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Advances in Water Resources Vol. 22, No. 7, pp 711±728, 1999
Ó 1999 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0309-1708/99/$ ± see front matter

Numerical methods for reactive transport on
rectangular and streamline-oriented grids
Olaf A. Cirpkaa,c,*, Emil O. Frindb & Rainer Helmiga,d
a
Institut f
ur Wasserbau, Universit
at Stuttgart, Germany
Department of Earth Sciences, University of Waterloo, Waterloo, ON N2L 3G1, Canada
c
Department of Civil and Environmental Engineering, Stanford University, Stanford, CA 94305-4020, USA
d
Institut f
ur Computeranwendungen im Bauingenieurwesen, Technische Universit
at Braunschweig, Pockelsstr. 3, 38106 Braunschweig,

Germany
b

(Received 1 October 1997; revised 1 August 1998; accepted 13 November 1998)

Coupling advection-dominated transport to reactive processes leads to additional
requirements and limitations for numerical simulation beyond those for non-reactive transport. Particularly, both monotonicity avoiding the occurence of
negative concentrations, and high-order accuracy suppressing arti®cial di€usion,
are necessary to study accurately the reactive interactions of compounds transported in groundwater. These requirements are met by non-linear Eulerian
methods. Two cell-centered Finite Volume schemes are presented for the simulation of advection-dominated reactive transport. The ®rst scheme is based on
rectangular grids, whereas the second scheme requires streamline-oriented grids
the generation of which is explained in an accompanying paper. Although
excellent results for conservative transport are obtained by the scheme for rectangular grids, some arti®cial transverse mixing occurs in the case of multi-component transport. This may lead to erroneous reaction rates if the compounds
interact. The transport scheme for streamline-oriented grids, on the other hand,
avoids arti®cial transverse mixing. A quantitative comparison is given by two test
cases. A conservative tracer simulation for a ®ve-spot con®guration in a heterogeneous aquifer shows a high coincidence of the breakthrough curves obtained
for the two methods, whereas a test case of two reacting compounds shows signi®cant di€erences. In this test case, a rate of convergence with respect to the
overall reaction rates lower than ®rst-order is calculated for the rectangular
grid. Ó 1999 Elsevier Science Ltd. All rights reserved
Key words: Transport modeling, Grid orientation, Arti®cial di€usion, Finite

volume schemes.

of the fate and behavior of contaminants in the subsurface, the assessment of related risks and the prediction of the future evolution of contamination scenarios.
The occurence of chemical interactions requires
mixing of the reacting compounds on the local scale.7
Dependent on boundary conditions and the masstransfer properties of the reactants, unsucient mixing
may lead to a strong limitation of the chemical reactions
compared to idealized mixed reactors. Hence, the accurate determination of local-scale mixing processes is a
key issue for the investigation on the interactions between advective±dispersive transport and reactive processes.

1 INTRODUCTION
In groundwater systems, dissolved compounds undergo
advective-dispersive transport, mass-transfer processes
and chemical transformations, both biotic and abiotic.
Numerous studies have appeared in the literature in
recent years investigating the interactions between these
processes by experimental as well as numerical means.
These studies have greatly enhanced the understanding

*

Corresponding
ford.edu

author.

E-mail:

ocirpka@hydro2.stan711

712

O. A. Cirpka et al.

In this context, numerical simulation of multi-component reactive transport is a powerful tool for coupling
reactive processes, studied in detail in lab-scale experiments, to hydrogeological mechanisms, the parameters
of which are generally retrieved by ®eld-scale experiments. This provides the opportunity for comparing
reactive behavior on di€erent scales. In order to achieve
reliable results, it must be ensured that the mixing of the
compounds approximated by the numerical methods
chosen re¯ects the hydrogeologic parameters rather than

numerical errors such as arti®cial di€usion. Therefore
accurate methods for the simulation of reactive transport are needed.
One of the fundamental requirements for the accurate
simulation of reactive transport is that numerical oscillations are not acceptable because they result in negative
concentrations which lead to unstabilities in the calculation of reactive processes9. This is in contrast to nonreactive transport simulations where small oscillations
are unproblematic. Small oscillations typically occur
when linear high-order methods are applied to advection-dominated transport problems. It follows that a
high order of accuracy alone does not guarantee the
correct and stable solution of reactive transport problems. Conversely, linear low-order methods may introduce signi®cant arti®cial di€usion thus leading to an
overestimation of local-scale mixing processes. Therefore, traditional linear transport schemes are not good
choices for simulating advection-dominated transport of
interacting compounds.
In this paper two numerical methods are presented
for the solution of such problems. The ®rst method is
based on the ¯ux-corrected transport (FCT) scheme3 and
formulated for rectangular grids. It may be extended to
curvilinear grids by coordinate transformation, but it
does not require any special orientation of the grid. The
second scheme is based on the slope limiter or MUSCL
approach32 and requires streamline-oriented grids which

are discussed in an accompanying paper8.
The paper is organized as follows: A brief review of
existing FVM schemes for advection-dominated transport is given in Section 2. In Section 3 the general approach underlying both schemes is explained. In
Section 4 the two numerical schemes for conservative
transport are described. Some aspects of solving the
reactive sub-problem and its coupling to advective-dispersive transport are discussed in Section 5. Finally the
two schemes are compared by application to two test
cases for conservative and mixing-controlled reactive
transport in Section 6.

2 REVIEW OF FINITE VOLUME METHODS FOR
ADVECTION-DOMINATED TRANSPORT
Most numerical schemes for multi-component reactive
transport are combinations of existing numerical meth-

ods for conservative transport and for reactions in
mixed systems, respectively. Since the equations describing reactive processes are based on concentrations,
the most common way to solve these equations for
spatially variable domains is to divide the domain into
control volumes and calculate the reactive processes

independently in each of them. It is a natural choice to
use the same spatial discretization for transport, that is
applying a Finite Volume Method (FVM). This has
been done by various authors18,17 and is adopted in the
present study. As an alternative, the Finite Element
Method (FEM) may be used for transport thus requiring the simulation of reactive processes on a nodal basis.20,33 Strictly Lagrangian12 and Eulerian±Lagrangian
schemes34 have also been used for reactive transport, but
most of these schemes included spatial redistribution of
the calculated masses after solution of the transport subproblem. This spatial redistribution causes undesirable
arti®cial di€usion. In most studies the extent of arti®cial
di€usion has not been determined.
Accurate modeling of multi-component reactive
transport requires the underlying transport scheme to be
neither oscillative nor arti®cially di€usive. Oscillations
may cause negative concentrations which do not re¯ect
any physical behavior. Furthermore, common equations
for the description of (bio)reactive processes such as the
Michaelis±Menten terms are discontinuous in the negative concentration range. As a consequence, negative
concentrations caused by spurious oscillations may lead
to serious stability problems. Oscillations are suppressed

if the transport scheme preserves the monotonicity of
the concentration distribution.
On the other hand, reactive interactions of compounds may be limited by insu€ucient mixing of the
compounds. Arti®cial di€usion introduced by the
scheme for advective transport may lead to an overestimation of mixing and related reactions rates. Therefore
arti®cial di€usion must be minimized. Since longitudinal
mixing is often much stronger than transverse, arti®cial
di€usion may be tolerated to a higher extent for the
longitudinal than for the transverse direction. It is well
known that linear, monotonic Eulerian schemes are at
most ®rst-order accurate15 and therefore di€usive. As a
consequence, these schemes are not well suited to reactive transport problems sensitive to di€usive mixing.
The requirement of both monotonicity and secondorder accuracy has led to the development of non-linear
FVM schemes refered to as total variation diminishing
(TVD) or high resolution methods22. Some of these
schemes are based on the reconstruction of concentration distributions within cells such as in the case of the
slope limiter (MUSCL) method32 or the essentially nonoscillatory (ENO) method.16,6 In contrast to these
schemes, the ¯ux-limiter approach31 is based on limiting
the so-called anti-di€usive contributions of the secondorder Lax±Wendro€ ¯uxes such that no new extrema
occur. The limiter functions used for the slope-limiter

Numerical methods for reactive transport on rectangular and streamline-oriented grids
and ¯ux-limiter approaches, respectively, are identical.
They are dependent on the concentration distribution: in
smooth regions no limitation of the Lax±Wendro€
¯uxes is necessary, whereas near discontinuities the
¯uxes approach those achieved by upstream di€erentiation. Le Veque21 interpreted the limited anti-di€usive
¯uxes as correction waves. In the present study, the
slope-limiter approach is applied in the scheme on
streamline-oriented grids (Section 4.2).
The ®rst non-linear TVD methods were formulated
for explicit time-integration and applied only to onedimensional domains. Unfortunately, the extension to
multi-dimensional applications was not straightforward.
In ®rst approaches multi-dimensional problems were
solved by directional splitting.22 As an alternative the
TVD schemes could be written in a semi-discrete form.
Higher-order results were achieved either by Crank±
Nicolson integration in time thus requiring linearization
or by explicit multi-step methods such as the Runge±
Kutta scheme. The latter was applied e.g. in the multidimensional ENO scheme of Casper and Atkins.6

LeVeque24,23 presented an explicit multi-dimensional
extension of the correction-wave approach. However,
the limiter function he applied did not include the in¯uences of diagonally positioned cells and could not
guarantee monotonicity.
A very attractive alternative to the introduction of
non-linear limiter functions is the Flux-Corrected
Transport (FCT) approach3,35. The FCT method has
been developed by Boris and Book3 and was one of the
®rst high-order monotonic methods. The ®rst extension
to multi-dimensions was done by Zalesak.35 In contrast
to other high-resolution methods the scheme could be
transfered to FEM schemes as well.26,25 In the present
study a version of the FCT method is applied in the
scheme on rectangular grids (Section 4.1).
Note that all of the above-mentioned schemes apply
exclusively for the stabilization of advective transport. It
was supposed that dispersive transport always leads to
additional stabilization. However, this is only the case if
the dispersion tensor is a diagonal matrix. As will be
shown in Section 4.1 oscillations due to dispersive

transport occurs for full dispersion tensors the principal
directions of which di€er from those of the grid.

3 GENERAL APPROACH
In the following, two di€erent cell-centered FVM
schemes for two-dimensional modeling of transport will
be presented, the ®rst of which is used for calculations
on rectangular grids with varying grid spacings, whereas
for the second scheme streamline-oriented grids consisting of quadrilateral elements are used. For both
transport schemes the underlying ¯ow-®eld is solved by
the mixed-hybrid FEM. The method for streamlineoriented grid-generation is explained by Cirpka et al.8

713

In the current setup of both schemes the reactive
transport problem is solved by an operator-split approach. Advective transport is solved for each compound by a non-linear explicit method which is followed
by implicit calculation of di€usive/dispersive transport:
!
X
1

a
d
Fk;m …t† ‡ Fk;m …t‡Dt†
c~k …t ‡ Dt† ˆ ck …t† ÿ
/k Rk
m
…1†

in which k determines the cell of interest and m denotes
a
all cells connected to cell k via the advective ¯uxes Fk;m
d
and the di€usive ¯uxes Fk;m , respectively. Note that
these ¯uxes are expressed as total mass transfered over
the entire timestep, the outward direction being positive. c~k …t ‡ Dt† is the intermediate concentration at time
t ‡ Dt considering exclusively advective-dispersive
transport. Eqn. 1 is solved for each mobile compound.
Reactive processes are solved sequentially in a separate
step without implicit feedback to the transport calculation:
ck …t ‡ Dt† ˆ ~ck …t ‡ Dt† ‡

t‡Dt
Z

rk dt

…2†

t

in which ck …t ‡ Dt† is the vector of all compounds considered in cell k at time t ‡ Dt and rk is the vector of
reactive source/sink terms which may contain non-linear
terms. The operator-split coupling has been compared by
the authors to implicit coupling methods including direct coupling and iterative two-step coupling.9 No signi®cant di€erences were found. The modi®cations which
are necessary for the implementation of implicit coupling schemes will be discussed in Section 5.

4 CALCULATION OF CONSERVATIVE TRANSPORT
The main di€erences for numerical simulation of
transport on streamline-oriented grids compared to arbitrarily oriented ones may be explained by the transport equation of a reacting compound which is given for
two dimensions in streamline coordinates by eqn (3) and
in arbitrarily oriented coordinates by eqn (4):


oci o…qci † o
oci
‡
ÿ
…qal ‡ /Dm †
/Ri
ot
on
on
on


o
oci
ˆ /ri ;
…qat ‡ /Dm †
…3†
ÿ
og
og
/Ri

oci o…qx ci † o…qy ci †
‡
‡
ÿ r
…/Drci † ˆ /ri ;
ot
ox
oy

…4†

where Ri is the retardation coecient for compound i, ri
is a source±sink term due to reactive processes which

714

O. A. Cirpka et al.

vanishes in the numerical methods for transport because
of the operator-split approach, / is the porosity, ~
q is the
speci®c discharge vector with its directional components
qx and qy and its absolute value q, n is the spatial coordinate in the direction of ¯ow and g transverse to it, al
and at are the longitudinal and transverse dispersivities,
respectively, and Dm is the molecular di€usion coecient. The full dispersion tensor D in the x; y-coordinate
system can be evaluated by transformation of coordinates:
2 qx qx al ‡qy qy at
3
qx qy …al ÿat †
‡ /Dm
j~
qj
j~
qj
5:
…5†
/D ˆ 4
qx qy …al ÿat †
qy qy al ‡qx qx at
‡ /Dm
j~
qj
j~
qj

Comparing eqn (3) with eqn (4), it is obvious that
advective transport is quasi-one-dimensional on
streamline-oriented grids whereas advective ¯uxes occur
into both principal directions of arbitrarily oriented
grids. This is illustrated in Fig. 1. Therefore using
streamline-oriented grids simpli®es the stabilization of
advective transport, since one-dimensional approaches
can directly be applied. In the scheme presented this is
the slope limiter method32.
Second, the dispersion tensor is diagonal in streamline coordinates whereas it is a full tensor in arbitrarily
oriented. As a consequence, no cross-di€usion terms
need to be evaluated on streamline-oriented grids thus
guaranteeing second-order accuracy for two-dimensional di€erentiation by a ®ve-point stencil. In contrast
to this, nine-point di€erentiation is required to approximate the ¯uxes related to the full dispersion tensor on
arbitrarily oriented grids. This is illustrated in Fig. 2.
As will be shown, adopting streamline-oriented grids
does not only simplify the discretization of advective±
dispersive transport but also avoids adding arti®cial
transverse di€usion which may be necessary in order to
achieve monotonicity on an arbitrarily oriented grid.
This is of particular interest for the transport simulation
of interacting compounds in which reaction rates are
dependent on mixing of the compounds. On the other
hand, the streamline-oriented grid-generation scheme8 is
restricted to steady-state ¯ow®elds, and the extension to

Fig. 1. Di€erences in the approximation of advective transport
on rectangular versus streamline-oriented grids.

Fig. 2. Di€erences in the approximation of dispersive transport
on rectangular versus streamline-oriented grids.

three-dimensional applications may be rather complicated.
4.1 Calculation of transport on a rectangular grid
For the calculation of transport on a rectangular grid
the Flux-Corrected Transport method (FCT)3,35 based on
a cell-centered Finite Volume discretization was chosen.
The basic idea is to combine a low-order monotonic
method and a high-order oscillatory method by means
of a predictor-corrector loop, in which the low-order
scheme acts as predictor. First a time step is solved independently for both methods. In the proceeding steps,
high-order ¯uxes are limited in such a way that no new
extrema with respect to the low-order solution and the
solution of the previous time step occur. Hence monotonicity of the low-order solution is preserved. In the
following sections the chosen low-order and the highorder methods for dispersive and advective transport,
respectively, as well as the implementation of the FCT
method, are explained.
4.1.1 Dispersive transport
In the classical block-centered FVM, the ¯uxes are
evaluated at the centers of the interfaces and multiplied
by the area of the interface. For two-dimensional rectangular cells of constant thickness Dz this yields the
following mass transfered due to dispersive ¯uxes within
a timestep of size Dt:

oc 
x
xx
Fd …i ‡ 1=2; j† ˆ ÿ DtDzDyj …/D †i‡1=2;j 
ox i‡1=2;j

oc 
;
ÿ DtDzDyj …/Dxy †i‡1=2;j 
oy i‡1=2;j

oc 
y
yy
Fd …i; j ‡ 1=2† ˆ ÿ DtDzDxi …/D †i;j‡1=2 
ox i;j‡1=2

oc 
ÿ DtDzDxi …/Dxy †i‡1=2;j 
;
ox i;j‡1=2

…6†

Numerical methods for reactive transport on rectangular and streamline-oriented grids

715

with i being the index in the x-direction and j in the ydirection. In the following …i; j† is the index to the cell in
row i and column j, …i ‡ 1=2; j† is the index to the
interface between the cells …i; j† and …i ‡ 1; j†, and
…i ‡ 1=2; j ‡ 1=2† is the vertex between the cells …i; j†,
…i ‡ 1; j†, …i; j ‡ 1† and …i ‡ 1; j ‡ 1†. The term ``¯ux'' is
used for the total mass transfered within a time step
rather than the mass transported per area and time. The
gradients occuring in eqn (6) perpendicular to an interface and parallel to it may be approximated by ®nite
di€erences:

oc 
2…ci‡1;j ÿ ci;j †
;


ox i‡1=2;j
Dxi ‡ Dxi‡1

oc 
…ci;j‡1 ‡ ci‡1;j‡1 ÿ ci;jÿ1 ÿ ci‡1;jÿ1 †
;


oy i‡1=2
Dyjÿ1 ‡ 2Dyj ‡ Dyj‡1

…7†

with similar expressions for the interface …i; j ‡ 1=2†.
For constant coecients in /D the ¯uxes may therefore
directly be evaluated by inserting eqn (7) or its equivalents into eqn (6). If the coecients di€er in the adjacent
cells, an averaging procedure is necessary. For the entries /Dxx and /Dyy it can be shown that the distanceweighted harmonic average re¯ects continuity of both
the concentration and the normal ¯ux components:
…/Dxx †i‡1=2;j ˆ
yy

…/D †i;j‡1=2

…Dxi ‡ Dxi‡1 †…/Dxx †i;j …/Dxx †i‡1;j
;
Dxi …/Dxx †i‡1;j ‡ Dxi‡1 …/Dxx †i;j

…Dyj ‡ Dyj‡1 †…/Dyy †i;j …/Dyy †i;j‡1
:
ˆ
Dyj …/Dyy †i;j‡1 ‡ Dyj‡1 …/Dyy †i;j

…8†

For the ¯uxes related to the o€-diagonal entries /Dxy ,
diculties arise from the choice of locations for their
evaluation. Inserting eqn (7) into eqn (6) yields a dependence between the cells …i ‡ 1; j ‡ 1† and …i; j†. As
illustrated by Fig. 3, the in¯uence of cell …i ‡ 1; j ‡ 1† on
cell …i; j† is evaluated at the edges …i ‡ 1=2; j† and
…i; j ‡ 1=2†, whereas the in¯uence of cell …i; j† on cell …i ‡
1; j ‡ 1† is evaluated at the edges …i ‡ 1=2; j ‡ 1† and
…i ‡ 1; j ‡ 1=2†. This may result in a non-symmetric
system of equations.
Arbogast et al.1 applied a combination of the trapezoidal and the mid-point rule in the evaluation of the
¯uxes, in order to guarantee symmetry. However, this is
only applicable for constant coecients in /D and
smooth grids. Arbogast et al.2 introduced Lagrangian
multipliers as additional unknowns on the edges if the
coecients in /D di€er in the two cells related to the
edge. Note that for most practical applications the ¯ow
®eld will be non-uniform throughout the entire domain.
As a consequnce, Lagrangian multipliers would be
necessary at almost every edge, ®nally resulting in a
mixed-hybrid FEM formulation based on the RT0
function space2. This system of equation contains approximately twice the number of unknowns than the

Fig. 3. In¯uence of cell …i; j† on cell …i ‡ 1; j ‡ 1† and vice versa
by cross-di€usion terms. Black dots: interfaces for cross-diffusive ¯uxes related to cell …i; j†, grey dots: interfaces for crossdi€usive ¯uxes related to cell …i ‡ 1; j ‡ 1†.

original one. Note that the approach of Arbogast et al.2
has been developed for groundwater ¯ow problems
rather than dispersive transport. For larger domains the
assumption of constant coecients is more realistic for
the hydraulic conductivity than the dispersion tensor.
By contrast, symmetric systems of equations are always achieved in the FEM regardless of the anisotropy
of dispersion tensors, even on deformed grids. Symmetric equations may be prefered due to their faster
solution. Assume a virtual element the corners of which
are identical to the cell midpoints as illustrated in
Fig. 4(a). Assuming further a constant value of /Dxy
which is evaluated at the vertex …i ‡ 1=2; j ‡ 1=2†, the
mass change due to the ¯uxes related to the o€-diagonal
entry in the dispersion tensor /Dxy is for the FEM:

Fig. 4. Evaluation of ¯uxes related to the o€-diagonal entry
Dxy at a vertex. (a) Di€usive and anti-di€usive ¯ux at the vertex
i ‡ 1=2; j ‡ 1=2. (b) Di€erentiation stencil for cell i; j and
choice of the points of evaluation: black dots: ¯uxes related to
Dyy , circles: ¯uxes related to Dxx , gray dots: ¯uxes related to
Dxy .

716

O. A. Cirpka et al.

Mxy
d …i

1=2Dx
Z j‡1
Z i‡1 1=2Dy

‡ 1=2; j ‡ 1=2† ˆ ÿDtDz

…rN†

For cell …i; j† there are now eight ¯uxes to be considered. The locations of evaluation are shown in
Fig. 4(b). These ¯uxes are calculated by:

T

ÿ1=2Dxi ÿ1=2Dyj



0
/Dxy

2

ci;j

Fdx …i ‡ 1=2; j†

3


6 c
7
/Dxy
6 i‡1;j 7
rN dx dy 6
7;
4 ci‡1;j‡1 5
0

ˆ ÿDtDzDyj …/Dxx †i‡1=2;j

ci;j‡1

ÿ/Dxy
6
DtDz 6 0
ˆ
6
2 4 /Dxy
0
2
3
ci;j
6 c
7
6 i‡1;j 7
6
7;
4 ci‡1;j‡1 5
2

/Dxy

0

/D

xy

0
ÿ/Dxy

0
ÿ/Dxy
0

2…ci‡1;j ÿ ci;j †
;
Dxi ‡ Dxi‡1

Fdy …i; j ‡ 1=2†
0

3

ˆ ÿDtDzDxi …/Dyy †i;j‡1=2

ÿ/Dxy 7
7
7
0 5
/Dxy

…9†

ci;j‡1

with clockwise numbering of the cells around the vertex
and the bilinear shape function N. In the FEM, the
approximation of the related temporal concentration
change is of course di€erent to the cell-centered FVM.
Only for the case of regular spacing in both directions
the related volumes are identical. For this particular case
eqn (9) can also be retrieved by the FVM.1 Nevertheless,
in the present study eqn (9) is also applied to grids with
irregular spacing.
Equation (9) may be interpreted as summation of two
diagonal di€usive ¯uxes across the vertex in which the
product of the di€usion coecient times the interfacial
area divided by the distance equals 1=2/Dxy for the ¯ux
between the lower left and the upper right cell and
ÿ1=2/Dxy for the ¯ux between the upper left and lower
right cell, respectively. From the de®nition of /Dxy in
eqn (5) it is clear that its value may be positive or negative. It vanishes only if the principal directions of the
grid and the dispersion tensor coincide. For all other
cases one relation between diagonally positioned cells
will be of a di€usive nature but with a negative di€usion
coecient. This is refered to as anti-di€usive relation. It
is important to notice that anti-di€usion does not occur
in nature, leads to unphysical sharpening of concentration distributions and may cause oscillations in the vicinity of discontinuities.
Because of the anti-di€usive ¯uxes, the approximation of full dispersion tensors on arbitrarily oriented
grids destroys the monotonicity of the scheme. That is,
even if a TVD method such as those listed above is
chosen for advective transport, the accurate approximation of dispersive ¯uxes by nine-point di€erentiation
may still lead to over- or undershooting of the concentration distribution. In contrast to this, the dispersion
tensor is a diagonal matrix on perfectly streamline-oriented grids, and therefore monotonic ®ve-point stencils
may be applied without the introduction of cross-diffusive errors.

2…ci;j‡1 ÿ ci;j †
;
Dyj ‡ Dyj‡1

Fdxy …i ‡ 1=2; j ‡ 1=2†
…10†
DtDz
…/Dxy †i‡1=2;j‡1=2 …ci‡1;j‡1 ÿ ci;j †;
ˆÿ
2
Fdyx …i ‡ 1=2; j ‡ 1=2†
DtDz
…/Dxy †i‡1=2;j‡1=2 …ci;j‡1 ÿ ci‡1;j †;
ˆ
2
in which Fdxy …i ‡ 1=2; j ‡ 1=2† is directed from cell …i; j†
to cell …i ‡ 1; j ‡ 1† and Fdyx …i ‡ 1=2; j ‡ 1=2† from cell
cell …i; j ‡ 1† to cell …i ‡ 1; j†. In this scheme, /Dxy is to
be evaluated at the vertex. For each cell a di€erent value
may be approximated at this location. The e€ective
value is calculated by volume-weighted harmonic averaging of the cell-related values:
…/Dxy †i‡1=2;j‡1=2
Vi;j ‡ Vi;j‡1 ‡ Vi‡1;j ‡ Vi‡1;j‡1
ˆ Vi;j
V
V
V
‡ …/Di;j‡1
‡ …/Di‡1;j
‡ …/Di‡1;j‡1
xy †
xy †
xy †
…/Dxy †
i;j

i;j‡1

i‡1;j

:

…11†

i‡1;j‡1

It appears physically reasonable that diagonal ¯uxes
only appear if the coecient is of same sign in all related
cells. Otherwise the diagonal ¯uxes are deleted. Note
that, in contrast to /Dxx and /Dyy , harmonic averaging
is not a direct result from continuity considerations.
In the framework of the FCT method, the low-order
sub-scheme must preserve monotonicity. Otherwise the
FCT solution is not monotonic. As a consequence, in
the low-order scheme the anti-di€usive diagonal ¯uxes
need to be eliminated, whereas the di€usive diagonal
¯uxes may remain. In terms of a truncation error analysis this leads to a second-order (di€usive) error of the
low-order method. In the high-order method both types
of diagonal ¯uxes are accounted for, since monotonicity
is not required for this sub-scheme. The low-order and
high-order di€usive ¯uxes Ld and Hd , respectively, are
therefore de®ned by:
Lxd …i ‡ 1=2; j† ˆ Hdx …i ‡ 1=2; j† ˆ Fdx …i ‡ 1=2; j†;
Lyd …i; j ‡ 1=2† ˆ Hdy …i; j ‡ 1=2† ˆ Fdy …i; j ‡ 1=2†;
Hdxy …i ‡ 1=2; j ‡ 1=2† ˆ Fdxy …i ‡ 1=2; j ‡ 1=2†;
Lxy
d …i ‡ 1=2; j ‡ 1=2†
( xy
Fd …i ‡ 1=2; j ‡ 1=2†
ˆ
0

if …/Dxy †i‡1=2;j‡1=2 > 0;
if …/Dxy †i‡1=2;j‡1=2 < 0;

Hdyx …i ‡ 1=2; j ‡ 1=2†; ˆ Fdyx …i ‡ 1=2; j ‡ 1=2†;

Numerical methods for reactive transport on rectangular and streamline-oriented grids
Lyx
d …i ‡ 1=2; j ‡ 1=2†
( yx
Fd …i ‡ 1=2; j ‡ 1=2†
ˆ
0

717

if …/Dxy †i‡1=2;j‡1=2 < 0;
if …/Dxy †i‡1=2;j‡1=2 > 0:
…12†

Indices are identical to those of eqn (10).
4.1.2 Advective transport
For the approximation of advective transport, the correction-wave approach of LeVeque24 has been modi®ed
and implemented into the FCT framework. The introduction of correction terms in eqns (15), (17) and (20) is
written in pseudo-code: The corrected ¯uxes are on the
left side of these equations, whereas the unmodi®ed
¯uxes appear with the same notation on the right side.
Low-order method. As low-order approximation, the
multi-dimensional upwind method for explicit time-integration as presented by Colella 10 is applied. Point of
departure is the dimensional upwind method which
yields for positive velocity components qx and qy , de®ned at the interfaces of the cells:

Fig. 5. Principle of the Corner Transport Upwind method
(CTU) 10. Fluxes originated from cell i; j.
y‡
with the correction factor Ci‡1;j
of cell …i ‡ 1; j† related
to the outward-directed ¯ux qyi‡1;j‡1=2 as de®ned by:

…13†

in which Lxa …i ‡ 1=2; j† is the low-order advective mass
¯ux into the x-direction from cell …i; j† to cell …i ‡ 1; j†
and Lya …i; j ‡ 1=2† is the equivalent ¯ux into y-direction
from cell …i; j† to cell …i; j ‡ 1†, respectively. At this state,
no diagonal advective ¯uxes are de®ned:
Lxy
a …i ‡ 1=2; j ‡ 1=2†
yx
La …i ‡ 1=2; j ‡ 1=2†

ˆ 0;

…14†
ˆ 0;
in which the indices are identical to those of the dispersive ¯uxes. As illustrated in Fig. 5, the ¯ux originated
from cell …i; j† may be interpretated as shifting the area
of cell …i; j† along the characteristics and redistributing
the mass to the intersecting cells. As a consequence, a
fraction of the ¯ux from cell …i; j† is oriented to the diagonally positioned downstream cell …i ‡ 1; j ‡ 1†, partially via cell …i ‡ 1; j† and partially via cell …i; j ‡ 1†. The
latter may be of importance for non-uniform ¯ow ®elds.
Assuming that cell …i ‡ 1; j† is downstream of cell …i; j†
the consideration of the diagonal ¯ux from cell …i; j† to
cell …i ‡ 1; j ‡ 1† via cell …i ‡ 1; j† leads to the following
corrections:
xy
Lxy
a …i ‡ 1=2; j ‡ 1=2† ˆ La …i ‡ 1=2; j ‡ 1=2†
y‡
;
‡ ci;j ui‡1=2;j Ci‡1;j
y‡
Lxa …i ‡ 1=2; j† ˆ Lxa …i ‡ 1=2; j† ÿ ci;j ui‡1=2;j Ci‡1;j
;

Lya …i

‡ 1; j ‡ 1=2† ˆ

Lya …i

‡ 1; j ‡ 1=2†

y‡
ÿ ci‡1;j ui‡1=2;j Ci‡1;j
;

…15†

y‡
ˆ
Ci‡1;j

Dt
qy
:
2/i‡1;j Dyj i‡1;j‡1=2

…16†

Similar results are obtained if cell …i; j ‡ 1† is assumed
to be downstream of cell …i; j† and cell …i ‡ 1; j ‡ 1†
downstream of cell …i; j ‡ 1†:
xy
Lxy
a …i ‡ 1=2; j ‡ 1=2† ˆ La …i ‡ 1=2; j ‡ 1=2†
x‡
‡ ci;j vi;j‡1=2 Ci;j‡1
;

Lxa …i ‡ 1=2; j ‡ 1† ˆ Lxa …i ‡ 1=2; j ‡ 1†
ÿ

…17†

x‡
;
ci;j‡1 vi;j‡1=2 Ci;j‡1

x‡
Lya …i; j ‡ 1=2† ˆ Lya …i; j ‡ 1=2† ÿ ci;j vi;j‡1=2 Ci;j‡1
;
x‡
of cell …i; j ‡ 1† related
with the correction factor Ci;j‡1
to the outward-directed ¯ux qxi‡1=2;j‡1 as de®ned by:
x‡
Ci;j‡1
ˆ

Dt
qx
:
2/i;j‡1 Dxi i‡1=2;j‡1

…18†

In the equations above and in all further discussions
of transverse propagation, only the case is considered in
which both qx and qy are positive. Any other case including those of switching signs may be treated by
symmetry. Note that in the original CTU formulation10
terms similar to cross-di€usion expressions are used to
correct the ¯uxes into the principal directions rather
than directly considering diagonally oriented ¯uxes.
High-order method. Colella10 suggested that the consideration of the diagonal ¯uxes in the CTU approach
may already be viewed as an approximation of the
mixed derivatives occuring in the second-order Taylor
expansion of the transport equation. LeVeque24,23

718

O. A. Cirpka et al.

mentioned that this approximation is slightly di€erent to
the one obtained by the Lax±Wendro€ scheme19.
However, both authors claim second-order accuracy if
only the directional second-order derivatives are added
to the CTU solution. This results in the following highorder ¯uxes:

…19†
LeVeque24,23 suggested that the transverse propagation of the low-order ¯ux may be applied as well to the
second-order correction terms. He did this by modifying
cross-di€usion terms. These terms may again be reformulated so that diagonal ¯uxes occur directly. This
yields to the following additional modi®cations related
to the correction term Bxi‡1=2;j :

Although the resulting scheme is almost oscillation-free,
it is not yet monotonic. Furthermore no limitation of
anti-di€usive ¯uxes due to cross-di€usion is conducted.
As a consequence, in the present method the secondorder corrector terms are taken as advective contribution to the anti-di€usive ¯uxes considered in the FCT
context.
4.1.3 Flux-corrected transport
The implementation of the FCT methods for two-dimensional applications follows mainly the approach of
Zalesak35. In contrast to the original formulation,
however, we assume that di€erences between high-order
and low-order ¯uxes do not only occur in the advective
terms but also in the di€usive ones since the discretization of cross-di€usion leads to anti-di€usive ¯uxes which
need to be eliminated in the low-order scheme.
Another modi®cation of the present implementation
concerns the direct consideration of diagonal ¯uxes both
in the advective and di€usive terms. This consideration
yields a FCT formulation which is similar to the scheme
developed for FEM discretizations26.
First the mass ¯uxes of the high-order and low-order
method H and L, respectively, are evaluated independently from each other according to the procedures
described in Section 4.1.1 for the di€usive contribution
and Section 4.1.2 for the advective, respectively. Total
¯uxes are calculated by summation of the di€usive and
advectice contributions. From these ¯uxes so called antidi€usive ¯uxes A are evaluated for all edges and diagonals:

Haxy …i ‡ 1=2; j ‡ 1=2† ˆ Haxy …i ‡ 1=2; j ‡ 1=2†

Ax …i ‡ 1=2; j† ˆ H x …i ‡ 1=2; j† ÿ Lx …i ‡ 1=2; j†;

y‡
ui‡1=2;j Bxi‡1=2;j Ci‡1;j
;

Ay …i; j ‡ 1=2† ˆ H y …i; j ‡ 1=2† ÿ Ly …i; j ‡ 1=2†;

‡

Hax …i ‡ 1=2; j† ˆ Hax …i ‡ 1=2; j†

Axy …i ‡ 1=2; j ‡ 1=2† ˆ H xy …i ‡ 1=2; j ‡ 1=2†

y‡
ÿ ui‡1=2;j Bxi‡1=2;j Ci‡1;j
;

Haxy …i ‡ 1=2; j ‡ 1=2† ˆ Haxy …i ‡ 1=2; j ‡ 1=2†

ÿ Lxy …i ‡ 1=2; j ‡ 1=2†;
Ayx …i ‡ 1=2; j ‡ 1=2† ˆ H yx …i ‡ 1=2; j ‡ 1=2†

x‡
‡ vi;j‡1=2;j Bxi‡1=2;j Ci;j‡1
;
x‡
Hay …i; j ‡ 1=2† ˆ Hay …i; j ‡ 1=2† ÿ vi;j‡1=2;j Bxi‡1=2;j Ci;j‡1
:

…20†
Similarly the following diagonal corrections are performed for the correction term Byi;j‡1=2 :
Haxy …i ‡ 1=2; j ‡ 1=2† ˆ Haxy …i ‡ 1=2; j ‡ 1=2†
x‡
;
‡ ui‡1=2;j Byi;j‡1=2 Ci;j‡1

Hax …i

‡ 1=2; j† ˆ

Hax …i

‡ 1=2; j† ÿ

x‡
ui‡12;j Byi;j‡1 Ci;j‡1
;
2

Haxy …i ‡ 1=2; j ‡ 1=2† ˆ Haxy …i ‡ 1=2; j ‡ 1=2†
x‡
‡ vi;j‡1=2 Byi;j‡1=2 Ci;j‡1
;

Hay …i; j

‡ 1=2† ˆ

Hay …i; j

‡ 1=2† ÿ

x‡
vi;j‡1=2 Byi;j‡1=2 Ci;j‡1
:

…21†
In the work of LeVeque24,23 the second-order corrector terms are limited by dimensional ¯ux-limiters.

…22†

ÿ Lyx …i ‡ 1=2; j ‡ 1=2†;
with Axy …i ‡ 1=2; j ‡ 1=2† being directed from cell i; j to
cell i ‡ 1; j ‡ 1 and Ayx …i ‡ 1=2; j ‡ 1=2† from cell i; j ‡ 1
to cell i ‡ 1; j. These anti-di€usive ¯uxes are individually
xy
and
limited by speci®c factors Ti‡1=2;j , Ti;j‡1=2 , Ti‡1=2;j‡1=2
yx
Ti‡1=2;j‡1=2 :
Ax;c …i ‡ 1=2; j† ˆ Ti‡1=2;j Ax …i ‡ 1=2; j†;
Ay;c …i; j ‡ 1=2† ˆ Ti;j‡1=2 Ay …i; j ‡ 1=2†;
Axy;c …i ‡ 1=2; j ‡ 1=2†
xy
ˆ Ti‡1=2;j‡1=2
Axy …i ‡ 1=2; j ‡ 1=2†;

…23†

Ayx;c …i ‡ 1=2; j ‡ 1=2†
yx
ˆ Ti‡1=2;j‡1=2
Ayx …i ‡ 1=2; j ‡ 1=2†:

The corrector solution is evaluated by summing all
corrected anti-di€usive ¯uxes belonging to a cell to the
low-order solution:

Numerical methods for reactive transport on rectangular and streamline-oriented grids
Dci;j ˆ Ax;c …i ÿ 1=2; j† ÿ Ax;c …i ‡ 1=2; j†
‡ Ay;c …i; j ÿ 1=2† ÿ Ay;c …i; j ‡ 1=2†
‡ Axy;c …i ÿ 1=2; j ÿ 1=2†
ÿ Axy;c …i ‡ 1=2; j ‡ 1=2†

…24†

‡ Ayx;c …i ÿ 1=2; j ‡ 1=2†
ÿ Axy;c …i ‡ 1=2; j ÿ 1=2†;
l
c^fct
i;j ˆ ci;j ‡

Dci;j
;
Vi;j

in which Vi;j ˆ Ri;j Dxi Dyj Dz/i;j is the e€ective volume of
cell i; j and cli;j is the low-order solution.
The limitation procedure listed in Appendix A is a
straightforward extension of Zalesak's35 method including diagonal contributions.
4.2 Calculation of transport on a streamline-oriented grid
The second scheme for transport calculations presented
is based on streamline-oriented grids containing quadrilateral cells. The method of grid-generation is explained in detail by Cirpka et al.8. The transport scheme
presented is an adaption of the principal direction (PD)
technique13,14 to cell-centered Finite Volumes. In order
to stabilize advective transport, a slope-limiter22 method
is used.
4.2.1 Approximation of gradients in streamline-oriented
elements
Spatial gradients along the direction of the streamlines
and orthogonal to it are required for the approximation
of dispersive ¯uxes as well as for the slope-limiter
method described below. Fig. 6(a) shows a perfectly
streamline-oriented curvilinear quadrilateral the edges
of which are orthogonal to each other. However, the

719

bilinear elements generated by the grid-generator8 are
no longer perfectly adopted to the streamlines (see
Fig. 6(b)). Particularly, perfect orthogonality of bilinear
element edges can only be achieved in regions of parallel
¯ow. By a transformation of coordinates both types of
elements may be transformed to a unit square in local
coordinates (see Fig. 6(c)). The direction of ¯ow may be
from the left-hand side to the right-hand side.
For the curvilinear element the e€ective lengths for
the evaluation of gradients parallel and orthogonal to
the direction of streamlines would equal the length of
the curves along the streamline and the pseudopotential
line, respectively, starting from point C and ending at
the edges of the element. Both the streamline and the
pseudopotential line through point C divide the element
into pieces of equal area. As illustrated in Fig. 6(c), these
lines point, due to the streamline-oriented grid-generation procedure, into the direction of the local coordinates n and g, respectively. Therefore point C may be
referred to as center of gravity with respect to the local
coordinates. The bilinear approximation leads in global
coordinates to linear lines of action which may not be
exactly orthogonal to the edges. Their lengths are evaluated by the following procedure:
· Determine the center of gravity C with respect to
the local coordinates …nC ; gC †.
· Transform the local coordinates of point C as well
as of the intermediate points I …nC ; 0†, II …1; gC †, III
…nC ; 1† und IV …0; gC † into global coordinates.
· Determine the distances between the intermediate
points and point C in global coordinates.
In the following the distances mentioned will be referred to either as the half apparent lengths lkp;k =2 and
lkk;q =2 of element k with respect to the longitudinal interfaces …p; k† and …k; q†, respectively, or as the half apparent widths wkk;m =2 and wkh;k of element k with respect
to the transverse interfaces …k; m† and …h; k†, respectively.
In this context, the elements p and q are directed

Fig. 6. Approximation of gradients oriented into the direction of ¯ow and orthogonal to it. (a) Perfectly streamline-oriented element; (b) Approximation by a bilinear quadrilateral; (c) Quadrilateral in local coordinates. C: Center of gravity with respect to the
local coordinates n; g.

720

O. A. Cirpka et al.
in which J is the Jacobian for the transformation of
coordinates. The integrals in eqn (28) can be evaluated
analytically. Transformation into global coordinates is
done following the isoparametric concept.

Fig. 7. Apparent widths and lengths of element k. Direction of
¯ow: cells k ! i ! m.

from element k into the direction of ¯ow, whereas the
elements h and m are directed into the direction of
pseodopotential lines. For illustration see Fig. 7.
The longitudinal gradient rck;q between cells k and q
may now evaluated by:
rck;q ˆ

2…cq ÿ ck †
:
lkk;q ‡ lqk;q

…25†

The procedure for the transverse gradient rck;m between cells k and m is equivalent:
rck;m ˆ

2…cm ÿ ck †
:
wkk;m ‡ wmk;m

…26†

The local coordinates …nC ; gC †of point C may be calculated by division of the ®rst areal moments with respect to the local directions MnA1 and MgA1 , respectively,
by the area of the element:
nC ˆ

MnA1
;
A

gC ˆ

MgA1
;
A

…27†

with
Aˆ

Z1 Z1
0

MnA1

ˆ

0

n det… J† dn dg;

Z1 Z1

g det… J† dn dg;

0

MgA1

ˆ

0

dl
ˆ DtDzwlk;q …qk;q al ‡ /Dm †rck;q ;
Fk;q

…29†

dt
Fk;m
ˆ DtDzltk;m …qe at ‡ /Dm †rck;m ;

…30†

in which the gradients rck;q and rck;m are de®ned in
eqns (25) and (26), respectively, and qk;q is the volumetric ¯ux across the longitudinal interface between
elements k and q. Eqn. (30) includes the e€ective volumetrix ¯ux qe for a transverse interface. Since the
volumetric ¯uxes are de®ned at the longitudinal rather
than the transverse interfaces, qe is evaluated by arithmetic averaging of the ¯uxes at the two longitudinal
interfaces belonging to each element followed by distance-weighted harmonic averaging of the element-related ¯uxes:
qe ˆ

det… J† dn dg;

Z1 Z1

4.2.2 Dispersive transport
As stated in the previous section and illustrated in
Fig. 6, the approximation of the streamline-bounded
cells by quadrilaterals leads to some non-orthogonality.
Applying strict transformation of coordinates to the
dispersion tensor would yield o€-diagonal entries. This
artefact is only due to the linearization of the element
edges introduced in the last step of the grid-generation
procedure8. In the preceeding steps the streamlines and
pseudopotential lines were tracked accurately leading to
a multi-point approximation of the element edges. If
these lines would have been kept in the description of
the elements, the dispersion tensor would be a diagonal
matrix in local coordinates. Therefore the authors assume that the neglection of cross-di€usion terms is justi®ed also for the quadrilateral elements used since this
only corrects for an error introduced by a preceeding
step of simpli®cation.
For this purpose the interfaces and the concentration
gradients derived in the previous section are treated as if
they were orthogonal which would be the case for
curvilinear elements. Introducing eqns (25) and (26)
into the de®nition of dispersive ¯uxes leads to the
evaluation of the di€usive ¯uxes across longitudinal
interfaces eqn (29) and transverse interfaces eqn (30),
respectively:

…28†

…wkk;m ‡ wmk;m †qk qm
;
qk wmk;m ‡ qj wkk;m

…31†

in which qk and qm are the arithmetic averaged volumetric ¯uxes in the elements k and m, respectively.

0

0

4.2.3 Advective transport
For the calculation of advective transport on streamlineoriented grids the slope limiter method is applied32. This

721

Numerical methods for reactive transport on rectangular and streamline-oriented grids
scheme is an extension of Godunov's explicit method15.
For given cell-averaged concentrations, a piecewise linear innercell distribution of the concentration is reconstructed. This concentration distribution is used to solve
exactly for the Riemann problem of advective transport.
Finally the cell-related concentrations are computed by
spatial averaging. It is obvious that this approach preserves monotonicity if the reconstruction of the innercell
concentration distributions does not lead to new extrema.
De®ning a slope sk inside of cell k, the mass transported over the interface k; q within a timestep is given
by:
a
Fk;q

~ k;q qk;q DtDz
ˆw

cnk

‡

snk

lkk;q ÿ vk;q Dt
2

!!

;

…32†

~ k;q is the width of the interface between cells
in which w
k and q. Note that in eqn (32) vk;q ˆ qk;q =/ is the
seepage velocity whereas qk;q is the volumetric ¯ux. The
crucial point of the method is the choice of the slope.
Some natural choices shown in Fig. 8 would be the
linear interpolation to the upstream or downstream cell
dwn
midpoints sup
lin and slin , respectively, and the slopes
yielding the values of the up- or downstream cells didwn
rectly at the interface sup
max and smax . For non-rectangular quadrilaterals these natural slopes may be
calculated by:

sdwn
klin ˆ
sdwn
kmax ˆ
sup
klin ˆ

2…cq ÿ ck †
;
lkk;q ‡ lqk;q
2…cq ÿ ck †

;
max lkk;q ; lqk;q

2…ck ÿ cp †
;
lpp;k ‡ lkp;k

sup
kmax ˆ

…33†

2…ck ÿ cp †

:
max lpp;k ; lkp;k

In the implementation of the slope limiter method
for streamline-oriented grids, a modi®cation of Roe's
Superbee limiter29 is used. The choice of the innercell
slope is given by:
up
if sdwn
lin
slin < 0 then s ˆ 0;
dwn
up
elseif jsup
max j < jslin j then s ˆ smax ;
up
dwn
elseif jsdwn
lin j > jslin j then s ˆ slin ;

…34†

up
dwn
elseif jsup
lin j < jsmax j then s ˆ slin ;

else
s ˆ sdwn
max :
Note that eqn (34) leads to a non-linear dependence
of the advective mass ¯ux approximated on the concentration distribution. However, since the scheme for
advective transport is fully explicit, this does not result
in signi®cant additional computational e€ort.

5 CALCULATION OF REACTIVE PROCESSES
AND COUPLING CONSIDERATIONS

Fig. 8. De®nition of slopes for element k. Open circles: averaged concentrations in elements p (upstream), k and q
(downstream).

For the solution of the system of di€erential-algebraic
equations (DAES) describing the reactive processes, the
solver DASSL is used27,4. DASSL is an adaptation of
Gear's sti€ method for systems of ordinary di€erential
equations to di€erential-algebraic systems which has
recently been modi®ed to include sparse matrices and
iterative solvers5. In an alternative approach the system
of algebraic equations (AES) may be decoupled from
the system of ordinary di€erential equations (ODES).
This has been applied by various authors18,20,33. For this
approach either the AES is solved ®rst and the resulting
concentrations are taken as initial conditions for the
solution of the ODES, or vice versa.
For certain types of reactive processes the operatorsplit coupling may not be accurate enough. This should
be checked for one-dimensional model problems before
transfering the scheme to multi-dimensional applications. If implicit coupling turns out to be necessary, the
transport schemes presented above need to be veri®ed.
These veri®cations will be explained in the following.
Two alternative schemes of coupling may be considered. In the iterative two-step method both the reactive processes and the advective-dispersive transport
are solved in independent steps. In contrast to the

722

O. A. Cirpka et al.

operator-split scheme the reactive source-sink term is
considered in the transport step as zero-order term,
whereas the impact of transport on the reactions is
considered in the reactive step as explicit source-sink
term. The terms of interaction are updated iteratively
until a de®ned convergence criterion is reached.
In this scheme the reactive source-sink terms are taken into account in the transport step. This requires the
¯uxes to be evaluated at the center of the timestep
(Crank±Nicolson integration). As a consequence, for the
method on rectangular grids neither the CTU approach
for the low-order method nor the second-order correction terms in the high-order method are applicable.
These schemes are restricted to explicit time integration.
However, using central di€erentiation in a Crank±
Nicolson integration scheme is already of second-order.
In the low-order method diagonal upwinding following
the scheme of Roe and Sidilkover30 may be applied. This
scheme is identical to the one of Rice and Schnipke28 but
transfered to the cell-centered FVM.
For the explicit slope limiter method as presented for
the transport calculations on a streamline-oriented grid,
no consideration of source-sink terms is possible. This
limitation may be overcome by reformulating the
scheme in a semi-discrete form (see e.g. in the book of
LeVeque22). Then again accurate and monotonic results
can be obtained using Crank±Nicolson time integration.
Note that, due to the non-linearity of the slope limiter
method, any (semi)-implicit time integration requires
linearization which may be performed by a Newton
method.
Comparing the FCT method and the slope limiter
method in the context of (semi)-implicit time integration, it is obvious that the FCT method is more ecient
since linearization is restricted to a single predictorlimiter-corrector loop whereas a Newton method for
linearization of the slope limiter method may require
several iterations. In contrast to this, the slope limiter
method is more ecient in an explicit time integration
scheme since the FCT method requires to solve every
timestep by two di€erent methods before applying the
limiter procedure.
The iterative two-step method for coupling as discussed above guarantees the coupling error to be in a
prede®ned range. However, the decoupled treatment of
the reactive and advective-dispersive parts of the problem may lead to non-convergent iterations if the discretization in time is not chosen in an adaptive manner.
Furthermore the scheme may be inecient due to double linearization, once by the two-step method and once
within the method for solving the reactive sub-problem.
Therefore direct coupling, solving the reactive and the
advective-dispersive terms at once, may be attractive.
Direct coupling can be performed by discretizing
advective-dispersive transport only in time and adding
the resulting ODES to the reactive DAES thus leading
to a very large DAES of the order ncomp nnode which may

be solved by a solver such as DASSL. This requires the
transport discretization to ®t into the method-of-lines
approach. The semi-discrete slope limiter method meets
this requirement whereas the FC