Numerical Schemes to Solve Advective Contaminant Transport Problems with Linear Sorption and First Order Decay

Transcription

1 Numerical Schemes to Solve Advective Cotamiat Trasport Problems with Liear Sorptio ad First Order Decay Adré Luís Brasil Cavalcate PhD, Associate Professor, Uiversity of Brasilia, Brazil, Departmet of Civil ad Evirometal Egieerig (correspodig author) Jorge Gabriel Zorberg PhD, Professor, The Uiversity of Texas at Austi, Uited States, Departmet of Civil, Architectural, ad Evirometal Egieerig ABSTRACT Hyperbolic equatios are related to advectio problems i which dissipative pheomea are miimal or may be cosidered egligible. There are a umber of evirometal problems ivolvig this type of equatio. This paper provides ew, efficiet schemes to solve problems of cotamiat trasport of a soluble cotamiat through saturated soil uder statioary flow i which reversible equilibrium cotrolled by sorptio ad irreversible decay are cosidered. The solutio of this equatio usig classical methods, such as Fiite Differeces ad Lax-Wedroff, is typically foud to produce dissipatio ad other spurious effects that are purely umerical. The Cubic Iterpolated Pseudo-particle (CIP) method is used to elimiate these umerical errors. Specifically, the CIP approach is used to provide stable umerical predictios to the advectiosorptio-decay equatio eve i cases where the solutio ivolves discotiuities i the cocetratio profile. Comparisos of the umerical predictios obtaied usig this method for the case of oe-dimesioal problems of kow aalytical solutio show a clear superiority of the CIP approach i relatio to other umerical approaches. KEYWORDS: Cotamiat trasport; Advectio; Liear sorptio; First order decay; Numerical schemes, CIP method. INTRODUCTION May problems i evirometal egieerig ivolve advectio pheomea. Perhaps the best kow of such problems is the trasport of solutes by fluids percolatig i porous media. This problem is govered by differet mechaisms, icludig advectio, hydrodyamic dispersio, sorptio ad decay (Freeze & Cherry, 979; Va Geutche, 99; Bear & Cheg, 200). Amog the multiple mechaisms that are relevat for the proper represetatio of the pheomeo, advectio is the trasport mechaism whose umerical implemetatio ofte leads to problems. This is because the solutio to the advectio process is fudametally differet tha that of the hydrodyamic dispersio, sorptio ad decay processes. Specifically, while the latter processes ted to smooth the cocetratio profile, the advectio process teds to maitai it ad

2 Vol. 2 [206], Bud ofte results i discotiuities i the cocetratio profile. Cosequetly, special attetio should be paid to the implemetatio of umerical techiques to solve the trasport equatios if the advective trasport domiates the problem. Amog the various trasport mechaisms, advectio is the oe by which the solute is carried by the fluid, geerally water, with or without iteractio with the porous media. I problems ivolvig advective trasport without iteractio with the porous media, the cotamiatio frot is ofte abrupt, advacig at a velocity equal to the average liear velocity (v) of the percolatig fluid. I additio, the advective trasport i this case does ot lead to alteratios i the peak cocetratio. The effective percolatio velocity of the fluid ca be computed as the discharge velocity (defied usig Darcy s law) divided by the porosity of the medium. The hydraulic coductivity of the soil is a fudametal parameter for the solute trasport problem, sice it quatifies the media resistace to the movemet of the percolatig fluid ad, cosequetly, of the dissolved solute. Accordigly, advectio represets solute trasport iduced by the presece of a hydraulic gradiet. Hyperbolic equatios gover advective problems i which dissipative pheomea are either ot preset or ca be cosidered egligible. The solutio of such problems requires the kowledge of iitial ad boudary coditios. The geeral mathematical formulatio of a advective problem, i oe-dimesioal space, is described by the followig hyperbolic equatio: ct + vp cx = 0 () where v p is the effective percolatio velocity. The idepedet variable c is the solute cocetratio, with cx ad c t deotig the cocetratio derivatives i relatio to distace x ad time t, respectively. The term v p.c x i Eq. () is ofte referred to as the advective term. Ultimately, Eq. () represets the trasport of solute with cocetratio c for icreasig distace x. Sice this equatio does ot cotai a dissipative term, c xx, the solute is just trasported alog x, without udergoig ay alteratio from time t 0 to time t 0 + t. The absece of dissipative mechaisms implies that ay discotiuity i the iitial coditios should propagate to the solutio at ay time. Accordigly, the solutio to this trasport problem should admit discotiuous cocetratio profiles, ad the umerical method adopted to solve the hyperbolic equatios should be able to deal efficietly with such discotiuities. The iitial value problem, or Cauchy problem, for the advectio equatio ivolves fidig a fuctio c(x, t) i the semi-space D = {(x, t)/ t 0, - < x < }, that satisfies Eq. () ad a particular set of iitial ad boudary coditios. The solutio to this problem is ofte ot a cotiuous or sufficietly differetiable fuctio. This paper presets the Cubic Iterpolated Pseudo-particle (CIP) method as a ew tool to deal with problems ivolvig advectio processes with liear sorptio ad first order decay. While the CIP method has bee previously used to solve other hyperbolic equatios, it has ever bee used to solve cotamiat trasport problems ivolvig both advective ad decay mechaisms. Specifically, this study ivestigates this problem usig the CIP approach to provide stable umerical predictios to the advectio-sorptio-decay equatio. Numerical predictios obtaied usig this method will be compared agaist aalytical results as well as agaist umerical predictios obtaied usig other traditioal umerical schemes such as Fiite Differeces ad Lax-Wedroff.

3 Vol. 2 [206], Bud ADVECTIVE CONTAMINANT TRANSPORT WITH LINER SORPTION ISOTHERM AND FIRST ORDER DECAY The mass coservatio equatio ca be derived cosiderig the mass balace of a cotamiat movig through a oe-dimesioal Represetative Elemetary Volume (REV) i a saturated udeformable porous media (Bear, 972), as follows: ( ) J = c + r (2) A x t where J A is the advective mass flux of cotamiat i pore fluid, with (J A ) x deotig its derivative i relatio to the spatial coordiate x, is the porosity of the soil, r is the rate of cotamiat mass sik per uit volume, ad t is the time. The advective mass flux is give by: J = v c (3) A p The mass sik rate represets the sorptio of the cotamiat oto the soil skeleto, ad it ca be represeted by: r = r S (4) d t where r d is the soil dry desity ad S is the mass of sorbed cotamiat per uit mass of dry soil, with S t deotig its derivative i relatio to time t. Combiig Eqs. (3) ad (4) ito Eq. (2), ad assumig that the soil porosity is spatially ivariat, results i the oe-dimesioal advectio trasport equatio with sorptio: v c = c + ρ S (5) p x t d t Applyig the chai rule to Eq. (5) leads to the oe-dimesioal advective trasport equatio with reversible equilibrium cotrolled by sorptio: ( ρ ) v c = + S c (6) p x d c t Cosiderig the case of advective trasport with liear sorptio, the rate of chage of sorbed cotamiat with respect to the cotamiat cocetratio remais costat, as follows (Freeze & Cherry, 979):

4 Vol. 2 [206], Bud S c = K (7) where K d is the equilibrium partitioig coefficiet of the cotamiat betwee the fluid ad solid phases. with, Substitutig Eq. (7) ito Eq. (6) leads to the liear advective trasport equatio: d vp ct + cx = 0 (8) R R ρ K d d = + (9) where R is the retardatio coefficiet. Cosiderig the effect of a liear first-order irreversible decay, actig simultaeously with liear reversible sorptio, the sik term ca be rewritte as: ( ρ ) ( λ ) R= + K c + c (0) d d t where λ is the rate coefficiet for the first-order decay. Combiig Eqs. (3) ad (0) ito Eq. (2) leads to the advective trasport equatio with liear sorptio ad first order decay, as follows: vp ct + cx + λ c= 0 () R ANALYTICAL SOLUTION FOR THE ADVECTIVE EQUATION WITH LINEAR SORPTION ISOTHERM AND FIRST ORDER DECAY The aalytical solutio of Eq. () i the semi-space D, subjected to a iitial coditio c(x, 0) = c 0 (x), is give by: v p c xt c0 x t t R (, ) =.exp( λ ) where c 0 (x) is a fuctio describig the iitial cocetratio profile for ay x ad t = 0. If the trasport occurs i a fiite domai (e.g. x [x L, x R ]) ad if v p /R > 0, a boudary coditio of the form c(x L,t) = c L (t) must be specified, where c L (t) is a kow boudary coditio. I cotamiat trasport problems, this boudary coditio specifies the cocetratio at the outflow boudary. Whe the liear sorptio ad reactive decay are ot cosidered, R equals ad λ equals 0. Cosequetly, the solutio preseted i Eq. (2) turs ito: (2)

5 Vol. 2 [206], Bud (, ) 0 ( p ) c xt = c x vt (3) which is the traditioal aalytical solutio for the advectio problem. I fact, i the absece of ay other term, ad for a costat effective percolatio velocity v p, it is ot ecessary to use umerical methods to solve this equatio. Noetheless, this problem was selected to illustrate the occurrece of urealistic dissipative compoets that appear i the umerical solutios predicted whe usig some classical umerical methods. This problem will also allow compariso of the aalytical solutio to the umerical predictios obtaied usig the CIP method. As a example, cosider the problem of a costat wave propagatig at a velocity v p = m/s, alog x, with the iitial coditio give by (Fig. ): 0mg/L, 0m x < m c0 ( x) = 0.5 mg/l, m x 2 m 0mg/L, x > 2m (4) Figure : Iitial coditio (t = 0 s) for the advective example problems. As the iitial costat wave propagates due to the advectio as the oly trasport mechaism, it will ot suffer ay dissipatio or retardatio (Fig. 2a). However if sorptio ad decay pheomea are cosidered (e.g. cosiderig R = 2 ad λ = 0. s - ), the iitial costat wave will suffer retardatio ad dissipatio due to these processes, as illustrated i Figs. 2(b), 2(c) ad 2(d).

6 Vol. 2 [206], Bud Figure 2: Aalytical solutios for the advective propagatio problem: (a) without sorptio or decay (v = m/s; R = ; λ = 0 s - ), (b) without decay but with sorptio (v = m/s; R = 2; λ = 0 s - ), (c) with decay but without sorptio (v = m/s; R = ; λ = 0. s - ), ad (d) with decay ad sorptio (v = m/s; R = 2; λ = 0. s - ). The overall solutios (x, t, c(x,t)), for the four cases, are also show i the Fig. 3 usig threedimesioal plots. Specifically, Fig. 3(b) allows observig the three-dimesioal represetatio of the aalytical solutio of the advective pheomeo with sorptio ad without decay whe compared with the results i Fig. 3(a), where oly the advective pheomeo is observed. As ca be observed, the sorptio pheomeo results i a retardatio processes, but it does ot cause dissipatios i the cocetratio. Fig. 3(c) allows observig the effect of the decay pheomeo, which results i a dissipatio process but it does ot cause retardatio. Fially, Fig. 3(d) illustrates the effect of advectio whe both sorptio ad decay are relevat processes.

7 Vol. 2 [206], Bud Figure 3: Three-dimesioal represetatio of the aalytical solutios for the advective propagatio problem: (a) without sorptio or decay (v = m/s; R = ; λ = 0 s - ), (b) without decay but with sorptio (v = m/s; R = 2; λ = 0 s - ), (c) with decay but without sorptio (v = m/s; R = ; λ = 0. s - ), ad (d) with decay ad sorptio (v = m/s; R = 2; λ = 0. s - ). NUMERICAL SOLUTIONS OF ADVECTION PROBLEMS USING TRADITIONAL METHODS As previously discusses, hyperbolic equatios such as that represeted by Eq. () ivolve propagatio of the iitial coditio of the relevat variable (i.e. c) at a costat velocity that may ot suffer physical dissipatio mechaisms. The iitial coditio may preset discotiuities, which i this case should propagate without showig dissipatio. The solutio may show erroeous dissipatio treds if the problem is ot adequately treated umerically. Whe the problem ivolves physical dissipatio mechaisms such as those represeted by Eq. (), the umerical results could preset dissipative treds that are a combiatio of actual physical dissipatio mechaisms ad erroeous umerical spurious results. The most commo umerical treatmet for problems such as that represeted by Eq. () ivolves adoptig a forward scheme i time ad a cetral scheme i the space domai. Despite its simplicity, the stability of this method caot be guarateed with a simple criterio. A more stable solutio usig Fiite Differeces may be obtaied by adoptig a forward scheme i time

8 Vol. 2 [206], Bud ad a backwards differece i space, leadig to the followig fully explicit algorithm (Xiao, 996; Ziekiewicz & Morga, 2006): with, ( ) c = C c + C c (5) + k r k r k C r t = vp (6) x where C r is kow as the Courat umber ad, accordig to vo Neuma (Smith, 985), a stability criterio ca be defied whe C r. Usig this umerical approach, the exact solutio, c + k = c k-, ca oly be obtaied whe C r =. Also, the solutio at a upstream locatio k- of a give poit k at time, c k-, should be trasported to the poit k at time +, c + k, without ay dissipatio. While there is sufficiet iformatio to compute c + k without umerical istabilities i cases with C r <, the exact solutio is ot predicted accurately. The profile c(x, t) is similar to that observed for problems ivolvig dissipative pheomea, although this dissipative tred i the solutio is purely umerical ad udesirable. Fig. 4(a) shows the umerical results for the previously discussed problem with iitial coditios defied by Eq. (4) ad with the exact solutio beig show i Fig. 2(a). The predictio i Fig. 4(a) used a Courat umber C r = 0.5, with the propagatio of the iitial coditio beig show for times t = 4, 8, 2, 6 s. Numerical dumpig of the iitial coditio ca be clearly oticed, with decreasig amplitudes of the costat wave as the umerical solutio propagates i time. The dissipative tred observed i Fig. 4(a) is the product of umerical errors, which should be avoided sice the real problem does ot ivolve dissipatio mechaisms. Sice the umerical solutio should approximate better the exact solutio illustrated i Fig. 2(a), a ovice i the use umerical techiques may be tempted to improve the umerical solutio by usig smaller time itervals t. This would, i fact, resultig a decrease of the Courat umber Cr ad provide a sigificatly poorer umerical approximatio. Fig. (4b) illustrates the results for the same problem ad at the same times (t = 0, 4, 8, 2, 6 s) but usig C r = 0.25 (Fig. 4b). The smaller time itervals (ad smaller C r ) led to more proouced udesirable umerical results. Aother classical method to solve umerically the advective problem is that proposed by Lax & Wedroff (960). I this case, the umerical approximatio for the advectio equatio (Eq. ()) is expressed as: with, ( ) ( ) ( ) c = p C c + p C c + p C c (7) + k r k 2 r k 3 r k+ p ( Cr) = Cr ( Cr + ) (8) 2 ( ) ( ) ( ) p2 Cr = Cr + Cr (9) p3 ( Cr) = Cr ( Cr ) (20) 2 where C r is the Courat umber ad p k (C r ) are the polyomials give i Eqs. (8), (9), (20) ad illustrated i Fig. 5.

9 Vol. 2 [206], Bud Figure 4: Numerical solutio of the advectio equatio (v = m/s; R = ; λ = 0 s - ) usig the Fiite Differeces (a) usig C r = 0.5, ad (b) C r = Figure 5: Polyomials pk(cr) of Lax-Wedroff Method For a Courat umber C r =, the Lax-Wedroff polyomials correspod to p =, p 2 =p 3 =0. I this case, the exact aalytical solutio, c k + = c k-, is obtaied as was also the case whe usig Fiite Differeces. However, umerical dissipatio is also observed whe usig the Lax- Wedroff method if values of C r < are assumed. This is illustrated i Fig. 6(a), which shows the solutio for the previous problem, ivolvig a costat wave iitial coditio represeted by Eq. (4), at differet times (t = 0, 4, 8, 2, 6 s). The solutio was obtaied usig Eq. (7) with C r = 0.5. A erroeous dissipatio tred is also obtaied whe usig the Lax-Wedroff method, although it appears less proouced tha that computed whe usig the Fiite Differeces method. O the other had, a differet type of umerical perturbatio is observed i the Lax- Wedroff solutio, which results i erroeous oscillatios precedig the peak values. As the solutio propagates i time, the peaks i the solutio are observed to decrease while the oscillatios are observed to icrease with icreasig x. Fig. 6(b) illustrates the effect of the Courat umber i predictios obtaied usig the Lax- Wedroff umerical approach. The solutio is show for the same times (t = 0, 4, 8, 2, 6 s) as those i Fig. 6(a), but usig the Lax-Wedroff scheme (Eq. (7)) with C r = Dissipatio also occurs i this case, but results show decreasig peaks for decreasig values of the Courat umber. However, these lower peaks seems to be compesated with larger oscillatios

10 Vol. 2 [206], Bud precedig the peaks. The solutio also lags behid for icreasig time, with this effect beig more proouced for smaller values of C r. Figure 6: Numerical solutio of the advectio equatio (v = m/s; R = ; usig Lax-Wedroff method, (a) usig Cr = 0.5, ad (b) usig Cr = ) = 0 s NUMERICAL SOLUTIONS OF ADVECTION PROBLEMS USING CIP METHOD The cubic iterpolated pseudo-particle method, CIP, proposed by Takewaki et al. (985) ad Takewaki & Yabe (987), is used here to fid a approximate solutio c(x, t) to advectio problems that are also characterized by sorptio ad decay. Although the CIP method has bee previously used to solve hyperbolic equatios (Yabe & Aoki, 98; Kikuchi, et al., 2007), use of the CIP is developed i this study to solve the advective cotamiat trasport with liear sorptio ad first order decay, i.e., the advectio-sorptio-decay equatio. The umerical evaluatios preseted i this paper were implemeted i Mathematica codes. For a give costat advectio velocity v p > 0, the solutio should propagate the iitial coditio, displacig the solutio by a amout equal to x = v p. t durig the time iterval t. For this to happe, the followig coditio is adopted: ( ) cxt (,) c x xt, t (2) This coditio is also illustrated i Fig. 7(a). I the CIP method (Yabe & Aoki, 99), the discrete solutio c k at time for a mesh of odes x k i the space domai x is smoothed by approximatig a Hermite cubic polyomial C(x) i the iterval x betwee successive poits [x k-, x k ]. The geeral form of the polyomial is give by: 3 2 ( ) ( ) ( ) C( x) = a x x + b x x + d x x + e (22) k k k k k k k The space derivative of the cubic Hermite polyomial is give by: 2 ( ) ( ) C'( x) = 3 a x x + 2 b x x + d (23) k k k k k The CIP method forces the polyomial approximatio ad its derivatives, C(x) ad C (x), to match the discrete values, c(x, t) ad c (x, t), at the extremes of each space iterval [x k-, x k ] (Fig. 7b):

11 Vol. 2 [206], Bud = ( ) = ad (, ) ( ) cx (, t) C x c k k k c x t = C x = c (24) k k k = ( ) = ad ( ) cx (, t) C x c k k k c ( x, t ) = C' x = c' (25) k k k Figure 7: Hypothesis of CIP Method: (a) for the fuctio C(x) ad, (b) for the fuctio C' (x). Cosiderig the approximatio preseted i Eq. (24) ad usig Eqs. (22) ad (23), the coefficiets dk ad ek ca be determied as: d = c ad ek = ck (26) k k Noticig that x= x x = v t ad applyig the approximatio preseted i Eq. (25) to k k p the odes values i Eqs. (22) ad (23), it is possible to determie the other coefficiets ak ad b as: k a b ( c k + c k ) 2( ck ck ) = + x k 2 3 x ( c ) ( k ck c k + c k ) 3 2 = x k 2 After havig determied all coefficiets, the discrete values of c(x, t) ad c (x, t) may be propagated to the ext time step +, as follows: x k k p k p k p k (27) (28) c = a ( v t) + b ( v t) + c ( v t) + c (29) c = 3 a ( v t) + 2 b ( v t) + c (30) + 2 k k p k p k The CIP method should be used at time step for all odes i the space domai. This allows explicit determiatios of c ad c for the ext time step +. I this way, the iitial solutio at t = 0 h is propagated i time for as may time steps as ecessary. It should be oted that the CIP scheme requires ot oly the values of the polyomial fuctio i all odes as a iitial coditio, c k 0, but also the values of the derivatives, c k0. If these derivatives are ot explicitly defied by a

12 Vol. 2 [206], Bud cotiuous fuctio, c (x, 0) = f (x), it is still possible to estimate them usig the cetral fiite differeces of the iitial discrete fuctio values, as follows (Moriguchi, 2005): c c c 2 x k = k+ k The CIP method is expected to elimiate the dissipatio ad oscillatio umerical problems observed whe solvig the advectio equatio usig the Fiite Differeces ad Lax-Wedroff methods with Courat umbers less tha a uit. This is fudametal i order to achieve a efficiet discretizatio of the problem both i the time ad space domais. The solutio for the advectio problem with the iitial coditios defied by Eq. (4), was obtaied usig the CIP method ad a Courat umber C r = 0.5. The predicted results are show i Fig. (8) for differet times (t = 0, 4, 8, 2, 6 s). As show i the Fig. (8), the predicted results are propagated i time without loss of iformatio, dissipatio or oscillatios. The predicted results are cosidered to match well the aalytical solutio show i Fig. 2(a). Also, the results i Fig. 8 illustrate clear superiority of the CIP method i relatio to the predictios obtaied usig classical umerical techiques such as the Fiite Differeces method (Fig. 4) ad the Lax-Wedroff method (Fig. 6). (3) Figure 8: Numerical solutio of the advectio equatio (v = m/s; R = ; λ = 0 s - ) usig the CIP method ad Courat umber C r = 0.5. NUMERICAL SOLUTIONS USING TRADITIONAL METHODS FOR ADVECTION PROBLEMS WITH LINEAR SORPTION ISOTHERM AND FIRST ORDER DECAY Advectio problems with liear sorptio ad first order decay have bee covetioally solved umerically usig Fiite Differeces by adoptig a forward scheme i time ad a backward differece i space. This approach leads to the followig fully explicit algorithm (Xiao, 996; Ziekiewicz & Morga, 2006):

13 Vol. 2 [206], Bud C C c = λ t c + c R R + r r k k k Fig. 9(a) ad 9(b) show the umerical solutio of this problem obtaied usig Fiite Differeces ad a Courat umber of 0.5. The parameters used i these examples are the same as those used to obtai the exact solutio preseted i Fig. 2(c), 2(d) ad 3(c), 3(d). That is, values R = or 2 ad λ = 0. s - were adopted. (32) Figure 9: Numerical solutio of the advectio-sorptio-decay equatio with C r = 0.5, obtaied usig: (a) Fiite Differeces without sorptio or decay (v = m/s; R = ; λ = 0. s - ), (b) Fiite Differeces sorptio ad decay (v = m/s; R = 2; λ = 0. s - ), (c) Lax- Wedroff method without sorptio or decay (v = m/s; R = ; λ = 0. s - ) ad (d) Lax- Wedroff method with sorptio ad decay (v = m/s; R = 2; λ = 0. s - ). I this case, the validity of umerical solutios may be particularly difficult to assess eve whe the geeral tred of the exact solutio is kow. This is because of the difficulty to establish whether the dissipatio tred observed i Fig. 9(a) ad 9(b) are the product of actual physical pheomeo (e.g., sorptio, decay processes), or they are due to umerical errors. The approach proposed by Lax & Wedroff (960) was modified i this paper i order to be also used to solve this problem. I this case, the followig fully explicit algorithm ca be deduced to be applied to the advective pheomeo with liear sorptio ad first order decay: (,, λ) (,, λ) (,, λ) c = p C R c + p C R c + p C R c (33) + k 4 r k 5 r k 6 r k+

14 Vol. 2 [206], Bud with, (,, ) Cr 4 ( Cr p C R λ = λ t r ) 2R + R 2 2 (,, ) λ t C p C R λ = λ t 5 r + 2 R 2 r 2 (34) (35) (,, ) Cr Cr p C R λ = 6 r 2R R ( λ t) (36) Whe R = ad λ = 0 s -, Eqs. (32) ad (33) reduce to the advectio case preseted i Eqs. (5) ad (7). Fig. 9(c) ad 9(d) preset the results obtaied usig the Lax-Wedroff method after adoptig the same parameters used to defie the exact solutio show i Fig. 2(c), 2(d) ad 3(c), 3(d), (i.e., R = or 2, λ = 0. s - ). NUMERICAL SOLUTIONS USING CIP METHOD FOR ADVECTION PROBLEMS WITH LINEAR SORPTION ISOTHERM AND FIRST ORDER DECAY The CIP method was foud to efficietly solve advectio problems such as that represeted by Eq. (). This problem correspods to hyperbolic equatios for which dissipatio pheomea are ot preset or may be cosidered egligible. I order to use the CIP method to simulate the trasport model represeted by Eq. (), the problem should be solved by cosiderig two compoets: a advective (Lagragia) compoet, ad a o-advective (Euleria) compoet. Accordigly, Eq. () which govers advectio with sorptio ad decay processes was divided ito o-advective ad advective compoets. The equatio goverig the o-advective process is as follows: ct = λ c (37) while the equatio goverig the advective process is as follows: vp ct + cx = 0 (38) R

15 Vol. 2 [206], Bud The CIP method should be applied to the advective equatios, thus avoidig the itroductio of spurious umerical dissipatios i the solutio that result whe usig traditioal umerical schemes. The o-advective equatios may still be solved usig classical methods such as the Fiite Differeces Method (FDM). I this case the Fiite Differeces scheme for the oadvective compoet, Eq. (37), ca be writte as: k ( λ ) c + = t c (39) The aalytical solutio for this oe-dimesioal first order decay equatio is give by: k (, ) exp( λ ) c xt = c t (40) 0 Fig. 0(a) ad 0(b) show the results obtaied for the o-advective equatio (Eq. (37)) usig the FDM (C r = 0.5) ad the exact aalytical solutio (Eq.(40)), respectively. The umerical predictio shows a very good agreemet with the aalytical results. I fact, errors were foud to be less tha 0.%. After solvig the o-advective compoet (Eq. (37)), at a give time +, the solutio c + k, obtaied from Eq. (39), is used as the iitial coditio ck0 i the CIP method to solve the advective compoet. The derivative c k 0 is obtaied usig Eq. (3). The CIP method was used to fid the solutio of Eq. () after + time steps. Fig. presets the results obtaied usig the CIP method. The parameters adopted i this problem are the same as those used i the exact solutio preseted i Fig. 2(c), 2(d) ad 3(c), 3(d) (i.e., R = ad 2, λ = 0. s - ). Values of C r = 0.5 were used i the predictios. As show i Fig., the umerical predictio matches very well the aalytical solutio preseted i Fig. 2(c) ad 2(d). No decay or oscillatios due to umerical errors appear i the results usig the CIP method. The good quality of these predictios is i clear cotrast with that of the predictios show i Fig. 9, which had bee obtaied usig the FDM ad Lax Wedroff Methods (Fig. 9). Figure 0: Numerical solutio of the o-advective compoet (decay λ = 0. s - ): (a) obtaied usig FDM, with C r = 0.5 ad, (b) Aalytical solutio.

16 Vol. 2 [206], Bud Figure : Numerical solutio of the advectio-sorptio-decay equatio with C r = 0.5, usig the CIP Method: (a) without sorptio or decay (v = m/s; R = ; λ = 0. s - ), (b) with sorptio ad decay (v = m/s; R = 2; λ = 0. s - ). As discussed i the paper, the o-advective compoet of the trasport process is solved usig the fiite differece method. However, it should be oted that if this o-advective compoet meets the umerical stability criteria, the CIP method preseted i this work ca be easilyeasily exteded for use i ay other liear partial differetial equatios with a advective compoet. This icludes a dispersio compoet i Eq. (2), which would allow usig the CIP method to model the trasport of cotamiats that cosider processes other tha those cosidered i this paper. CONCLUSIONS This paper itroduces a ew umerical scheme, the Cubic Iterpolated Pseudo-particle (CIP) method, to solve the hyperbolic differetial equatio that represets advectio problems ivolvig liear sorptio ad first order decay processes. The method is used to provide umerical solutios to problems ivolvig advective, sorptio ad decay mechaisms, ultimately resultig i a ew, superior umerical approach to this problem. The beefits of the ew scheme is evaluated by comparig the umerical predictios agaist kow aalytical solutios ad other commo umerical schemes, icludig: (a) the explicit Fiite Differeces Method with forward derivatives i time ad backwards derivatives i space; ad (b) the Lax-Wedroff method. The various umerical solutios were compared to the exact aalytical solutio of trasport problems characterized by havig a costat wave as the iitial coditio. The solutio obtaied usig Fiite Differeces was foud to be coditioally stable but to preset spurious umerical dissipatio errors for schemes ivolvig Courat umber smaller tha oe. This udesirable dissipatio effect led to icreasig dumpig of the solutio for decreasig Courat umbers. Therefore, the solutio was foud to rapidly deteriorate whe usig small time itervals. The solutio obtaied usig the Lax-Wedroff method was foud to provide a slightly better approximatio tha that provided by the Fiite Differeces method. However, this method was foud to also produce spurious umerical effects. They icluded solutios with erroeous oscillatios ad retardatios. These udesirable effects were foud to be more proouced for smaller Courat umbers.

17 Vol. 2 [206], Bud Ultimately, oly whe selectig the value of Courat umber C r =, the Fiite Differeces ad Lax-Wedroff methods were foud to produce results that compare well with the exact solutio. However, selectio of this value of C r results i iterdepedet time ad space discretizatios. Such a ideal discretizatio ca oly be achieved i simple oe-dimesioal problems with equal itervals i the space domai. The CIP method was foud to successfully overcome the umerical problems observed whe usig the other two umerical schemes. Specifically, the iitial costat wave was foud to propagate i time ad space without udue dissipatios or oscillatios. The good match of the umerical solutio agaist aalytical results was foud to be idepedet of the adopted Courat umber. Cosequetly, this method was foud to allow highly efficiet ad idepedet discretizatios i time ad space. Overall, the CIP method shows sigificat potetial for applicatios i evirometal problems. This is because may problems have at least oe compoet i their goverig equatios that correspods to advective mechaisms. I these cases, the results of this study show that the goverig equatios ca be solved usig the CIP method without itroducig spurious umerical effects. ACKNOWLEDGEMENTS The authors ackowledge the support of the followig agecies: the Natioal Coucil for Scietific ad Techological Developmet (CNPq-Project ), the Coordiatio for the Improvemet of Higher Level Persoel (CAPES-Project 43/4-5), the Natioal Sciece Foudatio (CMMI ), the Uiversity of Brasilia ad the Uiversity of Texas at Austi for fudig this research. REFERENCES. Bear, J. (972) Dyamics of Fluids i Porous Media, Elsevier, New York, 764 p. 2. Bear, J. & Cheg, A.H.-D. (200) Modelig Groudwater Flow ad Cotamiat Trasport, Series Theory ad Applicatios of Trasport i Porous Media, Vol. 23, 834 p. 3. Freeze, R.A. & Cherry, J.A. (979) Groudwater. Pretice Hall, Ic., U.S. 604 p. 4. Lax, P.D. & Wedroff, B. (960) Systems of Coservatio Laws, Comm. Pure Appl. Math. Vol. 3 pp MR 22: Kikuchi, T., Noborio, K., Katamachi, K., Abe, Y. (2007) The Simulatio for Solute Trasport i Soil usig CIP Method, Systems Modellig ad Simulattig. Theory ad Applicatios, Asia Simulatio Coferece 2006, p Moriguchi, S. (2005) CIP-Based Numerical Aalysis for Large Deformatio of Geomaterials, PhD Thesis, Gifu Uiversity, Japa, 23 p. 7. Smith, G. D. (985) Numerical Solutio of Partial Differetial Equatios: Fiite Differece Methods, 3rd ed., pp Takewaki, H & Yabe, T. (987) The cubic-iterpolated pseudo particle (CIP) method: applicatio to oliear ad multi-dimesioal hyperbolic equatios, J. Comput. Phys. Vol. 70, pp

18 Vol. 2 [206], Bud Takewaki, H., Nishiguchi, A. & Yabe, T. (985) Cubic iterpolated pseudo-particle method (CIP) for solvig hyperbolic-type equatios, J. Comput. Phys. Vol. 6, pp Va Geutche, M.T. (99) Recet progress i modellig water flow ad chemical trasport i the usaturated zoe, Proc. 20th Geeral Assembly of the Iteratioal Uio of Geodesy ad Geophysics, Viea, Austria, IAHS, Istitute of Hydrology, Walligford, UK, pp Xiao, F. (996) Numerical Schemes for Advectio Equatio ad Multilayered Fluid Dyamics, PhD Thesis, Tokyo Istitute of Techology, Tokyo. 2. Yabe, T. & Aoki, T. (99) A Uiversal Solver for Hyperbolic Equatios by Cubic Polyomial Iterpolatio, Comput. Phys. Commu. Vol. 66, pp Ziekiewicz, O.C. & Morga, K. (2006) Fiite Elemets & Aproximatio, Dover Publicatios, Ic., Mieola, New York, p ejge