Compatible algorithms for coupled flow and transport

Comput. Methods Appl. Mech. Engrg. 193 (2004) 2565–2580

www.elsevier.com/locate/cma

Compatible algorithms for coupled flow and transport q

Clint Dawson *, Shuyu Sun, Mary F. Wheeler

Center for Subsurface Modeling-C0200, Texas Institute for Computational Engineering and Sciences, The University of Texas at Austin,

Austin, TX 78712, USA

Received 10 April 2003; received in revised form 7 November 2003; accepted 18 December 2003

Abstract

The issue of mass conservation in numerical methods for flow coupled to transport has been debated in the literature

for the past several years. In this paper, we address the loss of accuracy and/or loss of global conservation which can

occur when flow and transport schemes are not compatible. We give a definition of compatible flow and transport

schemes, with emphasis on two popular types of transport algorithms, the streamline diffusion method and discon-

tinuous Galerkin methods. We then discuss several different approaches for flow which are compatible with these

transport algorithms. Finally, we give some numerical examples which demonstrate the possible effects of incompati-

bility between schemes.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Flow; Transport; Mass conservation; Streamline diffusion method; Discontinuous Galerkin methods

1. Introduction

The transport of chemically reactive species arises in a number of important applications. Specific

examples include groundwater contamination, water quality modeling and air quality modeling. This

physical process is described by advection–diffusion–reaction systems.The advection and diffusion of chemical species are governed by a velocity field, which is generally given

by a flow model. The flow model is application-dependent, and may be described, for example, by Darcy

flow in the subsurface, a hydrodynamic model in shallow water or the Navier–Stokes equations. In each of

these models, the velocity field satisfies a continuity equation (conservation of mass equation) of the form

r � u ¼ f ; ð1Þwhere u is the velocity field, and f is an external source/sink function.

qThe first author was supported in part by NSF grant DMS-0107247.* Corresponding author. Tel.: +1-512-475-8627; fax: +1-512-471-8694.

E-mail address: [email protected] (C. Dawson).

0045-7825/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2003.12.059

mail to: [email protected]

2566 C. Dawson et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) 2565–2580

We will consider transport equations for each chemical species of the form

oð/ciÞot

þr � ðuci � DðuÞrciÞ ¼ f~ci þ Riðc1; . . . ; cnÞ; ð2Þ

where ci denotes the concentration of species i, i ¼ 1; . . . ; nc; nc is the number of chemical species, DðuÞ is a(possibly) velocity-dependent diffusion/dispersion tensor which is symmetric and positive semi-definite, / is

a volumetric factor such as porosity, and Ri is a chemical reaction term. The concentration ~ci is usually

specified at sources (where f > 0) and ~ci ¼ ci at sinks ðf < 0Þ. In some cases, there could be feedback fromthe transport model to the flow model; that is, u ¼ uðc1; . . . ; cncÞ.

It is well known that in many cases (2) is advection-dominated, which can lead to steep concentration

gradients. Moreover, these concentration fronts may be made even steeper by the presence of chemical

reactions. Therefore, when solving (2) numerically, it is essential that the numerical method preserve these

steep gradients with minimal oscillation and numerical diffusion. The transport method should also be

provably accurate, at least for smooth solutions, and satisfy global conservation of chemical mass.

In recent years, there has been discussion in the literature about ‘‘locally conservative’’ methods for flow

and transport; see, for example [3,6,10,15]. Local conservation refers to conservation of mass or species overa control volume or an element in a finite element or finite difference grid. Local conservation combined with

flux continuity guarantees global conservation of a numerical scheme. For a number of transport schemes,

local conservation is a by-product of the fact that these schemes use discontinuous approximating spaces

combined with numerical fluxes to model advection-dominated transport. When modeling flow on its own,

local conservation, in the sense of satisfying (1) locally, may or may not be important. However, when

coupling flow with transport, how the flow model handles (1) can be quite important, and can directly affect

the accuracy and conservation properties of the transport method.

In this paper, we address the numerical modeling of coupled flow and transport. Our goal is to determine‘‘compatible’’ numerical methods for flow and transport. That is, we wish to determine the minimal

requirements on the flow algorithm to maintain certain accuracy and conservation properties of the

numerical method used for transport. In order to fix ideas, we will consider three transport algorithms: the

Streamline Diffusion (SD) method [4,12,13], and two variants of the discontinuous Galerkin (DG) method,

the local discontinuous Galerkin method (LDG) [7–9] and the primal discontinuous Galerkin methods

[14,16,22]. The SD method uses a standard, continuous Galerkin formulation. Both DG methods use

discontinuous approximating spaces defined over each element. These methods are all globally conserva-

tive, stable and accurate given the true velocity field u. However, they may lose these properties, dependingon how one approximates u. In particular, our primary result in this paper is that appropriately satisfying

(1) numerically is crucial to preserving the accuracy, stability and global conservation properties of these

transport methods.

The paper is organized as follows. In the next section, we briefly describe the SD and DG transport

algorithms and discuss their accuracy and conservation properties. In Section 3, we discuss several flow

algorithms for the specific case of an elliptic, stationary flow problem, with emphasis on how these algorithms

approximate (1). These include the standard continuous Galerkin method, the mixed finite element method,

and DG methods based on the primal and dual forms of the flow equation. In particular, we discuss thecompatibility of these algorithms with the SD and DG transport schemes. Finally, in Section 4, we give some

numerical results which further illuminate our findings, followed by conclusions and further discussion.

2. The streamline diffusion and discontinuous Galerkin transport algorithms

In order to simplify the discussion, we consider the following single transport equation:

ct þr � ðuc� DrcÞ ¼ f~c; ðx; tÞ 2 X� ð0; T �; ð3Þ

C. Dawson et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) 2565–2580 2567

defined on an open, fixed domain X 2 Rd with smooth boundary oX. Let n denote the unit outward normal

to oX. We assume oX is divided into inflow and noflow/outflow regions

CI ¼ fx 2 oX : u � n < 0g; ð4Þ

CO ¼ fx 2 oX : u � nP 0g: ð5Þ

On these boundaries we assume the boundary conditions,

ðcu� DrcÞ � n ¼ cIu � n; CI � ð0; T �; ð6Þ

ðDrcÞ � n ¼ 0; CO � ð0; T �: ð7Þ

We also impose the initial condition

cðx; 0Þ ¼ c0ðxÞ; x 2 X: ð8ÞWe discretize X by a finite element partition Th of elements Xe with diameter he. Let h denote the

maximal element diameter. Let Dt > 0 denote a time step, with tn ¼ nDt, n ¼ 0; 1; . . . Denote by Sn the

space–time slab X� ½tn; tnþ1Þ. We discretize Sn by a space–time finite element partition TnDt;h. Assuming

Dt ¼ OðhÞ, the maximal element diameter in TnDt;h is also OðhÞ.

We denote by ð�; �ÞR the standard L2 inner product over domain R. Surface integrals are denoted by h�; �iR.

2.1. The SD method

The SD method is based on the following reformulation of (3). Using (1), we write

r � ðucÞ ¼ u � rcþ fc ð9Þ

and

ct þ u � rc�r � ðDrcÞ ¼ f ð~c� cÞ; ðx; tÞ 2 X� ð0; T �: ð10ÞLet V n

h denote the space of continuous, piecewise linear functions in space and piecewise linears in time

defined on the partition TnDt;h of the slab Sn. Note that these functions may be discontinuous from one slab

to the next. Therefore, we denote by

v�ðtnÞ ¼ lims!0�

vðx; tn þ sÞ: ð11Þ

The SD diffusion method then is to find C 2 V nh satisfying, for n ¼ 0

ðC�ð�; 0Þ � c0ð�Þ; vþð�; 0ÞÞX ¼ 0; v 2 V 0h ð12Þ

and for each n ¼ 0; 1; . . .,

ðCt þ u � rC; vþ dðvt þ u � rvÞÞSn þ ðDrC;rvÞSn þ hu � nðcI � CÞ; viCI�ðtn;tnþ1Þ

þ ðCþð�; tnÞ � C�ð�; tnÞ; vþð�; tnÞÞX ¼ ðf ðeC � CÞ; vÞSn ; v 2 V nh : ð13Þ

The parameter d is typically chosen to be OðhÞ, but can also be chosen depending on the size of D [12].

2.2. The OBB method

The Oden–Babu�ska–Baumann discontinuous Galerkin method uses completely discontinuous approx-

imating spaces to approximate c. Let


Wh ¼ fw : wjXe2 PkðXeÞg;

where Pk denotes the space of complete polynomials of degree kP 1. We also need some additionalnotation. For adjacent elements Xþ

e and X�e with unit outward normals n�, let w� denote the trace of w on

the face e between X�e from the interiors of the elements. Define the average f�g and jump s � t for x 2 e as

follows:

fwg ¼ ðw� þ wþÞ=2; ð14Þ

swt ¼ wþnþ þ w�n�: ð15ÞFurthermore, for x 2 e, define the upwind value of w as follows:

wu ¼ w�; u � n� > 0;wþ; u � nþ > 0:

�ð16Þ

Let Ei denote the set of all interior element faces in the finite element mesh.

The OBB method (in continuous time) is then defined as follows. Find Cð�; tÞ 2 Wh satisfying

ðCð�; 0Þ � c0;wÞX ¼ 0; w 2 Wh ð17Þand for t > 0

ðCt;wÞX � ðuC � DrC;rwÞX þ hCuu� fDrCg; swtiEiþ hfDrwg; sCtiEi

þ hcIu � n;wiCI

þ hCu � n;wiCO¼ ðf eC ;wÞX; w 2 Wh: ð18Þ

2.3. The LDG method

The local discontinuous Galerkin method also uses completely discontinuous approximating spaces. The

method defines two auxiliary variables

~z ¼ �rc; ð19Þ

z ¼ D~z ð20Þand approximates z and ~z by functions Z and eZ in the space

Wh ¼ ðWhÞd : ð21ÞFor functions v 2 Wh we define

svt ¼ v � nþ þ v � n�

analogous to (15). The scheme differs from the OBB method in the way that the diffusion term is handled.Advection is handled in exactly the same way as in the OBB method.

The LDG method is defined as follows. Find Cð�; tÞ 2 Wh, Z, eZ 2 Wh, satisfying

ðCð�; 0Þ � c0;wÞX ¼ 0; w 2 Wh ð22Þand for t > 0

ðCt;wÞX � ðuC þ Z;rwÞX þ hCuuþ Z�; swtiEiþ hcIu � n;wiCI

þ hCu � n;wiCO¼ ðf eC ;wÞX; w 2 Wh;

ð23Þ

ð eZ ; vÞX � ðC;r � vÞX þ hCþ; svtiEiþ hC; v � nioX ¼ 0; v 2 Wh; ð24Þ


ðZ;~vÞX ¼ ðD eZ ;~vÞX; ~v 2 Wh: ð25ÞWe note that, because eZ and v are discontinuous, eZ can be eliminated element by element in terms of C in

(24). Similarly, Z can be eliminated element by element in terms of eZ by (25). Substituting into (23), we

obtain a system in C only.

2.4. Accuracy and global conservation

The schemes outlined above have all been proven to be stable, and through a priori error analysis, to beaccurate for smooth solutions c [8,12,16]. In particular, they all satisfy error estimates of the form

max06 t6 T

kc� CkL2ðXÞ 6Khp; ð26Þ

for some exponent pP 1 depending on the polynomial degree, and K a constant independent of h. Thisanalysis assumes the true velocity u is known. An analysis of the DG method with approximate velocity Ucan be found in [10].

Furthermore, each method is globally conservative in the following sense. Integrating (3) over X� ð0; tkÞand applying the boundary and initial conditions, we findZ

Xcðx; tkÞdxþ

Z tk

0

ZCO

cu � ndsdt ¼ZXc0ðxÞdx�

Z tk

0

ZCI

cIu � ndsdt þZ tk

0

ZXf~cdxdt: ð27Þ

For the SD method for example, setting v � 1 on Sn, n ¼ 1; . . . ; k and using the fact that r � u ¼ f , we findfrom (13),

ZXC�ðx; tkÞdxþ

Z tk

0

ZCO

Cu � ndsdt ¼ZXc0ðxÞdx�

Z tk

0

ZCI

cIu � ndsdt þZ tk

0

ZXf eC dxdt: ð28Þ

Similar statements hold for the OBB and LDG methods.Now we ask, what happens to the accuracy and global mass conservation properties of these methods if

u � U . In particular, we ask

1. Is the scheme still zeroth-order accurate; that is, if the solution c is identically a constant, do the methods

reproduce c?2. Is the scheme still globally conservative in the sense of (28) (with U replacing u)?

With respect to question 1, we note that if the initial, boundary and source data are all equal to aconstant c, then c � c for all time. The ability to reproduce a constant may seem trivial, but it is important

in many transport applications. In particular, it is often the case that the solution c is constant over large

parts of the domain for long periods of time. If the transport scheme can no longer reproduce a constant

when U � u, then spurious sources and sinks can be created in the transport solution, which can lead to

numerical inaccuracy (overshoot and undershoot) of the solution.

We now address questions 1 and 2 above with respect to each of the transport schemes.

First, we note that when using a numerical method to compute u, it is possible that the computed

velocity may not have a uniquely defined normal component across each element face, particularly if U isdiscontinuous. However, all of our transport schemes rely on knowing U � n on certain element faces. We

will assume in the discussion below that such a quantity is known on each edge, either as part of the flow

computation or by postprocessing U , and to be consistent with our discussion in Section 3, we denote this

quantity as bU � n.


2.5. SD method with approximate u

We consider now (13) with U replacing u; that is

ðCt þU � rC; vþ dðvt þU � rvÞÞSn þ ðDrC;rvÞSn þ h bU � nðcI � CÞ; viCI�ðtn;tnþ1Þ

þ ðCþð�; tnÞ � C�ð�; tnÞ; vþð�; tnÞÞX ¼ ðf ðeC � CÞ; vÞSn ; v 2 V nh : ð29Þ

It is easily seen that if c0 ¼ cI ¼ eC ¼ c, then C � c satisfies (29) independent of U . Thus, since C is unique,

the SD method is always zeroth-order accurate.

To check for global conservation, set v � 1 in (29), then

ðCt þU � rC; 1ÞSn þ h bU � nðcI � CÞ; 1iCI�ðtn;tnþ1Þ þ ðCþð�; tnÞ � C�ð�; tnÞ; 1ÞX ¼ ðf ðeC � CÞ; 1ÞSn : ð30Þ

Summing on n and using (12) we find

ðC�ðx; tkÞ; 1ÞX þXk�1

n¼0

½ðU ;rCÞSn þ h bU � n; cI � CiCI�ðtn;tnþ1Þ� ¼ ðc0; 1ÞX þXk�1

n¼0

ðf ; eC � CÞSn : ð31Þ

In order to obtain the analogue of (28), U and bU should satisfy

�ðU ;rvÞSn þ h bU � n; vioX�ðtn;tnþ1Þ ¼ ðf ; vÞSn ; v 2 V nh : ð32Þ

That is, setting v ¼ C in (32) and substituting into (31) gives (28) with u replaced by U . Setting v � 1, we

obtainZ tnþ1

tn

ZoX

bU � ndsdt ¼Z tnþ1

tn

ZXf dxdt; ð33Þ

which is a statement of ‘‘global conservation’’ of the flow field U; (32) is a more stringent requirement.

2.6. The OBB and LDG methods with approximate u

The OBB method with U replacing u is given by

ðCt;wÞX � ðUC � DrC;rwÞX þ hCu bU � fDrCg; swtiEiþ hfDrwgsCtiEi

þ hcI bU � n;wiCI

þ hC bU � n;wiCO¼ ðf eC ;wÞX; w 2 Wh: ð34Þ

Setting w � 1 and integrating in time, we findZXCðx; tkÞdxþ

Z tk

0

ZCO

C bU � ndsdt ¼ZXc0ðxÞdx�

Z tk

0

ZCI

cI bU � ndsdt þZ tk

0

ZXf eC dxdt: ð35Þ

Similarly, setting w � 1 in (23), we obtain the same result. Thus, both the OBB and LDG methods are

globally conservative, in the sense of (35), independent of how U is computed.Now we check to see whether these schemes are zeroth-order accurate. First, we check to see whether or

not C ¼ cI ¼ c satisfies (34). The left side of (34) becomes

ch�ðU ;rwÞX þ h bU ; swtiEi

þ h bU � n;wioXi: ð36Þ

For this term to be equal to

cðf ;wÞX;which is the right-hand side of (34) in this case, we need (for c 6¼ 0)


�ðU ;rwÞX þ h bU ; swtiEiþ h bU � n;wioX ¼ ðf ;wÞ; w 2 Wh: ð37Þ

Note that if w ¼ 1 on element Xe and w ¼ 0 elsewhere, we obtainZoXe

bU � nds ¼ZXe

f dx; ð38Þ

which is the usual meaning of a ‘‘locally conservative’’ flow field. However, (37) is a stronger statement of

local conservation, since it must hold not only for piecewise constants, but for all functions in the space Wh.

For the LDG method, setting C ¼ c in (24), we find by the divergence theorem that eZ � 0 and thus by

(25), Z � 0. Then setting C ¼ cI ¼ c in (36) we obtain exactly (36). Thus, for the LDG method to be zeroth-

order accurate, we also need (37) to be satisfied. We note that the usage of the conservation form for the

LDG and the OBB methods results in the loss of the zeroth-order accuracy in general cases.

3. Numerical methods for flow

The results of the last section can be summarized as follows. For the SD method to be globally con-

servative, the approximate flow field U must be ‘‘globally conservative’’ in the sense of (32). For the OBB

and LDG methods to be zeroth-order accurate, the flow field U must be ‘‘locally conservative’’ in the sense

of (37). Thus, if the flow field satisfies (37), we say it is compatible with the SD method. If it satisfies (37), we

say it is compatible with the OBB and LDG methods. In this section, we discuss various flow methods and

how they relate to these conditions.To fix ideas, we consider the following model which arises in single phase flow in porous media. Find

velocity u and pressure p satisfying

u ¼ �Krpr � u ¼ f

�x 2 X; ð39Þ

with boundary conditions

p ¼ gD; CD; ð40Þ

u � n ¼ gN � n; CN ; ð41Þ

where CD [ CN ¼ oX. Here K is permeability or hydraulic conductivity and is related to the properties of

the porous medium.

3.1. LDG method

The LDG method can be applied also to the flow equation (39) [5]. Define discontinuous, piecewisepolynomial approximating spaces W F

h and WFh � ðW F

h Þd similar to Wh and Wh in the previous section. The

LDG method consists of finding U � u, U 2 WFh , and P � p, P 2 W F

h satisfying

ðK�1U ; vÞX � ðP ;r � vÞX þ hbP ; svtiEiþ hbP ; v � nioX ¼ 0; v 2 WF

h ; ð42Þ

�ðU ;rwÞX þ h bU ; swtiEiþ h bU � n;wioX ¼ ðf ;wÞ; w 2 W F

h : ð43Þ

The ‘‘numerical fluxes’’ bP and bU must be determined. In their simplest form they are defined on a face e by

bP ¼ fPg; e 2 Ei; ð44Þ


bP ¼ P ; e 2 CN ; ð45Þ

bP ¼ gD; e 2 CD ð46Þand

bU ¼ fUg þ rsPt; e 2 Ei; ð47Þ

bU ¼ gN ; e 2 CN ; ð48Þ

bU � n ¼ U � nþ rðP � gDÞ; e 2 CD; ð49Þwhere the parameter r > 0 is chosen to be Oðh�1Þ. We see immediately by (37) and (43) that the LDG

method for flow with Wh � W Fh is compatible with the OBB and LDG methods for transport.

The LDG flow method is also compatible with the SD method, if formulated by extending the flow

approximating space W Fh to be defined on the space–time mesh Tn

Dt;h, and changing the integrals in (32) and

(43) to integrals over the space–time slab Sn. Then, by (32) and (43), the LDG method for flow with V nh � W F

h

is compatible with the SD method for transport. More precisely, the LDG method for flow using discon-

tinuous, piecewise polynomials defined on the space–time mesh TnDt;h is compatible with the SD method, as

long as the piecewise polynomials used to define V nh are contained in the flow space.

3.2. The discontinuous Galerkin methods

The discontinuous Galerkin family of flow methods includes the OBB scheme, the Nonsymmetric

Interior Penalty Galerkin (NIPG), the Symmetric Interior Penalty Galerkin (SIPG) and the Incomplete

Interior Penalty Galerkin (IIPG) methods, all of which can be applied to solve (39) [2,11,17,20,22]. In these

methods, we seek an approximation P 2 W Fh satisfying

ðKrP ;rwÞX � hfKrPg; swtiEi� hðKrP Þ � n;wiCD

þ sformhfKrwg; sPtiEi

þ sformhðKrwÞ � n; P iCDþ hrsPt; swtiEi

þ hrP ;wiCD

¼ ðf ;wÞX � hgN � n;wiCNþ sformhðKrwÞ � n; gDiCD

þ hrgD;wiCD; w 2 W F

h ; ð50Þ

where the parameter r is positive for SIPG, IIPG and NIPG and r ¼ 0 for the OBB method, and sform ¼ 1

for OBB and NIPG, sform ¼ �1 for SIPG, and sform ¼ 0 for IIPG. Note that r is optimally Oðh�1Þ for SIPG,

IIPG and NIPG, that is

r ¼ jh; j ¼ Oð1Þ;

but j is not required to be ‘‘sufficiently large’’ for NIPG; see [19]. Defining a velocity U by

U ¼ �KrP ; Xe; ð51Þ

bU ¼ �fKrPg þ rsPt; Ei; ð52Þ

bU ¼ gN ; CN ; ð53Þ

bU � n ¼ �ðKrPÞ � nþ rðP � gDÞ; CD; ð54Þ

then (50) can be written, similar to the LDG method, as


�ðU ;rwÞX þ h bU ; swtiEiþ h bU � n;wioX þ sformhfKrwg; sPtiEi

¼ ðf ;wÞ þ sformhðKrwÞ � n; gD � P iCD; w 2 W F

h : ð55Þ

Thus, the OBB, NIPG and SIPG methods do not quite satisfy (37) or (32). However, the IIPG method for

flow with Wh � W Fh is compatible with the OBB and LDG methods for transport. Moreover, the IIPG method

for flow using discontinuous, piecewise polynomials defined on the space–time mesh TnDt;h is compatible with

the SD method, as long as the piecewise polynomials used to define V nh are contained in the flow space.

3.3. The standard Galerkin method

Another approach for flow which can be made compatible with the SD method is to use a standardGalerkin finite element method to compute an approximation to p. Here, we would seek P 2 V n

h \ fv : v ¼gD on CDg satisfying

ðKrP ;rvÞSn þ hgN � n; viCN�ðtn;tnþ1Þ ¼ ðf ; vÞSn ð56Þ

for all v 2 V nh with v ¼ 0 on CD. Defining U ¼ �KrP on Sn and bU ¼ gN on CN , we have

�ðU ;rvÞSn þ h bU � n; viCN�ðtn;tnþ1Þ ¼ ðf ; vÞSn : ð57Þ

Thus, we almost have (32) except that we do not have a flux defined on CD. We can however, postprocess

and obtain a flux on CD.

Let V nh;D be the set of functions in V n

h which are nonzero on CD. Then, find a 2 V nh;D satisfying

ha; viCD�ðtn;tnþ1Þ ¼ ðf ; vÞSn þ ðU ;rvÞSn � h bU � n; viCN�ðtn;tnþ1Þ; v 2 V nh;D: ð58Þ

Define the flux bU � n ¼ a on CD � ðtn; tnþ1Þ. Combining (57) and (58) we obtain (32).

3.4. The mixed finite element method

Finally, we briefly mention another popular method for solving (39), the mixed finite element method(MFE) [18]. The MFE method solves for approximations to both u and p simultaneously, similar to the

LDG method. However, in the MFE method the velocity space WFh � Hðdiv;XÞ; thus, functions in this

space have continuous normal component across element faces. The pressure space Wh satisfies

r �WFh ¼ Wh, and generally consists of discontinuous, piecewise polynomials. Define

WFh;g ¼ WF

h \ fv : v � n ¼ g � n on CNg:

Then, in the MFE, we seek U 2 WFh;gN

and P 2 W Fh satisfying

ðK�1U ; vÞX � ðP ;r � vÞX ¼ hgD; v � niCD; v 2 WF

h;0; ð59Þ

ðr �U ;wÞX ¼ ðf ;wÞX; w 2 W Fh : ð60Þ

By the continuity of the normal flux U � n, one can integrate (60) by parts to obtain

�ðU ;rwÞX þ hU � n; swtiEiþ hU � n;wiCD

¼ ðf ;wÞX � hgN � n;wiCN: ð61Þ

Thus, the MFE method is compatible with the OBB and LDG transport methods if Wh 2 W Fh , and it is

compatible with the SD method if mixed finite element spaces can be constructed on the partition TnDt;h

such that V nh � W F

h .


4. Numerical results

In this section, we present some numerical results examining the compatibility of the various flow and

transport schemes outlined above.

We first consider the one-dimensional transport equation

ct þ ðucÞx � Dcxx ¼ fc; 0 < x < 1; t > 0 ð62Þwith u ¼ cosðpx=2Þ and f ¼ ux ¼ �p=2 sinðpx=2Þ. D is chosen to be 0.001. We choose K ¼ 1 in (39) with

Dirichlet boundary conditions pð0Þ ¼ 0 and pð1Þ ¼ �2=p. The flow direction then is from left to right, and

we specify inflow concentration cI at x ¼ 0. We consider the LDG and SD methods for transport, coupled

to the LDG and standard Galerkin methods for flow.

In the LDG method, we take piecewise linear approximations for both flow and transport. Thus, (37) is

satisfied. In computing flow, the penalty r ¼ 1=h. For (62), we integrate in time using a second order,

explicit Runge–Kutta method and apply a slope limiter at each step in the computation [1].For the standard Galerkin method for flow, the velocity bU has been computed on interior faces using a

postprocessing method described in [3], which gives a locally conservative flux in one space dimension in the

sense of (38). However, this postprocessed flow field is still not compatible with the LDG transport scheme

using piecewise linears, because it only satisfies (37) for piecewise constant functions w.In our first experiment, we test the LDG method for a constant solution of c � 1, obtained by setting the

inflow concentration cI and initial condition c0 both equal to one. We discretize the interval ½0; 1� with 50

grids blocks, solve the flow equation with f , K, pð0Þ and pð1Þ as described above, and then run a transport

simulation up to time T ¼ 0:5. Here we are testing the ability of the LDG method to propagate a constantexactly for many time steps. As seen in Fig. 1, when using LDG for flow we obtain the constant solution

c � 1. When using standard Galerkin for flow, we seem some overshoot and undershoot of the profile. This

exhibits the fact that flow fields which do not satisfy the compatibility condition (37) can introduce spurious

sources and sinks into the LDG transport solution.

Next, we consider the problem of propagating a concentration front through the domain. For this

example, we take the inflow concentration cI ¼ 1 and the initial condition c0 ¼ 0:1. We discretize the

interval ½0; 1� with 50 grid blocks. For the SD diffusion method, we take a constant space time mesh with

Dt ¼ h. Thus, for the SD method, we only need to solve for the flow field once at the beginning of the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.9

1

x

c

Fig. 1. LDG for transport with constant solution c � 1; dashed line is with LDG for flow, solid line is with standard Galerkin for flow.


simulation. We have also implemented a shock-capturing algorithm described in [12], whereby we replace Din (62) by bD ¼ maxðD; c2h2jct þ ucxjÞ and d ¼ c1 maxð0;Dt � bDÞ. The parameters c1 and c2 were chosen to

be 0.1. The solutions obtained by the SD method at T ¼ 0:5 with the LDG and standard Galerkin flow

methods are given in Fig. 2. Note that both flow methods give virtually identical transport solutions,

namely a traveling front joining two constant regions where c ¼ 1 and c ¼ 0:1. Moreover, they are both

compatible with the SD method, and the mass conservation errors are within computer roundoff.

The LDG method is applied to the same problem using LDG and standard Galerkin for flow. These

results are given in Fig. 3. Note that using standard Galerkin for flow causes some mild overshoot and

undershoot in the constant regions behind and ahead of the front. This inaccuracy is again caused by thefact that the standard Galerkin method for flow is not compatible with the LDG method for transport.

In Fig. 4, we directly compare the two transport methods for the problem above, using the LDG method

for flow in both schemes. The SD method has slight overshoot and undershoot near the head and tail of the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

x

c

Fig. 3. LDG for transport, with LDG for flow (+) and standard Galerkin for flow ()).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

x

c

Fig. 2. SD for transport, with LDG for flow (+) and standard Galerkin for flow ()).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

x

c

Fig. 4. Comparison of SD ()) and LDG (+) for transport, with 50 spatial elements.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

x

c

Fig. 5. Comparison of SD ()) and LDG (+) for transport, with 100 spatial elements.


front. The LDG method shows no oscillations, but is more numerically diffusive. Refining the grid to 100elements, both solutions improve, with the LDG solution still more numerically diffusive, see Fig. 5.

The next test involves coupling the SD method with the MFE method for flow. As noted above, the

MFE method can be made compatible with the SD method with the appropriate choice of spaces, in

particular, by approximating p in the space of piecewise linears. A more popular MFE, however, involves

approximating p by piecewise constants. In fact, this approach has been shown to be equivalent to a cell-

centered finite difference method for (39) [21]. However, this ‘‘lowest order’’ MFE method is not compatible

with the SD method, or for that matter, with the LDG or OBB transport methods. For the SD method, this

could lead to mass conservation errors. We again consider the problem of propagating a front through thedomain using the SD method with the lowest order MFE for flow, only this time we take as our initial

condition c0 ¼ 0. We compute the solution up to time T ¼ 0:75 and monitor the total mass conservation

errors for different mesh spacings. These errors are given in Table 1, where we see that while the mass error

is certainly small, it is not zero. In fact the mass error appears to be approaching zero with a rate of h2.

Table 1

Total mass errors for the SD method with the lowest order MFE method for flow

h Mass error

0.1 )0.00120.05 )0.000320.025 )0.000079

Fig. 6. Flow result from OBB.

Fig. 7. Flow result from IIPG.


Fig. 8. Transport result from OBB based on flow result from OBB.


The final example involves coupling OBB for transport with the IIPG method for flow (OBB–IIPG), andcomparing it to coupling OBB for flow with OBB for transport (OBB–OBB). We consider (3) and (39) on

the two-dimensional domain X ¼ ð0; 1Þ2. In this case, D and K are constant diagonal tensors with Dii ¼ 10:0and Kii ¼ 10:0. Initial concentration c0 and in-flow concentration cI are 1.0 uniformly. The flow boundary is

divided into CN ¼ ðf0g � ð0; 1ÞÞ [ ðð0; 1Þ � f1gÞ and CD ¼ oX n CN . The boundary pressure gD ¼ 0 on

ðð0; 1Þ � f0gÞ and gD ¼ 100 on ðf1g � ð0; 1ÞÞ. We know in this case that the exact solution is c � 1. The

simulation is run until time T ¼ 1, and backward Euler is used to march in time with Dt ¼ 0:01. A uniform

Fig. 9. Error of transport result from OBB based on flow result from OBB.


16 · 16 mesh and the space of discontinuous piecewise polynomials of total degree two are used for solvingboth transport and flow. The penalty parameter in the IIPG method is chosen according to the formula

r ¼ r0r2K1=2=h, where r is the order of the local polynomial space, K is the scalar permeability, h is the

element size and r0 ¼ 1:0� 10�10.

The flow results from using the OBB and IIPG methods are shown in Figs. 6 and 7, respectively. The

velocity fields from the two methods seem to be quite similar. However, the transport result using the

velocity from the OBB method has a significant error (see Fig. 8 for the concentration profile at t ¼ 1 and

Fig. 9 for its error), while the transport result using the velocity from the IIPG method has no error up to

machine precision. Clearly, the property of zeroth-order accuracy of the combined OBB–IIPG methodresults in essentially no error in this example, and the violation of this property by the combined OBB–OBB

method results in significant error accumulation for concentration.

5. Conclusions

In this paper, we have examined the compatibility of various flow and transport methods, based on using

either continuous or discontinuous finite element methods. It was found that the SD method is alwayszeroth-order accurate whereas both the OBB and the LDG methods are always globally conservative for

transport, independent of how velocity is computed. The global conservation of the SD method and the

zeroth-order accuracy of the OBB and the LDG methods for transport hold only when a compatible

algorithm for flow is used. We have also shown that the LDG, the IIPG and the MFE methods for flow can

be compatible with the SD and the LDG methods for transport, under certain conditions. The NIPG, the

OBB and the SIPG for flow are not compatible with the SD and the LDG methods for transport. We have

demonstrated, through numerical examples, how incompatible methods may produce erroneous solutions,

or result in loss of mass. Though we have concentrated primarily on transport in porous media, the ideaspresented here extend to other application areas, including water quality modeling.

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Compatible algorithms for coupled flow and transport

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