characteristic-galerkin methods for contaminant transport ...(3.5) a e,kisi. i=1 eachcomponent si is...

Characteristic-Galerkin methods for contaminanttransport with nonequilibrium adsorption kineticsCitation for published version (APA):Dawson, C. N., Duijn, van, C. J., & Wheeler, M. F. (1994). Characteristic-Galerkin methods for contaminanttransport with nonequilibrium adsorption kinetics. SIAM Journal on Numerical Analysis, 31(4), 982-999.https://doi.org/10.1137/0731052

DOI:10.1137/0731052

Document status and date:Published: 01/01/1994

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https://research.tue.nl/en/publications/characteristicgalerkin-methods-for-contaminant-transport-with-nonequilibrium-adsorption-kinetics(5164d436-9b75-41aa-8ce2-c7974dd4f3e8).html

SIAM J. NUMER. ANAL.Vol. 31, No. 4, pp. 982-999, August 1994

() 1994 Society for Industrial and Applied Mathematics002

CHARACTERISTIC-GALERKIN METHODS FOR CONTAMINANTTRANSPORT WITH NONEQUILIBRIUM ADSORPTION KINETICS*

C. N. DAWSONt, C. J. VAN DUIJN, AND M. F. WHEELERt

Abstract. A procedure based on combining the method of characteristics with a Galerkin finiteelement method is analyzed for approximating reactive transport in groundwater. In particular,equations modeling contaminant transport with nonlinear, nonequilibrium adsorption reactions areconsidered. This phenomenon gives rise to non-Lipschitz but monotone nonlinearities which com-plicate the analysis. A physical and mathematical description of the problem under consideration is

given, then the numerical method is described and a priori error estimates are derived.

Key words, method of characteristics, Galerkin finite elements, contaminant transport

AMS subject classifications. 65N15, 65N30

1. Introduction. In this paper, we describe a characteristic-Galerkin finite ele-ment method (CGFEM) for modeling contaminant transport with nonlinear, nonequi-librium adsorption kinetics. The CGFEM, also known as the modified method ofcharacteristics, Lagrange-Galerkin, or Euler-Lagrange method, has been used exten-sively in the modelling of linear and nonlinear flows; see, for example, [10], [14], [15],[16]. The method was first analyzed in [4] for advective flow problems in one spacedimension, and improvements and extensions of these estimates were derived in [5].These estimates were proved primarily for linear problems; however, certain types ofsmooth nonlinearities were also considered.

Here we consider the application of the CGFEM to a nonlinear system of equa-tions which arises in contaminant transport, and derive an a priori error estimate.The primary difficulty in these equations is the presence of possibly non-Lipschitznonlinearities, which require special treatment in the analysis. The presence of suchnonlinearities also reduces the regularity of the solution; thus, the expected rates ofconvergence are possibly suboptimal when approximating by piecewise polynomials.

Error estimates for a Galerkin finite element procedure for solving these typesof equations have recently been derived in [1]. The approach given there involvesapproximating the solution to a regularized problem, obtained by replacing the non-Lipschitz function (given by (3.9) below) with a Lipschitz approximation , andallowing e to approach zero. The estimates obtained for this procedure in the norm

L(0,T; L2()) appear to give the same or a slighly improved rate of convergencethan that derived below for the CGFEM, depending on the choice of e. However, theauthors are able to obtain a better rate of convergence in the L2(0, T; L2()) norm.Numerical comparisons of the two approaches remains to be done. We note that ourapproach is better suited for convection-dominated transport.

In the next section, we give some basic notation. In 3 and 4, the physicalproblem is described, and existence and uniqueness of weak solutions, and regularityof solutions are discussed. In 5, we describe the application of the CGFEM method,and in 6, the method is analyzed, assuming optimal regularity of the solution.

Received by the editors September 24, 1992; accepted for publication (in revised form) July 27,1993.

Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77251(cl+/-nt@r+/-ce.edu, mfw@r+/-ce.edu). These authors were supported in part by grants from the Depart-ment of Energy.

SDepartment of Mathematics, Delft University of Technology, Delft, the Netherlands(vanduijn@twi, tudelft, nl).

982

CHARACTERISTIC-GALERKIN FOR CONTAMINANT TRANSPORT 983

2. Notation. For Y a measurable space or space-time domain, let Lp(y), 1 <p < oc, denote the standard Banach space on Y, with norm I1" IILP(Y)" For f abounded spatial domain in ]Rd, 1 < d < 3, denote by W2k(f) the standard Sobolevspace on f with norm I1" Ilk" We denote the L2(f) norm by I1" II.

Let [, ] c [0, T] denote a time interval, where T > 0 is a fixed constant, and letX X() denote a normed space. Denote by I1" IILp(a,;X) the norm of X-valuedfunctions f with the map t Ill(’, t)llx belonging to LP(a, ).

Letting QT (0, T], we denote by V2’(QT) the Banach space consisting ofelements having a finite norm

We denote by W’1 (QT) the Hilbert space with scalar product

(u, V)w,l(Qr) =/Qr (uv + Vu Vv + uv)dxdt.

For [0,x3) -- [0, oc), the notation e CP([0, oc)), p e (0,1), means isHhlder continuous in its argument with exponent p. We denote by Ca’(QT), wherea and fl are positive numbers, the standard Hhlder space defined on QT [12]. Here crepresents smoothness in space and/3 represents smoothness in time.

3. Statement of the problem. When chemical species are dissolved in ground-water they may undergo adsorption or exchange processes on the surface of the porousskeleton. Knowledge about the influence of these chemical processes on the transportof the solutes when the groundwater is moving is of fundamental importance in un-derstanding, for instance, how pollutants spread in space and time through the soil.

Below we present the mathematical formulation for a one-species system in whichthe chemicals undergo nonequilibrium adsorption reactions. Certain types of two-species systems of binary ion exchange can also be put into this framework. In par-ticular, this is the case when a conservation property allows for the reduction to aone-species system. In [8], details of this reduction are given, as well as a fundamentaldiscussion of adsorption processes in porous media, and related references.

The domain is occupied by a porous material through which an incompressiblefluid, say water, flows. The related specific discharge l(m/s), with components qi, i1,..., d, and length I1, satisfies the equation

0e0-7 + v. q 0,

which expresses conservation of fluid volume. In this equation, 0 (dimensionless)denotes the water content.

In what follows we shall consider 0 and q as given quantities which are determinedindependently of the concentration of the dissolved chemicals. Thus, we implicitlyassume here that concentrations occur only at tracer levels and hence do not influencethe flow.

Let c(mol/m3) denote the concentration of adsorbate in solution and A(mol/kgporous medium) the adsorbed concentration. Conservation for the chemical speciesgives the equation

984 C.N. DAWSON, C. J. VAN DUIJN, AND M. F. WHEELER

(3.2) -- (0c + pA) + V (qc- ODVc) O,Ot

in which p p(x) (kg porous medium/m3) denotes the density of the porous medium(bulk density), and D(m2/s) the hydrodynamic dispersion matrix which incorporatesthe effects of molecular diffusion and mechanical (velocity-dependent) dispersion. Inmost transport models for porous media D takes the form (e.g., see, [2])

qiqj i,j- 1 d,(3.3) ODij {0Dmol -f- CT]q]} ij - (aL aT) Iq]

where Dmol(m2/s) is the molecular diffusivity of the adsorbate in the fluid (incorpo-rating the tortuosity effect) and aT(m) and aL(m) are the transverse and longitudinaldispersion lengths, respectively.

Next we turn to the adsorption process. In this paper we assume that the reactionsare relatively slow compared to the flow of the fluid. This makes it necessary toconsider nonequilibrium adsorption. In addition, the adsorbent surface of the grainsmay be heterogeneous. Consider a subdivision of a representative grain surface (i.e.,rescaled grain surface, e.g., the unit sphere) into rn chemically different collections ofadsorption sites corresponding to i E {1,..., m}; let hi be their relative size at therepresentative grain surface and si the corresponding adsorbed concentration. Then

m

(3.4) E Ai 1 (Ai > O)i--1

and

m

(3.5) A E ,kisi.i=1

Each component si is related to the dissolved concentration c through an adsorptionreaction that is described by the first-order equation

8i(3.6) 0--- kfi(c, with e {1,...,m}.

In these equations k > 0 (l/s) is the rate constant, and fi is the rate functiondescribing the adsorption reactions at sites i. In principle, k and 1,...,/m couldbe spatially dependent; however, for simplicity, we assume they are constant in spaceand time and k is independent of i. For the rate function in (3.6) we use the explicitform

(3.7) f,(c, e

which, in a heuristic approach, is widely used in contaminant transport models; e.g.,see [3]. The functions i in (3.7) are called the adsorption isotherms. They are theadsorbed concentrations in the equilibrium, i.e., fast reaction case, as k cx. Typicalexamples are the Langmuir isotherm

glc(3.8) (c)

1 + K2c’ K1, K2 > O,


and the Freundlich isotherm

(3.9) (c)=K3cp, K3>0, p>0.

In the last example one usually takes p E (0, 1]. For p < 1, the nonlinearities arenot Lipschitz continuous up to c 0. This results in the finite speed of propagationproperty for the concentration c as c $ 0, and thus gives rise to a free boundary asthe boundary of the support of c.

Thus, together with the transport equation (3.2), with A given by (3.5), we haveto solve for the m ordinary differential equations (ODEs)

(3.10) 0--- k (i(c)- si) with i e {1,... ,m}.

We consider (3.1), (3.2), (3.5), and (3.10) in the cylinder QT a (0, T].For all unknowns c and si we need to specify initial conditions. Thus

(3.11) c(.,0) co and si(.,O) soi in 12,

for i {1,..., m}. In addition we prescribe for c conditions along the boundaryq 0 of . Letting a -ft. q, we distinguish an inflow boundary $1 where a >_ 0and an outflow/no-flow boundary $2 where a < 0. Here $1 LJ $2 S and fi denotesthe outer unit normal. Then we impose

(ODVc- Ctc)" t F on S1T S1 X (0, T],

(ODVc). fi 0 on S2T $2 (0, T].

In (3.12), the function F F(x,t), with (x,t) S1T, denotes the flux of soluteentering the flow domain across $1 at time t > 0.

Equations (3.1)-(3.13) define the Contaminant Transport Model, which we shallrefer to as Problem CTM. In defining a weak solution or a weak formulation forProblem CTM we follow the usual definition for linear problems; e.g., see [12] and[11]. In particular, we use the space v2 tT). Then we have the following definition.

DEFINITION 1. The functions c, si QT --* [0, oo) for i e {1,..., m} form a weaksolution of Problem CTM if the following holds:

(i) c e V’(QT), ,(c) E L2(QT) for i e {1,... ,m};(ii) s, (Osi/Ot) e L2(QT) for i e {1,... ,m};(iii)

i=1(3.14)

+ fQr (ODVc-ClC)’Vl=s,r Frl+s2r acrl,

for all 7 W’1 (QT) which vanish at t T;

(iv) (Os/Ot) k {i(c) s}, i e {1,..., m}, (x, t) e QT;(v) c(-, 0) co and si(., 0) s0i, e {1,..., m}, (x, t) e QT.


Throughout this paper we take S to be piecewise smooth. With respect to the co-efficients and functions appearing in Definition 3.1 we shall assume that the followingstructural and regularity conditions are satisfied; see also [6] and [11].

(Hla) For each E {1,..., m} the isotherms i: [0, oo) - [0, o) are nondecreas-ing;

(Hlb) There are constants , # > 0 such that for Rd, (x, t) QT,

[[2 <_ T(OD)(x t) <_

the Dij are measurable in QT, OtDij L(QT) for i,j 1,...,d, and OD is sym-metric;

(Hlc) There exists 0 > 0 such that

0(x, t) _> O0 for (x, t) QT,

O, 09 L(QT);(Hld) qi L(QT) for 1,... ,d, V. L(QT), 1" exists in the sense of

trace and 1" L(ST), O and q satisfy equation (3.1);(Hie) F L(S1T), co, soi >_ O, and co, soi L2(f) for i E {1,..., m};(Hlf) p L(f), p _> p. > 0, for some positive constant p..

4. Some analytical observations. Weak solutions of Problem CTM, in thesense of Definition 3.1, were studied in [6] and [11]. The main, and nonstandard, diffi-culty for this problem lies in the fact that one of the isotherms i may be nonsmoothat c 0. This happens, for instance, when it is of Freundlich type; see (3.9). Thena situation may occur where the set (c > 0} spreads at finite speed through the flowdomain . A free boundary or interface arises as the boundary of the support of c,i.e., O{c > 0}. Across the free boundary c will have limited smoothness, even thoughthe coefficients in (3.2) may be C. The free boundary aspect of the problem wasstudied for a one-dimensional flow situation in [6] and further, for the special case oftravelling wave solutions, in [7] and [9].

To prove uniqueness and stability for weak solutions only the monotonicity of theisotherms i is required. There are three essential steps needed, which we outlinebriefly below. In a later section on convergence estimates, they will reappear indiscrete form.

Let cl -c2 and i si- s2i, i e {1,..., m), where (c, si) and (c2, s2i) aretwo weak solutions of Problem CTM. First, set

(4.1) y(x, t)0, t e (T, T],

e (0,

where T e (0, T). Note that /is not in W21’1 (QT), because of the lack of smoothnessof Cl and c2 in time, and because of the discontinuity at t T. Thus, it is not a validtest function for (3.14). This difficulty can be circumvented, however, by introducingSteklov means as described in [12]. Using the Steklov mean of r/in (3.14) leads to anexpression that contains the term

m 0i--1


Next, multiplying (3.10) by r/defined by (4.1) gives

kA >

where we have used the monotonicity of the isotherms. Then (4.2) can be estimatedfrom below by

m

(4.4) k/Q pE Aifli"T i=0

Thirdly, in (3.14), we set

I 0, t e (T,T],(4.5) l(x, t) r

(x, s)ds, t e (0, T).

This gives an expression which contains a term similar to (4.4). After some techni-calities and a Gronwall argument we obtain [6], [11] the following theorem.

THEOREM 4.1. Let hypothesis (H1) be satisfied. Then Problem CTM has a uniqueweak solution.

At the expense of some additional conditions it is possible to extend the unique-ness proof and to obtain a Lipschitz stability result for the difference in the V’(QT)norm. Assume

0O(H2a) q and are independent of time;

(H2b)0

-:-(OD) is positive definite a.e. in QT and .0 < 0 in f.

Then we obtain [6], [11] Theorem 4.2.THEOREM 4.2. Let (cl,sli) and (c2, s2i), i E {1,...,m}, denote the weak so-

lution of Problem CTM corresponding to the data {col, so,i, F } and {co2, s02i, F2},respectively, and assume hypotheses (H2) hold. Then there exists a constant C > 0such that

{ m }(4.6) Io a2[Q C IIv01 a02ll + IIp( o , so,)ll +i=1

Remark. Using expression (3.3) for the hydrodynamic dispersion matrix, hy-potheses (H2) are obviously fulfilled for the case of stationary water distribution (0)and flow (q).

Existence of weak, strong, and classical solutions was also established in [6] and[11]. Here we make some remarks regarding classical solutions. When the isothermssatisfy, in addition to the monotonicity (Hla), the conditions (for i E {1,... ,m})

(4.7) i(0)=0, i(s)>0 fors>O,

(4.s) i e CP([O, oo)) N cloc((O, oo)) for some p e (0, 1)


(such as the Freundlich isotherm (3.9)), and if the coefficients and initial-boundarydata in Problem CTM are sufficiently smooth (e.g., co E C2+p(=), 8oi CP(),F acf with cf _> 0 and ci Loo(qlT), and other technical conditions), then

c e C2+p’I+P/2(QT) and s, e CP’P+I(QT).

Note that if one of the isotherms is of Freundlich type (3.9), then (4.9) is the optimalglobal regularity. Even if the coefficients in (3.2) (0, , q, and D) were C, this wouldonly imply

For future reference, we note that (4.9) implies that c is continuously differentiable intime, and twice continuously differentiable in space, ct and the second-order spatialderivatives of c are Hhlder continuous in time with exponent 2 and in space withexponent p, and the first-order spatial derivatives of c are Hhlder continuous in timewith exponent 2

5. The CGFEM for non-equilibrium adsorption. In this section we dis-cuss the numerical approximation of solutions to (3.2), (3.10) by the CGFEM. Forsimplicity, assume m 1 and s s, so that A1 1 and A s. We will make thefollowing assumptions.

(H3a) The data and coefficients are sufficiently smooth so that (4.9) holds.(H3b) The isotherm satisfies (Hla), (4.7), and (4.8).(H3c) S $2, with DVc. n 0 on 8.(H3d) 0 and q are independent of time.

(H3e) D D(q) is symmetric and positive definite.For convenience, we will assume Problem CTM is t-periodic; i.e., we assume all

functions involved are spatially ft-periodic. This assumption is reasonable since theno-flow boundary conditions (H3c) are generally treated by reflection, and we areprimarily interested in interior flow patterns and not boundary effects.

Let 0 to < t < < tM T be a given sequence, with Atn tn -tn-1.Define fn (.) f(., tn). Let h > 0 and Mh be a finite-dimensional subspace ofW(t)consisting of continuous, piecewise polynomials of degree at most one on a quasi-uniform mesh of diameter less than or equal to h.

In the CGFEM, we approximate c(x, t") and s(x, t") by functions Cn and S inMh. Writing (3.2) in nondivergence form and applying (3.1), we obtain

(5.1) OCt A- pst A- q" VC- V" (ODVc) O.

Let T denote the unit vector in the direction (q, 0) and set

(5.2) (Iql + IOl)/.

Then

(5.3) c Oct + (1" Vc

and (5.1) can be written as

(5.4) c V. (ODVc) +pst O.


Let

()(.) --,and let ](x) =_ f(&) for a given function f. Approximate c by the backwarddifference

Let

and

cr(x, tn) O(x) c(x, tn) c(, tn-l)At

(x) ()(x) ()() (’t-)At

() () () -()At

S’-S’- )(5.14) P At ,v k(p((C) S), v), v E Mh.

Equaiions (5.13) and (5.14) are obtained by multiplying (5.9) and (5.10) by testfunctions X and v in Mh, respectively, integrating (5.9) by parts, and substituting Cn

for c and S’ for sn.Existence and uniqueness of C and Sn follow from standard results found in

[13]. Let I denote the dimension of Mh, and let {j(x)}= denote the standard nodal

}j=l andbasis. Note that (5.13) and (5.14) are equivalent to finding vectors Cn {C I

n_ n I cn I n I{S }=, where (x) andE=l C (x), S C SO-j=l S) Cj(x), withdetermined by (5.11) and (5.12), and for n _> 1,

(5.13) + (ODVC, VX) O, X Mh,

(5.8)

Then (5.4) can be written

an n- 8n 8n_l(5.9) 0

AtV. (ODVc) + p At + aN + pw O,

and (3.10) can be written

8n 8n-1

(5.10) P At kP((c) s) pw"

Initially, set CO o Mh, where (x, t) is the 0-weighted L2-projection definedby

(5.11) ((., t), X) (c(., t), X) X e Mh.

rthermore, set S o Mh,where (x, t) is the &weighted L2-projection

(5.12) (pS(., t), X) (ps(. t), X), X e Mh.S Mh byFor n 1, 2,..., M, define C Mh,

0At + p At


(A1 + atB)e + ( + kAt)-IkAt(Cn)

(5.15) t:n-1 + (1 + kAt)-1kAtA28-,(5.16) (1 + kAt)A2$ A2,n-1 -+- kAt(C).Here (C) {(p(Cn),j) I}j=, A, 1, and A2 are mass matricies with entries(X, X), (;, Xj), and (PX, Xj), respectively, and B is the stiffness matrix withentries (t?DVx, VXj). Note that (5.15) can be written in the form F(C) O.Since A + AtB is positive definite and the components of (Cn) are continuous andmonotone increasing, F is continuous and monotone increasing, and the existence anduniqueness of Cn follows. The existence and uniqueness of Sn then follows from (5.16)since A2 is also positive definite.

6. Error estimates. In this section we analyze the method given by (5.11)-(5.14). In the arguments that follow, K will denote a generic positive constant ande a small positive constant, independent of h and At. We will also employ the well-known inequality

1 a2 e b2ab <_ -e + - a,b,e ll, e > O.

A standard argument used in finite element analysis of parabolic equations is tocompare the approximate solution to an elliptic projection [17]. This technique leadsto optimal rates of convergence in the norm L(0, T; L2(ft)) as long as the solutionis sufficiently smooth. In particular, when approximating by piecewise polynomialsof degree one, one must have that ct lies in the space L2(0, T; W22(a)) to obtain h2

accuracy in space. In the problem considered here, we are not guaranteed this muchsmoothness on the solution; in particular, we will only assume the smoothness givenby (4.9). Thus, we will compare our approximate solutions to the L2-projections givenby (5.11) and (5.12). This will reduce the provable rate of convergence from h2 to h.

Let C- (, c- (, where is given by (5.11), and let S- , whereis given by (5.12). Subtract (5.9) from (5.13), and (5.10) from (5.14), and use (5.11)and (5.12) to obtain

At X + (ODV, VX) + At X

X) + (ODV", VX)

+ 0 ,x + X - Mh,

n _/n-1 )(6.2) At ,v k(p((C") (cn) n), v) + (pwn, v) v e Mh,

where an is given by (5.7) and wn by (5.8).We will derive an error estimate by taking X Cn in (6.1), multiplying by At, and

summing on n. We must first obtain a lower bound for the term in (6.1) in terms


of , , w, and a. To begin, following the uniqueness arguments given in [6], we setv Cn in (6.2) and note that by the monotonicity of :

-) k(((C") (")),

k(pn, n) 4- (pwn, n)

k(((C) ()), C )

+ k(((C) ()),)

k(pZ=, =) + (", =)

> -k(p’, ’) + k(p((C’) ()),)

(6.3) / (pw’, ’).

Thus

P zx, = zxt > [-}(pZ=, =) + (p((c=) (=)),=)] a,n:l n--1

M

n--1

Next, we seek a lower bound for the term on the right side of (6.4).To this end, we consider (6.1) with a discrete time-integral test function in analogy

with (4.5). Let X -]n eat in (6.1), multiply the result by At, and sum on n,n 1,..., M. First, we observe

n=l e=n n=l e=n e=n+l

M

(6.)n-l

by summation by parts (where the sum from n 4- 1 to M is understood to be zero ifn M) and because o O. Similarly, since o O,

(6.6)n=l =n n=l

rthermore,

1+ vC. vCzxt;


thus

E DVCn, VelAr At=- OD VCnAt,n=l .=n n=l n=l

1M

(6.7) 4- - E(ODVn, vCn)At2.n----1

Using (6.6), (6.1) with X-- Y--n tAt, (6.5), and (6.7), we obtain

M

n=l =n

kE 0At EAt + ODVn, Vtat

=n =n

( )+ ODV(n, VCtAt--n

+ an +0 At

+ 0At + pwn’ At At

1kE (ocn, cn) + _(ODVCn, vcn)At

n-’-I

+ ODV(n, VelAr=n

4- 0,n-1

At + Pwn’-EAt At--n

(6.8) + 0DEVCnAt, EVCnAtn=l n=l

By assumption (Hlb), we can bound the last term in (6.8) from below, replacing ODby . Substituting (6.8) into (6.4), we obtain the desired lower bound for the termof (6.1).


Setting X n in (6.1), multiplying by At and summing on n, and using thebounds given by (6.4) and (6.8), we obtain

M{(0At

n=l() + (ODV( V(’)

[ 1(ODV" VCn)At] ) At+k (0C,C)+

+ VAt, vct + C/t, CAtn=l n=l n=l n=l

< (p(() (c)),) + (,C) + t

Cn--1 (n--1 n)+ (ODV’, V(’) + 0At

+ k an, eAt + 0At eAt

=n =n

e=n

+ pw",EAt At

ku

"At, E ("At+ n--1 n--1

(6.9) T1 -[" "’"-[’- Tll

We now estimate the terms T1 through Tll.By the Hblder continuity of , and Hblder’s inequality

M

T (p((")n=l

M

n--1

M (L)l/qn--1


where

1 1-+-=1.q r

2Choose q ; then

r1

1_ p _<2,2

M M

T < Kh2 E (I]]]2 + [[n[[2)2/ At + Kh-2EI]n]]2L(a>At

M M

< Kh2E[[ln[[p + [[n[12/]At + Kh-2EI[’[[2L(a)Atn--I

M M

Consider

M

T (,)tM

Note that

()(, t) -() Cn(X)At

1 f(,t")()’(’ t) ()X-/ ce(s, t)d

[cr (x, tn) c(s, t)] d’,

where AT V/Ix [2 + At2, and, for x fixed, ce denotes the directional derivativeof c in the direction of the constant vector (0(x), (x)). For (s, t) between (, tn-l)and (x, tn), using the Hblder continuity of ct and cx in space and time (see (4.9)), wefind

Ic(s, t) c(x, t")l I()((, t) (, t")) + (). (w(s, t) V(x,

< KAtP/2.


Thus

M

tAt I’ (x) dx

M

<_ KAtp + K-II"IIAt.n---1

For a function g(x) e W21(), we note that

IIg .011 < KAtllVgll,

and using the fact that (On, X) 0 for X E Mh, we obtain

T= At

At

M M

< K--IIV"-IIAt + KYII"IIAt.n=l n=l

Moreover,

M

n-’-’l

M M

rt=l n---1

Similar to the estimate for T3,

T_<y Atn--1

M M

n=l n--1


and

Similar to the estimate for T2,

)T=ky a, (tAt Atn=l t=n

=k a, (At- At Atn=l =1 =1

< K/ktp + e

M

M

+KE=1

Similar to the estimates for T3, T4, and T6,

M

_< K E IIV"-llA +M

M

+KEn-1

)T8= E ODV’, V(At Atn=l

M

:1

M

+K En--1

Before estimating T9, we note that, for a spatially periodic, L2 function g, it isshown in [4] that

-1

(6.10) _< KIIgll.


Thus

n--1

M

_KE At-1

At

M

n--I

+ K E *Atn=l =1

For the estimate of Tlo, consider

st(., t)8n 8n--1

At

A-- [st(’, t) st(’, t)] dt.--1

By the HSlder continuity of st, we obtain

1 ft. Kit_tlPdtA- [st(., t) st(., t)] dt < -< KAtp.

Thus

)rlO=E ,ECat tn--1

<_ KAt2p + e At2 M

/KEn--I

Finally,

Tll-- -- ntt, nttn=l n=l

iIn--I n--I

Combining the estimates for T1-Tll with (6.9), and using the fact that

0At

n=l

n At -(0M


and the estimate

I:(’, t)ll + hi I:(’, t)ll _< Khellc( ., t)ll(here we use the quasi-uniformity assumption on the mesh), we find

M

[ I(DVn,Vn)At]At1(8M’2 cM) + E (OVen’ VCn) + k( 8, ’)+n=l

M M

< Khll’ll’zxt + K--II’IIZXt + K At2p + KAtp + Kh2

n-1 n--1

M M

+ IICIIAt + K En=l n=l

n-1

=1

2

(6.11)M

We now hide terms multiplied by e, and define

a (C,C)+

The L2 stability of Cn can be demonstrated using essentially the same argumentsgiven above; i.e., we set X and v Cn in (5.13) and (5.14), and use the monotonicity of. We then set X ,=n CeAt in (5.13), multiply the result by At and sum on n, andsum by parts. The result is that IICnll <_ KIICII for each n, where g is independentof h and At. Thus, by the L2 stability of ’, we have

IICII _< K.

Then, (6.11) implies

M

gM_K(h2 + At) +

Applying Gronwall’s lemma and the triangle inequality, we obtain the following result:THEOREM 6.1. Assume (nla)-(nlf), (4.7)-(4.9), and (n3a)-(n3e)hold, then

max I1"-C"ll <_ K(At/ + h).

Remark. When O and q are time dependent (hence D is time dependent), thisintroduces additional terms into the summation by parts arguments which lead to


(6.5) and (6.7). In particular, we obtain terms involving discrete time differenceson and OD. In this case, to derive error estimates we must assume these discretedifferences satisfy assumptions analogous to those given in (H2b). To our knowledge,these assumptions do not have any physical justification.

REFERENCES

[1] J. W. BARRETT AND P. KNABNER, Finite element approximation of transport of reactivesolutes in porous media part 1: Error estimates for non-equilibrium adsorption processes,to appear.

[2] J. BEAR, Dynamics o] Fluids in Porous Media, Elsevier, New York, 1972.[3] G. H. BOLT, Soil chemistry B. Physico-chemical models, Elsevier, Amsterdam, 1979.[4] J. DOUGLAS, JR. AND W. F. RUSSELL, Numerical methods for convection-dominated diffusion

problems based on combining the method of characteristics with finite element or finitedifference procedures, SIAM J. Numer. Anal., 19 (1982), pp. 871-885.

[5] C. N. DAWSON, T. F. RUSSELL, AND M. F. WHEELER, Some improved error estimates forthe modified method of characteristics, SIAM J. Numer. Anal., 26 (1989), pp. 1487-1512.

[6] C..]. VAN DUIJN AND P. KNABNER, Solute transport through porous media with slow adsorp-tion, in Free Boundary Problems: Theory and Applications, Vol. 1, K. H. Hoffman and J.Sprekels, eds., Longman, New York, 1990, pp. 375-388.

[7] Solute transport in porous media with equilibrium and non-equilibrium multiple-siteadsorption: Travelling waves, J. Reine Angewandte Math., 415 (1991), pp. 1-49.

[8] Travelling waves in the transport o] reactive solutes through porous media: Adsorptionand binary ion exchange-part I, Transport in Porous Media, 8 (1992), pp. 167-194.

[9] Travelling waves in the transport o reactive solutes through porous media: Adsorptionand bionary ion exchange part II, Transport in Porous Media, 8 (1992), pp. 199-226.

[10] R. E. EWING, T. F. RUSSELL, AND U. F. WHEELER, Convergence analysis of an approxi-mation of miscible displacement in porous media by mixed finite elements and a modifiedmethod of characteristics, Comput. Methods Appl. Mech. Engrg., 47 (1984), pp. 73-92.

[11] P. KNABNER, Mathematische modelle fiir den transport gelLster stoffe in sorbierenden porLsenmedien, Habilitationsschrift, Universitit Augsburg, Germany, 1988.

[12] O. A. LADYZHENSKAYA, V. A. SOLONNIKOV, AND i. N. URAL’CEVA, Linear and quasilinearequations of parabolic type, Trans. Math. Monographs, Vol. 23, American MathematicalSociety, Providence, RI, 1968.

[13] J. M. ORTEGA AND W. C. RHEINBOLDT, Iterative solution of nonlinear equations in severalvariables, Academic Press, New York, 1970.

[14] O. PIRONNEAU, On the transport-diffusion algorithm and its application to the Navier-Stokesequations, Numer. Math., 38 (1982), pp. 309-332.

[15] T. F. RUSSELL, Time-stepping along characteristics with incomplete iteration for a Galerkinapproximation of miscible displacement in porosu media, SIAM J. Numer. Anal., 22 (1985),pp. 970-1013.

[16] E. SOLI, Convergence analysis of the Lagrange-Galerkin method for the Navier-Stokes equa-tions, Report 86/3, Oxford University Computing Laboratory, Oxford, 1986.

[17] M. F. WHEELER, A priori L2 error estimates for Galerkin approximations to parabolic partialdifferential equations, SIAM J. Numer. Anal., 10 (1973), pp. 723-759.

characteristic-galerkin methods for contaminant transport ...(3.5) a e,kisi. i=1 eachcomponent si is...

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