A finite analytic method for solving the 2-D time-dependent advection–diffusion equation with time-invariant coefficients q Thomas Lowry a, * ,1 , Shu-Guang Li b a Sandia National Laboratories, P.O. Box 5800, MS 0735, Albuquerque, NM 87185, USA b Department of Civil and Environmental Engineering, Michigan State University, A135 Engineering Research Complex, East Lansing, MI 48824, USA Received 28 April 2004; received in revised form 28 September 2004; accepted 15 October 2004 Abstract Difficulty in solving the transient advection–diffusion equation (ADE) stems from the relationship between the advection deriv- atives and the time derivative. For a solution method to be viable, it must account for this relationship by being accurate in both space and time. This research presents a unique method for solving the time-dependent ADE that does not discretize the derivative terms but rather solves the equation analytically in the space–time domain. The method is computationally efficient and numerically accurate and addresses the common limitations of numerical dispersion and spurious oscillations that can be prevalent in other solu- tion methods. The method is based on the improved finite analytic (IFA) solution method [Lowry TS, Li S-G. A characteristic based finite analytic method for solving the two-dimensional steady-state advection–diffusion equation. Water Resour Res 38 (7), 10.1029/ 2001WR000518] in space coupled with a Laplace transformation in time. In this way, the method has no Courant condition and maintains accuracy in space and time, performing well even at high Peclet numbers. The method is compared to a hybrid method of characteristics, a random walk particle tracking method, and an Eulerian–Lagrangian Localized Adjoint Method using various degrees of flow-field heterogeneity across multiple Peclet numbers. Results show the IFALT method to be computationally more efficient while producing similar or better accuracy than the other methods. Ó 2004 Published by Elsevier Ltd. Keywords: Contaminant transport; Groundwater; Advection–diffusion; Laplace transform 1. Introduction Numerical methods are categorized as one of three general types, Eulerian, Lagrangian, and mixed Eule- rian–Lagrangian (E–L). Eulerian methods attempt to solve the ADE directly on a fixed grid. Common exam- ples of this type of method include finite-difference and finite-element methods. However, Eulerian methods are generally susceptible to numerical dispersion or spu- rious oscillations, especially when advection dominates. Lagrangian methods track a contaminant plume through a given velocity field using particles to represent discrete packets of solute mass. Dispersion is modelled by adding a random component to the particle trajec- tory at each time step. Concentrations are recovered at the end of the simulation by summing the mass in each cell (the number of particles) and dividing by the cell volume. Random walk models are representative of this class [2–4]. Lagrangian methods are useful for locating the spatial extent of a contaminant plume or for deriving concentration fields for plumes of small extent. However 0309-1708/$ - see front matter Ó 2004 Published by Elsevier Ltd. doi:10.1016/j.advwatres.2004.10.005 q This research was jointly sponsored by the National Science Foundation under grants BES–9811895 and EEC–0088137 and the New Zealand Foundation for Research, Science, and Technology, under contract #LVLX0006. * Corresponding author. Tel.: +1 505 284 9735; fax: +1 505 844 7354. E-mail addresses: [email protected](T. Lowry), lishug@egr. msu.edu (S.-G. Li). 1 Formerly with Lincoln Environmental, Lincoln, New Zealand. Advances in Water Resources 28 (2005) 117–133 www.elsevier.com/locate/advwatres
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Advances in Water Resources 28 (2005) 117–133
www.elsevier.com/locate/advwatres
A finite analytic method for solving the 2-D time-dependentadvection–diffusion equation with time-invariant coefficients q
Thomas Lowry a,*,1, Shu-Guang Li b
a Sandia National Laboratories, P.O. Box 5800, MS 0735, Albuquerque, NM 87185, USAb Department of Civil and Environmental Engineering, Michigan State University, A135 Engineering Research Complex, East Lansing, MI 48824, USA
Received 28 April 2004; received in revised form 28 September 2004; accepted 15 October 2004
Abstract
Difficulty in solving the transient advection–diffusion equation (ADE) stems from the relationship between the advection deriv-
atives and the time derivative. For a solution method to be viable, it must account for this relationship by being accurate in both
space and time. This research presents a unique method for solving the time-dependent ADE that does not discretize the derivative
terms but rather solves the equation analytically in the space–time domain. The method is computationally efficient and numerically
accurate and addresses the common limitations of numerical dispersion and spurious oscillations that can be prevalent in other solu-
tion methods. The method is based on the improved finite analytic (IFA) solution method [Lowry TS, Li S-G. A characteristic based
finite analytic method for solving the two-dimensional steady-state advection–diffusion equation. Water Resour Res 38 (7), 10.1029/
2001WR000518] in space coupled with a Laplace transformation in time. In this way, the method has no Courant condition and
maintains accuracy in space and time, performing well even at high Peclet numbers. The method is compared to a hybrid method
of characteristics, a random walk particle tracking method, and an Eulerian–Lagrangian Localized Adjoint Method using various
degrees of flow-field heterogeneity across multiple Peclet numbers. Results show the IFALT method to be computationally more
efficient while producing similar or better accuracy than the other methods.
The term ft is the forward particle travel time from the upstream source node (i � 2, j � 1 in this case) to the element boundary, and bt is the backward
particle travel time from node Pi,j to the element boundary.
122 T. Lowry, S.-G. Li / Advances in Water Resources 28 (2005) 117–133
2.5. Inversion of the laplace transformation
The IFALT method utilizes the Laplace inversion
algorithm developed by DeHoog et al. [34] due to its
performance in the area of discontinuities (sharp con-
centration fronts), and the fact that the inverse formany values of time can be obtained from one set of
Laplace parameter evaluations. The form of the De-
Hoog et al. algorithm used in this research was imple-
mented by Neville in 1989 and later modified by
McLaren in 1991 to allow inversion one nodal point
at a time. The FORTRAN code for the DeHoog algo-
rithm was obtained directly from Sudicky and McLaren
(in 1999) for this research with no significant changes tothe 1991 form.
The inverse Laplace transform, modified from the
general form to specify concentration, is given by [22]
Cðx; y; tÞ ¼ 1
2pi
Z aþi1
a�i1ept eCðx; y; pÞdp ð13Þ
By manipulating the real and imaginary parts of (13), an
alternative expression is formed:
Cðx; y; tÞ ¼ eat
p
Z 1
0
fRe½eCðx; y; pÞ� cosxt
� Im½eCðx; y; pÞ� sinxtgdx ð14Þ
where Re and Im denote the real and imaginary parts oftheir arguments and a and x are defined below.
If we discretize Eq. (14) using a trapezoidal rule with
a step size of p/T, we obtain the following approx-
imation:
Cðx; y; tÞ � eat
T
(1
2Re½eCðx; y; aÞ�þ
X2Nþ1
k¼0
Re½eCðx; y; pÞ� cosxt
�X2Nþ1
k¼0
Im½eCðx; y; pÞ� sinxt)
ð15Þ
where x = ip/T. Eq. (15) is the basis of the Fourier
inversion method first used by Dubner and Abate [35]
and later improved by others [36–39]. Here the complex
concentrations serve as the Fourier coefficients.
The infinite series in Eq. (15) have been truncated to
2N + 1 terms, which introduces truncation error into the
inversion process. An expression for the error term com-
pared to (2N + 1) ! 1 is given by Crump [37] fromwhich the parameter a can be evaluated. It is given as
a = l � ln(Er)/2T, where l is the order of C(x,y, t) such
that jC(x,y, t)j 6 Melt with M being constant. The term,
Er, is defined [37] as the relative error (Er = E/Melt) and
E is an error term that arises since the Fourier coeffi-
cients are not exact but are approximations usingeCðx; y; pÞ. Sudicky [27] suggests that l = 0, Er = 10�6,
and T = 0.8tmax are adequate for most transport prob-lems and recommends using a = �ln(E 0)/1.6tmax, where
E 0 is the maximum tolerable relative error and tmax is
the maximum time of the simulation.
The complete procedure involves calculatingeCðx; y; pkÞ, [k = 0 . . . 2N] for each value of pk and a sin-
gle value tmax. Once this array is evaluated, inversion at
any time 0.1tmax < t < tmax can then be performed. For
t < 0.1tmax, the absolute error term becomes unmanage-able due the averaging effect of Fourier series at discon-
tinuities (e.g. C(x,y, t) at t = 0) [37]. It is convenient to
T. Lowry, S.-G. Li / Advances in Water Resources 28 (2005) 117–133 123
save the complex concentration array and perform the
inversion as part of a post processing procedure.
2.6. Dimensional and p-space discretization
There are two types of errors associated with theinversion of the LT, approximation error and truncation
error. The first type of error is of a general nature with
respect to any LT solution method that utilizes a Fou-
rier approximation in the inversion routine, and is due
to the approximation of the Fourier series in Eq. (15).
As discussed above, it is controlled by the selection of
the a parameter. In reality, approximation error is minor
in comparison to other sources of error. However, withregards to truncation error there are specific issues asso-
ciated with the IFALT method that are not present in
other LT methods.
If we consider a single element in the IFALT domain
and re-write Eq. (8), assuming no reaction, dispersion,
or source terms we geteCi;j ¼ eCp0e�pDn=U ð16Þ
where Dn is the distance along the streamline from the
central nodal point to the element boundary and U is
the average velocity along the streamline over the dis-
tance Dn. Noting that p = a + ib and b = kx/i = kp/T,equation (16) can be re-written aseCi;j ¼ eCp0e
�aDn=U ðcosðkpDn=ðUT ÞÞ þ i sinðkpDn=ðUT ÞÞÞð17Þ
From Eq. (17) it can be seen that the value of the com-
plex valued concentration at the central nodal point is
periodic in both space and successive values of pk[k = 0 . . . 2N + 1]. The spatial period is proportional to
U and T, and inversely proportional to k. Over k, theperiod is proportional to U and T, and inversely propor-
tional to Dn. This information can be used to determine
appropriate spatial discretization as well as the number
of p values, or what we call p-space discretizations, to
solve for the concentration. Specifically, for the spatial
period, we get
UðDnÞ � 2TUk
ð18Þ
and over k the period is
UðkÞ � 2TUDn
ð19Þ
The periods are given as approximations since the coef-
ficient of the exponent in Eq. (16), eCp0 , is complex val-
ued and thus also periodic, which effectively reduces
the periodicity of eCi;j. With respect to the IFALT meth-od the value of Dn/U is usually small as compared to T,
meaning that consideration must be given to both the
p-space and spatial discretization. For sharp-edged
plumes in a smooth velocity field, good results are
generally obtained using values of T/(DsN) < 3, where
2N + 1 is the number of p evaluations and Ds = Dn/U.
For irregular plumes, the period is reduced by the peri-
odic coefficient so that values of T=ðDsNÞ < 5 are suffi-
cient and where Ds is the domain average of Ds. Forplumes undergoing dispersion, or for soft-edged plumes,these rules can be significantly relaxed. As an example,
for sharp-edged plumes undergoing pure advection, with
travel distances of 250 grid cells (T � 250Ds), values ofN from 25–80 will produce sufficient accuracy. In prac-
tice, we have found values of N from 10–30 to be ade-
quate for most transport problems.
3. Examples and comparisons
Three different hypothetical examples are simulated.
The first example simulates a step function input from
the left hand boundary of a rectangular modelling do-
main, as it moves left to right through a uniform velocity
field at three different Peclet numbers, 300, 120, and 20.
This 1-D example enables comparison to an analyticalsolution and tests the ability of the IFALT method to
model sharp concentration fronts undergoing various
levels of dispersion. Additional simulations with this
configuration are performed with varying p-space and
grid-space discretization to show the sensitivity of the
method to these two parameters. The second example
simulates transport of a Gaussian source plume through
a randomly generated heterogeneous flow field with twodegrees of heterogeneity; one with a log-conductivity
variance of 1.5 and the other at 0.5. Transport is simu-
lated at three different Peclet numbers, 300, 120, and 20.
The third example simulates a Gaussian source plume
through a deterministic sinusoidal velocity field, where
the velocity in the x-direction is given as a constant
and the velocity in the y-direction is given as a sine func-
tion dependent on the x-position in the domain. As theplume moves through the domain, it periodically
deforms and reforms, allowing for direct com-
parison to the initial condition at each sine-wave period.
This example assumes no dispersion or molecular
diffusion.
Comparisons are made to three other numerical
methods and either an analytical solution (Examples 1
and 3) or a high-resolution numerical solution (Example2). For the first example, the three additional numeri-
cal methods are: a finite difference Laplace transform
method (FDLT), a hybrid method of characteristics
(HMOC) [40,41], and an Eulerian–Lagrangian localized
adjoint method (ELLAM) [7,42–47]. For the second
example comparison is made to a random walk parti-
cle tracking method (RW) [4,48] instead of the FDLT
method. The third example compares only the ELLAMand the IFALT methods. Each method is briefly ex-
plained below.
124 T. Lowry, S.-G. Li / Advances in Water Resources 28 (2005) 117–133
The FDLT method uses an upwinding finite differ-
ence method in space, and the LT method in time. This
allows comparison of the IFALT method to a solution
method that is accurate in time but not in space.
The RWmethod is based on [4] and is the same as the
explanation in the introduction. The initial distributionof particles is random within each cell with the number
of particles determined by dividing the user given solute
mass in each cell by the particle mass. RW methods are
free from numerical dispersion so they provide an excel-
lent means to determine the shape and extent of a
plume.
The HMOC method is part of the original MT3D
transport model package [40] and uses the method ofcharacteristics (MOC) in areas of high concentration
gradients and the modified method of characteristics
(MMOC) in areas of low concentration gradients. In
this way, the HMOC method reduces the numerical dis-
persion common with the MMOC method by utilizing
the computationally heavy, yet much more accurate
MOC method only when needed. The HMOC method
is very accurate under most transport conditions. How-ever, even with the inclusion of the more computation-
ally easy MMOC method in the low gradient areas,
high computational costs are still an issue.
The ELLAM scheme used here is the finite-volume
implementation that is part of the MOC3D transport
package [45]. The ELLAM was first introduced in
1990 [7] and due to its sound conceptual basis, is has
undergone significant expansions and development sincethat time, with applications to many practical problems.
As the name implies, it is a �high-resolution� Eulerian–Lagrangian method that solves an integral form of the
ADE by tracking mass associated with fluid volumes
through time [49]. It then separately solves for disper-
sion on a fixed grid in space. Because of its theoretical
foundation, mass conservation is inherent in the EL-
LAM as well as its ability to handle complicated bound-ary functions. It also has the ability to handle large time
steps with Courant conditions �1, which makes it very
computationally efficient as compared to other time-
stepping methods. However, under certain circum-
stances, it can show non-physical oscillations [47].
Simulations were performed on a 2.56Ghz Pentium-4
computer with 1.0Gb of RAM. All codes were compiled
under Compaq Visual Fortran with the default maxi-mum optimizations. Where appropriate, absolute con-
vergence for each solution was set at 5e�5.
3.1. Example 1—square pulse source
The first example uses a heaviside boundary function
on the left hand side of a rectangular modelling domain.
The domain is 225m by 100m with 1m grid spacing inboth directions. The initial and boundary conditions
are given by
Cðx; y; 0Þ ¼ 0 x P 0
Cð0; y; tÞ ¼C0 0 6 t 6 s
0 t > s
�Cð1; y; 0Þ ¼ 0 t P 0
where �1 6 y 6 1, t is time, s is the length of thesource pulse, and C0 is the magnitude of the pulse con-
centration. For this example, s = 25 days and
C0 = 100mg/l. The rest of the boundaries are designated
as advective flux boundaries with inflow concentrations
of zero and outflow concentrations equal to the simu-
lated concentration at the boundary. The longitudinal
dispersivity is al = 0.003333, 0.008333, and 0.05m to
produce Peclet numbers (Pe) of 300, 120, and 20, respec-tively. The simulations are numbered example 1a
(Pe = 300), 1b (Pe = 120), and 1c (Pe = 20). The lateral
and transverse to longitudinal dispersivity ratio is set
to 0.1. The analytical solution used as comparison in