Engineering Analysis with Boundary Elements 26 (2002) 571–575
www.elsevier.com/locate/enganabound
High-order fundamental and general solutions of convection –diffusion
equation and their applications with boundary particle method
W. Chen
Simula Research Laboratory, P.O. Box 134, 1325 Lysaker, Norway
Received 8 October 2001; revised 22 February 2002; accepted 31 March 2002
Abstract
In this study, we presented the high-order fundamental solutions and general solutions of convection – diffusion equation. To demonstrate
their efficacy, we applied the high-order general solutions to the boundary particle method (BPM) for the solution of some inhomogeneous
convection –diffusion problems, where the BPM is a new truly boundary-only meshfree collocation method based on multiple reciprocity
principle. For the sake of completeness, the BPM is also briefly described here. q 2002 Elsevier Science Ltd. All rights reserved.
Keywords: High-order fundamental solution; High-order general solution; Boundary particle method; Radial basis function; Meshfree; Multiple reciprocityBEM
1. Introduction
The high-order fundamental solutions play a pivotal role
in the multiple reciprocity-BEM (MR-BEM) [1]. On the
other hand, the boundary particle method (BPM) recently
introduced by the present author [2], also requires the use of
the high-order general solutions. In addition, an underlying
relationship between the radial basis function (RBF) and
Green integral identity highlights the importance of these
operator-dependent solutions [3]. A kernel RBF-creating
strategy [3,4], which employs these high-order fundamental
or general solutions as the RBF, is presented for the dual
reciprocity BEM [5], boundary knot method [6], and
Kansa’s method [7].
To the author’s best knowledge, the explicit high-order
fundamental solutions are now available only for the Laplace
[1], Helmholtz and modified Helmholtz operators [8]. Based
on the underlying relationship between the convection–
diffusion equation and the modified Helmholtz equation [9],
we derive the high-order fundamental and general solutions
of the convection –diffusion equation [10]. In this study, we
will give a systematic description of this recent advance. In
addition, to demonstrate the efficacy of these high-order
solutions, we apply them to the BPM for the solution of
some inhomogeneous convection – diffusion problems.
The BPM is developed based on the multiple reciprocity
principle [2,10], which uses either high-order nonsingular
E-mail address: [email protected] (W. Chen).
general solutions or singular fundamental solutions to
evaluate high-order homogeneous solutions, and then their
sum approximates the particular solutions. Like the MRBEM, the BPM does not require any inner nodes for
inhomogeneous problems and therefore is a truly boundaryonly technique. However, unlike the MR-BEM, the BPM
does not need to generate more than one interpolation
matrix which tremendously reduces computing effort and
memory requirements. In addition, the BPM is essentially
meshfree, spectral convergence, easy-to-use, and symmetric
technique.
The rest of this paper is organized into three sections. We
discuss the difference between the singular fundamental
solution and nonsingular general solution and then introduce the high-order fundamental and general solutions of
convection –diffusion equation in Section 2. The BPM is
briefly described in Section 3, and followed by numerical
testing of the convection – diffusion high-order general
solutions with the BPM to some 2D inhomogeneous
convection – diffusion problems under complicated geometries in Section 4.
2. High-order fundamental and general solutions of
convection – diffusion operator
The general solution u # satisfies
R{u# } ¼ 0;
0955-7997/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 9 5 5 - 7 9 9 7 ( 0 2 ) 0 0 0 2 6 - 7
ð1Þ
572
W. Chen / Engineering Analysis with Boundary Elements 26 (2002) 571–575
where R is a differential operator. The general solution at
origin is equal to a limited value rather than zero and infinity.
In contrast, the fundamental solution u p has to satisfy
R{up } þ Di ¼ 0;
ð2Þ
where Di represents the Dirac delta function which goes to
infinity at the origin point xi and is equal to zero elsewhere.
According to the above definitions, the essential distinctness
between the general and fundamental solution of a differential
operator is that the former is nonsingular, while the latter is
singular.
The above high-order general solutions of the convection– diffusion equation are also validated by mathematical
deduction approach with the help of computer algebraic
package ‘Maple’. Namely, first we prove the zero-order
general solution, and then, it is found that R{u#m } consists
only of ðm 2 1Þth or lower-order general solutions of
operator R. Therefore, the m-order general solution is
proved.
2.2. High-order fundamental solutions
2.1. High-order general solutions
The fundamental solution of steady convection –
diffusion equation is determined by
Consider a homogeneous isotropic region V. The steady
convection –diffusion equation is given by
D72 uðxÞ 2 vk·7uðxÞ 2 kuðxÞ ¼ 2Di ;
D72 uðxÞ 2 kv·7uðxÞ 2 kuðxÞ ¼ 0;
where Di represents the Dirac delta function at a source
point i. Following the preceding procedure for high-order
general solutions, we can easily derive the m-order
fundamental solution
in V;
ð3Þ
where kv denotes a velocity vector, D is the diffusivity
coefficient, k represents the reaction coefficient. The variable
u can be interpreted as temperature for heat transfer
problems, concentration for dispersion problems [5].
Assuming that convective velocity kv and diffusivity D are
constant, we have an exponential variable transformation
[9]
nk·kr
uðxÞ ¼ exp
wðxÞ;
ð4Þ
2D
where wðxÞ is an intermediate field variable, and kr is the
distance vector between the source and field points.
Convection – diffusion equation (Eq. (3)) can be accordingly
rewritten as a modified Helmholtz equation
7 2 w 2 t2 w ¼ 0
ð5Þ
in V;
where
in V;
upm ðrÞ ¼ Am eðkv·krÞ=2D ðtrÞ2n=2þ1þm Kn=221þm ðtrÞ;
ð9Þ
n $ 2;
ð10Þ
where K represents the modified Bessel function of the
second kind. As in the modified Helmholtz case, the highorder fundamental solution of the convection– diffusion
equation has the singularity order of ðr 22n Þ except the 2D
case where the only singularity occurs in the zero-order
fundamental solution. For instance, the singularity for 3D
case is always r 21 irrespective of the order of fundamental
solution. In other words, the singularity order has nothing to
do with the order of a fundamental solution.
2.3. Evaluating fundamental and general solutions
2
t ¼ ½ðlkvl=2DÞ þ k=D
1=2
ð6Þ
:
According to Itagaki’s high-order fundamental solutions of
the modified Helmholtz equation [8], it is very easy to get
the corresponding higher-order general solutions, i.e.
w#m ðrÞ ¼ Am ðtrÞ2n=2þ1þm In=221þm ðtrÞ;
n $ 2;
ð7Þ
2
where Am ¼ Am21 =ð2mt Þ; A0 ¼ 1; n is the dimension of the
problem; m denotes the order of general solution; r is the
Euclidean distance norm; I represents the modified Bessel
function of the first kind, which can be transformed to the
modified spherical Bessel function when n ¼ 3: We found
that those higher-order fundamental solutions are also
established for more than three dimensions.
Through an inverse transformation process, we derive the
high-order general solutions of the convection –diffusion
equation
u#m ðrÞ ¼ Am eðkv·krÞ=2D ðtrÞ2n=2þ1þm In=221þm ðtrÞ;
n $ 2;
ð8Þ
where t is defined by formula (6).
It is worth pointing out that we can greatly simplify the
above-given standard forms of high-order solutions involving some Bessel functions. Thus, the computing effort for
them could be trivial. For example, the zero-, first-, and
second-order general solutions of 3D convection– diffusion
equation are, respectively, simplified via computer algebra
package ‘Maple’ as
pffiffi
A0 2 ðvk·krÞ=2D sinhðtrÞ
#
u0 ðrÞ ¼ pffiffi e
;
ð11Þ
p
tr
pffiffi
A1 2 ðkv·krÞ=2D r coshðtrÞ 2 sinhðtrÞ
ffiffi e
;
u#1 ðrÞ ¼ p
p
tr
u#2 ðrÞ
ð12Þ
pffiffi
A2 2 ðkv·krÞ=2D r2 sinhðtrÞ þ 3 sinhðtrÞ 2 3r coshðtrÞ
;
¼ pffiffi e
p
tr
ð13Þ
where sinh and cosh, respectively, denote the sinehyperbolic and cosine-hyperbolic functions.
W. Chen / Engineering Analysis with Boundary Elements 26 (2002) 571–575
3. Boundary particle methods
In Section 4, the above-derived high-order general
solutions of the convection –diffusion equation are applied
with the BPM to some 2D inhomogeneous problems. For
the completeness and a very recent origin of the BPM, we
give a brief description of the method in this section.
Consider a steady convection – diffusion system
D72 uðxÞ 2 vk·7uðxÞ 2 kuðxÞ ¼ f ðxÞ;
uðxÞ ¼ RðxÞ;
x , Su ;
›uðxÞ
¼ NðxÞ;
›n
x , ST ;
x [ V;
ð14Þ
ð15Þ
ð16Þ
where n is the unit outward normal. The system solution can
be split as
u ¼ u0h þ u0p ;
ð17Þ
where u0h and u0p are the zero-order the homogeneous and
particular solutions, respectively. The multiple reciprocity
principle assumes that the particular solution can be
approximated by higher-order homogeneous solutions, i.e.
u ¼ u0h þ u0p ¼ u0h þ
1
X
um
h;
ð18Þ
m¼1
where superscript m is the order index of the homogeneous
solution. Through an incremental differentiation operation
via convection – diffusion operator, we have successively
higher-order boundary differential equations
8 0
0
>
< uh ðxi Þ ¼ Rðxi Þ 2 up ðxi Þ;
ð19aÞ
0
0
>
: ›uh ðxj Þ ¼ Nðx Þ 2 ›up ðxj Þ ;
j
›n
›n
8 n
o
n
o
0 1
>
>
R uh ðxi Þ ¼ f ðxi Þ 2 R0 u1p ðxi Þ
>
<
n
o
n
o
ð19bÞ
0 1
0 1
>
›
f
ðx
Þ
2
R
u
ðx
Þ
›
R
u
ðx
Þ
p j
j
h j
>
>
:
¼
;
›n
›n
8
n
o
n21 n
n22
n21 n
>
>
u
ðx
Þ
¼
R
{f
ðx
Þ}
2
R
u
ðx
Þ
;
R
f
g
h
p
i
i
i
>
<
n
o
n
o
>
› Rn22 {f ðxj Þ} 2 Rn21 unp ðxj Þ
›Rn21 unh ðxj Þ
>
>
:
¼
;
›n
›n
n ¼ 2; 3; …;
ð19cÞ
where Rn { } denotes the nth order of convection –diffusion
equation, say R1 { } ¼ RR0 { }; and R0 { } ¼ R{ }; i and j
are knots indices, respectively, on Dirichlet and Neumann
boundary. The m-order homogeneous solution can be
approximated in terms of the m-order general solution u#m
basis function
um
h ðxÞ ¼
L
X
k¼1
bmk u#m ðrk Þ;
ð20Þ
573
where L is the number of boundary nodes, and k denotes the
index of source points on boundary. bk is the desired
coefficients. Note that we can use the fundamental solution
upm instead of general solution u#m in Eq. (20). However, the
use of fundamental solution requires a fictitious boundary
outside physical domain as encountered in the MFS [11].
Collocating Eqs. (19a) – (19c) at all boundary knots in
terms of representation (Eq. (20)), we have the BPM
boundary discretization equations:
9
L
>
X
>
0 #
0
>
bk u0 ðrik Þ ¼ Rðxi Þ 2 up ðxi Þ
>
>
=
k¼1
ð21aÞ
¼ b0 ;
>
0
#
L
X
>
›up ðxj Þ >
0 ›u0 ðrjk Þ
>
¼ Nðxj Þ 2
bk
>
›n
›n ;
k¼1
9
L
n
o
n
o
X
>
>
>
b1k R0 u#1 ðrik Þ ¼ f ðxi Þ 2 R0 u1p ðxi Þ
>
>
=
k¼1
n
o ¼ b1 ;
n
o
L
>
› f ðxj Þ 2 R0 u1p ðxj Þ >
›R0 u#1 ðrjk Þ
X
>
>
>
¼
b1k
;
›
n
›
n
k¼1
ð21bÞ
9
L
n
o
n
o
X
>
>
>
bnk Rn21 u#n ðrik Þ ¼ Rn22 {f ðxi Þ} 2 Rn21 unp ðxi Þ
>
>
=
k¼1
n
o
L
>
› Rn22 {f ðxj Þ} 2 Rn21 unp ðxj Þ >
X
>
›Rn21 u#n ðrik Þ
>
>
¼
bnk
;
›n
›n
k¼1
¼ bn ;
n ¼ 2; 3; …:
ð21cÞ
Whenever the mixed boundary conditions are present, the
present BPM interpolation matrix is unsymmetric even for
self-adjoint operator. By analogy with Fasshauer’s Hermite
RBF interpolation [12], the present author also proposed a
symmetric BPM formulation for self-adjoint operator [2].
Since the convection– diffusion equation is not self-adjoint,
this study simply uses the present unsymmetric BPM
scheme.
As in the MR-BEM [1], the successive process is
truncated at some order M, namely,
n o
RM21 uM
ð22Þ
p ¼ 0:
The practical solution procedure is a reversal recursive
process:
bM ! bM21 ! · · · ! b0 :
ð23Þ
It is noted that due to
n
o
Rn21 u#n ðrÞ ¼ u#0 ðrÞ;
ð24Þ
the coefficient matrices of all successive equations (Eqs.
(21a) – (21c)) are the same, i.e. let
Q bn ¼ b n ;
n ¼ M; M 2 1; …; 1; 0:
ð25Þ
574
W. Chen / Engineering Analysis with Boundary Elements 26 (2002) 571–575
The LU decomposition algorithm is suitable for this task.
Then we can employ the obtained expansion coefficients b
to calculate the BPM solution at any knot, i.e.
uðxi Þ ¼
M X
L
X
bnk u#n ðrik Þ
ð26Þ
n¼0 k¼1
4. Numerical validations
The tested case is a 2D steady convection –diffusion
problem described by
D72 u 2 kv·7u 2 ku ¼ e2hðxþyÞ ð2 2 4hx þ 2sxÞ
ð27Þ
with the boundary conditions of the Dirichlet type (Eq. (15))
and Neumann type (Eq.p(16)),
where D ¼ 1; vx ¼ vy ¼ 2s;
ffiffiffiffiffiffiffiffiffiffi
k ¼ 3s2 =2; h ¼ ðs þ s2 þ 2kÞ=2: The exact solution is
u ¼ x2 e2hðxþyÞ :
ð28Þ
The Peclet number Pe is defined as
Pe ¼
lvklLc
;
D
ð29Þ
where Lc denotes a characteristic length. The Peclet number
indicates the relative importance of convection versus
diffusion.
Fig. 1 illustrates the 2D irregular geometry. Unless
specified Neumann boundary conditions at x ¼ 0 and y ¼ 0
edges, the otherwise boundary are all of the Dirichlet type.
Equally spaced knots were applied on the boundary. The L2
norms of relative errors are calculated based on the
numerical solutions at 460 inner and boundary nodes. The
absolute error is taken as the relative error if the absolute
value of the solution is less than 0.001.
Fig. 1. Configuration of 2D irregular geometry.
Table 1
L2 norm of relative errors for 2D inhomogeneous convection– diffusion
problem
L
Pe ¼ 24
L
Pe ¼ 72
25
9:0 £ 1023
33
1:2 £ 1022
33
1:4 £ 1023
41
4:5 £ 1022
41
3:6 £ 1024
49
2:2 £ 1023
49
3:6 £ 1024
57
4:5 £ 1024
Table 1 displays the experimental results, where L
represents the number of the used boundary knots. It is seen
that the BPM scheme based on the high-order general
solutions produces stable solutions with increasing number
of boundary nodes for the present inhomogeneous convection– diffusion problems. In particular, the BPM solutions
are found very accurate even with a small number of
boundary nodes. It is stressed that we did not use any inner
nodes here. For higher Peclect number problems, the BPM
solutions become unstable due to very poor condition of the
BPM interpolation matrix. As in the BEM, some preconditioning techniques could be used to handle this issue, which
is beyond the present study. Hence one can conclude that the
presented high-order general solutions of convection –
diffusion equation do work well in this numerical
experiment.
Acknowledgments
This research is supported by Norwegian Research
Council.
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