ARTICLE IN PRESS
Engineering Analysis with Boundary Elements 32 (2008) 395–412
www.elsevier.com/locate/enganabound
The Eulerian–Lagrangian method of fundamental solutions for
two-dimensional unsteady Burgers’ equations
D.L. Younga,, C.M. Fana, S.P. Hua, S.N. Atlurib
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: +886 2 23626114.
E-mail address: dlyoung@ntu.edu.tw (D.L. Young).
0955-7997/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.enganabound.2007.08.011
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
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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
(n+1) Δt
Y
(n) Δt
(n-λ) Δt
C
A
X
(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.
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397
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 9.00.
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.
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D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) 395–412
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.
ARTICLE IN PRESS
3.29E05
4.77E05
6.38E04
5.43E05
2.99E04
4.77E05
6.45E04
2.13E03
2.99E04
3.03E03
6.38E04
2.13E03
5.04E05
1.0000
0.9986
0.9248
0.9986
0.9238
0.9986
0.9235
0.1803
0.9238
0.1794
0.9248
0.1803
0.0040
1.0000
0.9985
0.9241
0.9985
0.9241
0.9985
0.9241
0.1824
0.9241
0.1824
0.9241
0.1824
0.0041
6.08E06
1.95E04
9.45E04
9.11E05
1.75E03
1.95E04
2.99E03
1.18E04
1.75E03
1.82E04
9.45E04
1.18E04
1.96E06
0.9997
0.9820
0.5000
0.9820
0.5000
0.9820
0.5000
0.0180
0.5000
0.0180
0.5000
0.0180
0.0003
(2)
subject to the initial conditions:
uðx; y; t0 Þ ¼ f1 ðx; yÞ
ðx; yÞ 2 O,
(3)
vðx; y; t0 Þ ¼ f2 ðx; yÞ
ðx; yÞ 2 O
(4)
0.9959
0.8176
0.0759
0.8176
0.0759
0.8176
0.0759
0.0015
0.0759
0.0015
0.0759
0.0015
0.0000
1.70E04
1.05E03
6.60E05
7.88E05
2.55E04
1.05E03
5.18E04
3.93E06
2.55E04
3.16E06
6.60E05
3.93E06
2.40E07
qv
qv
qv
1 q2 v q2 v
þu þv ¼
þ
qt
qx
qy Re qx2 qy2
Luðx; y; tÞ ¼ f3 ðx; y; tÞ
6.07E04
4.66E04
1.59E06
3.79E04
9.98E07
4.66E04
8.07E06
2.00E07
9.98E07
1.95E08
1.59E06
2.00E07
3.92E08
0.9961
0.8186
0.0759
0.8177
0.0756
0.8186
0.0753
0.0015
0.0756
0.0015
0.0759
0.0015
0.0000
0.9997
0.9822
0.5009
0.9821
0.4983
0.9822
0.4970
0.0179
0.4983
0.0178
0.5009
0.0179
0.0003
Analytical solution
|ERROR|
ELMFS
Analytical solution
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
qu
qu
1 q2 u q2 u
þu þv ¼
þ
,
(1)
qt
qx
qy Re qx2 qy2
Analytical solution
|ERROR|
2. Governing equations
|ERROR|
ELMFS
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.
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.
f1 ðx; yÞ, f2 ðx; yÞ, f3 ðx; y; tÞ and f4 ðx; y; tÞ are all known
functions. Re is the Reynolds number, and t0 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.
ELMFS
0.9532
0.2694
0.0067
0.2686
0.0067
0.2694
0.0067
0.0001
0.0067
0.0001
0.0067
0.0001
0.0000
0.9526
0.2689
0.0067
0.2689
0.0067
0.2689
0.0067
0.0001
0.0067
0.0001
0.0067
0.0001
0.0000
and the boundary conditions:
Analytical solution
t ¼ 1.00
t ¼ 0.75
t ¼ 0.50
399
ðx; yÞ 2 qO;
(5)
Lvðx; y; tÞ ¼ f4 ðx; y; tÞ
ðx; yÞ 2 qO;
(6)
0.1
0.1
0.1
0.3
0.3
0.5
0.5
0.5
0.7
0.7
0.9
0.9
0.9
0.1
0.5
0.9
0.3
0.7
0.1
0.5
0.9
0.3
0.7
0.1
0.5
0.9
y
3. Numerical method
x
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)
t ¼ 1.25
ELMFS
|ERROR|
D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) 395–412
The two-dimensional Burgers’ equations, Eqs. (1) and (2),
can be transferred to the following two characteristic
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400
diffusion equations using the ELM:
Du
1 q2 u q2 u
¼
,
þ
Dt Re qx2 qy2
Dv
1 q2 v q2 v
¼
þ
,
Dt Re qx2 qy2
(7)
(8)
where the total or material derivative including the
convective term is defined as [25,30]
D
q
q
q
¼ þu þv .
Dt qt
qx
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
1 2
¼
r Gð~
x; t; ~
x; tÞ þ dð~
x ~
xÞdðt tÞ,
qt
Re
(10)
where Gð~
x; t; ~
x; tÞ is the fundamental solution of the linear
diffusion equation. ~
x ¼ ðx; yÞ and ~
x ¼ ðx; 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.
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D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) 395–412
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
401
Eq. (11), as
uð~
x; tÞ ¼
N
X
aj Gð~
x; t; ~
xj ; tj Þ,
(12)
bj Gð~
x; t; ~
xj ; tj Þ,
(13)
j¼1
j~
x~
xj2 =½4ð1=ReÞðttÞ
e
Gð~
x; t; ~
x; tÞ ¼ d=2 Hðt tÞ,
4pð1=ReÞðt tÞ
(11)
vð~
x; tÞ ¼
N
X
j¼1
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,
where N is the number of source point. aj and bj 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.
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D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) 395–412
½Au fag ¼ ½f u ,
(14)
½Av fbg ¼ ½f v .
(15)
The components of [Au] and [Av] are the representation
of the fundamental solutions. ff u g is the combination of
f1 ðx; yÞ and f3 ðx; y; tÞ, and ff v g is the combination of
f2 ðx; yÞ and f4 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 ¼ ðn þ 1ÞDt can 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:
u¼
dx xnþ1 xn
¼
,
dt
Dt
nþ1
v¼
n
dy y
y
¼
,
dt
Dt
n
nþ1
n
nþ1
x ¼x
y ¼y
u Dt,
v Dt.
(16)
(17)
(18)
(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 ¼ ðn þ 1ÞDt thus can be acquired.
This procedure can be repeated until either the terminal
time or steady-state solution is achieved.
1.0E-001
Maximum absolute error
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 ¼ mR.
By observing the diffusion fundamental solution, it is noted
that the temporal difference (tt) 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):
1.0E-002
1.0E-003
u
Δt = 0.01
v
Δt = 0.01
Δt = 0.05
Δt = 0.05
Δt = 0.005
Δt = 0.005
1.0E-004
1.0E-005
0
0.5
1
1.5
2
2.5
Time
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).
1.0E+000
1.0E-001
Maximum absolute error
402
1.0E-002
1.0E-003
u
1.0E-004
v
λ (Δt)=0.5R
λ (Δt)=0.5R
λ (Δt)=1R
λ (Δt)=1R
λ (Δt)=5R
λ (Δt)=5R
λ (Δt)=10R
λ (Δt)=10R
1.0E-005
0
0.4
0.8
1.2
1.6
2
Time
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).
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4. Numerical results
and the analytical solution is [16]
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.
uðx; y; tÞ ¼
4.1. Problem 1
The first validation problem is the unsteady Burgers’
equation in one variable which is described as below:
qu
qu
qu
1 q2 u q2 u
þu þu ¼
þ
(20)
qt
qx
qy Re qx2 qy2
403
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)
0.10
0.50
0.90
0.30
0.70
0.10
0.50
0.90
0.30
0.70
0.10
0.50
0.90
0.10
0.10
0.10
0.30
0.30
0.50
0.50
0.50
0.70
0.70
0.90
0.90
0.90
0.62481
0.59420
0.56708
0.62481
0.59420
0.65543
0.62480
0.59420
0.65543
0.62480
0.68261
0.65543
0.62480
0.62481
0.59420
0.56708
0.62480
0.59420
0.65543
0.62480
0.59420
0.65543
0.62480
0.68261
0.65543
0.62481
3.59E07
2.05E07
1.53E07
3.26E07
1.81E07
3.04E07
2.96E07
1.75E07
3.15E07
2.76E07
3.13E07
3.10E07
3.07E07
0.61525
0.58540
0.55984
0.61525
0.58540
0.64628
0.61525
0.58540
0.64628
0.61525
0.67481
0.64628
0.61525
0.61526
0.58540
0.55984
0.61526
0.58540
0.64628
0.61527
0.58540
0.64629
0.61527
0.67482
0.64629
0.61526
3.10E06
3.57E06
2.15E06
7.62E06
6.44E06
5.38E06
1.14E05
5.91E06
1.13E05
1.24E05
4.77E06
1.02E05
6.57E06
0.58716
0.56127
0.54113
0.58716
0.56127
0.61720
0.58716
0.56127
0.61720
0.58716
0.64817
0.61720
0.58716
0.58716
0.56127
0.54113
0.58717
0.56128
0.61720
0.58717
0.56128
0.61721
0.58717
0.64817
0.61721
0.58717
2.63E06
2.80E06
1.97E06
6.02E06
4.98E06
5.17E06
9.83E06
5.12E06
1.10E05
1.22E05
5.26E06
1.09E05
6.73E06
(b)
0.10
0.50
0.90
0.30
0.70
0.10
0.50
0.90
0.30
0.70
0.10
0.50
0.90
0.10
0.10
0.10
0.30
0.30
0.50
0.50
0.50
0.70
0.70
0.90
0.90
0.90
0.87520
0.90580
0.93292
0.87520
0.90580
0.84457
0.87520
0.90580
0.84457
0.87520
0.81739
0.84457
0.87520
0.87520
0.90580
0.93292
0.87520
0.90580
0.84457
0.87519
0.90580
0.84457
0.87519
0.81739
0.84457
0.87520
1.98E07
1.82E07
4.56E09
2.74E07
2.24E07
3.73E07
3.13E07
2.32E07
3.69E07
3.42E07
2.29E07
3.61E07
2.83E07
0.88475
0.91460
0.94016
0.88475
0.91460
0.85372
0.88475
0.91460
0.85372
0.88475
0.82519
0.85372
0.88475
0.88475
0.91461
0.94016
0.88474
0.91460
0.85372
0.88474
0.91460
0.85372
0.88474
0.82519
0.85372
0.88474
9.89E07
6.70E07
1.39E06
3.35E06
2.90E06
1.66E06
7.58E06
3.21E06
7.51E06
9.96E06
6.48E07
7.91E06
4.23E06
0.91284
0.93873
0.95887
0.91284
0.93873
0.88280
0.91284
0.93873
0.88280
0.91284
0.85183
0.88280
0.91284
0.91284
0.93873
0.95887
0.91284
0.93873
0.88280
0.91283
0.93873
0.88280
0.91283
0.85183
0.88280
0.91284
1.71E06
1.73E06
2.22E06
1.06E06
4.00E07
2.98E07
4.37E06
9.19E07
6.12E06
7.09E06
4.19E07
6.89E06
2.69E06
ARTICLE IN PRESS
D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) 395–412
404
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
1
,
4 4 1 þ expðð4x þ 4y tÞðRe=32ÞÞ
(22)
vðx; y; tÞ ¼
3
1
.
þ 4 4 1 þ expðð4x þ 4y tÞðRe=32ÞÞ
(23)
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 11 11
N ¼ 21 21
N ¼ 21 21
ELMFS
|ERROR|
ELMFS
|ERROR|
FDM [8]
|ERROR|
t ¼ 0.01
0.1
0.5
0.9
0.3
0.7
0.1
0.5
0.9
0.3
0.7
0.1
0.5
0.9
0.1
0.1
0.1
0.3
0.3
0.5
0.5
0.5
0.7
0.7
0.9
0.9
0.9
0.62305
0.50162
0.50001
0.62305
0.50162
0.74827
0.62305
0.50162
0.74827
0.62305
0.74999
0.74827
0.62305
0.62323
0.50141
0.50035
0.62306
0.50201
0.74863
0.62321
0.50143
0.74774
0.62302
0.74969
0.74854
0.62305
1.87E04
2.11E04
3.36E04
1.79E05
3.84E04
3.59E04
1.60E04
1.95E04
5.39E04
2.56E05
2.94E04
2.71E04
1.04E06
0.62229
0.50154
0.49995
0.62308
0.50164
0.74826
0.62307
0.50159
0.74827
0.62308
0.74992
0.74826
0.62269
7.58E04
8.37E05
6.01E05
2.98E05
2.17E05
1.27E05
2.68E05
3.68E05
7.19E06
3.53E05
6.74E05
1.29E05
3.62E04
0.62310
0.50161
0.50000
0.62311
0.50162
0.74827
0.62311
0.50162
0.74827
0.62311
0.74998
0.74827
0.62311
5.30E05
1.21E05
1.10E05
6.30E05
2.07E06
4.04E06
6.30E05
2.07E06
4.04E06
6.30E05
8.29E06
4.04E06
6.30E05
t ¼ 0.5
0.1
0.5
0.9
0.3
0.7
0.1
0.5
0.9
0.3
0.7
0.1
0.5
0.9
0.1
0.1
0.1
0.3
0.3
0.5
0.5
0.5
0.7
0.7
0.9
0.9
0.9
0.54332
0.50035
0.50000
0.54332
0.50035
0.74221
0.54332
0.50035
0.74221
0.54332
0.74995
0.74221
0.54332
0.53566
0.50150
0.50432
0.54905
0.50193
0.73665
0.54347
0.50025
0.74263
0.54331
0.75128
0.74218
0.54338
7.66E03
1.14E03
4.32E03
5.73E03
1.58E03
5.56E03
1.51E04
9.80E05
4.14E04
1.08E05
1.34E03
3.82E05
5.66E05
0.54241
0.50024
0.49999
0.54427
0.50030
0.74222
0.54366
0.50030
0.74220
0.54367
0.74991
0.74230
0.54377
9.12E04
1.17E04
7.65E06
9.48E04
5.40E05
4.84E06
3.39E04
5.60E05
1.51E05
3.45E04
3.92E05
8.93E05
4.44E04
0.54235
0.49964
0.49931
0.54207
0.49961
0.74130
0.54222
0.49997
0.74145
0.54243
0.74913
0.74201
0.54232
9.72E04
7.13E04
6.92E04
1.25E03
7.43E04
9.14E04
1.10E03
3.83E04
7.64E04
8.92E04
8.16E04
2.04E04
1.00E03
t ¼ 2.0
0.1
0.5
0.9
0.3
0.7
0.1
0.5
0.9
0.3
0.7
0.1
0.5
0.9
0.1
0.1
0.1
0.3
0.3
0.5
0.5
0.5
0.7
0.7
0.9
0.9
0.9
0.50048
0.50000
0.50000
0.50048
0.50000
0.55568
0.50048
0.50000
0.55568
0.50048
0.74426
0.55568
0.50048
0.49845
0.50142
0.50201
0.49020
0.49589
0.55469
0.49774
0.49878
0.56310
0.49998
0.74114
0.55848
0.50063
2.03E03
1.41E03
2.01E03
1.03E02
4.11E03
9.86E04
2.74E03
1.22E03
7.42E03
4.98E04
3.12E03
2.81E03
1.44E04
0.50012
0.49996
0.49995
0.50042
0.49999
0.55516
0.50041
0.49999
0.55587
0.50045
0.74416
0.55637
0.50051
3.59E04
3.95E05
4.57E05
6.05E05
1.53E05
5.15E04
7.31E05
1.18E05
1.95E04
3.45E05
9.21E05
6.95E04
2.69E05
0.49983
0.49930
0.49930
0.49977
0.49930
0.55461
0.49973
0.49931
0.55429
0.49970
0.74340
0.55413
0.50001
6.52E04
7.03E04
7.00E04
7.12E04
7.03E04
1.07E03
7.52E04
6.93E04
1.39E03
7.82E04
8.56E04
1.55E03
4.72E04
ARTICLE IN PRESS
D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) 395–412
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
pffiffiffi ¼ mR. 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
405
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 11 11
N ¼ 21 21
N ¼ 21 21
ELMFS
|ERROR|
ELMFS
|ERROR|
FDM [8]
|ERROR|
t ¼ 0.01
0.1
0.5
0.9
0.3
0.7
0.1
0.5
0.9
0.3
0.7
0.1
0.5
0.9
0.1
0.1
0.1
0.3
0.3
0.5
0.5
0.5
0.7
0.7
0.9
0.9
0.9
0.87695
0.99838
0.99999
0.87695
0.99838
0.75173
0.87695
0.99838
0.75173
0.87695
0.75001
0.75173
0.87695
0.87678
0.99860
0.99966
0.87694
0.99799
0.75137
0.87679
0.99857
0.75227
0.87698
0.75031
0.75145
0.87695
1.75E04
2.20E04
3.28E04
1.72E05
3.84E04
3.52E04
1.59E04
1.95E04
5.39E04
2.53E05
2.99E04
2.73E04
2.96E06
0.87750
0.99836
0.99988
0.87694
0.99837
0.75164
0.87692
0.99836
0.75175
0.87694
0.74993
0.75170
0.87723
5.51E04
1.84E05
1.06E04
1.56E05
5.37E06
8.13E05
2.92E05
1.46E05
2.46E05
1.52E05
7.84E05
2.97E05
2.74E04
0.87688
0.99837
0.99998
0.87689
0.99838
0.75172
0.87689
0.99838
0.75173
0.87689
0.75001
0.75173
0.87689
7.30E05
7.93E06
9.00E06
6.30E05
2.07E06
5.96E06
6.30E05
2.07E06
4.04E06
6.30E05
1.71E06
4.04E06
6.30E05
t ¼ 0.5
0.1
0.5
0.9
0.3
0.7
0.1
0.5
0.9
0.3
0.7
0.1
0.5
0.9
0.1
0.1
0.1
0.3
0.3
0.5
0.5
0.5
0.7
0.7
0.9
0.9
0.9
0.95668
0.99965
1.00000
0.95668
0.99965
0.75779
0.95668
0.99965
0.75779
0.95668
0.75005
0.75779
0.95668
0.96467
0.99878
0.99592
0.95127
0.99834
0.76366
0.95667
0.99990
0.75763
0.95669
0.74897
0.75786
0.95661
7.99E03
8.67E04
4.07E03
5.41E03
1.30E03
5.87E03
3.55E06
2.52E04
1.58E04
1.61E05
1.09E03
6.92E05
6.97E05
0.95717
0.99952
0.99974
0.95551
0.99952
0.75748
0.95621
0.99957
0.75760
0.95632
0.74984
0.75769
0.95630
4.93E04
1.24E04
2.60E04
1.16E03
1.23E04
3.01E04
4.73E04
8.09E05
1.86E04
3.58E04
2.15E04
9.33E05
3.79E04
0.95577
0.99827
0.99861
0.95596
0.99827
0.75699
0.95685
0.99903
0.75723
0.95746
0.74924
0.75781
0.95777
9.08E04
1.38E03
1.39E03
7.18E04
1.38E03
7.96E04
1.72E04
6.17E04
5.56E04
7.82E04
8.14E04
2.40E05
1.09E03
t ¼ 2.0
0.1
0.5
0.9
0.3
0.7
0.1
0.5
0.9
0.3
0.7
0.1
0.5
0.9
0.1
0.1
0.1
0.3
0.3
0.5
0.5
0.5
0.7
0.7
0.9
0.9
0.9
0.99952
1.00000
1.00000
0.99952
1.00000
0.94432
0.99952
1.00000
0.94432
0.99952
0.75574
0.94432
0.99952
1.00191
0.99902
0.99845
1.01020
1.00454
0.94571
1.00268
1.00164
0.93727
1.00043
0.75928
0.94188
0.99977
2.40E03
9.72E04
1.55E03
1.07E02
4.54E03
1.38E03
3.16E03
1.64E03
7.06E03
9.14E04
3.54E03
2.45E03
2.49E04
0.99946
0.99980
0.99978
0.99938
0.99984
0.94450
0.99941
0.99984
0.94387
0.99937
0.75558
0.94345
0.99938
6.02E05
1.95E04
2.18E04
1.42E04
1.56E04
1.70E04
1.07E04
1.56E04
4.56E04
1.47E04
1.66E04
8.72E04
1.42E04
0.99826
0.99860
0.99861
0.99820
0.99860
0.94393
0.99821
0.99862
0.94409
0.99823
0.75500
0.94441
0.99846
1.26E03
1.40E03
1.39E03
1.32E03
1.40E03
3.95E04
1.31E03
1.38E03
2.35E04
1.29E03
7.44E04
8.50E05
1.06E03
ARTICLE IN PRESS
D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) 395–412
406
0.64
Analytical solution
0.60
1.0E-002
u (0.5,0.5,t)
Maximum absolute error
1.0E-001
1.0E-003
u (k=1%)
ELMFS
[TSVD] tol=10-5
0.56
-6
[TSVD] tol=10
-8
[TSVD] tol=10
u
k=10-2
k=10-3
k=10-4
k=10-5
1.0E-004
[DLSQRR] tol=10
-8
0.52
k=0
1.0E-005
0.48
0
0.4
0.8
1.2
1.6
2
0
0.4
0.8
Time
1.6
2
Time
1.0E-001
1.04
Analytical solution
1.00
1.0E-002
v (0.5,0.5,t)
Maximum absolute error
1.2
1.0E-003
v
k=10-2
k=10-3
k=10-4
k=10-5
k=0
1.0E-004
0.96
u (k=1%)
ELMFS
[TSVD] tol=10-5
[TSVD] tol=10-6
0.92
-8
[TSVD] tol=10
[DLSQRR] tol=10-8
0.88
1.0E-005
0
0.4
0.8
1.2
1.6
2
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 21 21. 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:
3
1
uðx; y; 0Þ ¼ ð1 þ kÞ 4 4½1 þ expðð4x þ 4yÞðRe=32ÞÞ
ðx; yÞ 2 O;
0.84
0
Time
ð24Þ
0.4
0.8
1.2
1.6
2
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).
vðx; y; 0Þ ¼ ð1 þ kÞ
ðx; yÞ 2 O;
3
1
þ
4 4½1 þ expðð4x þ 4yÞðRe=32ÞÞ
ð25Þ
3
1
uðx; y; tÞ ¼ ð1 þ kÞ 4 4½1 þ expðð4x þ 4y tÞ=ðRe=32ÞÞ
ðx; yÞ 2 qO;
ð26Þ
3
1
vðx; y; tÞ ¼ ð1 þ kÞ þ
4 4½1 þ expðð4x þ 4y tÞ=ðRe=32ÞÞ
ðx; yÞ 2 qO;
ð27Þ
ARTICLE IN PRESS
D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) 395–412
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
105 to 102. The maximum absolute errors of u and v are
shown in Fig. 10 for different k. When the amplitude of k is
smaller than 102, 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-001
1.0E-002
407
can be used to successfully analyze problems with
moderate noise level up to k ¼ 102 . 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 102 . 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
1.0E-003
u
1
k=1%
1.0E-004
[TSVD]
tol=10-4
[TSVD]
tol=10-5
[TSVD]
tol=10-6
[TSVD]
tol=10-8
[TSVD]
Ω
r2
r1
0
-10
tol=10
c2
[DLSQRR] tol=10-8
c1
Γ
1.0E-005
0
0.4
0.8
1.2
1.6
2
Time
-1
-2.8
Maximum absolute error
1.0E-001
-2
1
1.0E-002
1
2
1.0E-003
6
7
v
k=1%
1.0E-004
tol=10
[TSVD]
tol=10-5
[TSVD]
tol=10-6
[TSVD]
tol=10-8
[TSVD]
tol=10-10
0
9
3
1.0E-005
0.4
8
-4
[TSVD]
[DLSQRR] tol=10-8
0
1
-1
0
(r1 = r2 = 1, c1 = (0,0) , c2 = (−1.8,0) )
0.8
1.2
1.6
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).
11
5
14
10
4
2
Time
12
13 14 15
-1
-0.8
0
1
Fig. 13. (a) Computational domain of problem 3 (b) Distribution of some
specific points of problem 3 for comparison.
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408
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.
4.3. 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.
ARTICLE IN PRESS
D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) 395–412
409
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 2.00.
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
ARTICLE IN PRESS
D.L. Young et al. / Engineering Analysis with Boundary Elements 32 (2008) 395–412
410
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)
0.00
0.70
0.70
0.00
0.90
0.00
0.45
0.38
0.45
0.00
0.50
0.00
0.07
0.00
0.08
0.90
0.50
0.50
0.90
0.00
0.50
0.25
0.00
0.25
0.50
0.00
0.08
0.00
0.08
0.00
0.68858
0.70428
0.64035
0.56113
0.56113
0.58698
0.67628
0.65359
0.64035
0.58698
0.58698
0.63105
0.62988
0.61856
0.61856
0.68863
0.70430
0.64041
0.56129
0.56130
0.58700
0.67629
0.65359
0.64037
0.58699
0.58700
0.63107
0.62989
0.61858
0.61858
4.88E05
1.81E05
5.45E05
1.68E04
1.70E04
1.44E05
4.27E06
4.21E06
2.02E05
7.03E06
1.44E05
1.82E05
1.28E05
1.65E05
2.36E05
0.68122
0.69828
0.63086
0.55433
0.55433
0.57851
0.66808
0.64437
0.63086
0.57851
0.57851
0.62149
0.62031
0.60907
0.60907
0.68128
0.69827
0.63114
0.55475
0.55459
0.57859
0.66819
0.64453
0.63113
0.57887
0.57859
0.62157
0.62042
0.60919
0.60917
6.35E05
1.87E06
2.83E04
4.15E04
2.60E04
8.22E05
1.04E04
1.56E04
2.75E04
3.56E04
8.22E05
8.11E05
1.04E04
1.22E04
9.40E05
0.65561
0.67645
0.60183
0.53701
0.53701
0.55568
0.64054
0.61525
0.60183
0.55568
0.55568
0.59292
0.59183
0.58157
0.58157
0.65584
0.67648
0.60212
0.53738
0.53736
0.55613
0.64080
0.61560
0.60221
0.55616
0.55613
0.59339
0.59228
0.58205
0.58207
2.23E04
3.41E05
2.89E04
3.67E04
3.53E04
4.59E04
2.57E04
3.46E04
3.80E04
4.83E04
4.59E04
4.66E04
4.55E04
4.78E04
4.94E04
(b)
0.00
0.70
0.70
0.00
0.90
0.00
0.45
0.38
0.45
0.00
0.50
0.00
0.07
0.00
0.08
0.90
0.50
0.50
0.90
0.00
0.50
0.25
0.00
0.25
0.50
0.00
0.08
0.00
0.08
0.00
0.81142
0.79572
0.85965
0.93887
0.93887
0.91302
0.82372
0.84641
0.85965
0.91302
0.91302
0.86895
0.87012
0.88144
0.88144
0.81138
0.79572
0.85961
0.93872
0.93874
0.91301
0.82372
0.84641
0.85963
0.91301
0.91301
0.86894
0.87011
0.88143
0.88143
3.52E05
7.99E06
4.21E05
1.50E04
1.37E04
8.04E06
8.14E07
2.98E06
1.50E05
5.04E06
8.04E06
1.06E05
7.62E06
9.72E06
1.35E05
0.81878
0.80172
0.86914
0.94567
0.94567
0.92149
0.83192
0.85563
0.86914
0.92149
0.92149
0.87851
0.87969
0.89093
0.89093
0.81873
0.80175
0.86894
0.94532
0.94546
0.92142
0.83183
0.85550
0.86893
0.92117
0.92142
0.87845
0.87959
0.89082
0.89085
5.47E05
2.86E05
2.05E04
3.48E04
2.07E04
6.75E05
8.35E05
1.30E04
2.15E04
3.19E04
6.75E05
6.80E05
9.19E05
1.08E04
7.76E05
0.84439
0.82355
0.89817
0.96299
0.96299
0.94432
0.85946
0.88475
0.89817
0.94432
0.94432
0.90708
0.90817
0.91843
0.91843
0.84419
0.82355
0.89796
0.96269
0.96271
0.94394
0.85924
0.88446
0.89787
0.94391
0.94394
0.90669
0.90779
0.91802
0.91801
1.95E04
5.04E06
2.09E04
2.98E04
2.76E04
3.82E04
2.16E04
2.88E04
3.00E04
4.19E04
3.82E04
3.91E04
3.84E04
4.05E04
4.13E04
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.
ARTICLE IN PRESS
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411
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)
0.00
0.70
0.70
0.00
0.90
0.00
0.45
0.38
0.45
0.00
0.50
0.00
0.07
0.00
0.08
0.90
0.50
0.50
0.90
0.00
0.50
0.25
0.00
0.25
0.50
0.00
0.08
0.00
0.08
0.00
0.75000
0.75000
0.73048
0.50000
0.50000
0.74950
0.74996
0.74765
0.73048
0.50047
0.50047
0.68122
0.67149
0.56571
0.56571
0.74992
0.75001
0.73155
0.49995
0.49999
0.74951
0.74995
0.74765
0.73068
0.50047
0.50048
0.68127
0.67147
0.56573
0.56567
7.56E05
7.01E06
1.07E03
5.45E05
9.02E06
6.85E06
5.36E06
7.59E08
2.03E04
1.29E06
1.74E05
5.67E05
1.39E05
1.83E05
3.90E05
0.74998
0.75000
0.67965
0.50000
0.50000
0.74772
0.74981
0.73948
0.67965
0.50010
0.50010
0.59074
0.58021
0.51790
0.51790
0.75011
0.74979
0.67929
0.50013
0.50004
0.74789
0.74982
0.73990
0.68394
0.50014
0.50008
0.59524
0.58433
0.51827
0.51831
1.29E04
2.12E04
3.60E04
1.28E04
3.73E05
1.75E04
1.00E05
4.22E04
4.29E03
4.32E05
1.65E05
4.49E03
4.13E03
3.77E04
4.14E04
0.74833
0.74996
0.50574
0.50000
0.50000
0.62500
0.73104
0.54332
0.50574
0.50000
0.50000
0.50131
0.50108
0.50018
0.50018
0.74946
0.75184
0.51385
0.49996
0.50082
0.63329
0.73304
0.54915
0.50772
0.49980
0.50087
0.50242
0.50213
0.50176
0.50070
1.13E03
1.88E03
8.10E03
3.74E05
8.17E04
8.29E03
2.00E03
5.82E03
1.98E03
1.99E04
8.67E04
1.12E03
1.05E03
1.58E03
5.22E04
(b)
0.00
0.70
0.70
0.00
0.90
0.00
0.45
0.38
0.45
0.00
0.50
0.00
0.07
0.00
0.08
0.90
0.50
0.50
0.90
0.00
0.50
0.25
0.00
0.25
0.50
0.00
0.08
0.00
0.08
0.00
0.75000
0.75000
0.76952
1.00000
1.00000
0.75050
0.75004
0.75235
0.76952
0.99953
0.99953
0.81878
0.82851
0.93429
0.93429
0.74993
0.74992
0.76834
0.99996
0.99985
0.75051
0.75005
0.75235
0.76932
0.99955
0.99953
0.81873
0.82853
0.93427
0.93433
7.63E05
7.61E05
1.18E03
4.05E05
1.42E04
1.30E05
5.50E06
2.34E08
2.03E04
1.24E05
1.95E06
5.63E05
1.28E05
2.03E05
3.98E05
0.75002
0.75000
0.82035
1.00000
1.00000
0.75228
0.75019
0.76052
0.82035
0.99990
0.99990
0.90926
0.91979
0.98210
0.98210
0.74988
0.75029
0.82075
0.99999
0.99996
0.75214
0.75018
0.76012
0.81616
0.99998
0.99993
0.90478
0.91568
0.98176
0.98169
1.35E04
2.88E04
4.00E04
4.49E06
4.03E05
1.44E04
8.44E06
4.01E04
4.19E03
8.07E05
3.34E05
4.48E03
4.11E03
3.47E04
4.15E04
0.75167
0.75004
0.99426
1.00000
1.00000
0.87500
0.76896
0.95668
0.99426
1.00000
1.00000
0.99869
0.99892
0.99982
0.99982
0.75059
0.74792
0.98637
1.00052
1.00213
0.86745
0.76705
0.95320
0.99471
1.00276
1.00188
1.00030
1.00062
1.00096
1.00200
1.08E03
2.12E03
7.89E03
5.24E04
2.13E03
7.55E03
1.91E03
3.47E03
4.59E04
2.76E03
1.88E03
1.61E03
1.71E03
1.14E03
2.18E03
Acknowledgment
The National Science Council of Taiwan is greatly
appreciated for providing financial support of this research
under Grant no. NSC 94-2611-E-002-007. Prof. DL Young
also likes to thank the support of the University of
California at Irvine when he was a visiting scholar.
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