An explicit unconditionally stable numerical solution of the advection problem in irrotational flow fields

Transcription

1 WATER RESOURCES RESEARCH, VOL. 40, W06501, doi:10.109/003wr00646, 004 An xplicit unconditionally stabl numrical solution of th advction problm in irrotational flow filds Alssandra Bascià Dpartmnt of Mchanics and Matrials, Mditrranan Univrsity of Rggio Calabria, Rggio Calabria, Italy Tullio Tucciarlli Dpartmnt of Hydraulic Enginring and Environmntal Applications, Univrsity of Palrmo, Palrmo, Italy Rcivd Sptmbr 003; rvisd 13 Fbruary 004; accptd March 004; publishd 5 Jun 004. [1] A nw mthodology for th Eulrian numrical solution of th advction problm is proposd. Th mthodology is basd on th consrvation of both th zro- and th first-ordr spatial momnts insid ach lmnt of th computational domain and lads to th solution of svral small systms of ordinary diffrntial quations. Sinc th systms ar solvd squntially (on lmnt aftr th othr), th mthod can b classifid as xplicit. Th proposd mthodology has th following proprtis: (1) it guarants local and global mass consrvation, () it is unconditionally stabl, and (3) it applis scond-ordr approximation of th concntration and its fluxs insid ach lmnt. Limitation of th procdur to irrotational flow filds, for th -D and 3-D cass, is discussd. Th rsults of thr 1-D and -D litratur tsts ar compard with thos obtaind using othr tchniqus. A nw -D tst, with radially symmtric flow, is also carrid out. INDEX TERMS: 189 Hydrology: Groundwatr hydrology; 183 Hydrology: Groundwatr transport; 1831 Hydrology: Groundwatr quality; 330 Mathmatical Gophysics: Numrical solutions; KEYWORDS: advction, numrical tchniqus, transport Citation: Bascià, A., and T. Tucciarlli (004), An xplicit unconditionally stabl numrical solution of th advction problm in irrotational flow filds, Watr Rsour. Rs., 40, W06501, doi:10.109/003wr Introduction [] Th numrical simulation of advction procsss is a crucial issu for many groundwatr modling applications. This is bcaus (1) many transport problms can b rducd to thir advctiv componnt, which is affctd by th only uncrtainty of th flow fild, and () th mor gnral advction-diffusion quation is oftn solvd by splitting tchniqus, in which th solution of th diffusiv componnt is usually th asist [Abbott, 1979; Holly and Prissmann, 1977]. [3] In spit of th importanc of th advction problm, computational difficultis rmain for its numrical solution, mainly for -D and 3-D cass, whr nonstructurd grids and irrgular lmnts ar usd. [4] Th availabl mthods can b classifid as Eulrian and Lagrangian. Eulrian mthods comput th unknown function at th nods or at th lmnts of a computational msh fixd in spac, aftr spatial and tim discrtization of th PDE. It is wll known that th classical Eulrian finit diffrnc or finit lmnt mthods provid numrical solutions affctd by numrical diffusion or oscillations [Blla and Grnny, 1970; Gray and Pindr, 1983; Vnzian, 1984]. A rduction of numrical diffusion can b obtaind by valuating th spatial drivativs starting from th function valus at distant grid points, as in th QUICK and QUICKEST procdur [Lonard, 1979], that can b coupld to a limiting algorithm lik ULTIMATE Copyright 004 by th Amrican Gophysical Union /04/003WR00646 W06501 [Lonard, 1991], aiming at liminating spurious oscillations. A popular Eulrian approach is th so-calld Rung- Kutta discontinuous Galrkin (RKDG) mthod [Cockburn and Shu, 1998]. Th RKDG combins a picwis linar discontinuous finit lmnt spatial approximation of th unknown function with a tim discrtization that guarants th so-calld total variation diminishing (TVD) proprty, that is an incrasing (in tim) spatial rgularity of th solution. Aftr a spatial discrtization of th quation that guarants approximat Rimann fluxs along th lmnt discontinuitis, th proposd tim discrtization provids, with simpl low-ordr matrix oprations, a systm of ODEs that can b solvd, for givn tmporal stp, using a Rung- Kutta high-ordr accurat schm. Othr mthods ar th stramlin upwind Ptrov-Galrkin [Brooks and Hughs, 198], th Taylor-Galrkin [Dona, 1984] and th Galrkinlast squars schms [Hughs t al., 1989]. If a fully implicit tim discrtization is usd in Eulrian mthods, no limitation xists for th choic of th tim stp; howvr, this tchniqu rquirs th solution of larg nonsymmtric algbraic systms for ach tim stp, with a fast growth of th numrical ffort with th numbr of lmnts. Most of th rcntly adoptd Eulrian mthods us xplicit tim discrtization, ar scond-ordr accurat, but hav limitations on th siz of th Courant numbr, that must always b takn smallr than on, corrsponding to th wll-known Courant-Fridrichs-Lwy (CFL) condition. An xcption is an algorithm for th finit diffrnc solution of th 1-D cas, proposd by Ponc t al. [1979]. Th CFL condition fulfillmnt dos not produc a larg incrmnt of th numrical ffort in structurd mshs, bcaus it is not 1of18

2 W06501 BASCIÀ AND TUCCIARELLI: EXPLICIT UNCONDITIONALLY STABLE SOLVER W06501 affctd by th numbr of lmnts. Nvrthlss, it can limit th fficincy of th algorithm in nonstructurd mshs, obtaind by automatic msh gnrators. In this cas, th xistnc of vn a singl small lmnt can rquir th us of a small tim stp for all th lmnts, with a strong incrmnt of th computational ffort and a potntial loss of accuracy du to th us of vry small Courant numbrs in som parts of th domain. Th Courant numbr can b hld within th stability limit using diffrnt grid rfinmnt tchniqus in computational lmnts with changing vlocity valus [Soby, 1984], but this lad to vry larg tim-consuming procdurs, spcially in th -D and 3-D cass. [5] In th Lagrangian approachs, th computational grid is not fixd in spac, moving along th charactristic lins with th sam vlocity of th flow fild. Thn, it is ncssary to accuratly track th fluid particls and to valuat thir trajctoris during thir motion. Th particl tracking tchniqu is oftn th critical point of th procdur [Olivira and Baptista, 1998]. Both forward and backward tracking tchniqus hav bn proposd in th past, but th sarch for an accurat mthod is still in progrss [Bnsabat t al., 000; Pokrajac and Lazic, 00]. An xampl of Lagrangian approach is th scond momnt mthod, originally dvlopd for th cas of air pollution [Egan and Mahony, 197] and subsquntly applid also to shallow flow [Nassiri and Babarutsi, 1997]. Th disadvantag of th Lagrangian approachs is that thy ar difficult to b applid to -D and 3-D problms with irrgular boundaris and htrognous domains. Morovr, many Lagrangian tchniqus do not guarant mass consrvation. [6] In th smi-lagrangian approachs for th solution of th advction-diffusion problm, th us of Lagrangian tchniqus is rstrictd to th advctiv componnt and limitd to th tim stp usd for th solution of th nxt diffusiv problm. Th mthod of charactristics (MOC) solvs th advctiv problm by locating, at ach tim stp, th foot of th charactristic lin nding in ach nod of a fixd grid. This mans that a potntially dissipativ intrpolation of th known valus of th surrounding grid nods must b prformd. Many xampls of similar mthods can b found in th litratur, lik th two-point fourth-ordr intrpolation [Holly and Prissmann, 1977], th finit lmnt charactristic [Wang t al., 1988], th minimax charactristics [Li, 1990] mthods, as wll as th cubicsplin intrpolation [Schohl and Holly, 1991], or th charactristic Galrkin schm, rcntly dvlopd for th 3-D cas also [Kaazmpur-Mofrad and Ethir, 00]. Th MOC basd tchniqus hav th advantag of allowing th us of larg tim stps and prsrving th solution from oscillations and numrical diffusion, but thy sldom guarant local and global mass balanc [Chilakapati, 1999]. Volum tracking tchniqus [Van Lr, 1977; Ridr and Koth, 1998] follow th volum volution of th mass initially prsnt insid ach lmnt of th msh and guarant its consrvation. Th smi-lagrangian approachs, also known as Eulrian-Lagrangian approachs, ar nowadays probably th most popular tools for th solution of th advction problm quations [Clia t al., 1990; Haly and Russll, 1993]. [7] Rcntly, som grid-fr mthods hav bn proposd, basd on th intrpolation of scattrd data [Bhrns and Isk, 00], but thy suffr of th sam limitations of th Lagrangian mthods. [8] In th following sction an Eulrian procdur is proposd, that mbracs svral advantags of th Eulrian and Lagrangian approachs. It is xplicit, mass consrvativ, but also unconditionally stabl with rspct to th Courant numbr. It adopts a scond-ordr approximation of th unknown advctd function in ach computational lmnt and shows an avrag convrgnc ordr qual to in 1-D numrical tsts with smooth initial concntration. Th procdur is rstrictd to th cas of irrotational flow filds, as it is mor xtnsivly discussd in sction 4. In sction th proposd algorithm is prsntd for th 1-D cas, in sction 3 it is xtndd to th -D cas. Extnsion from th -D to th 3-D cas is straightforward. Som bnchmark problms ar solvd in ach sction, in ordr to show th faturs of th mthod.. Algorithm in th 1-D Cas [9] Th 1-D advction quation of th unknown concntration function ¼ 0; whr u is th known vlocity and x and t ar spac and tim indpndnt variabls. Equation (1) is dfind in th 0 x L, 0 t T intgration domain. A uniqu solution xists givn th initial c(x, 0) and th upstram boundary c(0, t) condition. [10] Divid th spac domain in N computational lmnts, with lngth Dx = L/N. Assum a vlocity u constant in ach lmnt. Call x 1, x th coordinats ( 1)Dx, Dx of th two xtrm x valus of th th lmnt. Divid also th tim domain in N t tim stps, with xtnsion Dt = T/N t. Call t k th tim lvl at th nd of th kth tim stp (t k = kdt). Sort all th lmnts along th downstram dirction. If u is positiv vrywhr, th ordrd squnc is I 1 =1, I =,..., I N = N. [11] Assum a picwis spatial linar approximation c (x, t) of th function c insid ach lmnt at any tim (Figur 1). Obsrv that th function c (x, t) can b discontinuous at th nod btwn two lmnts. St c 1 (t) =c (x 1, t), c (t) =c (x, t). Assum also a known low-ordr polynomial tim approximation x 1 (t) of th concntration at th scond nod of th upstram lmnt 1 from tim lvl t k to tim lvl t k+1. A third-ordr polynomial is usd in th implmntd cod, that is: X ¼ X ;0 þ X ;1 t þ X ; t þ X ;3 t 3 ; whr X is a vctor with componnts x 1 and x, that ar th tim approximations of c at th two nods of lmnt. [1] It is possibl to discrtiz, in ach lmnt, th PDE (1) in a systm of two ODEs. Th first quation is drivd from th mass balanc btwn th ntring flux, th laving flux and th avrag concntration insid th lmnt. Th scond quation is drivd from th consrvation of th spatial first-ordr momnt insid th lmnt. Th momnt is computd with rspct to th scond nod of ð1þ ðþ of18

3 W06501 BASCIÀ AND TUCCIARELLI: EXPLICIT UNCONDITIONALLY STABLE SOLVER W06501 Mrging quation (4) in quation (8) you obtain: c 0 t ðþ þ c 0 1 ðþ¼ t x1 ðþ t c 1 ðþþc t ðþ t 3 u Dx : ð9þ [14] Equation (3) can b coupld with quation (9) and solvd in th c 1 (t), c (t) unknowns. Th systm can b writtn in normal form as: c 0 1 ¼ 4x1 c u 3c 1 Dx ; ð10þ c 0 ¼ x1 c þ u 3c 1 Dx ; ð11þ Figur 1. Picwis linar approximation in spac of th concntration c in th 1-D cas. lmnt, with spac coordinat x. Th first quation is givn by: dc 1 ðþþc t ðþ t dt ¼ c 0 1 t ðþþc0 ðþ t x1 t ðþc ðþ t u Dx whr u is th vlocity in lmnt. Call M th momnt of th linar approximation of c insid lmnt with rspct to its downstram nd. Its valu is, at any tim, M ¼ c ðþdx t þ c 1 ðþc t ðþ t Dx : 3 [13] Its chang in tim is du to th variation of both c 1 (t) and c (t). Ths two variations ar not indpndnt, but ar linkd by th advction quation (1). At tim lvl t + dt th momnt of th ral concntrations corrsponding to th picwis approximation of c at tim t is qual to (s Figur ): ð3þ ð4þ and in matrix form as: dc dt ¼ A c þ bðþ: t ð1þ [15] According to th polynomial form of th approximatd concntration in quation (), b(t) is also a vctor of th form: bðþ¼b t 0 þ b 1 t þ b t þ b 3 t 3 : ð13þ [16] S in th Appndix A th rlationship btwn A, b and x 1, u, Dx. Th solution of systm (1) at th nd of th tim stp is givn by: c ¼ a 1 c 1 þ a c þ v0 þ v 1 Dt þ v Dt þ v 3 Dt 3 ; ð14þ whr a 1 and a ar two arbitrary cofficints, that hav to b chosn according to th initial concntration nod valus, c 1 and c ar th two solutions of th homognous quations associatd to systm (1) and v 0, v 1, v, v 3 ar four vctors that can b computd aftr substitution of th right-hand sid of quation (14) in quation (1). Matrix A M ðt þ dtþ ¼ Dx 0 c ðþþc t 1 ðþc t ðþ t s Dx ðs u dtþ To obtain th idntity: ds þ u x 1 ðþdt t Dx þ Odt : ð5þ M ðt þ dtþ ¼ M ðt þ dtþ; ð6þ th following quation has to b solvd: dm dt ¼ M ðt þ dtþm ðþ t : ð7þ dt Mrging quations (4) and (5) in quation (7) you obtain: dm dt ¼ Dxu x 1 ðþ t c 1 t ðþþc ðþ t : ð8þ Figur. Evaluation of th spatial momnt of c in th 1-D cas. 3of18

4 W06501 BASCIÀ AND TUCCIARELLI: EXPLICIT UNCONDITIONALLY STABLE SOLVER W06501 Figur 3. Tim cubic approximation of th laving flux. Figur 4a. Charactristic lin through th point (x i, t k+1 ). can hav ithr two ral or two complx conjugat ignvalus. In th first cas call l 1, l, and u 1, u th corrsponding ignvalus and ignvctors; in th scond cas, call l r, l i, and u r, u i th ral and th imaginary part, rspctivly, of th conjugat ignvalus and of th corrsponding ignvctors. Th solutions of th homognous quations at th nd of th tim stp ar, rspctivly: or c 1 ¼ u 1 l 1 Dt c ¼ u l Dt ð15aþ c 1 ¼ l r Dt u r cos l i Dt u i sin l i Dt c ¼ l r Dt u r sin l i Dt þ u i cos l i Dt ð15bþ : [17] Obsrv that matrix A is not a function of th concntration, thrfor it can b computd and factorizd only onc for ach vlocity fild distribution. This also holds for th homognous solutions (15) that can b stord for ach lmnt bfor starting th tim marching computations. [18] Vctors v 0, v 1, v and v 3 can b obtaind by comparing th trms with th sam tim xponnt in th polynomial part of quation (1). This lads to th squntial solution of th following linar systms: according to th typ of homognous solution, to th systm: or c 0 ¼ a 1u 1 þ a u þ v0 ; c 0 ¼ a 1u r þ a u i þ v 0 : ð17aþ ð17bþ Onc th ODEs ar solvd, th concntration at th nods of th lmnt can b approximatd by th cubic polynomials X by maintaining th initial and th final valus, as wll as th man (in tim) valu of th concntration and of thir first-ordr momnts (Figur 3). Th xact consrvation of th man concntration x j at ach lmnt nod guarants th mass balanc in th lmnt and th global mass consrvation; x j can b obtaind by tim analytical intgration of th solution of quation (1). Th man of th first-ordr momnts can b mor asily stimatd by numrical intgration. S in Appndix B th quations to b solvd for th stimation of th cofficints of polynomial (). [0] According to th scond-ordr approximation of th concntrations, ngativ concntration fluxs laving th A v 3 ¼b 3 ð16aþ A v ¼b þ 3v 3 ð16bþ A v 1 ¼b 1 þ v ð16cþ A v 0 ¼b 0 þ v 1 : ð16dþ [19] As alrady pointd out, cofficints a 1, a in quation (14) ar computd by forcing th solution to honor th givn initial concntration valus c 0. This lads, 4of18 Figur 4b. Possibl foot location of th charactristic lin (thick sgmnts).

5 W06501 BASCIÀ AND TUCCIARELLI: EXPLICIT UNCONDITIONALLY STABLE SOLVER W06501 Figur 5. Gaussian concntration tst for th 1-D algorithm. lmnt can b computd whil solving quation (1). Ngativ flux corrction can b simply carrid out according to th following procdur: aftr stimation of th polynomial cofficints of th approximating concntration, valuat th minimum valu within th rang 0 t Dt and th corrsponding tim t min. If th minimum concntration x j (t min ) is ngativ, comput th root a of th following quation: and st x ;0 j ð1 aþx ;0 j min 0; x j ¼ ð1 aþx j ðt min Þþax j ð18þ þ ax j ; x;i j ð1 a Þx ;i j i ¼ 1; ; 3: ð19þ Th corrctd polynomial concntrations do not altr th man valu. Thy hav only positiv valus if th man valu is gratr than zro. If th man valu is ngativ th corrctd polynomial is a ngativ constant valu. Mor complx procdurs ar rquird to always guarant positiv fluxs. Aftr computation of th unknown cofficints X,0, X,1, X,, X,3 is complt, a nw systm of ODEs can b solvd for th downstram lmnt with indx +1. [1] Consistncy and unconditional stability of th algorithm hav bn provd in th 1-D cas using Fourir analysis, assuming a first-ordr approximation of th concntration insid th lmnt, a constant in tim laving flux and computing th analytical solution of th rsulting ODE along th givn tim stp [Tucciarlli and Fdl, 000]. [] A usful insight about th qustion, in th cas of scond-ordr approximation of th concntration, is givn by th obsrvation of th charactristic lin passing through any x i, t k+1 point of th computational domain. Obsrv in Figur 4a that for larg Courant numbrs th foot of th charactristic lin, at tim t k, falls vry far from th original point. Th numrical stimat of concntration c at point x i, t k+1 is possibl if an initial valu is known for th ODE associatd to th charactristic lin; thrfor it is impossibl to valuat th concntration in xplicit form only as a function of th concntrations at th points x i1, t k and x i, t k. This is possibl only for th point x 1, t k+1, bcaus in this cas th foot of th charactristic lin is locatd along th tim axis and th boundary valu is known (Figur 4b). Th basic ida of th proposd algorithm is to stimat, along with th unknown valu at point x 1, t k+1, also a low-ordr tim approximation of th concntration c at th sam distanc, from tim t k to tim t k+1. Aftr this, it is possibl to solv th problm at point x, t k+1 using th computd tim approximation as initial valu for th nw ODE problm, and so on for all th unknown concntrations at tim lvl t k+1. Th algorithm statd in sction 1 applis this ida in intgral form, to guarant mass consrvation. Th Tabl 1. Convrgnc Numrical Tst for th 1-D Advction of a Gaussian Concntration Wav Cou = 0.4 Cou = 0.96 Cou =.4 Numbr of Elmnts Stp Lngth, m Avrag Error Ordr Avrag Error Ordr Avrag Error Ordr E-03 a E E E E E E E E-05 a Rad 1.78E-03 as of18

6 W06501 BASCIÀ AND TUCCIARELLI: EXPLICIT UNCONDITIONALLY STABLE SOLVER W06501 Figur 6. Unitary squar concntration tst for th 1-D algorithm. us of low-ordr spatial and tim approximations supprsss th propagation of th high-frquncy rror componnts of th stimatd fluxs from on computational lmnt to th othrs. [3] Th algorithm is tstd first for th following litratur cas [Yh, 1990]: givn th initial condition! ð cx; ð 0Þ ¼ xp x x 0Þ ; ð0þ s 0 u = 0.5 m/s, Dx = 00 m, Dt =96s,s 0 = 64 m, x 0 = 000 m and a msh with 65 lmnts, comput th concntration distribution aftr 100 tim stps. [4] Th Courant numbr of th xampl is 0.4. Initial condition (0) has bn assignd to all th lmnt nods; th final concntrations shown in Figur 5 ar th avrag concntrations computd at th nod shard by two connctd lmnts. Obsrv in Figur 5 th tst rsults obtaind using Courant numbrs qual to 0.96 and.4. With th two Courant numbrs smallr than on, numrical diffusion is almost th sam and producs a pak rduction from 1.0 to With th Courant numbr gratr than on, th pak is rducd from 1.0 to 0.84, but th shap of th concntration distribution rmains almost th sam as in th prvious cass. Also, no instabilitis occur. Yh [1990] rports a pak rduction from 1.0 to 0.5 for th tst rsults obtaind using th Ptrov-Galrkin mthod and a Courant numbr qual to 0.4. [5] A numrical convrgnc tst has bn carrid out using th sam xampl and a msh dnsity ranging from 3 to 104 lmnts. Th rats of convrgnc hav bn infrrd from th valus of th avrag rror, dfind as: E ¼ 1 N X N ¼1 1 X n n j¼1 c computd j c xact j! ; ð1þ whr n is th numbr of th nods of ach lmnt (n =in th 1-D cas, n = 3 in th -D cas). A powr dpndncy btwn th avrag rror and th grid siz has bn considrd, and th xponnt has bn assumd as th convrgnc ordr. Tabl 1 shows th ordr of convrgnc, for diffrnt Courant numbrs, computd from on msh dnsity to th nxt on. Obsrv that th ordr of convrgnc in this cas is gratr than two. [6] A scond tst has bn carrid out for th sam domain and th sam vlocity fild, using as initial concntration condition a unitary squar wav whos width is 400 m. Th cntr of th squar wav is initially locatd at 1800 m from th origin of th domain. Th final concntrations aftr a priod T = 9600 s, obtaind for diffrnt valus of th Courant numbr and Dx = 1.5 m, ar shown in Figur 6. Th shap of th solution is almost th sam for Tabl. Convrgnc Numrical Tst for th 1-D Advction of a Block-Shapd Concntration Wav Cou = 0.4 Cou = 1.0 Cou =.4 Numbr of Elmnts Stp Lngth, m Avrag Error Ordr Avrag Error Ordr Avrag Error Ordr E E E E E E E E E-03 6of18

7 W06501 BASCIÀ AND TUCCIARELLI: EXPLICIT UNCONDITIONALLY STABLE SOLVER W06501 Figur 7. Initial concntration for th -D first tst cas. Figur 8b. Tst 1: Computational msh with th hypotnus paralll to th flow dirction. th thr diffrnt valus of th Courant numbr, approximating quit wll a squar wavform vn whn th Courant numbr is largr than on. Th numrical disprsion causs th smoothing of th vrtical fronts of th concntration puls and local maxima and minima aris around th sharp fronts, but again no instabilitis ar prsnt. Th maximum valu is locatd at th bginning of th unitary wav, and its valu is for Cou = 0.4, for Cou = 1.0 and for Cou =.4. Th minimum valu is locatd at th nd of th squar wav and incrass for incrasing valus of th Courant numbr; it is qual to 0.049, and 0.00, rspctivly for Cou = 0.4, Cou = 1.0 and Cou =.4. Th convrgnc rat has bn valuatd, obtaining th valus shown in Tabl. Bcaus of th difficultis in rproducing th sharp fronts, th ordr of convrgnc is much smallr than in th prvious tst cas. 3. Extnsion of th Algorithm to th -D Cas [7] An important rquirmnt for th application of th algorithm is th possibility of sorting th lmnts in ordr to know th tim approximation of th ntring fluxs bfor ach systm of ODEs is solvd. This is always possibl, in -D and 3-D cass, if a scalar potntial xists such that th dirction of its gradint is opposit to th flow dirction. In this cas, if th lmnts ar ordrd from th highst to th lowst potntial valu, th ntring fluxs ar always known from th prvious solution of th lmnts with highr potntial. For groundwatr transport problm, th scalar potntial is givn by th pizomtric had, dfind as usual as th sum of th topographical lvation and th prssur hight [d Marsily, 1986]. [8] Assum th us of a triangular msh for th flow fild computation. If th finit lmnt mthod is usd and a constant vlocity is stimatd insid ach lmnt, th watr fluxs through th common sid of two adjacnt lmnts computd using th two lmnt vlocitis ar not ncssarily th sam. To fix this inconsistncy, ach lmnt of th original msh can b dividd in four sublmnts and th vlocitis can b changd in th sublmnts surrounding ach nod to prsrv th flux continuity [Kinzlback and Cords, 199]. If th finit volum mthod is usd, thr fluxs ar computd at th sids of ach lmnt, whr a vlocity vctor has to b stimatd. In th stady stat cas this can b don by solving a linar systm whr th fluxs of th u, v componnts of th vlocity ar st Figur 8a. Tst 1: Solution aftr T = 18000s (msh 1). 7of18 Figur 9a. Tst 1: Solution aftr T = s (msh ).

8 W06501 BASCIÀ AND TUCCIARELLI: EXPLICIT UNCONDITIONALLY STABLE SOLVER W06501 Figur 9b. Tst 1: Computational msh with th hypotnus orthogonal to th flow dirction. qual to th computd ons along two of th thr sids of th lmnt. According to th stady stat assumption, also th third flux of th vlocity vctor will b qual to th computd on. [9] Th most svr limitation of th finit lmnt mthod is that th pizomtric had changs linarly insid ach lmnt; th cll whr th mass balanc is nforcd is a polygon dfind by th cntrs of all th triangls surrounding ach nod. In -D problms, fluxs ar always dirctd from th clls of th nods with highr to th clls of th nods with lowr pizomtric had only if obtus triangls ar missing. Morovr, analytical intgration of th concntration spatial momnt insid th polygon is awkward. In th finit volum mthod, th mass balanc cll is th triangl itslf and th flux is always dirctd from th lmnt with highr to th lmnt with lowr pizomtric had. In th following, w assum th flow fild to b known from th solution of a finit volum problm and from th subsqunt stimation of a singl vlocity vctor insid ach lmnt. Figur 10. Tst 1: Solution aftr T = s for diffrnt Courant numbrs and msh 1. 8of18

9 W06501 BASCIÀ AND TUCCIARELLI: EXPLICIT UNCONDITIONALLY STABLE SOLVER W06501 Figur 11. Tst 1: Solution aftr T = s and msh 1 (Dx = 50m). Figur 1. Initital concntration for th -D scond tst [30] Givn th following -D advction ¼ 0; ðþ whr u and v rprsnt th x and y vlocity componnts, call I 1,..., I N th squnc of triangular lmnts ordrd according to th corrsponding pizomtric had. Assum th following picwis linar approximation of th concntration insid ach lmnt: c ¼ X Ni c i ; ð3þ whr N i is th Galrkin shap function of x and y coordinats and is th indx of th lmnt whr x and y ar locatd. Mass consrvation can b guarantd by th following ODE: d c i þ c þ c 3 ¼ F ðþfl t ðþ t dt 3 s ; ð4þ whr s is th ara of lmnt, F and Fl ar, rspctivly, th ntring and laving concntration fluxs. Th x and y momnts can b asily stimatd as functions of th x and y coordinats of th lmnt nods, i.., Mx ¼ s My ¼ X X N i c i c i j¼1;3 X N X y j a ij; a ij ¼ s 6 i x i ¼ X if i ¼ j; c i j¼1;3 X x j a ij a ij ¼ s 1 if i 6¼ j ð5þ ð6þ [Huyakorn and Pindr, 1983]. Th sam procdur usd in th 1-D cas can b applid hr in ordr to obtain a tim drivativ of th first-ordr momnts (5) qual to th drivativ obtaind from th solution of quation (). Th lmnt momnts, in th x and y dirction at tim t + dt, ar qual to: Mx ðt þ dtþ ¼ c ½ðx þ u dtþšds þ X d i s L i X ð1 d i Þ f i c xdl i ; My ðt þ dtþ ¼ c ½ðy þ v dtþšds þ X d i s L i X ð1 d i Þ f i c ydl i ; L i L i f i x mðx; y; tþxdl i ð7aþ f i x mðx; y; tþydl i ð7bþ whr u and v ar th componnts of th vlocity insid th lmnt in th x and y dirction, d i is qual to 1 or 0 if th flux is, rspctivly, ntring or laving th lmnt, L i is th lngth of th ith sid of lmnt, that is th sid following th ith nod in countrclockwis dirction, f i is th mass flux (positiv if laving th lmnt) pr unit lngth through th sam sid of th lmnt and x m is th approximatd concntration of th ntring flux. In th linar intgral, c rprsnts th concntration at points of th ith sid of lmnt. Linar intgrals in quation (7) rprsnt th momnt fluxs and can b asily stimatd Tabl 3. Two-Dimnsional Tst 1: Convrgnc Numrical Tst Cou x = 0.15 Cou x = 1.0 Cou x =.5 Numbr of Elmnts Stp Lngth, m Avrag Error Ordr Avrag Error Ordr Avrag Error Ordr E E E E E E E E E-03 9of18

10 W06501 BASCIÀ AND TUCCIARELLI: EXPLICIT UNCONDITIONALLY STABLE SOLVER W06501 Tabl 4. Paramtr Valus for Tst x, m y, m C s,m Pak Pak assuming c as a function of th concntrations at th nods of th ith sid of lmnt and x m as a function of th concntrations at th nods of th corrsponding sid of th upstram lmnt. This provids, for th x and y spatial momnt fluxs: L i L i f i c xdl i ¼ L i f i c j L i s þ c i L i s L i x j L i s þ x i L i s L i ds 3 c ¼ f j c i x j x i i L i þ c j x i þ c 4 i x j 5; ð8aþ 3 f i c ydl i ¼ L i f i c j L i s þ c i L i s L i y j L i s þ y i L i s L i ds 3 c ¼ f j c i y j y i i L i þ c j y i þ c 4 i y j 5 ð8bþ 3 whr j is th indx of th nod following th ith nod in countrclockwis dirction. Subtracting momnts (5) from (7) and dividing by dt, you gt: dmx dt ¼ X u c i s N X ð1 d i Þ i ds þ X d i L i f i c xdl i L i f i x mðx; y; tþxdl i ð9aþ dmy dt Figur 13b. ¼ X v c i s X ð1 d i Þ Ni ds þ X d i L i Tst : Computational msh. L i f i c ydl i : f i x mðx; y; tþydl i ð9bþ [31] Th avrag in spac concntration c _ and th momnts givn by quation (5) can b writtn in matrix form as: _ c B 11 B 1 B B C A ¼ B 1 B B B 3 C My B 31 B 3 B 33 c 1 c c 3 1 C A ð30þ B 11 ¼ B 1 ¼ B 13 ¼ 1 3 ; B 1 ¼ s 1 x 1 þ x þ x 3 ; B ¼ s 1 x 1 þ x þ x 3 ; B 3 ¼ s 1 x 1 þ x þ x 3 B 31 ¼ s 1 y 1 þ y þ y 3 ; B 3 ¼ s 1 y 1 þ y þ y 3 ; B 33 ¼ s 1 y 1 þ y þ y 3 : ð31þ Tabl 5. Tst : Comparison Btwn th Proposd Procdur and Othr Numrical Schms a Schm Max c Min c RMS Figur 13a. Tst : Solution aftr T = 9600 s. 10 of 18 Proposd procdur Exact solution First-ordr upwind schm Lapfrog schm Lax-Wndroff schm Six-point schm SOWMAC Proposd procdur (Dx/) a Dx = 00 m.

11 W06501 BASCIÀ AND TUCCIARELLI: EXPLICIT UNCONDITIONALLY STABLE SOLVER W06501 Figur 14. Tst : Solution aftr T = 9600 s for diffrnt avrag Courant numbrs (Dx = 100 m). [3] Diffrntiating quation (30), multiplying both sids by th invrs of th squar matrix B and substituting th momnt drivativs with th right hand sid of quations (4) and (9), th systm of ODEs can b writtn in normal form as: 0 dc dt B 11 B 1 B dc B dt ¼ B 1 B B B 3 C dc A 3 B 31 B 3 B 33 0dt X d i s f i x mdl i X 1 1 d i s L f i c dl i i L i _ c s u þ X d i f i x mxdl i X ð1 d i Þ f i c xdl i L ; i L i _ c s v þ X d i f i x mydl i X ð1 d i Þ f i c ydl C i A L i L i ð3þ that can also b xprssd using th sam matrix notation of quations (1) and (13) (s Appndix A for th matrix A and vctor b cofficint xprssions). [33] Owing to asymmtry, matrix A can hav ithr thr ral or on ral and two conjugat ignvalus and ignvctors. In th first cas call l 1, l, l 3 and u 1, u, u 3 th ral ignvalus and ignvctors, in th scond cas call l 3 and u 3 th ral ignvalu and ignvctor, l r, u r and l i, u i Tabl 6. Tst : Maximum and Minimum Concntrations and Root Man Squar Error for Diffrnt Avrag Courant Numbrs a Cou x ¼ Cou y Max c Min c RMS E E E E E E E a Dx = 100 m. 11 of 18

12 W06501 BASCIÀ AND TUCCIARELLI: EXPLICIT UNCONDITIONALLY STABLE SOLVER W06501 Tabl 7. Two-Dimnsional Tst : Convrgnc Numrical Tst Cou x = 0.15 Cou x =1.0 Cou x =.5 Numbr of Elmnts Stp Lngth, m Avrag Error Ordr Avrag Error Ordr Avrag Error Ordr E E E E E E E E E-04 th ral and th imaginary part of th two conjugat ignvalus and ignvctors. Th solution of systm (1) at t = Dt is givn by: c ¼ a 1 c 1 þ a c þ a 3c 3 þ v0 þ v 1 Dt þ v Dt þ v 3 Dt 3 ; whr c 1 and c ar givn by quations (15) and c 3 ¼ l 3 Dt u 3 : ð33þ ð34þ [34] Th first thr trms of quation (33) rprsnt th solution of th homognous part of quation (1). Vctors v 0, v 1, v and v 3 hav to b stimatd by substitution of solution (33) in quations (1), according to th squnc of quation (16). Cofficints a 1, a and a 3 ar computd by forcing th solution of quation (1) to honor th initial concntration valus at th thr nods of th lmnt, according to th systm: c 0 ¼ a 1u 1 þ a u þ a 3u 3 þ v0 c 0 ¼ a 1u r þ a u i þ a 3 u 3 þ v0 : or ð35þ [35] Th man (in tim) concntrations at th thr nods of th lmnt can b computd aftr th solution of th systm of ODEs (1), to obtain th polynomial cofficints shown in Appndix B. S also in Appndix C th psudocod of th algorithm for th -D cas. [36] Th -D algorithm is tstd using two litratur cass and a third on, in which th analytical solution is known. In th thr xampls th plottd final concntrations at nods ar valuatd as th avrag of th concntration valus obtaind at all th lmnt nods with th sam x-y location. [37] Th first litratur cas [Nassiri and Babarutsi, 1997] is th uniform advction of a block-shapd adimnsional concntration puls, assuming a squar spac domain and a vlocity forming a 45 angl with th domain axs. Th block is initially locatd nar th origin (s Figur 7) and its valu is qual to 1. [38] In Figurs 8a, 8b, 9a, and 9b th rsults obtaind aftr a tim T = s, using two diffrnt mshs of isoscls right-angld triangls, ar shown. Th first and th scond msh hav th hypotnus, rspctivly, paralll and orthogonal to th flow dirction. Th lngth of th smallr lmnt sid is Dx = 100 m and th adoptd tim stp is Dt = 150 s. Th valu of th vlocity componnts is u = v = 0.1 m/s. Th corrsponding Courant numbrs in th x and y dirction ar valuatd trough th following quations: ar smoothd, as a consqunc of th numrical disprsion, but no oscillations occur, rgardlss of th msh orintation. Rsults ar similar to thos obtaind using th Hrmit schm [Holly and Prissmann, 1977], in which th block shap is not prsrvd and th final valu of th pak is ovrstimatd by 0%, whil th us of th oscillation-fr mthod HLPA [hu, 1991] producs a pak undrstimation of 7%. Th sam tst is carrid out by Stfanovic and Stfan [001], comparing th rsult obtaind using two smi-lagrangian schms basd on cubic splin intrpolation [Branski and Holly, 1986] and on cubic Hrmit intrpolation [Holly and Prissmann, 1977]. Both schms produc ovrstimatd final paks (by 7% and 1%, rspctivly) and spurious minima. All th tsts ar carrid out for Courant numbrs lss than 1. [40] Obsrv that all th mthods usd for comparison ar applid with th us of quadrilatral lmnts forming a rgular msh. Rsults obtaind using th proposd procdur with incrasing Courant numbrs and th first triangular msh ar shown in Figur 10: obsrv that no instabilitis occur also with Courant numbrs gratr than 1, vn if numrical diffusion incrass. Morovr, a strong improvmnt can b obtaind by halving th lngth of th lmnt sids and multiplying by four th siz of th tim stp, laving basically unchangd th total computational ffort. S in Figur 11 th rsults obtaind in this cas with th proposd mthod using th first triangular msh. [41] W prformd th numrical convrgnc tst and obtaind th valus shown in Tabl 3 for diffrnt Courant numbrs and th first triangular msh. Th avrag rror has bn valuatd through quation (1) using n = 3. Th valus obtaind ar smallr than on and similar to th valus of th analogous 1-D cas, du to th numrical Cou x ¼ udt Dx ; Cou y ¼ vdt Dy ð36þ and ar both qual to [39] Th final concntration pak is ovrstimatd, a small minimum ariss and th sharp fronts of th block 1 of 18 Figur 15. Initial concntration for th -D third tst cas.

13 W06501 BASCIÀ AND TUCCIARELLI: EXPLICIT UNCONDITIONALLY STABLE SOLVER W06501 Figur 16. Tst 3: Comparison btwn analytical and numrical rsults. difficultis arising also in th -D cas whn trying to rproduc sharp fronts. [4] Th scond tst cas [Komatsu t al., 1997] is th uniform advction of a two-gaussian pak adimnsional concntration on an indfinit two-dimnsional domain (s Figur 1). Again, th vlocity vctor is orintd at 45 with rspct to th axs, and its componnts ar u = v = 0.5 m/s. Th two-gaussian initial concntration at point P(x, y) is 13 of 18

14 W06501 BASCIÀ AND TUCCIARELLI: EXPLICIT UNCONDITIONALLY STABLE SOLVER W06501 Tabl 8. Tst 3: Prformanc Indicators at Diffrnt Itrations Numbr of Itrations Max c Min c RMS valuatd as: cp; ð 0Þ ¼ C 1 xp PP! 1 þ C xp PP! s 1 s ð37þ whr P 1 and P ar th initial locations of th two paks, C 1 and C ar th pak valus and s 1, s rprsnt th standard dviations of th two distributions. Th distribution paramtrs usd in th tst ar outlind in Tabl 4. [43] In Figur 13a th rsult obtaind, aftr a tim T = 9600 s, using a rgular quilatral triangular msh (s Figur 13b) with Dx = 00 m and Dt = 100 s ar shown. Th avrag Courant numbrs ar dfind as: Cou x ¼ p udt ffiffiffiffi ¼ s 4p ffiffi udt 3 ; Cou Dx y ¼ p vdt ffiffiffiffiffi ¼ s 4p ffiffi vdt : ð38þ 3 Dx and ar qual to 0.38 in both dirctions. In th first row of Tabl 5 th maximum and minimum concntration valus ar shown. Th root man squar rror with rspct to th xact solution is also valuatd, as a global masurmnt of th rsult quality. In Tabl 5 th prvious prformanc indicators ar compard with indicators of th xact solution and with th rsults of som othrs numrical procdurs [Komatsu t al., 1997]. Th proposd procdur shows a good capability of rproducing th maximum valu; it succds in avoiding spurious minima and rproducing th shap of th concntration, providing th smallst valus of minimum and RMS. As in th prvious xampl, vn bttr rsults can b obtaind by halving th siz of th lmnt sids and multiplying by four th original tim stp. S th corrsponding prformanc indicators in th last row of Tabl 5. S also, in Figur 14 and Tabl 6, th rsults obtaind with th mor dns msh using diffrnt valus of th avrag Courant numbr. [44] Th numrical convrgnc tst givs th rsults shown in Tabl 7 for diffrnt valus of th avrag Courant numbr. Obsrv that in this cas th ordr of convrgnc is always gratr than two. [45] Th last tst cas is th advction of a known Gaussian adimnsional concntration puls in a nonuniform flow fild. Th flow fild has radial symmtry, with a timconstant flow rat xtractd at a point of an indfinit confind two-dimnsional aquifr of constant thicknss l and porosity w. Assuming th origin of axs as th xtraction point, th vlocity vctor at a distanc r from th origin is dirctd according to th radial dirction and its valu is: v r ¼ Q prw l ; ð39þ whr Q is th xtractd flow rat. Th adoptd vlocity fild is affctd by an stimation rror, which is function of th msh dnsity. [46] Equation () can b rformulatd on th radial dirction lading to th following þ ¼ 0: Th xact solution can b asily found as: cr; ð tþ ¼ c ð40þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! r þ Qt ; 0 : ð41þ pwl In th tst w assum Q =1m 3 /s, l =10m,w = 0.1. Th initial concntration at a point P is valuatd as: cr; ð 0Þ ¼ C 0 xp PP! 0 s 0 ð4þ in which r is th radial abscissa of point P. Th pak location P 0 has coordinats x 0 = 3.5 m and y 0 = 3.6 m, and w assum C 0 = 1 and s 0 = m (s Figur 15). Th numrical procdur has bn carrid out using a rgular quilatral triangular msh with Dx = 1 m and a tim stp Dt = 100 s. Bcaus of th non uniform vlocity fild, th avrag Courant numbrs vary from on point to th anothr. Th rang of avrag Courant numbrs in x and y dirctions is [ ]. [47] In Figur 16 th rsults of th numrical procdur ar compard with th analytical solution, at diffrnt itrations. Again, no instabilitis occur and th undrstimation of th pak valu is small. In Tabl 8 th maximum and minimum valus ar shown, togthr with th RMS valu, at diffrnt tims. Obsrv that th RMS maintains a small valu throughout th itrations. Th numrical convrgnc tst prformd in this cas givs th rsults shown in Tabl Mass Consrvation and Irrotational Vlocity Fild Limitation [48] Elmnt concntrations, at th nd of ach tim stp, ar givn by th analytical solution of th systm of ODEs givn by th consrvation diffrntial quations of th mass, as wll as of th x and y first-ordr spatial momnts. Bcaus of this, th local mass balanc is satisfid for givn stimation of th total ntring flux, that is: F ¼ c_ ðt þ DtÞ _ c ðþ t t þ Fl ; D ð43þ whr c _ (t) and c _ (t + Dt) ar th man (in spac) concntration valus at th bginning and at th nd of th tim stp, F is th givn tim avrag of th total ntring flux and Fl is th tim avrag of th computd total laving flux of lmnt. Entring fluxs ar st qual to Tabl 9. Two-Dimnsional Tst 3: Convrgnc Numrical Tst Numbr of Elmnts Stp Lngth, m Avrag Error Ordr E E E of 18

15 W06501 BASCIÀ AND TUCCIARELLI: EXPLICIT UNCONDITIONALLY STABLE SOLVER W06501 th approximation of th laving fluxs of th upstram lmnts by quations (A) and (A4) (s Appndix A). Equation (B7) (s Appndix B) guarants that th man (in tim) lmnt concntrations, stimatd by th solution of th systm of ODEs, ar qual to th man of th approximatd ons. Bcaus th ntring volumtric flux is, for ach sid of ach lmnt, qual to th flux laving from th common sid of th nxt upstram lmnt, this also implis that, for th sam sid, th man of th assignd ntring flux is qual to th man of th computd laving flux of th nxt upstram lmnt. Bcaus of this, th sum of quations (43) provids: X F u u ¼ X c ðt þ DtÞ c ðþ t þ X Dt Fl d d ð44þ whr F u is th total man flux ntring in ach upstram boundary lmnt and Fl d is th total man flux laving from ach downstram boundary lmnt. Equation (44) guarants th global mass consrvation. [49] Th proposd algorithm has bn dvlopd for th solution of problms whr vlocity is proportional to th gradint of a scalar potntial and vorticity is zro. Th rlationship btwn vorticity (or rotationality) and potntial can b found in any introductory book of watr wav mchanics, lik th txt of Dan and Dalrympl [199]. Vlocity filds displaying vorticity, lik shallow watr bodis with high-frquncy wavs or vlocity filds producing scours around pirs of a bridg cannot b tratd according to th proposd numrical schm. On th othr hand, bcaus th mthodology rquirmnt is to hav fluxs always moving from points with highr to points with lowr scalar potntial, th tchniqu can also b applid if th mor gnral condition v ¼KðHÞ gradh ð45þ holds, whr v is th vlocity vctor, H is th scalar potntial and K is a smipositiv dfinit matrix. This implis that th algorithm can b applid for th simulation of all th groundwatr transport problms whr th Darcy s law holds, with saturatd or unsaturatd, isotropic or anisotropic porous mdium. [50] A scalar potntial also xists in shallow watr flow filds if th inrtial trms ar nglctd in th dpthavragd form of th Navir-Stoks quations, calld Saint-Vnant quations. This simplification of th momntum quation can b adoptd in all th flow-routing problms whr th upstram flow wav has a larg nough tim priod [Tsai, 003]. An application of th arly vrsion of th algorithm to this problm, using a picwis constant approximation of th unknown watr dpth, is givn by Noto and Tucciarlli [001] for th 1-D ntwork cas and by Tucciarlli and Trmini [000] for th -D cas. In both th groundwatr transport and shallow watr applications th scalar potntial is th pizomtric had. In th first cas pizomtric hads ar known from th prvious solution of th flow problm and in th scond cas thy hav to b itrativly computd using a fractional stp mthodology. Th fractional stp mthodology splits th quations in a nonlinar advctiv componnt, which is solvd with th proposd tchniqu, and in a linar diffusiv componnt that is solvd with a standard Galrkin mthod. 5. Conclusions [51] Th proposd algorithm for th numrical solution of th advction problm has th following appaling proprtis: (1) unconditional stability, () scond-ordr approximation of th unknown concntration within ach computational lmnt, and (3) local and global mass consrvation. Th 1-D and -D tsts suggst a computational accuracy similar to othr xplicit scond-ordr mthods, but th unconditional stability of th algorithm allows th us of non structurd mshs and th choic of a tim stp basd on th avrag siz of th lmnts and th avrag norm of th vlocity. This should mak th algorithm comptitiv about th tim computation rquird for ach tim stp. In th -D cas (s th psudocod in Appndix C) th solution of fiv factorizd linar systms of ordr thr for ach lmnt ar ndd, along with th computation of th corrsponding right hand sids. Of cours, a prliminary work is rquird for th factorization of th matrics and th lmnt ordring according to thir scalar potntial. Furthr improvmnt of th rsults, in th cas of initially discontinuous functions, can b obtaind with th us of a function limitr as will b discussd in a futur work. Th major limitation of th algorithm is th nd of a scalar potntial for th vlocity fild, which includs, howvr, quit larg classs of nvironmntal and nginring problms. Anothr limitation is th srial structur of th computations, which impairs, in th 1-D cas, th us of paralll computing; this limitation is partially avoidd in -D and 3-D cass, whr svral lmnts can b solvd simultanously along diffrnt flux pips. Appndix A: Cofficints of th Linar Diffrntial Systm [5] Th matrix and vctor lmnts in quation (1) ar On-dimnsional cas Two-dimnsional cas A 1;1 ¼ 3u Dx A ;1 ¼ 3u Dx A 1; ¼ u Dx A ; ¼ u Dx ða1þ b 1 ¼ 4u Dx x1 b ¼ u Dx x1 : ðaþ A i;j ¼ X3 m¼1 ð B Þ 1 i;m U m;j; ða3þ whr cofficints of matrix B ar givn by quation (31). Call jp and jm th nods following and prcding nod j in countrclockwis dirction; th cofficints of matrix U ar th following: U 1;j ¼ d jf j L j þ d jm f jm L jm s ; ða4þ 15 of 18

16 W06501 BASCIÀ AND TUCCIARELLI: EXPLICIT UNCONDITIONALLY STABLE SOLVER W06501 U ;j ¼ d j f x j L jp x j j x jp d jm f jm 3 L jm x j x jm þ x jm þ u s 3 3 ; ða5þ U 3;j ¼ d j f y j L jp y j j y jp 3 d jm f y jm L j y jm jm þ y jm þ v s 3 3 : ða6þ [53] Th vctor cofficints of th polynomial quation (13) hav componnts b n i ¼ X3 j¼1 ðb Þ 1 i;j V j;n n ¼ 0; 1; ; 3: ða7þ [54] Call x m,n j and x m,n jp th polynomial approximation of th concntrations at th nods of th upstram lmnt m sharing th sid with nods j and jp, or th assignd upstram boundary concntration. Elmnts of matrix V ar th following: V ;n ¼ X3 j¼1 V 3;n ¼ X3 V 1;n ¼ X3 x m;n jp j¼1 j¼1 1 d j f j L j x m;n jp x m;n j 1 d j x jp x j 3 1 d j f j L j x m;n j y jp y j 3 x m;n j þ xm;n jp þ xm;n jp þ x m;n jp f j L j s ; ða8þ x j þ x m;n j y j þ x m;n j x jp y jp ; ða9þ : ða10þ [56] In th -D cas th following quations complt th st (B1): c 3 ¼ l 3 Dt u 3 l 3 Dt ; ðbþ [57] Th following quation guarants th quality of th initial valu (t =0): X ;0 ¼ c 0 : ðb3þ [58] Th following quation guarants th quality of th final valu (t = Dt): X ;0 þ X ;1 Dt þ X ; Dt þ X ;3 Dt 3 ¼ c : ðb4þ [59] Th following quation guarants th quality of th man valu: X ;0 þ X ;1 Dt= þ X ; Dt =3 þ X ;3 Dt 3 =4 ¼ a 1 c 1 þ a c þ v 0 þ v 1 Dt= þ v Dt =3 þ v 1 Dt 3 =4: ðb5þ [60] Th following quation approximats th first-ordr momnts: X ;1 Dt = þ X ; Dt 3 =3 þ X ;3 Dt 4 =4 þ X ;4 Dt 5 =5 ¼ MX : ðb6þ [61] In th -D cas quation (B5) is rplacd by th following: X ;0 þ X ;1 Dt= þ X ; Dt =3 þ X ;3 Dt 3 =4 ¼ a 1 c 1 þ a c þ a 3c 3 þ v 0 þ v 1 Dt= þ v Dt =3 þ v 1 Dt 3 =4: ðb7þ Appndix B: Polynomial Approximation [55] Th man valus of th homognous solution of quation (1) in th 1-D cas ar givn by: or c 1 ¼ u 1 Dtl l 1 Dt c ¼ u 1 Dtl l Dt ; c 1 ¼ l r l r Dt h Dt l þ i i l u r r sin l i Dt l i u i l r c ¼ l r l r Dt h Dt l þ i i l u i r sin l i Dt l i þu r l r l i l r cos l i Dt l i l r cos l i Dt sin l i Dt þ cos l i Dt h l r u r þ l i u i Dt l þ i i l r þ cos l i Dt sin l i Dt ðb1aþ l r u i l h i u r Dt l þ i l i: r ðb1bþ 16 of 18 Appndix C: Psudocod for th -D Cas [6] Solv th flow fild Bgin cycl for =1,..., N Comput u, v, s, f i, L i, i =1,...,3 Comput A (quations (A1) (A6)) Comput l 1, l, l 3, u 1, u, u 3 or l r, l i, l 3, u r, u i, u 3 Comput c 1, c, c 3 (quations (15) and (33)) and c 1, c, c 3 (quations (B1) and (B)), End cycl [63] Sort th lmnt according to th dcrasing pizomtric had. Call I th vctor of th ordrd lmnt indxs. Bgin cycl for k =1,..., N t Bgin cycl for i =1,..., N = I i Comput b 0, b 1, b, b 3 (Equations (A7) (A10)) Comput v 3 (quation (16a)) Comput v (quation (16b)) Comput v 1 (quation (16c)) Comput v 0 (quation (16d)) Comput a 1, a, a 3 (quation (34)) Comput c (quation (3)) at t = Dt

17 W06501 BASCIÀ AND TUCCIARELLI: EXPLICIT UNCONDITIONALLY STABLE SOLVER W06501 Notation Comput th vctor cofficints of th approximating concntrations of c : X,0, X,1, X,, X,3 (quations (B3), (B4), (B6), (B7)) Updat initial valus: c! c 0 End cycl End cycl A b(t) b 0, b 1, b, b 3 B c c c c i c 0 c 1, c, c 3 _ c c 1, c, c 3 C 0, C 1, C Cou x, Cou y Cou x, Cou y E F, Fl F, Fl F u Fl d H I I 1, I,..., I N K l L L i M M Mx, My matrix of th systm of ODEs. known vctor of th systm of ODEs. vctor cofficints of th third-ordr polynomial b. squar cofficint matrix. unknown concntration function. picwis linar approximation of c in th th lmnt. concntration vctor of th c at th nods of th th lmnt. valus of c at th ith nod of th th lmnt. vctor of th initial concntration valus in lmnt. solutions of th homognous problm coupld to th systm of ODEs. spatial avrag valu of concntration in lmnt. tim man valu of th homognous solutions at lmnt. adimnsional concntration pak valus. courant numbrs in th x and y dirctions; avrag Courant numbrs in th x and y dirctions. avrag rror btwn th xact and th computd solution. ntring and laving concntration fluxs. tim avrag valu of th total ntring and laving fluxs at lmnt. tim avrag valu of th total flux ntring in th upstram boundary lmnts. tim avrag valu of th total flux laving from th downstram boundary lmnts. scalar potntial. vctor of th ordrd lmnt indxs. sorting indxs of th N lmnts. smipositiv dfinit matrix; constant thicknss of an indfinit confind -D aquifr. domain spac lngth in th 1-D cas. lngth of th ith sid of th lmnt. momnt of th ral concntrations in th 1-D cas. momnt of th linar approximation of c insid lmnt with rspct to its downstram nd (1-D cas). momnts of th ral concntration at lmnt in x and y dirctions (-D cas). Mx, My momnts of th linar approximation of c insid lmnt (-D cas). MX vctor of th numrically stimatd first-ordr concntration momnts at th lmnt nods. n numbr of nods of ach lmnt. N numbr of computational lmnts. N t numbr of tim stps. N i Galrkin shap function of th ith nod insid th th lmnt. P gnric point of th spatial domain. P 0, P 1, P locations of th concntration paks. Q flow rat. r, s, x, y spatial abscissas. t tmporal abscissa. t min tim corrsponding to th minimum approximatd concntration. T domain tim lngth. u, v, v r known vlocity componnts in th x, y an r dirctions. u, v known componnts of th vlocity vctor at lmnt. u 1, u, u 3 ral ignvctors. u i imaginary part of a complx ignvctor. u r ral part of a complx ignvctor. U cofficint matrix for th valuation of A. v vlocity vctor. v 0, v 1, v, v 3 vctor cofficints of th zro, first, scond and third powr of t in th c solution. V cofficint matrix for th valuation of th b vctor cofficints. x i spatial coordinat of th gnric point i. x 0, y 0 coordinats of th initial location of th concntration paks. x i, y i coordinats of th ith nod of th lmnt. a numrical wighting cofficint. a ij numrical cofficints for th valuation of th spatial momnt of c. a 1, a, a 3 arbitrary cofficints of th solutions of th homognous systm. d i adimnsional cofficint qual to 0 or 1. Dx, Dy spatial stps. Dt tim stp. l 1, l, l 3 ral ignvalus. l i imaginary part of a complx ignvalu. l r ral part of a complx ignvalu. X polynomial tim approximation vctor of th concntrations at th nods of lmnt. x i lmnts of vctor X, i = 1,..., 3 (function of t). X,0, X,1, X,, X,3 vctor cofficints of X. x i man (in tim) valu of x i x m approximatd concntration of th ntring flux. 17 of 18