INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 40, 2493—2516 (1997)

ON OPEN BOUNDARIES IN THE FINITE ELEMENTAPPROXIMATION OF TWO-DIMENSIONAL

ADVECTION—DIFFUSION FLOWS

FRANCISCO PADILLA

Instituto del Agua, ºniversidad de Granada, Rector ¸o& pez Argueta s/n, 18071 Granada, Spain

YVES SECRETAN AND MICHEL LECLERC

Institut National de la Recherche Scientifique - Eau 2800 rue Einstein, suite 105, Ste-Foy, Que&bec, Canada, G1V 4C7

SUMMARY

A steady-state and transient finite element model has been developed to approximate, with simple triangularelements, the two-dimensional advection—diffusion equation for practical river surface flow simulations.Essentially, the space—time Crank—Nicolson—Galerkin formulation scheme was used to solve for a givenconservative flow-field. Several kinds of point sources and boundary conditions, namely Cauchy and Open,were theoretically and numerically analysed. Steady-state and transient numerical tests investigated theaccuracy of boundary conditions on inflow, noflow and outflow boundaries where diffusion is important(diffusive boundaries). With the proper choice of boundary conditions, the steady-state Galerkin and thetransient Crank—Nicolson—Galerkin finite element schemes gave stable and precise results for advection-dominated transport problems. Comparisons indicated that the present approach can give equivalent ormore precise results than other streamline upwind and high-order time-stepping schemes. Diffusive bound-aries can be treated with Cauchy conditions when the flow enters the domain (inflow), and with Openconditions when the flow leaves the domain (outflow), or when it is parallel to the boundary (noflow).Although systems with mainly diffusive noflow boundaries may still be solved precisely with Openconditions, they are more susceptible to be influenced by other numerical sources of error. Moreover, thetreatment of open boundaries greatly increases the possibilities of correctly modelling restricted domains ofactual and numerical interest. ( 1997 by John Wiley & Sons, Ltd.

KEY WORDS: finite elements; advection—diffusion; open boundaries; boundary conditions; open conditions; Cauchyconditions

INTRODUCTION

For the past two decades, the resolution of practical advection—diffusion problems with anEulerian approach has been the source of numerous innovative numerical techniques. Thishas been so essentially because the application of the standard space—time centered Crank—Nicolson—Galerkin scheme to the transport equation shows stable and precise solutions onlywhen the mesh Peclet and Courant numbers do not exceed certain values within the solutiondomain. The treatment of convection-dominated transport problems leads to high Peclet andCourant numbers and therefore to non-physical oscillations and errors.

To prevent such oscillations, two general classes of numerical schemes can be found in theliterature. The first class of early ‘upwind’ schemes relies on an apparent step-down of spatialaccuracy in the discretization process (‘upstream weighting methods’1). More recent upwind

CCC 0029—5981/97/132493—24$17.50 Received 3 January 1996( 1997 by John Wiley & Sons, Ltd. Revised 2 September 1996

methods use different types of weighting functions in the streamline direction to extend the rangeof application of the classical sharp front transport problem.2~4 In the second class, a step-up oftemporal and spatial accuracy is the basis of the remedies. In general, the proposed schemes leadto a higher-order space—time accuracy in the approximation of the convection-dominatedtransport problem.5~7

These two classes of schemes both avoid local mesh refinement near sharp fronts; instead, theyintroduce additional numerical calculations and diffusion to extend the application of theadvection—diffusion equation to higher ranges of Peclet numbers. It has been recently demon-strated that more sophisticated non-diffusive finite element schemes have some difficulties whenthe transport of sharp fronts must be simulated.8,9 However, to perform efficiently, these schemesrequire a rigorous check of Peclet and Courant numbers, which is no longer possible, or unique,in three-dimensional practical problems. To overcome the difficulties normally experiencedin the solution of the sharp front transport problem in the whole range of Peclet and Courantnumbers, Noorishad et al.8 suggested to introduce into the commonly used Crank—Nicolson—Galerkin scheme a very simple corrective function with the help of the diffusion tensor. Thiscorrective function can be interpreted as an extra longitudinal dispersion along the directions ofthe flow lines for which a corrective action is required, which acts only at the points and in thedirections that require stabilization. Consequently, results are stable and do not show overdiffu-sion signals.

Spatial oscillations in solutions to the transport equation are related to the inability ofa discrete mesh to resolve an arbitrarily steep gradient in the dependent variable. It has alreadybeen demonstrated10, 11 that steep gradients normally occur only in the vicinity of boundaries.Wiggly and unstable behaviour of the solution can be observed, and its corrupting effects preventnumerical convergence toward accurate and valid results. Consequently, the use of upwindingschemes1 is not always the appropriate alternative to obtain accurate and stable solutions.A better alternative would be to make a proper choice of the classical and new types of boundaryconditions for the sort of problem to solve.12,13 In the solution of the advection—diffusionequation, the proper choice of boundary conditions depends strongly on the characteristics of theboundary and the prescribed flow field, as well as on the finite element integration and type offormulation.14, 15 Therefore, rigorous physical and mathematical interpretation of conservativeand non-conservative flow fields, types of boundaries and formulation are required to make theproper combination of finite element formulation and boundary conditions for the advection-type problem.

The present paper provides stationary and transient theoretical and numerical informationconcerning the development of a finite element model to solve the two-dimensional solute trans-port equation for practical river flow problems. Essentially, the space—time Crank—Nicolson—Galerkin discretization is used with a conservative formulation. A simple linear approximationover each triangular element facilitates the direct analytical integration of the finite elementformulation. The matrix storage is avoided and the calculation time is greatly reduced byadequate factorization in the integration procedure as well as by diagonal preconditioning in theiterative solution algorithm GMRES (Generalized Minimal Residual16). Several kinds of boun-dary conditions (Dirichlet, Neumann, Cauchy and Open), are theoretically and numericallyexplored together with the conservative finite element formulation for different flow fields andtypes of boundaries. For a proper combination of boundary conditions, stationary and transientnumerical results show very accurate and stable solutions for various test problems. Comparisonsshow clearly that when a judicious choice of boundary conditions on diffusive boundaries ismade, the application of the basic Crank—Nicolson—Galerkin finite element method can beextended to comparable or even higher ranges of convection-dominated problems than those

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considered with more sophisticated and expensive numerical approaches (e.g. Douglas—Wang orTaylor—Galerkin).

ADVECTION—DIFFUSION EQUATIONS

The steady-state horizontal vertically integrated flow field necessary for the following model isprovided by the numerical simulation of the 2-D shallow-water equations (St.Venant equations),which delivers the velocity field as well as the depth and a diffusion tensor obtained througha formulation based on a zero-equation approach which uses the mixing length to define theturbulent viscosity.17, 18 This imposed steady-state velocity field can be either conservative, i.e. itsdivergence is locally and globally null, or non-conservative, i.e. certain water mass gains andlosses can be expected within the flow domain. Hence, some consequences and advantages can befound in considering two types of numerical models, one conservative and one non-conserva-tive.14 We will analyse the general time-dependent advection—diffusion equation under these twodifferent points of view.

The general transient equation for the conservative advection—diffusion and first-order decayof a dissolved constituent in a surface flow can be written in two dimensions for a horizontal-Cartesian co-ordinate system and for the variable HC as19

L (HC)

Lt#

L (uHC)

Lx#

L (vHC)

Ly#jHC!QHC0

!

LLxAKxx

HLC

LxB!LLxAK

xyH

LC

Ly B!LLyAK

yxH

LC

LxB!LLyAK

yyH

LC

Ly B"0 (1)

where C is the depth-averaged volumetric solute concentration, H is the local depth of the watercolumn, HC represents then the mass of solute per unit area of surface, u and v are thedepth-averaged current velocity parameters in the x and y directions, K

ijis the hydrodynamic

diffusion tensor, j is the first-order degradation rate, and QHC0 is the mass flow of solute injectedor abstracted per unit area of surface. Injections (abstractions) work as positive (negative) sources(sinks) of the solute.

Equation (1) is exact for a steady-state and conservative flow field. However, if the flow isnon-conservative, unexpected gains (losses) of water in the domain caused by the local non-zerodivergence of the flow field, will not change locally the mass of solute (but its concentration)since this gained (lost) water is theoretically pure. Nevertheless, when the water mass balanceis far from being null, unsuitable changes and steep gradients in concentration should beexpected.

The transient equation for the non-conservative advection—diffusion and decay of a dissolvedconstituent in a surface flow can be written in two dimensions and for the variable C as19,20

HLC

Lt#uH

LC

Lx#vH

LC

Ly#jHC!QHC0#QHC

!

LLxAKxx

HLC

LxB!LLxAK

xyH

LC

Ly B!LLyAKyx

HLC

LxB!LLyAK

yyH

LC

Ly B"0 (2)

where QHC indicates the mass flow of solute gained or lost per unit area of the surface.The non-conservative equation (2) will give more stable solutions for an imposed steady-state

and non-conservative flow field. In that case, unexpected gains or losses of water in the system,caused by the local non-zero divergence of the flow field, will change locally the mass of solute

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(but not its concentration) since the gained (lost) water has the same solute concentration as thesimulated medium. Therefore, even when the water mass balance is not null, changes in theconcentration and steep gradients will not be expected.

The degradation process jHC can be considered as an abstraction of the solute mass within thedomain. To assess the mass flow of solute abstracted per unit of surface, the model uses thefirst-order degradation rate coefficient j. The conservative and non-conservative models do notmake the difference in dealing with the mass rate of degradation; in fact, there are no volumechanges of water associated with degradation or decay processes. This approach to consideringthe decay of solutes in surface water is of course very simplified. It is well known that degradationand transient behaviour of dissolved chemical compounds in water is strongly dependent uponthe combined effects of biological, thermal, chemical and physical factors.21,22

INITIAL AND BOUNDARY CONDITIONS, SOURCES AND SINKS

Appropriate initial and boundary conditions are required to solve any of the time-dependentpartial differential equations derived in the previous section.

Initial conditions

Initial conditions for transient problems are expressed as following:

C(x, y, t0)"C

0(x, y ) on ) (3)

Apart from initial conditions for transient problems, several types of boundary conditions arepossible on the boundaries !"!

1#!

2#!

3#!

4of the whole domain ). It must be noted that

conservative and non-conservative boundary conditions are equally suitable. The proper choiceof the type of boundary conditions must also take into account the type of equation in order tooptimize the convergence, stability and accuracy of the solution.

Prescribed solutions (Dirichlet boundary condition)

Dissolved solute concentration can be imposed on the boundaries !1

or even within the flowdomain ).

C (x, y, t )"C1

on !1

or on parts of ) (4)

Dirichlet condition is usually prescribed on well known inflow boundaries. Nevertheless, we willshow later that for both steady-state and transient transport problems, a Cauchy condition is ingeneral more appropriate than a Dirichlet condition on inflow boundaries.

Prescribed diffusive flux (Neumann boundary condition)

The diffusive flux of the solute can be prescribed on the boundaries !2:

!AKxxH

LC

Lx#K

xyH

LC

Ly Bnx!AKyx

HLC

Lx#K

yyH

LC

Ly Bny"q

2on !

2(5)

where q2

is the prescribed diffusive flux of the solute, and (nx, n

y) are the direction cosines of the

outward pointing normal to !2. It is customary to name a Neumann boundary condition as the

natural condition given by the homogeneous version of equation (5), q2"0.

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Neumann boundary conditions can be imposed properly only on impermeable boundaries,when there is no fluid flow across the boundary and no fluid outside the domain. Otherwise, whenthe diffusive flux through the boundary is non-zero (and unknown), one has a diffusive permeableboundary. Therefore, Neumann (non-diffusive) conditions (q

2"0) on diffusive boundaries are

inappropriate. In spite of this, the diffusive flux of solute is very often considered as non-existenton outflow diffusive boundaries. Thus far, and to avoid inaccuracies in the calculated resultswhich will be introduced as soon as the numerical solutions reach the boundaries, the concernedboundaries need to be far enough from the zone of interest. In general, the calculated concentra-tions are overestimated.

For non-conservative flow fields, Neumann boundary conditions on impermeable bound-aries are compatible with the non-conservative equation. This is because local and unexpectedgains (losses) of water along the boundaries and inside the domain will change locally themass of solute, and consequently the gained (lost) water will remain at the concentrationlevel of the simulated medium. Then, solute concentration changes and steep gradients willnot be expected along the so-defined impermeable boundaries. However, Neumann boundaryconditions on defined impermeable boundaries are not compatible with the conservative formu-lation for non-conservative flow fields. Steep gradients and resulting oscillations would beexpected.

Prescribed convective-diffusive flux (Cauchy boundary condition)

Mixed boundary condition involving the convective and the diffusive flux can be imposed onthe boundaries !

3as

(uHCnx#vHCn

y)!AKxx

HLC

Lx#K

xyH

LC

Ly Bnx!AK

yxH

LC

Lx#K

yyH

LC

Ly Bny

"(uHC*nx#vHC*n

y) on !

3(6)

or equivalently as

!AKxxH

LC

Lx#K

xyH

LC

Ly Bnx!AK

yxH

LC

Lx#K

yyH

LC

Ly Bny

"(uHC*nx#vHC*n

y)!(uHCn

x#vHCn

y)"q

nHC*!q

nHC on !

3

where C* is the prescribed concentration of the solute in the influx fluid, and qnis the volumetric

water flux per surface unit through !3

for the prescribed flow field. This condition implies in itselfa discontinuity of the concentration and its derivatives at the boundary, because the gradient ofthe concentration outside the domain is assumed to be zero.23

Cauchy condition is best used to simulate a continuous feed solution, where the soluteis injected at a prescribed rate along the inflow convective—diffusive boundaries of the system(e.g. where one polluted river flows into another). If one treated this case as a Dirichletcondition, it would impose that the concentrations inside and outside be equal (C*"C),the solute concentration be equal to C*. Though for some steady-state problems, Cauchyand Dirichlet conditions can give similar results, from a numerical point of view, inflowboundaries for both steady-state and transient problems are best treated by the Cauchy con-dition. Later we will numerically verify these concepts about Cauchy conditions for two-dimensional transport problems.

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Non-prescribed diffusive flux (Open boundary conditions)

In order to establish the feasibility of the so-called Open boundary conditions,12 it is knownthat on permeable outflow diffusive boundaries (!

4), there is no discontinuity in the solution, and

consequently, its derivative must be continuous.

!AKxxH

LC

Lx#K

xyH

LC

Ly Bnx!AKyx

HLC

Lx#K

yyH

LC

Ly Bny

"!AKxxH

LC*

Lx#K

xyH

LC*

Ly Bnx!AK

yxH

LC*

Lx#K

yyH

LC*

Ly Bny

on !4

(7)

This means that along and across these outflow diffusive boundaries the solution is continuousbecause the diffusive flux within the domain equals the one outside of it.

In transport problems, the solution is continuous on outflow and noflow boundarieswhen these are typically defined as diffusive. This kind of boundary can be totally managedby the numerical resolution procedure, provided that it is formally incorporated into the weakformulation and in the steady-state and transient solution schemes related to the finite elementmethod.

Open conditions can only be applied along outflow or noflow boundaries when the fluid existsoutside the limit of the domain. If the diffusive flux is leaving the domain, then it can beconsidered by the numerical model. Therefore, Open conditions can always be applied on outflowdiffusive boundaries, close enough to the zone of interest, if the discretized model is able to respectthe required accuracy in the calculated results, even when the numerical solutions reach this typeof boundary. Later we will prove this numerically for two-dimensional transport problems.

For steady-state and non-conservative flow fields, Open conditions on outflow and noflowdiffusive boundaries are also compatible with the non-conservative equation. This is because theunexpected gains or losses of water will change locally the mass of solute, allowing also diffusionalong the boundaries, and consequently, the concerned fluid would retain a similar concentrationin the simulated medium. Then (in this scenario), solute concentration changes and thereforesteep gradients will not be expected along the permeable open boundaries. On the contrary, Openconditions on diffusive boundaries are not compatible with the conservative formulation fornon-conservative and steady-state flow fields. Again, steep gradients and therefore oscillationswould be expected.

Sources and sinks

In the two-dimensional numerical model, injection or abstraction of solute can be prescribed aspoint or distributed sources and sinks. With the conservative model, the eventual volumes ofwater coming with the dissolved solutes in the sources and sinks need to be included in the givenflow field. Therefore, the transport model must be specified with the mass flow of solutes injectedor abstracted per unit area of surface (QCH0). However, if the given flow field does not take intoaccount the volumes of water injected or abstracted by the sources and sinks, then the transportmodel needs to be non-conservative. This implies that it also needs the specification of the volumeflow of water injected or abstracted per unit of surface (QH).

In the present surface water transport model, the above numerical approach for pointor distributed sources is designed to analyse pollution problems, e.g. contamination comingfrom the polluted groundwater or waste waters originating from industrial and urbansewage systems.

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FINITE ELEMENT FORMULATION

A standard weighted residual approach with Galerkin-type weighting functions is used todetermine approximate solutions to the transient transport equations (1) or (2) under theappropriate initial and boundary conditions. Full details of the involved solution procedures willnot be given here; however, an outline of our numerical scheme for the conservative equation (1) ispresented below to provide some ideas on how a specific differential equation with boundaryconditions and sources were dealt with.

The weighted residual approach requires the residue (1) to be orthogonal to properly selectedtest functions ¼(x,y):

P) CL (HC)

Lt#

L (uHC)

Lx#

L (vHC)

Ly#jHC!QHC0

!

LLxAK

xxH

LC

LxB!LLxAK

xyH

LC

Ly B!LLyAK

yxH

LC

Lx B!LLyAK

yyH

LC

Ly BD¼d)"0 (8)

The second-order diffusion terms are integrated by part, leading to the weak form of (8). Thecontour integral arising from this integration procedure provides the natural boundary condi-tions of our system in regard to diffusive fluxes.

P)¼

L(HC)

Ltd)#P)

¼AL (uHC)

Lx#

L(vHC)

Ly Bd)

#P)¼ (jHC!QHC0) d)

#P) CL¼Lx AK

xxH

LC

Lx#K

xyH

LC

Ly B#L¼Ly AK

yxH

LC

Lx#K

yyH

LC

Ly BDd)

!Q!¼CAKxx

HLC

Lx#K

xyH

LC

Ly Bnx#AKyx

HLC

Lx#K

yyH

LC

Ly BnyDd!"0 (9)

Discretization

In the finite element method, the computational domain and the unknowns are represented byappropriate shape functions N (x, y) on a spatial grid composed of a finite number of subdomainsof simple geometrical shape, called the finite elements.24

The standard Galerkin method corresponds to the choice of the weighting functions ¼ (x, y)as the shape functions N(x, y) over the elements. Introduced into (9), this leads globallyto the following algebraic system of equations for the state variable HC (conservativeformulation):

[M]GL (HC)

Lt H#MK (HC)N"MFN (10)

The global system (10) is the sum or assemblage over the elements of the elementary algebraicsystems. Its components are expressed below in terms of the elementary weak form e, where

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N, i

expresses a partial derivative with respect to i.

[M])"+eP)e

MNNSNT dxdy (Mass)

MKN)"+eP)e

MNNSN,x

TMuHCN dxdy#+eP)e

MNNSN,y

TSvHCNdx dy (Advection)

#+eP)e

MNNjSNTMHCN dxdy (Degradation)

#+eP)e

(MN,x

NSNTMHNSNTMKxx

NSN,x

T

#MN,x

NSNTMHNSNTMKxy

NSN,y

T

#MN,y

NSNTMHNSNTMKyx

NSN,x

T

#MN,y

NSNTMHNSNTMKyy

NSN,y

T)GHC

H Hdx dy (Diffusion)

!+eQ!e

MNNSNTMqnHCN d!

3(Cauchy boundaries)

!+eQ!e

((MNNSNTMHNSNTMKxx

NSN,x

T

#MNNSNTMHNSNTMKxy

NSN,y

T)nx

#(MNNSNTMHNSNTMKyx

NSN,x

T

#MNNSNTMHNSNTMKyy

NSN,y

T)ny)G

HC

H Hd!4

(Open boundaries)

MFN)"+eP)e

MNNSNTMQHC0Ndxdy (Sources and Sinks)

!+eQ!e

MNNSNT MqnHC*Nd!

3(Cauchy boundaries)

where [ ] represent a matrix set, and S T and M N represent a row and a column vector, respectively.It is worthy to note that in order to formally incorporate Open boundary conditions, all the

elements of the domain having at least a node at the boundary will contribute to the termaccounting for open boundaries in the above finite element formulation.

It must also be noted that for the conservative variable of state HC, the advection—diffusionequation permits the choice of a type of factorization for certain terms in order to interpolatedifferent nodal values of the conservative form of the equation. This choice diminishes thecalculations and, in general, increases very little the errors associated to the numerical approxi-mation.27 Obviously, precision of the approximation can be enhanced by avoiding the factoriz-ation of terms especially when parameters are very variable.

2500 F. PADILLA, Y. SECRETAN AND M. LECLERC

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Space discretization and integration strategy

For computer efficiency, we selected a linear element fulfilling the minimal global and localcontinuity requirement, the triangular first-order Lagrange reference element with three nodes(T3). This permits the analytical integration of the terms of the elementary weak form, leading toa more efficient and a better vectorization of the computer code.25,26 In fact, with this element, itis possible to describe conveniently any geometrical shape, as well as to have only one type ofreference element, which tremendously simplifies the computer program structure. Moreover, thissimple triangle is well suited for dynamical mesh refinement.

When concerning point sources and sinks, the surface affected by the solute injected orabstracted depends on the type of element used in the numerical discretization. In the case oflinear triangular elements, this surface is equal to 1

3of the surface of the elements connected with

the nodes where the sources and sinks are prescribed.

SOLUTION METHOD

Steady-state and transient solution schemes

The advection—diffusion equation can be solved for either steady state or transient problems.For steady-state conditions, the global matrix form (10) is rewritten as

MK (HC)N"MFN (11)

For time-dependent problems, the semi-implicit Euler finite difference scheme has been foundto provide good results. Under that scheme the linear advection—diffusion system can beexpressed as

[M]MHCt#*tN!MHC

tN

*t"(a!1)MK(HC

t)N!aMK(HCt#*t )N#(1!a)MF

tN#aMFt#*tN (12)

With a"1, it is the fully implicit first-order backward scheme; with a"0·5, it is theCrank—Nicolson scheme, semi-implicit and second-order accurate. The resultant space—time Crank—Nicolson—Galerkin method gives, in general, better solutions for most types ofequations.8

Solution algorithm

The generalized minimal residual GMRES is a Galerkin method onto a Krylov subspace. Anorthonormal basis on the Krylov subspace is generated with Gramm—Schmidt algorithm.GMRES minimizes the residual of the algebraic system. Obviously, the dimension of the subspacewill influence the convergence of the method. GMRES iterative algorithm was proposed by Saadand Schultz16 for solving non-symmetric linear systems. The method has been applied success-fully to non-linear systems.28~30 It does not need computation and storage of the global matrixand therefore it is very suitable for mesh refinement.

For an efficient practical calculation, the dimension of the Krylov subspace is very smallas compared to the order of the global matrix of the system. This dimension governs notonly the precision but also the memory required. We generally used approximately 25 vectorsof Krilov for all sizes of linear and non-linear systems, but restarted the resolution a numberof times depending if the transport problem is stationary or transient, and mainly dispersiveor advective.

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The convergence of the iterative method is influenced by the conditioning of the matrix. Allsuccessful applications required an efficient preconditioning.31 Bearing in mind, the possibleparallelization of the code, we used diagonal preconditioning. The preconditioning matrix canthen be the diagonal lumped mass matrix or a diagonal approximation of the global matrix ofthe system.26

NUMERICAL RESULTS

After introducing several kinds of boundary conditions in the conservative Crank—Nicolson—Galerkin finite element formulation for a two-dimensional transient advection—diffusion prob-lem, some numerical illustrations will show the performance of the proposed model.

The following steady-state and transient numerical tests were mainly used to analyse thebehaviour of the present formulation for a judicious choice of the above-mentioned boundaryconditions. Most tests have characteristics that correspond to well-known analytical solutions tostudy the performance of numerical strategies for advection—diffusion problems. Results of othernumerical models as well as calculated errors served also to analyse the solutions of the presentmodel for the selected numerical tests.

¹est of convergence

A finite element solution with linear elements is convergent in the ¸2norm (the Euclidian norm

associated to the scalar product of the space of real values of square integrable functions definedon )) if

EerrorE0"EC!C

%9!#5E0"J:) (C!C

%9!#5)2d))ch2 (13)

where c is a constant independent of the grid size h. The convergence rate of the discrete systemcan then be controlled on a function of the grid size.26

A second-order elliptic problem has been considered in order to study the convergence rate ofthe conservative finite element approximation on the triangular element T3 for the steady-stateadvection—diffusion equation.

Kxx

L2C

Lx2#K

yy

L2C

Ly2!u

LC

Lx!v

LC

Ly"0 (14)

Taking as exact solution of the above differential equation (14), an analytical function that cannotbe represented exactly by the finite element approximation, we introduce a numerical error.Therefore, equation (15) (Figure 1(a)) was selected to perform the convergence test; it representsthe homogeneous solution of the system (14) under a proper choice of velocities (u, v) anddiffusivities (K

ij).

C(x, y )"e~x~y with u"!1; v"!1; Kxx"K

yy"1 (15)

The system was solved on a square domain of boundaries xi3 0[0, 1]. All the boundaries have

boundary conditions of the Dirichlet type corresponding to the analytic solution. The error hasbeen controlled on regular grids of sizes ranging from 3]3 nodes up to 33]33 nodes. Figure 1(b)shows in a log—log diagram the evolution of the ¸

2norm of the error as a function of the grid

size h. It can be seen as expected that the convergence rate of the discrete system is satisfactoryand it is also quadratic for the dependent variable C.

2502 F. PADILLA, Y. SECRETAN AND M. LECLERC

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Figure 1. Convergence of the approximation for the triangular element T3: (a) characteristics of the convergence test;(b) results of the convergence test for different mesh densities

Steady-state advection in a rotating flow field

In order to assess the precision of the transport model when convection is dominant, a steady-state problem representing the advection of a Gaussian profile in a rotating flow field has beenselected.32, 33

On a unit square domain of co-ordinates !0·5)(x, y ))#0·5 (Figure 2(a)), Dirichletconditions C"0 are classically suggested to be imposed along all the external boundaries.A Gaussian profile of concentration was prescribed within the domain.

C"cos2A2ny#n2B (16)

The diffusivity was numerically reduced to a negligible value of 10~6 m2/s. Considering then thequasi-absence of diffusion, the steady-state problem (14) becomes hyperbolic and the rotating

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Figure 2. Steady-state advection in a rotating flow field: (a) problem statement; (b) mesh and results for T6 element afterKhelifa (1992)

flow field generates the following analytical solution (Figure 2(b)):

C"

cos2A2nJx2#y2#n2B for 0)Jx2#y2)0·5

0 for 0·5)Jx2#y2)R

(17)

A uniform mesh with 29]29 nodes (1568 elements) was used to represent the analytical flowdomain (see Figure 2(b)).

Obviously, this numerical test could be solved with boundary conditions that are different fromthe Dirichlet imposed values (C"0) that are suggested by the classical problem statement. Infact, considering that the external boundaries are physically open, we have tried other boundarycondition schemes and formulation to assess the main differences in the calculated results (Two of

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Figure 3. Steady-state advection in a rotating flow field: (a) types of boundary conditions on the external boundaries;(b) results for ‘Dirichlet’ and ‘Cauchy—Open’ boundary conditions

these schemes are illustrated in Figure 3(a)). Results of numerous tests for conservative andnon-conservative formulations, show clearly that for this conservative flow field, the best numer-ical results are obtained when the Galerkin technique is applied to the conservative form of thetransport equation, and that Cauchy and Open boundary conditions are applied to inflow andoutflow open boundaries, respectively (Figure 3(b)).

We have indicated in Table I our calculated extreme values and errors as compared to theresults obtained by Khelifa33 for the quadratic Douglas—Wang finite element approach. In-dicated extreme values and errors correspond to: maximum and minimum values of calculatedsolutions, the error interval as evaluated by the sum of the absolute differences between calculatedand analytical maximum and minimum values, and the total error over the domain as calculatedby relationship (13). It must be noted that when pure advection or linear elements are considered(this numerical test), the streamline upwind Petrov—Galerkin (SUPG) method, the Galerkin least-squares (GLS) method and the Douglas—Wang method, behave identically. As Khelifa33 usedquadratic elements (triangles of six nodes T6 and rectangles of nine nodes Q9), a procedureknown to give the best results for the Douglas—Wang method,32 the results indicated herein for

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Table I. Steady-state advection of a Gaussian profile in a rotating flow field: numerical comparison of errorsand extreme values

Method ErrorBoundary Mesh interval

Formulation conditions Element nodes Max. Min. Total error

Non-conservative33 Douglas-Wang T6 29]29 #1·0008 !0·0091 $0·0099Dirichlet —

Non-conservative33 Douglas—Wang Q9 29]29 #1·0000 #0·0045 $0·0045Dirichlet —

Conservative Galerkin T3 29]29 #1·0269 !0·0361 $0·0630(this research) Dirichlet 0·0134

Conservative Galerkin T3 29]29 #1·0012 !0·0033 $0·0045(this research) Cauchy—Open 0·0027

the Douglas—Wang method must be in general more precise than those expected from the use oflinear elements in the SUPG or GLS methods.

Comparison of numerical differences shows on one hand, more precise results for the standardGalerkin (Cauchy-Open) T3 element of this approach than for the T6 element of the Douglas—Wang (Dirichlet) method. On the other hand, the Douglas—Wang (Dirichlet) Q9 element seems togive as good results as those obtained by the present Galerkin (Cauchy-Open) T3 of this research.Except for the too high calculated errors of the classical Galerkin (Dirichlet) approach, theestimated errors of all the others methods are much lower and, in a way, comparable. It can thenbe concluded that for two-dimensional steady-state advection-dominated problems, the Galerkinfinite element method, joined to a conservative formulation and linear shape functions on theelements, can give equivalent or more precise results than some streamline upwind methods. Forexample, this is the case for the Douglas—Wang method with higher-degree polynomial shapefunctions, provided that for the Galerkin method the choice of the boundary conditions isproperly adapted to the type of problem, namely Cauchy and Open conditions along inflow andoutflow open boundaries, respectively.

¹ransient advection of a rotating cone

As a severe test of the present formulation, we have investigated the rotating cone in a transientadvection-dominated problem. The test problem (Figure 4(a)) considers a cosine hill profileadvected in a two-dimensional rotating flow field (after Donea et al.34).

The domain consists of a unit square of co-ordinates !0·5)(x, y))#0·5, with homogene-ous Dirichlet boundary conditions imposed zero everywhere on the external boundaries. Theinitial condition is a hill profile with the following form:

C(x, y, 0)

"

1

4A1#cosAnx!5/30

0·2 BBA1#cosAnx#5/30

0·2 BB if AAx!5

30B2#Ay#

5

30B2

B)0·22

0 elsewhere

(18)

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Figure 4. Rotating cone: (a) statement of the numerical problem; (b) mesh, initial condition and analytical solution aftera rotation (from Reference 33)

Considering the absence of diffusion (10~7 m2/s as suggested for this particular numerical test),the rotating flow field (u"!y, v"x) should generate after one complete rotation (6·28 s or 200time steps of 2n/200 s each) a resulting state exactly the same as the initial condition (18) for thequasi-pure transient advection problem as following:

LC

Lt"!u

LC

Lx!v

LC

Ly(19)

A uniform mesh, with 31]31 nodes (1800 triangular elements), was used, as illustrated in Figure4(b), for carrying the initial and the analytical solutions.4 Similarly to the test of advection ina rotating flow field, this numerical test can be solved for boundary conditions that are differentfrom the Dirichlet conditions (C"0) as suggested in the classical problem statement.34 Consider-ing that the external boundaries are physically open, we have used a Crank—Nicolson—Galerkinscheme for the conservative transient advection equation, with Cauchy and Open conditions oninflow and outflow open boundaries, respectively.

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In Figure 5 we have illustrated the analytical and the calculated results after one completerotation of the flow field. Table II presents some comparisons with results obtained by Khelifa etal.4 and by Bezier,35 for the same problem solved with the Douglas—Wang and theTaylor—Galerkin methods, respectively. The data presented for these methods correspond to thebest results obtained, respectively, by the authors. Notice that TG2 denotes the Taylor—Galerkinmethod with a second-order time derivative.

Comparison of numerical differences shows more precise results for the Crank—Nicolson—Galerkin (Cauchy—Open) T3 element of this approach than for the Douglas—Wang T6 and theTaylor—Galerkin (TG2) Q4 elements. Nevertheless, the Douglas—Wang Q9 and the Taylor—Galerkin (TG2) T3 elements give better results than the present Crank—Nicolson—Galerkin(Cauchy—Open) approach. It can be concluded however that for the present numerical test, theestimated errors of all the methods are in a way comparable. Therefore, the Crank—Nicolson—Galerkin approach for two-dimensional transient advection-dominated problems, joined in thiscase to a conservative formulation and simple triangular elements, can give results which are

Figure 5. Comparison of calculated and analytical results after one complete rotation of a cone in a rotating flow field

Table II. Advection of a rotating cone: numerical comparison of errors and extreme values

Method ErrorBoundary Mesh interval

Formulation conditions Element nodes Max. Min. Total error

Non-conservative4 Douglas—Wang T6 31]31 #0·982 !0·27 $0·045Dirichlet —

Non-conservative4 Douglas—Wang Q9 31]31 #0·993 !0·019 $0·026Dirichlet —

Non-conservative35 Taylor—Galerkin (TG2) T3 31]31 #1·001 !0·024 $0·025Dirichlet —

Non-conservative35 Taylor—Galerkin (TG2) Q4 31]31 #0·982 !0·018 $0·036Dirichlet —

Conservative Crank—Nicolson—Galerkin T3 31]31 #1·003 !0·028 $0·031(this research) Cauchy—Open 0·0077

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similar to others recent time-accurate methods (e.g., the Douglas—Wang and the Taylor—Galerkinmethods with Dirichlet boundary conditions and higher-degree elementary approximations),provided that the choice of the boundary conditions along the so-defined open boundaries isproperly adapted to the type of problem, namely Cauchy and Open conditions on inflow andoutflow open boundaries, respectively. It is worthy to note that this numerical approach on openboundaries does not seem to have any restriction to the application of recent time-accuratemethods in order to improve even more the precision of present transient solutions.

Transient advection—diffusion of a continuous point source in an open system

In order to know better the range of applicability of this Crank—Nicolson—Galerkin approachto the transport equation, firstly a very severe numerical test has been conceived: the transientadvection—diffusion of a continuous point source closely surrounded by open boundaries. Themain objective is to evaluate the errors associated to the influence of the proximity of theprescribed boundary conditions12 by comparing the numerical results to the analytical solutions.

The domain consists of a unit square of co-ordinates !0·5)(x, y))#0·5, with physicallydefined open boundaries: inflow, noflow and outflow (Figure 6(a)). Considering an homogenous

Figure 6. Advection—diffusion of a continuous point source in an open system (I) diffusivities"0·005 and 0·05 m2/h.Scheme A: (a) scheme of the open analytical problem; (b) scheme of boundary conditions A; (c) analytic and calculated

solutions after 3 h (diffusivity"0·005 m2/h); (d) analytic and calculated solutions after 3 h (diffusivity"0·05 m2/h)

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flow field in the direction of the x-axis (liquid velocity of 1 m/h), the transient parabolicadvection—diffusion equation for a continuous and non-volumetric point source can be written asfollows:

HLC

Lt"K

xxH

L2C

Lx2#K

yyH

L2C

Ly2!uH

LC

Lx!vH

LC

Ly#QHC0 (20)

A solute point source (1 kg/d), which does not imply volume changes in the liquid phase(H"1 m), is injected at the co-ordinates x"!0·25, y"0. In order to assess the influence of theboundaries on the transient solution over the discrete domain, we use two diffusivity values(0·005 m2/h and 0·05 m2/h). Firstly, this test considered a uniform and symmetric mesh of 33]33nodes.

The analytical model36 allows diffusion and obviously advection through all the open bound-aries of the domain. Nevertheless, because of the previous tests, let us firstly consider Cauchyconditions (for C*"0) on inflow and Open conditions on outflow boundaries. At the presenttime, noflow boundaries will be treated as non-diffusive Neumann conditions. A scheme of thisnumerical problem (A) is illustrated in Figure 6(b). For this first boundary scheme, calculated andanalytical results after three hours are shown in Figures 6(c) and 6(d). We can easily notice thedifference in the numerical influences of the Neumann or non-diffusive lateral boundary condi-tions for the two diffusivity values. Well-known inaccuracies concerning the lack of meshrefinement around the point source can also be observed for the 1 ppm calculated concentrations.This aspect will be examined later in the present research.

In fact, the solutions obtained with the diffusivity value of 0·05 m2/h are strongly influenced bythe proximity of the lateral Neumann conditions. It must be noticed that these are noflowboundaries but open to diffusion anyway. Let us consider a second numerical scheme (B) in whichthe last Neumann conditions are treated as Open conditions (Figure 7(a)). For the same firstchoice of a regular symmetric mesh, we have illustrated in Figure 7(b) the analytical andcalculated results obtained after 3 h. Comparison of results between the two last boundaryconditions schemes (A and B; Figures 6(d) and 7(b)) indicates a clear improvement in the solutionsobtained by the second scheme (B). In order to further improve the results of the numericalproblem (B), we have obviously minimized the influence of the lateral boundaries by makinga local mesh refinement of the affected elements (Figure 7(c)). The results after 3 h are illustrated inFigure 7(d). It can be clearly noticed that the solution becomes improved all over in the domain.Gradients near all the external boundaries are much more correctly represented even for very lowconcentrations. Our various numerical tests indicate that the present Open conditions on noflowopen boundaries are sensitive, to a certain extent, to non-symmetric meshes and, as we have seen,to the size of the elements near the boundaries.

Open noflow boundaries are a direct outcome of the boundaries having the same directionthan the flow field. Let us now consider zigzag boundaries to this direction (Figure 8(a)).Obviously, the noflow boundaries become inflow and outflow and they need to be treated withCauchy (C*"0) and Open conditions, respectively, (Figure 8(b)). The spatial finite elementdiscretization of the domain is illustrated in Figure 8(c). For the present numerical scheme, Figure8(d) shows the comparison between the analytical and the calculated results after 3 h. It can beeasily observed that the calculated results can likely reproduce the exact solutions in a fully opendomain, even when the discretized boundaries does not follow necessarily the direction of the flowfield. These last solutions are in a way equivalent to the ones illustrated in Figure 7(d) fororiginally noflow lateral boundaries treated with Open conditions.

The main point of interest of these last numerical tests is the intimate relationship between theapplied boundary conditions and the calculated solutions all over in the numerical domain.

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Figure 7. Advection—diffusion of a continuous point source in an open system (II) diffusivity"0·05 m2/h. Scheme B:(a) scheme of boundary conditions B; (b) analytic and calculated solutions after 3 h; (c) new spatial finite element

discretization; (d) analytic and new calculated solutions after 3 h

A conservative formulation for a conservative flow field contributes of course to this. The point tonote is that open boundaries can actually be treated with Cauchy conditions when the fluid phaseenter the domain (inflow), and with Open conditions otherwise, i.e., when the flow field leaves thedomain (outflow) or is parallel to the open boundary (noflow).

For the prescribed flow field of this numerical test, the advection—diffusion equation (20) canbe, in general, classified as parabolic. Nevertheless, when the advection term becomes largecompared with diffusion in the direction normal to outflow boundaries, the advection—diffusionequation is mainly hyperbolic in this particular direction, with an attendant reduction in requiredboundary data.23 Cauchy and Open conditions on open inflow and outflow boundaries aresuitable and not very sensitive to other sources of error. When advection is negligibly smallcompared with diffusion in the direction normal to noflow diffusive boundaries, the advec-tion—diffusion equation is mainly parabolic in this particular direction. The consequence is theneed for an increase in required boundary conditions. It can then be concluded that for systems

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Figure 8. Advection—diffusion of a continuous point source in an open system (III) diffusivity"0·05 m2/h; Scheme C:(a) scheme of the open analytical problem; (b) scheme of boundary conditions C; (c) spatial finite element discretization;

(d) analytic and calculated solutions after 3 h

with mainly diffusive noflow open boundaries, Open conditions can still give accurate solutions;however, the numerical system can be more strongly influenced by others numerical sources oferror, e.g. asymmetric meshes, or the size of the elements near these noflow open boundaries. Inthe case of boundary elements that are too large, Open conditions could be unable to catchaccurately the computed gradients corresponding to the correct diffusive flux. Another importantfactor negatively affecting the accuracy of the evaluation of the diffusive flux at open boundaries isrelated to the shape functions used for elementary interpolation that do not satisfy continuityof derivatives at element boundaries, as it is the case for the linear triangular element (T3) ofthis research.

Continuous point source in an open channel

The calculated solutions of the last series of tests show some errors mainly associated withthe lack of an adapted mesh refinement near a prescribed point source. In order to prove this

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assumption, the main objective of the present test is to assess the accuracy of the model for meshrefinement near and around the source of contamination.

The present numerical test consists of a 12 km long, 2 km width and 8 m deep channel. Theflow field has a velocity of 1 m/s in the direction of the x-axis. A non-volumetric continuous pointsource of 1 kg/d is injected at 1·5 km downstream in the middle of the channel. The diffusivity iskept constant and equal to 0·659 m2/s. For this diffusion value, solutions over the open domain36

are not mainly affected by the diffusive lateral boundaries. Therefore, the actual open channel canbe represented in the numerical domain with Cauchy and Open conditions on upstream anddownstream boundaries, respectively, as well as with Neumann or Open conditions on noflowlateral boundaries.

As concerns the spatial finite element discretization, the size of the triangular elements inthe domain is based on a new technique of mesh refinement. The approach makes use of aLagrangian (Particle Tracking) method38 in order to compute a mesh adapted to the type ofproblem. For an originally very large spatial discretization (triangles of 500 m]750 m), thealgorithm calculates automatically a more appropriate mesh refinement around and downstreamof the continuous point source (Figure 9(a)). In this particular case, the retained criteria of element

Figure 9. Continuous point source in an open channel: (a) adapted numerical discretization; (b) surface influenced by thecontaminant point source; (c) comparison of calculated and analytical results after 3 h in the proximity of the point source

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partition is the following: the maximum number of particles (20) founded in an element, aftera number of particles (20 000) dropped during a certain time (3 h) was transported by theLagrangian method considered.

The advection—diffusion of the non-volumetric point source can be mathematically representedby the equation (20). Figure 9(b) illustrated the surface of influence of the contaminant injectionnear the point source after the mesh refinement has been made. As illustrated near the pointsource in Figure 9(c), comparison of calculated and analytic results after three hours shows veryaccurate solutions especially in the proximity of the source of contamination. It can also benoticed that steep gradients can be correctly represented by the numerical model.

CONCLUSIONS

The present research provides stationary and transient theoretical and numerical informationconcerning the development of a finite element model to approximate, with the aid of simpletriangular elements, the two-dimensional advection—diffusion equation. Essentially, the space—timeCrank—Nicolson—Galerkin formulation is used for conservative flow fields in practical surfaceflow simulations. Several kinds of point sources and boundaries conditions, namely Dirichlet,Neumann, Cauchy and Open, are theoretically and numerically analysed. As far as this isconcerned, stationary and transient numerical tests are explored for inflow, noflow and outflowopen boundaries. When a proper choice of formulation and boundary conditions is made, theGalerkin and the Crank—Nicolson—Galerkin finite element methods give stable and preciseresults for advection-dominated transport problems. Comparisons indicate in general that thepresent method can give equivalent or more precise results than others streamline upwind andhigh-order time-stepping schemes.

The main point of interest of this method is the intimate relationship between the appliedboundary conditions and the characteristics of the physically open boundaries in a discretedomain. The use of a conservative formulation for conservative flow fields contributes lightly tothe results. When the advection term becomes large compared with diffusion, in the directionnormal to the boundary, the transient advection—diffusion equation is mainly hyperbolic in thisparticular direction, with an attendant reduction in required boundary data. Open boundariescan then be treated with Cauchy conditions when the fluid phase enters the domain (inflow), andwith Open conditions otherwise, i.e. when flow field leaves the domain (outflow) or even when it isparallel to the boundary (noflow). Nevertheless, for the case of noflow open boundaries whenadvection is negligibly small compared with diffusion in the direction normal to the boundary,the advection—diffusion equation is mainly parabolic in this particular direction. The conse-quence is the need for an increase in required boundary conditions. Though systems with mainlydiffusive noflow boundaries can still be solved precisely with Open conditions, these systems arein a way more strongly influenced by others numerical sources of error, e.g. very deformed orasymmetric meshes, boundary elements that are too large, or shape functions for elementaryinterpolations which yield discontinuous derivatives at element boundaries. A critical combina-tion of these cases, for instance, could make that Open conditions fail to calculate precisely thediffusive flux across noflow open boundaries, leading to a badly conditioned numerical systemand compromissing the convergence of the solution.

Nevertheless to this concern, the applicability of Open boundary conditions is restricted to testcases for linear parabolic and hyperbolic problems. Typical numerical errors and oscillations,commonly associated with steady and transient Galerkin and Crank—Nicolson—Galerkin schemes,are not observed in the solutions when using the correct boundary conditions on open bound-aries. The choice and the treatment of open boundaries greatly increases the possibilities of

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correctly modelling restricted domains of actual and numerical interest. Moreover, this numericalapproach does not seem to have any particular restriction to the application of streamline upwindand high-order time-stepping schemes in order to improve even further the present steady andtransient solutions. New developments must also look after the applicability of Open boundaryconditions to non-linear equations, elliptic problems, test cases and actual events.

ACKNOWLEDGEMENTS

The authors are grateful to A. Khelifa, P. Boudreau and J.-L. Robert for their help and comments.The research work was fulfilled at the ‘Institut National de la Recherche Scientifique-Eau’,Quebec, Canada. Financial support comes in part from the CICYT: Science and TechnologyInter-Ministerial Commission, Spain (AMB94-0611 Project).

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