Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue3.1999:

Advances in Water Resources 23 (1999) 229±237

Hybrid Laplace transform ®nite element method for solving the
convection±dispersion problem
Li Ren a, Renduo Zhang
a

b,*

Department of Soil and Water Sciences, China Agricultural University, Beijing 100094, People's Republic of China
b
Department of Renewable Resources, University of Wyoming, Laramie, Wyoming 82071-3354, USA
Received 8 September 1998; accepted 18 March 1999

Abstract
It can be very time consuming to use the conventional numerical methods, such as the ®nite element method, to solve convection±
dispersion equations, especially for solutions of large-scale, long-term solute transport in porous media. In addition, the conventional methods are subject to arti®cial di€usion and oscillation when used to solve convection-dominant solute transport problems.
In this paper, a hybrid method of Laplace transform and ®nite element method is developed to solve one- and two-dimensional
convection±dispersion equations. The method is semi-analytical in time through Laplace transform. Then the transformed partial
di€erential equations are solved numerically in the Laplace domain using the ®nite element method. Finally the nodal concentration
values are obtained through a numerical inversion of the ®nite element solution, using a highly accurate inversion algorithm. The

proposed method eliminates time steps in the computation and allows using relatively large grid sizes, which increases computation
eciency dramatically. Numerical results of several examples show that the hybrid method is of high eciency and accuracy, and
capable of eliminating numerical di€usion and oscillation e€ectively. Ó 1999 Elsevier Science Ltd. All rights reserved.
Keywords: Hybrid method; Laplace transform; Finite element; Numerical diculties

1. Introduction
Convection±dispersion type equations are being
widely used to model solute transport in soils and
groundwater systems. However, owing to the particular
combination of hyperbolic and parabolic terms, serious
diculties, such as numerical di€usion and oscillation,
are often encountered in obtaining accurate numerical
solutions of the equations. Besides the numerical diculties, it would be very time consuming to use the
conventional numerical methods, such as the ®nite element method, to solve convection±dispersion equations, especially for solutions of large-scale, long-term
solute transport in porous media. A variety of numerical schemes have been developed to deal with the dif®culties and to improve numerical eciency. It has
been shown that combination of integral transforms
with numerical methods has advantages to overcome
the numerical problems and enhance computation eciency ([13,11]).

*

Corresponding author. Tel.: +307 766 5032; fax: +307 766 6403;
e-mail: renduo@uwyo.edu

Liggett and Liu [10] used the Laplace transform in
conjunction with the boundary element method to solve
the unsteady groundwater ¯ow equation. The method is
semi-analytical and semi-numerical in nature. The Laplace transform is ®rstly applied to the governing
equation as well as initial and boundary conditions
describing the physical problem. Then a numerical
method is employed in the Laplace domain to solve the
transformed partial di€erential equation. Finally the
solution of the original problem is obtained by inverting
the Laplace solutions numerically. In recent years, these
kinds of new numerical methods has been developed
and applied in the ®eld of heat transfer and solute
transport in subsurface ¯ow ([1±4,13,14]). Chen and
Chen [1±3] solved the transient heat conduction problems by combining the Laplace transformation with the
®nite di€erence method or ®nite element method. Sudicky and McLaren [14] applied the Laplace transform
Galerkin technique with the numerical inversion algorithm of Crump [5] for large-scale simulation of mass

transport in discretely fractured porous formations.
Recognizing the de®ciency of the Crump [5] algorithm,
Ren [11,12] developed a hybrid method of Laplace
transform and ®nite element (HLTFEM), using the

0309-1708/99/$ - see front matter Ó 1999 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 9 9 ) 0 0 0 1 3 - 5

230

L. Ren, R. Zhang / Advances in Water Resources 23 (1999) 229±237

algorithm by Honig and Hirdes [9] for numerical inversion of Laplace solutions. The new algorithm of
Laplace transform inversion was developed based on
the Fourier series approximation of Durbin [6]. The
algorithm combines procedures to diminish discretization error, accelerate the convergence of Fourier progression, and to select optimal parameters. Therefore,
the discretization and truncation errors of the inversion
algorithm by Honig and Hirdes [9] do not depend on
how to choose the free parameters. On the other hand,
the Crump [5] algorithm uses a di€erent method to

speed up the convergence of the Fourier series ([7,6]).
The method does not provide a consistent reduction of
truncation errors. Its computation eciency heavily
depends on the choice of the parameters, which is
somewhat arbitrary. The main disadvantage of the
Crump [5] algorithm is that the discretization and
truncation errors depend on the choice of the free parameters. For example, a choice of the parameters can
result in an arbitrarily small discretization error; however, at the same time the truncation error grows to
in®nity and vice versa. Fortunately, the problem of the
Crump [5] algorithm is resolved in the algorithm of
Honig and Hirdes [9]. Because of high accuracy of the
algorithm, the HLTFEM has been successfully used to
solve solute transport problems in the subsurface
([11,12]).
In this paper, the HLTFEM is further developed and
utilized to simulate one- and two-dimensional solute
transport under uniform ¯ow conditions. Examples are
analyzed to illustrate the numerical accuracy and eciency of the present method. The performance of the
method is evaluated against results from analytical solutions and other numerical methods.





oc
o
oc
oc
o
oc
oc
‡
ˆ
Dxx ‡ Dxy
Dyx ‡ Dyy
ot ox
ox
oy
oy
ox

oy
ÿ Vx

oc
oc
ÿ Vy ;
ox
oy

…4†

where c…x; y; t† is the concentration, x and y are spatial
coordinates ‰LŠ; Vx and Vy are the average pore velocities
‰ LT ÿ1 Š in x and y directions, respectively, and Dxx , Dxy ,
Dyx , Dyy are the components of the hydrodynamic dispersion tensor for an anisotropic medium. The boundary and initial conditions for Eq. (4) are
c…x; y; t† ˆ c1

…x; y† 2 C

c…x; y; 0† ˆ c0

…x; y† 2 X;

…5†

where C is the exterior boundary of the solution domain
X; c1 is the concentration speci®ed along C, and c0 is the
initial concentration distribution.
To remove the time derivatives from the governing
equations, the method of Laplace transform is utilized.
The Laplace transform of a real function f …t† and its
inversion are de®ned as
Z1
…6†
F …s† ˆ L‰f …t†Š ˆ eÿst f …t† dt;
0

1
f …t† ˆ L ‰F …s†Š ˆ
2pi

ÿ1

v‡i1
Z

est F …s†ds;

…7†

vÿi1

where s is the Laplace transform parameter and s ˆ v ‡
iw with v; w 2 R.
3. Formulation of ®nite element equations

2. Theory

3.1. One-dimensional solute transport problem

One-dimensional solute transport through a porous

medium can be described by the following governing
equation


oc
o
oc
o
ÿ …uc†;
ˆ
D
…1†
ot ox
ox
ox
with speci®ed boundary conditions

After taking the Laplace transform with respect to
time, Eqs. (1), (2a) and (2b) become
!

o
o~
c
o
ÿ …u~
D
c† ˆ s~
c ÿ c…x; 0†;
…8†
ox
ox
ox

c…0; t† ˆ c0 ;

…2a†

c…1; t† ˆ 0 t > 0
and the initial condition

…2b†

c…x; 0† ˆ 0

06x61

ÿ3

…3†

Here c…x; t† is the concentration ‰ ML Š, D the dispersion
coecient ‰L2 T ÿ1 Š, u the average pore velocity ‰ LT ÿ1 Š,
and c0 the initial concentration ‰MT ÿ3 Š.
For solute transport in a two-dimensional porous
medium under a steady water ¯ow condition, the governing equation is

c~ ˆ c0 =s at x ˆ 0;
c~ ˆ 0 at x ˆ 1:

…9†
…10†

In the ®nite element method, an approximate solution
for c~ is in the form of
c~ …x; s† ˆ

N
X
iˆ1

c~i …s†ui …x†:

…11†

Here ui is the linear interpolation function, N the
number of nodes in the grid, s the Laplace transformed
variable, and c~i …s† the concentration in s space at the
node points.

231

L. Ren, R. Zhang / Advances in Water Resources 23 (1999) 229±237

Applying the Galerkin procedure to Eq. (8) leads
#
"
!
P Z
X
o
o~
c
o
D
c † ÿ s~
c ‡ c…x; 0† ui …x† dx
ÿ …u~
ox
ox
ox
eˆ1
Le

ˆ 0;
…12†
where P is the number of elements that are joined to the
node i, and Le is the length of element e. Incorporating
Eq. (11) into Eq. (12) and applying Green's theorem to
reduce the order of the second derivative term yields a
system of algebraic equations as follows
Ei c~iÿ1 ‡ Fi c~i ‡ Gi c~i‡1 ˆ Hi
where

i ˆ 1; 2; . . . ; n;

…13†

sDx Diÿ1=2 uiÿ1=2
ÿ
ÿ
Dx
2
6

Fi ˆ

4sDx Diÿ1=2 Di‡1=2 uiÿ1=2 ui‡1=2
‡
‡
ÿ
‡
Dx
Dx
2
2
6

Gi ˆ

sDx Di‡1=2 ui‡1=2
‡
ÿ
Dx
2
2

…14†

oUj oUi
oUj oUi
oUj oUi
‡ Dxy
‡ Dyx
ox ox
oy ox
ox oy
)

oUj oUi
‡ sUj Ui c~j …s† ÿ c…x; y; 0†Ui dxdy ˆ 0
‡ Dyy
oy oy
‡ Dxx

…i ˆ 1; 2; . . . ; N †
…19†
T
Let ~cT ˆ …c~1 ; c~2 ;


; c~N † , then Eq. (19) has the matrix
form of
…20†

The elements of matrix ‰AŠ and vector F are expressed
as follows:
Z Z 
oUi oUj
oUi oUj
…e†
‡ Dxy
Dxx
Aij ˆ
ox ox
ox oy
…D†

oUi oUj
oUi oUj
oUj
‡ Dyy
‡ Vx Ui
oy ox
oy oy
ox

oUj
‡ Vy U i
‡ sUi Uj dx dy
oy

1 
ˆ
Dxx bi bj ‡ Dxy …bi cj ‡ ci bj † ‡ Dyy ci cj
4D


1ÿ
…e†
‡ Vx bj ‡ Vy cj ‡ Bij ;
6
where

Z Z
sD=6
when i ˆ j;
…e†
Bij ˆ s
Ui Uj dx dy ˆ
sD=12 when i 6ˆ j
‡ Dyx

Hi ˆ c…x; 0†Dx

in which Dx is the descretized size of the space.
3.2. Two-dimensional solute transport problem
After taking
becomes

the

o
o~
c
o~
c
Dxx ‡ Dxy
ox
ox
oy

!

‡

Laplace
o
oy

Dyx

transform,

o~
c
o~
c
‡ Dyy
ox
oy

Eq. (4)

!

o~
c
o~
c
c ÿ c…x; y; 0†
…15†
ÿ Vx ÿ Vy ˆ s~
ox
oy
in which c~ ˆ c~…x; y; s† is the transformed concentration.
The transformed boundary condition is
c~…x; y; s† ˆ c1 =s:
The approximate solution of c~ is de®ned by
N
X
c~j …s†Uj …x; y†:
jˆ1

…16†
…17†

Here Uj is the linear interpolation function and c~j …s† is
the transformed concentration at node j, and N is the
total number of nodes in the triangular element mesh.
Using the Galerkin procedure and applying Green's
theorem to Eq. (15), we have
Z Z (
o~
c
o~
c
o~
c oUi
Ui ‡ Vy
Ui ‡ Dxx
Vx
ox
oy
ox ox
X

o~
c oUi
o~
c oUi
o~
c oUi
‡ Dyx
‡ Dyy
oy ox
ox oy
oy oy
o
‡ s~
c Ui ÿ c…x; y; 0†Ui dx dy ˆ 0

‡ Dxy

…i ˆ 1; 2; . . . ; N †:

X

‰AŠ~cT ‡ F ˆ 0:

Ei ˆ

c~ …x; y; s† ˆ

Substituting Eq. (17) into Eq. (18), we obtain
Z Z (X
N 
oUj
oUj
‡ Vy Ui
Vx Ui
ox
oy
jˆ1

…21†
…22†

…D†

and
…e†
Fi

ˆÿ

Z Z

c…x; y; 0†Ui dx dy ˆ ÿc…x; y; 0†D=3:

…D†

Here D is the area of the triangular element e with nodes
i; j, and k numbered in the counterclockwise order and
ai ˆ xj yk ÿ xk yj ;
aj ˆ xk yi ÿ xi yk ;

ak ˆ xi yj ÿ xj yi ;

bi ˆ yj ÿ yk ;
bj ˆ yk ÿ yi ;

bk ˆ yi ÿ yj ;

ci ˆ xk ÿ xj ;
cj ˆ xi ÿ xk ;

ck ˆ xj ÿ xi :

…23†

The coordinates of nodes i; j and k are designated as
…xi ; yi †; …xj ; yj †, and …xk ; yk †, respectively. The element
matrices and vectors are formed sequentially and the
contributions to the global matrix are summed. The
matrix equation is completed with the boundary condition of Eq. (16). Solving Eqs. (13) and (20), we can obtain the transformed concentration values at the nodes.
4. Inversion of the Laplace transformation

…18†

For the ®nal solution of solute concentration, the remained task is to inverse the transformed concentrations.

232

L. Ren, R. Zhang / Advances in Water Resources 23 (1999) 229±237

Letting Lÿ1 denote the inverse transformation, from
Eq. (11) we have the one-dimensional solution as follows
N
N
h
i
X
X
ci …t†ui …x†;
…24†
Lÿ1 c~i …sk † ui …x† ˆ
c …x; t† ˆ
iˆ1

iˆ1

where ci …t† is the concentration at node i and time t. The
inversion algorithm requires knowledge of the value of
the transformed variable for di€erent values of
s ˆ sk ; k ˆ 1; 2; . . . ; as shown below (Eqs. (30)±(34)).
Similarly, from Eq. (17) the two-dimensional solution is
c …x; y; t† ˆ

N
X
iˆ1

N
h
i
X
Lÿ1 c~i …sk † Ui …x; y† ˆ
ci …t†Ui …x; y†:

ÿ F1 …v; t; T †;

iˆ1

…25†
The inversion process must be perfomed numerically.
The numerical inversion form of the Laplace transform
can be written ([1±3]):
F …s† ˆ

Z1

eÿvt f …t†‰ cos …wt† ÿ i sin …wt†Š dt

0

or
F …s† ˆ RefF …v ‡ iw†g ‡ ImfF …v ‡ iw†g:

Substituting Eq. (26) into Eq. (7) yields
Z1
1
f …t† ˆ
evt ‰ cos …wt† ‡ i sin…wt†Š‰RefF …S†g
2pi

…26†

‡ iImfF …s†gŠi dw
21
Z
evt 4
ˆ
…RefF …s†g cos…wt† ÿ ImfF …s†g sin …wt†† dw
2p
ÿ1

‡i

3

…ImfF …s†g cos …wt† ‡ RefF …s†g sin …wt†† dw5:

ÿ1

…27†

Combining Eqs. (26) and (27) leads
21 1
Z Z
vt
e 4
eÿvs f …s† cos …w…s ÿ t†† ds dw
f …t† ˆ
2p
ÿi

ÿ1 0

3

eÿvs f …s† sin …w…s ÿ t†† ds dw5:

kˆ1

Since the Fourier series in Eq. (30) can only be
summed to a ®nite number of terms, a truncation error
is introduced in the form of
"
 



1
evt X
kp
kp
Re F v ‡ i
cos
t
FT …M; v; t; T † ˆ
T kˆM‡1
T
T
 

 
kp
kp
sin
t ;
ÿ Im F v ‡ i
T
T
where M is the ®nite number of terms of the Fourier
series expansion and T is the half period of the Fourier
series approximating the inverse on the interval [0, 2T ].
Then the approximate solution of f …t† is ([9,1±3])

evt
1
ÿ RefF …v†g
fM …t† ˆ
2
T



 
M
X
kp
kp
‡
cos
t
Re F v ‡ i
T
T
kˆ0


#


M
X
kp
kp
sin
t
…33†
Im F v ‡ i
ÿ
T
T
kˆ0
or

ci …t† ˆ

ÿ1 0

Z1 Z1

…30†

where F1 …v; t; T † is the discretization error given by
1
X
F1 …v; t; T † ˆ
eÿ2vkT f …2kT ‡ t†:
…31†

…32†

ÿ1

Z1

Based on the method of Durbin [6] for the Fourier series
expansion, f …t† on the interval ‰0; 2T Š can be derived as
follows

evt
1
ÿ RefF …v†g
f …t† ˆ
T
2
 



1
X
kp
kp
Re F v ‡ i
cos
t
‡
T
T
kˆ0


#


1
X
kp
kp
sin
t
Im F v ‡ i
ÿ
T
T
kˆ0

…28†

In Eq. (28), sin …w…s ÿ t†† is an odd function of w;
therefore, the second integral is zero and the equation is
simpli®ed as
3
21
Z
evt 4
…RefF …s†g cos …wt† ÿ ImfF …s†g sin …wt††dw5:
f …t† ˆ
p
0

…29†


evt
1
ci …v†g
ÿ Ref~
2
T



 
M
X
kp
kp
cos
t
Re c~i v ‡ i
‡
T
T
kˆ0
 
#
 
M
X
kp
kp
sin
t :
Im c~i v ‡ i
ÿ
T
T
kˆ0

…34†

In the methods of Durbin [6] and Dubner and Abate [7],
v > 0 is chosen arbitrarily. However, an optimal choice
of the free parameters M and vT is essential not only to
increase accuracy but also to accelerate convergence of
the results. In this paper, the transform inversion algorithm of Honig and Hirdes [9] is employed to determine
the optimal value of v when the values of M and T are
®xed. Two methods are developed based on the

L. Ren, R. Zhang / Advances in Water Resources 23 (1999) 229±237

following two conditions, respectively, (1) the parameter
v is optimal if the absolute values of discretization and
truncation error are equal, and (2) the parameter v is
optimal if the sum of the absolute values of discretization and truncation errors is minimal.
For the numerical Laplace inversion, the only input is
the time t at which the concentration is required. Then
f …t† ˆ Lÿ1 ‰F …s†Š is computed based on the Laplace
transform F …s†. More speci®cally, the time values are
given by ([9])
tk ˆ T1 ‡ …TN ÿ T1 †k=…N ‡ 1†;

k ˆ 1; . . . ; N :

…35†

Here N is the number of t-values for which f …t† is to be
computed, T1 and TN are the lower and upper limits of
the interval in which f …t† is calculated. For instance, to
obtain concentration results at t1 ˆ 160 and t2 ˆ 320
day (Fig. 3), we selected T1 ˆ 0, TN ˆ 480 and N ˆ 2. In
the Laplace inversion (Eq. (33)), the chosen value t in
the time domain is related to a corresponding particular
value of s in the Laplace domain by
s ˆ v ‡ ikp=T ;

k ˆ 1; . . . ; M:

…36†

5. Numerical applications
5.1. Example 1
This example has become the standard test of a numerical scheme designed to solve the one-dimensional
convection±dispersion equation. The initial and boundary conditions are given by
c…x; 0† ˆ 0

t ˆ 0;

c…0; t† ˆ c0 t > 0;
c…1; t† ˆ 0 t > 0:

…37†

The analytical solution to this problem is ([16])





 ux 
c0
x ÿ ut
x ‡ ut
erfc p ‡ exp
c…x; t† ˆ
:
erfc p
2
D
2 Dt
2 Dt
…38†
To discuss di€erent transport modes, a local Peclet
number is used and de®ned by
Pe ˆ uDx=D:
…39†
A length x ˆ 30 m and the solution time t ˆ 15 d were
chosen such that at all times between 0 and t, the
boundary condition c…1; t† ˆ 0 was satis®ed. Other
parameters used include c0 ˆ 10 g/l, u ˆ 1 m/d, and
Dx ˆ 0:1 m. Di€erent dispersion coecients were used:
the range of D values from 0.1 to 0.001 m2 /d, corresponding to Pe ˆ 1 to 100. The HLTFEM solutions were
compared with numerical solutions of the ®nite element
method (FEM) and the analytical solution. As shown in
Fig. 1(a)±(c), the numerical solutions of the HLTFEM
and the FEM are very close to the analytic solution for
Pe < 2. For Pe P 2, the FEM produced breakthrough

233

curves more di€used than the analytical solution and the
numerical di€usion increased with Pe. In contrast, the
HLTFEM solution did not produce any numerical diffusion and oscillation for any Pe.
5.2. Example 2
This example was used to test the accuracy and robustness of the HLTFEM for simulating transport in a
two-dimensional aquifer under one-dimensional constant groundwater velocity ®eld. The governing equation and initial and boundary conditions are given by




oc
o
oc
o
oc
oc
‡
ÿV
ˆ
aV
Dy
ot ox
ox
oy
oy
ox
c…x; y; 0† ˆ 0

c…0; y; t† ˆ c0 …y†

oc 
ˆ0
ox xˆLx

oc 
ˆ0
oy yˆ0

oc 
ˆ 0;
oy yˆLy

c 2 ‰0; 1Š

…40†

where c ˆ c…x; y; t† is the concentration, a the longitudinal dispersivity …L†; V the average pore velocity, Dy
the e€ective molecular di€usivity in y direction, t the
time, and Lx and Ly the total lengths of seepage ®eld in
x and y directions, respectively.
The parameters used in this example are a ˆ 1:0 m,
V ˆ 0:1 m/day, Dy ˆ 8  10ÿ5 m2 /day, Lx ˆ 200 m and
Ly ˆ 2 m. The in¯ow boundary condition is expressed
by
8
y 6 0:4 m
0
oc=ox…1; y; t† ˆ 0
06y 65

…42†

Simulations were conducted in a studying domain of
10  5, whose lower half is shown in Fig. 4 because of
the symmetry of the problem at y ˆ 2:5. Two hundred
triangular elements …N ˆ 200† and 126 nodes were used
to discretize the computational domain.
Following Taigbenu and Liggett [15], we examined
three transport cases in the aquifer: dispersion dominant
transport (Pe ˆ 0.05), dispersion and convection transport (Pe ˆ 1.0), and convection dominant transport (Pe
ˆ 50). Other parameters used in the simulations for
these cases are listed in Table 1.
The numerical results from the HLTFEM and the
boundary element method (BEM) ([15]) were compared along with the analytical solutions. As an example, Fig. 5 presents concentration breakthrough
curves along the axis y ˆ 2:5 for Pe ˆ 50. The Courant
number of the BEM was 0.4. The ®gure shows a

L. Ren, R. Zhang / Advances in Water Resources 23 (1999) 229±237

235

Fig. 2. Concentration pro®les (Example 2, Dx ˆ 2 m and Dy ˆ 0:2 m) calculated with the HLTFEM, the Galerkin FEM, and the analytical solution
for (a) longitudinal (y ˆ 0) and (b) transverse (x ˆ 8) pro®les.

Fig. 3. Concentration curves (Example 2, Dx ˆ 8 m and Dy ˆ 0:2 m)
calculated with the HLTFEM, the alternation direction Galerkin element method (ADG), and the analytical solution for the longitudinal
pro®le …y ˆ 0†.

reasonable agreement between the HLTFEM results
and the analytical solution. There are both profound
numerical oscillation and di€usion in the BEM solution. The HLTFEM solution exhibits no numerical
di€usion and some numerical oscillation in the upwind
nodes.
The HLTFEM is an inherently non-time-marching
method. In other words, the method calculates nodal
concentrations at a speci®c time using one time step. In
contrast, the conventional FEM must compute nodal
concentrations through numerous intermediate time
steps until the required time is reached. In many cases,
the time step for the FEM has to be very small to
guarantee numerical convergence and accuracy. Therefore, the HLTFEM is a time-saving procedure to solve
long-time chemical transport problems in large ®eld
domains. Fig. 6 shows a typical relationship between the
required CPU time as a function of the total simulation
time for a ®xed node number. The CPU time of the
HLFEM is mainly used for numerical inversion,
therefore, more or less constant for any total simulation
time. However, the CPU time of the FEM increases

Fig. 4. Computational domain for the two-dimensional semi-in®nite ®eld (Example 3).

236

L. Ren, R. Zhang / Advances in Water Resources 23 (1999) 229±237

Table 1
Parameters of Example 3
Type of transport

Pe

u

Dxx ˆ Dyy

Dx ˆ Dy

Solution time, t

Dispersion dominant
Dispersion±convection
Convection dominant

0.05
1.0
50

1
1
1

10
0.5
0.01

0.5
0.5
0.5

4.0
4.0
4.0

Fig. 5. Longitudinal concentration pro®les (Example 3, y ˆ 2:5 m, and
Pe ˆ 50) calculated with the HLTFEM, the boundary element method
(BEM), and the analytical solution.

persion problems. Several application examples have
shown that the HLTFEM has great potential in modeling of solute transport in one- and two-dimensional
®elds under steady-state ¯ow conditions.
Compared with the conventional Galerkin ®nite element technique, the alternation direction Galerkin element method and the boundary element method, the
HLTFEM shows many advantages. The HLTFEM has
higher accuracy in simulating sharp solute fronts in
convection-dominant problems and overcomes numerical di€usion as well as oscillation e€ectively. The
HLTFEM is an inherently non-time-marching method
to calculate nodal concentrations at any speci®c time.
Therefore, the method eliminates the disadvantage of
the step by step computation in the time domain and
provides high computation eciency as well as low
computation cost. The HLTFEM is especially useful to
predict long-time soil and groundwater contamination
in large ®eld domains with stable and highly accurate
solutions.

Acknowledgements
The authors express appreciation to Professors W.
Zhang and Y. Zhang at Wuhan University of Hydraulic
and Electric Engineering for their guidance and encouragement.
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Fig. 6. A typical relationship between the required CPU time as a
function of the total simulation time for a ®xed node number.

approximately linearly or exponentially as the total
simulation time increases.

6. Conclusions
A hybrid Laplace transform ®nite element method
(HLTFEM) was developed for solving convection±dis-

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