDM [8]. In the second problem the study of the disturbance of noisy initial and boundary data is also taken into consideration. In order to demonstrate the flexibility of the ELMFS, the third problem is devoted to an irregular computational domain which is nontrivial by the conventional numerical methods Problem 1 The first validation problem is the unsteady Burgers equation in one variable which is described as below: qu qt þ u qu qx þ u qu qy ¼ 1 q 2 u Re qx 2 þ q2 u qy 2 (20) and the analytical solution is [16] uðx; y; tþ ¼ 1 1 þ expðreðx þ y tþ=2þ. (21) The computational domain is O ¼ fðx; yþ : 0pxp1; 0pyp1g. The numerical results of velocity for Re ¼ 1 are shown in Fig. 2. The evolutionary process can be observed in the figure and the absolute errors are depicted in Fig. 3. Since the convective term is not large in comparing with the diffusion term, the diffusion process varies smoothly. The absolute errors in Fig. 3 are quite small, so it is proven that ELMFS can handle the Burgers equation at low Reynolds number. Additionally, the numerical results in Re ¼ 20 are displayed in Fig. 4 and the absolute errors are depicted in Fig. 5. It is easy to find conspicuously a sharp gradient which moves with time in Fig. 4 as Re increases and the absolute errors in Fig. 5 also move with that front. The errors will occur near the sharp front in any numerical method and the absolute errors in this test are acceptable. The complete numerical results, absolute errors and analytical solutions are tabulated in Table 1. By observing the detailed comparison of numerical and analytical results, it is convinced that the proposed scheme is very simple, stable and accurate for the solutions of the Burgers equation. There is no iteration Table 2 Numerical solutions of (a) u and (b) v at different time levels in some specific points of problem 2 (Re ¼ 10, Dt ¼ 0.01, N ¼ 441) x y t ¼ 0.01 t ¼ 0.5 t ¼ 2.0 Analytical solution MFS ERROR Analytical solution MFS ERROR Analytical solution MFS ERROR (a) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 06 (b) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 06

10 404 D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) process required by the ELMFS as far as the nonlinear Burgers problem is concerned 4.2. Problem 2 The second validation problem is the unsteady Burgers equations in two variables, Eqs. (1) and (2). The analytical solutions can be obtained by the Cole Hopf transformation [4,5,7] and have been used as a test problem by Bahadir [8]: uðx; y; tþ ¼ 3 4 1, (22) 41þexpðð 4x þ 4y tþðre=32þþ vðx; y; tþ ¼ 3 4 þ 1. (23) 41þexpðð 4x þ 4y tþðre=32þþ The computational domain is O ¼ fðx; yþ : 0pxp1; 0pyp1g. The initial and boundary conditions are taken from the analytical solutions. The numerical results are shown in Fig. 6 when Re ¼ 100. It is noted that there are sharp gradients which move toward the same direction in the u and v distributions, respectively. The nonlinear term dominates the evolutionary process and there appears a wave-like profile in Fig. 6 at Re ¼ 100. The absolute errors are displayed in Fig. 7 and the same phenomenon is revealed clearly that errors will move with the fronts. The Table 3 Numerical solutions of u at different time levels in some specific points of problem 2 (Re ¼ 100, Dt ¼ 0.005) x y Analytical solution N ¼ N ¼ N ¼ ELMFS ERROR ELMFS ERROR FDM [8] ERROR t ¼ E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 05 t ¼ E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 03 t ¼ E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 04

11 D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) maximum absolute errors of u and v at three different time increments are shown in Fig. 8. When a smaller time step is used, the numerical accuracy is systematically improved. It seems that only accuracy instead of stability problem is involved in the selection of time step. At the previous section, we suggest that the time level of source points can be determined by the empirical function lðdtþ p ffiffi ¼mR. When m is set to 0.5, 1, 5 or 10 and R is equal to 2, the results of time history of maximum absolute errors are demonstrated in Fig. 9. The numerical solution with m ¼ 0:5 is the worst one and the result with m ¼ 1 is the best case in this numerical test. Therefore, we suggest choosing m ¼ 1 and all numerical results in this investigation are obtained by m ¼ 1. The numerical results in this study show that the empirical formula is very useful and provides a valuable guide to determine the optimal temporal location of the source point in the unsteady MFS. The theoretical study and more numerical tests of the proposed formula will be thoroughly examined in the future research. The detailed velocity results and associated errors at some specified points are listed in Table 2 for Re ¼ 10 and the proposed numerical scheme is very stable and accurate when the evolutionary process happened. The results of u and v components for Re ¼ 100 are listed in Tables 3 and 4, respectively, and the problem is also solved by FDM [8]. The numerical computations were preformed using Table 4 Numerical solutions of v at different time levels in some specific points of problem 2 (Re ¼ 100, Dt ¼ 0.005) x y Analytical solution N ¼ N ¼ N ¼ ELMFS ERROR ELMFS ERROR FDM [8] ERROR t ¼ E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 05 t ¼ E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 03 t ¼ E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 03

12 406 D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) E Analytical solution 1.0E Maximum absolute error 1.0E E-004 u k=10-2 k=10-3 k=10-4 k=10-5 k=0 u (0.5,0.5,t) u (k=1%) ELMFS [TSVD] tol=10-5 [TSVD] tol=10-6 [TSVD] tol=10-8 [DLSQRR] tol= E Time Time 1.0E Analytical solution 1.0E Maximum absolute error 1.0E E-004 v k=10-2 k=10-3 k=10-4 k=10-5 k=0 v (0.5,0.5,t) u (k=1%) ELMFS [TSVD] tol=10-5 [TSVD] tol=10-6 [TSVD] tol=10-8 [DLSQRR] tol= E Time Fig. 10. Time history of maximum absolute errors of (a) u and (b) v on the problem 2 with noisy data at Re ¼ 100 (Dt ¼ 0.005, N ¼ 441). uniform node distribution, with the number of nodes N ¼ 121 and 441, respectively. The last columns in both Tables 3 and 4 show the numerical results by FDM [8] with a uniform mesh By examining those results, the solution obtained by ELMFS is more accurate than FDM [8]. Even using only 121 coarse collocating points the ELMFS results have reached acceptable accuracy. Furthermore, we consider the problem with noisy initial and boundary data as follows: uðx; y; 0Þ ¼ð1þkÞ ðx; yþ 2O; ½1 þ expðð 4x þ 4yÞðRe=32ÞÞŠ ð24þ vðx; y; 0Þ ¼ð1þkÞ 4 þ 1 4½1 þ expðð 4x þ 4yÞðRe=32ÞÞŠ ðx; yþ 2O; ð25þ 3 uðx; y; tþ ¼ð1 þ kþ 4 1 4½1 þ expðð 4x þ 4y tþ=ðre=32þþš ðx; yþ 2qO; ð26þ 3 vðx; y; tþ ¼ð1þkÞ 4 þ 1 4½1 þ expðð 4x þ 4y tþ=ðre=32þþš ðx; yþ 2qO; Time Fig. 11. Time history of (a) u and (b) v at (0.5, 0.5) on problem 2 with noisy data at Re ¼ 100 (Dt ¼ 0.005, N ¼ 441). ð27þ

13 D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) where e is a uniformly distributed random number and 1pp1. And k is the amplitude of noise level. These random numbers are generated by the FORTRAN subroutine RANDOM_SEED. In our numerical experiment, we solve the problems with noise levels from k ¼ 10 5 to The maximum absolute errors of u and v are shown in Fig. 10 for different k. When the amplitude of k is smaller than 10 2, the results are accurate. For larger values of k (1%), the results are not as good as solutions with smaller k but still acceptable (within 1% error). In this test, the proposed ELMFS without regularization methods Maximum absolute error 1.0E E E E-004 u k=1% [TSVD] tol=10-4 [TSVD] tol=10-5 [TSVD] tol=10-6 [TSVD] tol=10-8 [TSVD] tol=10-10 [DLSQRR] tol= E Time can be used to successfully analyze problems with moderate noise level up to k ¼ The same conjecture was also observed when the MFS is used to solve the Laplace equations with the moderate noise level [32]. This demonstrates that the present ELMFS is superior to other numerical methods as far as dealing with moderate noise level is concerned. For larger noise disturbance we also consider the regularization methods to improve the accuracy of numerical results for k ¼ Marin et al. [33] indicated that more accurate results could be obtained if the singular value decomposition (SVD) technique was used. Fig. 11 shows the time evolution of u and v at (0.5, 0.5) by the truncated SVD (TSVD) and QR decomposition with the regularization. The results of TSVD are obtained by the NUMERICAL RECIPES [34] subroutine SVDCMP; and the results of QR decomposition are found by the FORTRAN subroutine DLSQRR. In addition, tol is a parameter to be assigned. For TSVD, it means the singular value smaller than tol of matrix is allowed to be zero. For DLSQRR, it means the tolerance tol used to determine the 1 0 Ω r 2 r 1 c 2 c 1 Γ 1.0E (r 1 = r 2 = 1, c 1 = (0,0), c 2 = ( 1.8,0) ) 1 1.0E Maximum absolute error 1.0E E-004 v k=1% [TSVD] tol=10-4 [TSVD] tol=10-5 [TSVD] tol=10-6 [TSVD] tol=10-8 [TSVD] tol=10-10 [DLSQRR] tol= E Time Fig. 12. Time history of maximum absolute errors of (a) u and (b) v on problem 2 for the amplitude of noise k ¼ 1% at Re ¼ 100 (Dt ¼ 0.005, N ¼ 441) Fig. 13. (a) Computational domain of problem 3 (b) Distribution of some specific points of problem 3 for comparison.

14 408 D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) subset of columns of matrix is included in the solution. Fig. 12 shows the maximum absolute errors of u and v by TSVD and QR decomposition (DLSQRR). Those figures indicate that the regularization by TSVD did not improve much numerical accuracy as DLSQRR did. Therefore, we conclude that the DLSQRR is a powerful algorithm to solve matrices in the regularization process. In summary, this ELMFS technique will produce accurate and stable solutions with or without regularization for the studied level of noise added into the data Problem 3 After validating the above two problems by analytic solutions with and without noise consideration, it is found that the ELMFS can handle the evolutionary process of the two-dimensional unsteady Burgers equations in the regular domain. Even in the second problem, the ELMFS will give more accurate results than the FDM. In order to demonstrate the flexibility of the meshless method, the computational domain is chosen as an irregular one as shown in Fig. 13(a). The Burgers equations in such an irregular domain are difficult to be handled by meshdependent methods, such as FDM or FEM. The analytical solution is the same as the one which is used in problem 2. The initial and boundary conditions are taken from the analytical solutions. The u, v results and absolute errors for Re ¼ 100 are present in Figs. 14 and 15, respectively. The fronts moved in the same direction as we expected and the absolute errors also moved with that front. To examine Fig. 14. Velocity profiles of problem 3 at different time levels (Re ¼ 100, Dt ¼ 0.005, N ¼ 364). (a) t ¼ 0.01; (b) t ¼ 0.50; (c) t ¼ 2.00.

15 D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) Fig. 15. Error profiles of problem 3 at different time levels (Re ¼ 100, Dt ¼ 0.005, N ¼ 364). (a) t ¼ 0.01; (b) t ¼ 0.50; (c) t ¼ more seriously, the velocities and absolute errors at some specific points, which are drawn in Fig. 13(b), are recorded in Tables 5 and 6 for Re ¼ 10 and 100, respectively. Those results are very accurate even in an irregular domain by inspecting these solutions in the tables. The proposed ELMFS can render the correct results in an irregular domain even using very coarse collocating points, and then it is proven that this method is a simple, stable and accurate scheme due to the features of meshless method. 5. Conclusions and discussions The unsteady nonlinear two-dimensional Burgers equations are analyzed by the ELMFS which is the combination of the ELM and the MFS. The two-dimensional quasilinear Burgers equations are converted to the characteristic diffusion equations by the ELM, and then the MFS is applied to the diffusion equations. Finally, the solutions of the Burgers equations can be obtained by performing the back-tracking scheme through the characteristics. The proposed numerical scheme, which is free from mesh generation and numerical integration, is a truly meshless method. Therefore, it is very easy to simulate the nonlinear Burgers problem in irregular domain with or without the disturbances of noisy initial and boundary data. In addition the unsteady MFS is applied in time space united system, so Laplace transform or difference discretization for time domain is not needed. Furthermore, through this

16 410 D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) Table 5 Numerical solutions of (a) u and (b) v at different time levels in some specific points of problem 3 (Re ¼ 10, Dt ¼ 0.005, N ¼ 120) x y t ¼ 0.01 t ¼ 0.5 t ¼ 2.0 Analytical solution ELMFS ERROR Analytical solution ELMFS ERROR Analytical solution ELMFS ERROR (a) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 04 (b) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 04 process we are able to extend the time-dependent MFS to solve nonlinear partial differential equations. In this article, the two-dimensional Burgers equations in one and two variables are analyzed by the proposed meshless method and the ELMFS analysis compares very well with the analytical solutions and FDM results. Hence, it is convinced that the proposed method could provide a simple, robust and reliable numerical tool for Burgers equations. Although the proposed ELMFS can be easily used to deal with the nonlinear Burgers equations, there are still some issues which have to be addressed at this stage. One of the issues is the stability of ELMFS which means the determinations of the time increment and the temporal location of the source points. Roughly speaking, the time increment of the ELMFS will be determined by a compromise of accurate schemes between the ELM (higher-order finite difference scheme will surely improve the accuracy) and the MFS (very accurate method). So there is no stability but only accuracy problem in the ELMFS. On the other hand, the temporal location of the source points can be settled by the proposed empirical formula. More detailed numerical and theoretical study of the stability and accuracy will be performed in the near future. Another relevant issue of ELMFS is the comparison of efficiency between the proposed method and conventional numerical methods. Though the proposed method outperforms conventional methods in the issue of mesh generation and numerical quadrature, and high powers to get accurate nonlinear solutions for dealing with the irregular domains and initial and boundary noise data by using very coarse collocating points, the full-populated matrices solvers are crucial for promoting efficiency of the ELMFS. It is too premature to draw a solid conclusion now to take into considerations of so many issues discussed above. Perhaps a thorough study to find an efficient matrices solver is of paramount importance to the popularity of the ELMFS and this investigation deserves more intensive research.

17 D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) Table 6 Numerical solutions of (a) u and (b) v at different time levels in some specific points of problem 3 (Re ¼ 100, Dt ¼ 0.005, N ¼ 364) x y t ¼ 0.01 t ¼ 0.5 t ¼ 2.0 Analytical solution ELMFS ERROR Analytical solution ELMFS ERROR Analytical solution ELMFS ERROR (a) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 04 (b) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 03 Acknowledgment The National Science Council of Taiwan is greatly appreciated for providing financial support of this research under Grant no. NSC E Prof. DL Young also likes to thank the support of the University of California at Irvine when he was a visiting scholar. References [1] Bateman H. Some recent researches on the motion of fluids. Mon Weather Rev 1915;43: [2] Burgers JM. A mathematical model illustrating the theory of turbulence. In: von Mises RV, von Karman TV, editors. Advanced in applied mechanics. New York: Academic Press; [3] Fletcher CAJ. Burgers equation: a model for all reasons. In: Noye J, editor. Numerical solutions of partial differential equations. New York: North-Holland Pub Co; [4] Cole JD. On a quasi-linear parabolic equation occurring in aerodynamics. Q Appl Math 1951;19: [5] Hopf E. The partial differential equation u t þ uu x ¼ m xx. Commun Pure Appl Math 1950;3: [6] Benton ER, Platzman GW. A table of solutions of the onedimensional Burgers equation. Q Appl Math 1972;30: [7] Fletcher CAJ. Generating exact solutions of the two-dimensional Burgers equation. Int J Numer Methods Fluids 1983;3: [8] Bahadir AR. A fully implicit finite-difference scheme for twodimensional Burgers equation. Appl Math Comput 2003;137: [9] Kutluay S, Bahadir AR, Ozdes A. Numerical solution of onedimensional Burgers equation: explicit and exact-explicit finite difference methods. J Comput Appl Math 1999;103: [10] Radwan SF. Comparison of high-order accurate schemes for solving the two-dimensional unsteady Burgers equation. J Comput Appl Math 2005;103: [11] Froncioni AM, Labbe P, Garon A, Camarero R. Interpolation-free space time remeshing for the Burgers equation. Commun Numer Methods Eng 1997;13: [12] Kutluay S, Esen A, Dag I. Numerical solutions of the Burgers equation by the least-squares quadratic B-spline finite element method. J Comput Appl Math 2004;167: [13] Chino E, Tosaka N. Dual reciprocity boundary element analysis of time-independent Burgers equation. Eng Anal Bound Elem 1998;21: [14] Kakuda K, Tosaka N. The generalized boundary element approach to Burgers equation. Int J Numer Methods Eng 1990;29:

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