A MARCHING IN SPACE AND TIME (MAST) SOLVER OF THE SHALLOW WATER EQUATIONS

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1 XXX Convgno di Idraulica Cosruzioni Idraulich - IDRA 6 Masr Class: Modlli numrici di corrni luviali su ondo isso ondo mobil A MARCING IN SPACE AND TIME () SOLVER OF TE SALLOW WATER EQUATIONS Cosanza Aricò Diarimno di Inggnria Idraulica d Alicazioni Ambinali, Univrsià di Palrmo Palrmo (IT) mail arico@idra.unia.i Kywords: shallow war, unsady low, numrical mhods, Eulrian mhods, racional im-s mhods, low rouing. SOMMARIO Nl rsn lavoro si rsna una nuova modologia r la soluzion numrica dll uazioni coml D D dl Sain Vnan. Ad ogni asso moral, il sisma di uazioni dirnziali all driva arziali (PDEs) ch govrnano il rocsso vin suddiviso, alicando una modologia dl asso moral razionao, in un sisma di rdizion conviva in un sisma di corrzion diusiva. Il sisma di rdizion vin a sua vola suddiviso in un sisma di rdizion conviva in un sisma di corrzion conviva, sulla bas dl valor di una unzion onzial arossimao. S il camo di moo amm un onzial scalar sao, la as di corrzion conviva scomar la soluzion dl sisma di corrzion conviva è la sssa dl sisma di rdizion conviva. I du sismi di rdizion corrzion conviva vngono risoli alicando la modologia (Marching in Sac and Tim). risolv r ogni lmno di calcolo un sisma di du (nl caso D) o r (nl caso D) uazioni dirnziali all driva oali (ODEs), usando r la discrizzazion moral una cnica a asso moral variabil, razion dl asso moral sclo. L cll di calcolo vngono ordina risol in bas a valori dlla unzion onzial arossimao dcrscni, duran la as di rdizion, crscni duran la as di corrzion. Il sisma di corrzion diusiva si risolv con una cnica imlicia, risolvndo un sisma linar bn condizionao, la cui maric ha l dimnsioni dll cll di calcolo, risula sarsa, simmrica dinia osiiva. Il modllo rooso ha mosrao incondizionaa sabilià riso al numro di Couran (CFL), non richid un raamno scial di rmini sorgni il coso comuazional risula roorzional al numro dgli lmni di calcolo. Sono sai condoi divrsi s numrici i risulai dl modllo sono sai conronai sia con soluzioni analiich, sia con dai srimnali, sia con alr soluzioni numrich orni da alri modlli. ABSTRACT A nw aroach is rsnd or h numrical soluion o h coml D and D Sain-Vnan uaions. A ach im s, h govrning sysm o Parial Dirnial Euaions (PDEs) is sli, using a racional im s mhodology, ino a convciv rdicion sysm and a diusiv corrcion sysm. Convciv rdicion sysm is urhr sli ino a convciv rdicion and a convciv corrcion sysm, according o a sciid aroimad onial. I a scalar ac onial o h low ild iss, corrcion vanishs and h soluion o h convciv corrcion sysm is h sam soluion o h rdicion sysm. A MArching in Sac and Tim () chniu is usd or h soluion o h wo sysms. solvs a sysm o wo (in h D cas) or hr (in h D cas) Ordinary Dirnial Euaions (ODEs) in ach comuaional cll, using or h im discrizaion a sl-adusing racion o h original im s. Th comuaional clls ar ordrd and solvd according o h dcrasing valu o h onial in h convciv rdicion s and o h incrasing valu o h sam onial in h convciv corrcion s. Th diusiv corrcion sysm is solvd using an imlici schm, ha lads o h soluion o a larg linar sysm, wih h sam ordr o h cll numbr, bu sars, symmric and wll condiiond. Th numrical modl shows uncondiional sabiliy wih rgard o h Couran numbr 7

2 Cosanza Aricò (CFL), ruirs no scial ramn o h sourc rms and a comuaional or almos roorional o h cll numbr. Svral ss hav bn carrid ou using analyical soluions, rimnal daa and numrical soluion givn by ohr numrical rocdurs roosd in liraur. INTRODUCTION Sain-Vnan (SV) uaions (or shallow war uaions) ar commonly alid or h simulaion o unsady shallow war lows (D Sain Vnan, 87). Boh lici and imlici numrical mhods hav bn nsivly usd or h comuaion o hir numrical soluion. Evn hough imlici schms ar sabl or CFL numbrs grar han, hy lad o h soluion o larg non linar algbraic sysms, and h comuaional or grows vry uicly along wih h numbr o lmns. Imlici Prissman schms show also ohr limiaions, scially or simulaions o ranscriical lows (Mslh al., 997). For hs rasons, mos o h algorihms rcnly roosd by h Auhors all in h class o lici mhods. Svral Godunov-y schms hav bn roosd or h non viscous orm o h SV uaions, whr sourc rm in h momnum uaion is zro (Alcrudo and Garcìa-Navarro, 99; Mingham and Causon, 998). A racional im s rocdur (Toro, 997) can b alid or h soluion o h shallow war uaions using Godunov-y schms whn sourc rms canno b nglcd. Th corrsonding non-homognous orm o h govrning uaions is solvd in wo sunial ss. During h irs s, h homognous roblm (wihou h sourc rm) is solvd. In h scond s, a s o ODEs sysms including sourc rms ar solvd sunially, on ar h ohr. A similar aroach is siml, bu on roducs oor soluions, scially in saionary or uasi-saionary cass (LVu, 998). In ac, whn h local im drivaiv o h low variabls is ngligibl, h saial drivaiv o h lu rms should acly balanc h sourc rm. Fracional s aroach in his cas can ail, sinc h soluion o h homognous roblm may lad o larg changs in h indndn variabls, which ar diicul o corrc by solving h ordinary dirnial uaions during h scond s. In h las dcad many Auhors hav dvlod Godunov-y schms or h soluion o balanc laws, including sourc rms. Th main ocus o hs schms is h balanc bwn h numrical lu and h sourc rm, whn h sam numrical discrizaion is alid. Brmudz and Vazuz-Cndon (994), using a hr-oins uwind schm wih s saial aroimaion ordr and uwind discrizaion o boh lu and sourc rms, inroducd h diniion o consrvaion rory. This rory is h accuracy rovidd by a numrical schm in h soluion o h sady-sa roblm rrsning war a rs (consan oal had and zro low ra in all h domain). I h consrvaiv rory is no saisid, acly or aroimaly, h roagaion o unhysical oscillaion is dcd also in non-saionary roblms. In h ar o Brmudz and Vazuz-Cndon (994) sourc rm is givn only by h boom slo in hydrosaic condiion. Vazuz-Cndon (999) ndd h diniion o consrvaion rory o h cas o uniorm low in rcangular scions. ubbard and Garcìa- Navarro () rovd his consrvaion rory or h nd ordr MUSCL schm wih lu limir. Uwinding h sourc rm is comuaionally nsiv, bcaus sourc rm has o b rocd on an ignvcors basis. LVu (998) dvlod an aroach whr h sourc rm is mbddd ino h wav-roagaion algorihm, avoiding h racional s inconvnin mniond bor. Th Auhor dind a Rimann roblm insid h cll o balanc h sourc rm and h lu gradin; h mhod rsrvs sady and uasi-sady low condiions. Saring rom h wor o Brmudz and Vazuz-Cndon (994), Vuovic and Soa () and Zic al. (4) dvlod an arly vrsion o h highr ordr schms (ENO and WENO), whr numrical lu rms acly balanc h sourc rms. Th main limiaion o hs numrical mhods roosd or balanc laws is ha hy hav bn dvlod or sciic orm o h sourc rms and miss gnraliy. Zhou al. () inroducd h surac gradin mhod (SGM) or h accura ramn o h bd slo rms in h shallow wars uaions. Th Auhors roosd a Godunov-y schm whr, in conras o convnional daa rconsrucion mhods basd on h consrvaiv variabls, h war surac lvl is chosn as h basis or daa rconsrucion. Th consrvaiv variabls ar accuraly 8

3 Th Schm or Shallow War Euaions rconsrucd by a nd ordr schm a cll inracs and h numrical lus ar comud by a Rimann solvr. Ingraion in im is rormd by mans o a rdicor and a corrcor s. Mor rcnly, h SGM has bn usd by h sam Auhors (Zhou al., ) o dal wih bd oograhy wih vrical ss. Th mhod is rrrd o as h surac gradin mhod or ss (SGMS). Boh SGM and SGMS roduc accura soluion ovr srucurd mshs. All hs mhods hav bn roosd or srucurd mshs and ar rsricd in hir abiliy o i irrgular and/or curvilinar boundaris. A oular aroach or h ramn o irrgular boundaris is h us o boundary-id grids which ma h boundary conour a coordina surac (ausr al., 985; Yang and su, 99). Th low uaions ar ransormd rom Carsian o curvilinar coordinas and hn aroimad using ini dirncs. A grid can b gnrad iraivly by mans o an lliic solvr (Thomson al., 974) or dircly by an algbraic chniu such as ransini inrolaion (Gordon al., 97). Th comliy o low and momnum uaions wrin in h ransormd coordinas incrass and discrizaion o h mrics ransormaion drivaivs could roduc nw aroimaion rrors. Alrnaivly o h boundary-id grids, Causon al. () ndd h Carsian cu cll aroach (Brgr al., 995; D Zuw and Powll, 99) o a Godunov-y schm basd on a MUSCL rconsrucion and suiabl aroima Rimann solvr. In h Carsian cu cll aroach, boundary conours ar cu ou o a bacground Carsian msh and clls arially or comlly cu ar signd ou or scial ramn in h comuaion o h gradins and rconsrucion o h low daa (Causon al., ). Sinc a cu cll may urn ou o b arbirarily small, numrical sabiliy ruirmns may orc a vry small im s. To ovrcom his roblm, a cll mrging chniu is imlmnd (Chiang al., 99). Th basic ida is o combin svral nighbouring clls oghr so ha inracs bwn mrgd clls ar ignord and wavs ravl in a nw combind largr cll wihou rducing h global im s. Th rmaindr o h low clls ar rad in a sraighorward mannr. Succssivly, Zhou al. (4) alid h Carsian cu cll aroach o h SGM and SGMS mhods. Triangular msh is gnrally h simls and mos convnin mhod or covring a D domain. An advanag o using riangular mshs is hir abiliy o gnra grids on arbirary gomris and o incras h numbr o clls in high-gradin rgions or in rgions o aricular inrs in h low ild. Rcnly, svral numrical schms hav bn roosd o solv h shallow war uaions ovr unsrucurd mshs. Anasasiou and Chan (997) dvlod a D dh ingrad nd ordr Godunovy schm, basd on a cll-cnrd ini volum uwind ormulaion. Th Ro s lu uncion is usd or h valuaion o h inviscid lus a cll inracs, solving a Rimann roblm in h dircion normal o h cll inrac. Th viscous rms ar comud using a nd ordr accura ini volum ormulaion. Tim ingraion is don according o a nd ordr razoidal imlici schm. ubbard (999) roosd a nd ordr MUSCL schm ovr unsrucurd mshs. Th Auhor solvs h homognous orm o h shallow war uaions and alis h Ro s aroima Rimann solvr a cll inracs or h lu simaion. Th Auhor uss a nd ordr Rung-Kua im sing or im ingraion. Jiwn and Ruun () alid a comosi ini volum ormulaion or h soluion o h homognous SV uaions. Th Auhors combin a La-Wndro and a La-Fridrich schm ino a muli-s comosi schm. Th La-Wndro schm is highly disrsiv and roducs unhysical oscillaions; on h oosi, h La-Fridrich schm is characrizd by numrical diusion. Th comosi schm combins hs wo mhods in a s-by-s schm o loi hir mris and rmov hir dicincis. Yoon and Kang (4) roos a nd ordr ini volum schm ovr unsrucurd riangular msh. Th Auhors aly h LL aroima Rimann solvr or lu simaion a h cll boundaris. To circumvn numrical insabiliis, h Auhors sli h sourc rm in h ricion and h boom slo comonns alying an oraor sliing chniu. Tim ingraion is mad by a rd ordr TVD- Rung-Kua mhod (Shu and Oshr, 988). In hs Godunov-y schms im s siz is limid by h Couran-Fridrichs-Lvy (CFL) condiion. In h rsn ar w urhr dvlod h rcn algorihm or h soluion o h D and D SV uaions. has bn alrady dvlod or h soluion o h siml convciv ransor 9

4 Cosanza Aricò roblm (Bascià and Tucciarlli, 4; Aricò and Tucciarlli, 6), h D and D shallow war roblm wih irroaional low ild (Tucciarlli and Trmini, ; Noo and Tucciarlli, 7), as wll as in h D shallow war roblm wih a mono-orind low (Tucciarlli, ). Th mhodology is no rsricd by h CFL condiion in h choic o h im s and, vn i h soluion o a larg linar sysm is ruird or ach im s, h corrsonding comuaional cos is ngligibl and h oal comuaional cos is almos roorional o h cll numbr. Th govrning uaion sysm is sli irs in a convciv comonn and in a diusiv comonn. Th convciv sysm is solvd alying a MArching in Sac and Tim () rocdur, whr h convciv lus ar solvd using an Eulrian aroach and h comuaional clls ar ruird o b ordrd and solvd according o a dcrasing scalar onial valu. Th corrcion s comus h diusiv corrciv lus by mans o h soluion o a larg linar sysm, ha has h ordr o h cll numbr, bu is sars, symmric and wll condiiond. Th onial is dind as a scalar whos gradin is oosi o h lu dircion. Bcaus h low ild comud solving h SV uaions has no a scalar onial, an aroimad onial has o b usd, along wih a scond corrcion s ha is aimd o comu h roaional lus. This aroimad onial is dind according o h low dircion insid ach lmn a h bginning o ach im s. In h ollowing o h ar, h alicaion o h numrical rocdur o h D cas will b shown irs and hn o h D cas. TE PROCEDURE IN TE CASE OF D MONO-ORIENTED FLOW Th D SV uaions in a channl wih non-rismaic scion can b wrin in h ollowing orm (igur ) (s or aml Abbo and Minns, 99): h n g g R 4 / S whr is h low dircion, is h im, g is h graviaional acclraion, h is h war dh, is h z low cross scion, is h sram low ra, S is h boom bd slo ( S whr z is h boom lvaion wih rsc o a rrnc lvl), R is h hydraulic radius and n is h Manning ricion coicin. E. () can also b wrin in h ollowing orm: g n g R whr is h oal war lvl, h z. Th unnown variabls in E. () and () ar h war cross scion (h) and h low ra. 4 / () () ()

5 Th Schm or Shallow War Euaions T h h (h) z z z Figur. Diniion sch Assum h ollowing gnral sysm o balanc uaions: ( U) U F B( U ) whr U is h unnowns vcor, F(U) is h numrical lu-vcor rm and B(U) is h sourc rm. Using a racional im s sragy, w s: F ( ) ( ) ( U) F ( U) F( U) F ( U), B( U) B ( U) B( U) B ( U) (4) (5) whr F (U) and B (U) ar suiabl numrical lu-vcor and sourc rms, urhr dind. Ar ingraion in im, sysm (4) can b sli in h wo ollowing sysms: / U U F d B d (6,a) U U / F d F B d B (6,b) whr F and B ar h man valus o h lu and h sourc rms as comud in h rdicion s. W call sysms (6,a)-(6,b) rscivly rdicion and corrcion sysm. U / and U ar h unnown variabls comud a h nd o rscivly h rdicion and h corrcion s. Obsrv ha, summing E. (6,a) and (6,b) h ingral o h original sysm (4) is ormally obaind. Th numrical corrcd soluion will b clos o h soluion o his on as ar as h dirnc bwn h lu and h sourc rms is ihr small or im-indndn. Th advanag o using ormulaion (6) insad o (4) is ha, wih a suiabl choic o h rdicion rms F (U) and B (U), ach o h wo sysms (6,a)-(6,b) can b much asir o solv han h original sysm (4). For h SV uaions soluion, w s: U, (7,a) U F, gn F gμ, B, B g S (7,b) 4 / R F, F, B, gn (7,c) R B g 4 / whr M is h momnum o h cross scion wih rsc o is o widh. Th oal had gradin in h B rm is comud a im lvl and is consan along h im s. For sa o simliciy,

6 Cosanza Aricò assum a mono-orind osiiv vlociy and a cll numraion incrasing rom o N in h downsram dircion (,,, N). Th argumn o h ingral rdicion sysm (6,a) is givn by: B (8) ha dirs rom h original on only in h im lvl o h oal had gradin includd in h sourc rm B. Th argumn o h ingral corrcion sysm (6,b) is: / R gn S g gm 4 B F F (9) Obsrv ha h dirnc bwn h lu gradin and h sourc rm gos o zro, in sysm (9), along wih h im disanc wih rsc o h iniial im lvl. This imlis ha boh h unnown changs and h rlaiv rrors go o zro along wih h im s siz. In sysm (9) w assum h dirnc bwn h gradin o h momnum lu and h gradin o h man valu o h corrsonding rm comud in h rdicion s F o b ngligibl. According o his hyohsis, ar siml maniulaions, sysms (8) and (9) can b wrin in uasilinar mari orm (Tucciarlli, ): B F B 4 / R gn S g A U A U () whr h Jacobians o sysms () ar rscivly: gh, A A () T h and T is h cross scion o widh. I is asy o show ha A ignvalus ar λ λ v, whr v is h vrically avragd low vlociy. Obsrv ha h soluion o sysm (8) is uivaln o h soluion o a singl non-linar convcion uaion, uncion o h had gradin a im lvl. A

7 Th Schm or Shallow War Euaions ignvalus ar λ h g, λ h g λ. This imlis ha in h corrcion sysm convciv lus ar missing. For hs rasons w call h rdicion and h corrcion sysms rscivly convciv rdicion sysm and diusiv corrcion sysm. Din B (U ) h rdicion sourc rms comud as uncion o h unnowns U (, ) in cll and F (U, U ) h rdicion lu bwn cll and. W ado a irs ordr saial aroimaion o h unnown variabls insid h clls. Ar ingraion in sac and im, E. () can b wrin as: ) convciv rdicion sysm: ( ) d /,,, N (,a) ( ),, / d d d B U,,, N (,b) ) diusiv corrcion sysm: ( ) ( ) ( ) /, F, F d U U U U (,a) ( ) d R gn S g d M M g /,, / 4 B (,b) whr M, is h cross scion momnum a h inrac bwn clls and.. Soluion o h convciv rdicion sysm Convciv rdicion sysm is solvd using a MArching in Sac and Tim () rocdur. Bcaus o h sign o h ignvalus o h Jacobian A, i is ossibl o solv ach E. () i a olynomial aroimaion o h volum and h momnum lus nring rom h usram cll is always nown. This condiion can b guarand i h low dircion has a consan orinaion or, mor gnrally, i a scalar onial iss. In his cas, i is always ossibl o ordr and solv sunially h clls on ar h ohr. In h cas o mono-orind low, assuming a cll ind incrasing in h downsram dircion, E. () can b solvd as a sandard ODEs sysm, ingrad rom im o im, ha is: () d d, φ (4,a) () 4 g R g n d d, /, ξ (4,b) In E. (4), h rms φ -, () and ξ -, () ar a olynomial (nown) aroimaion o rscivly h volum and h momnum lu nring cll rom cll -. To solv E. (4) a Rung-Kua mhod, wih sl-adaing im s, is alid (Nag Library Manual, 5). Ar h ODEs in cll ar solvd, olynomial aroimaions ruird in cll can b comud as aroimaion o h volum and h

8 Cosanza Aricò momnum lus laving cll along im s. Th basic ida o h numrical chniu is o comu h soluion, wihin a givn im s, by marching in sac along h lu dircion hrough all h comuaional domain. Th mass and momnum lus nring h irs cll ar givn by h usram boundary condiions, as br sciid in a ollowing scion. Obsrv ha i is ossibl o solv sunially h ODEs sysms, wihou any rsricion on h siz o h im s, bcaus h characrisic lin o h roblm (8) is orind according o h vlociy dircion and h vlociy sign has bn assumd o b consan. This is uivaln, ar soluion o cll rom im o im, o ransla h boundary condiion (hic sgmn in igur ) a h inrac bwn cll and, o g h iniial valu o h characrisic curv carrying on h soluion o cll and so on hrough all h comuaional domain. Bcaus o his, h mhod can b classiid, ohr han marching in im, also marching in sac. marching in sac cll cll marching in im i Figur. Th cor o h marching in sac and im rocdur.. Polynomial im aroimaion A rd ordr aroimaing in im olynomial has bn chosn or h nring volum and momnum lus. Th corrsonding coicins ar obaind by sing h iniial, h inal and h avrag valus o h olynomials uals o h comud ons. This imlis: φ, /, φ φ φ (5,a),,, φ, φ, φ,, (5,b) ξ, m,,,,, /, ξ ξ ξ m (6,a) whr ξ, ξ, ξ, m, (6,b) m, / (7) Th avrag laving volum lu, is comud by alying h cll mass balanc, ha is: 4

9 Th Schm or Shallow War Euaions, / ( ), (8). Du o h non-linariy o h momnum balanc uaion, h avrag valu o h laving momnum lu m, is comud via numrical im ingraion o h valus comud by E. (7). Th soluion a h Gauss oins is obaind by a C inrolaion o h soluion valus roducd by h Rung-Kua mhod usd or h soluion o h ODEs sysm... Toal had gradin comuaion In h convciv rdicion sysm h oal had gradin o cll a h inrac wih cll downsram along h low dircion a h bginning o ach im s is comud as:,, N- (9) In h las lmn, according o h hyohsis o consan saial had gradin along h im s, i is assumd h had chang o b ual o h usram lmn on. From his assumion h ollowing rlaionshis hold: / / ( h N ) ( h ) / / / N N N, N N () whr / N is h downsram boundary lu rdicd a h nd o h im s... Small war dhs comuaion Th ricion and h momnum lu rms simaion, in h momnum balanc uaion, is subc o larg rrors whn war dh is clos o zro. This uncrainy acs h rsul sabiliy mainly in h dcrasing ar o h hydrograh, bcaus h small war dh valus nd o rmain and urhr rduc or a larg comuaional im. To dal wih his wll-nown roblm, h ollowing rocdur has bn alid. A minimum war dh valu h m, ngligibl or racical alicaions, is irs slcd. Call m h corrsonding cross scion. W assum h Manning rlaionshi and h momnum lu diniion o hold only or h > h m. In h ohr cas, hs wo rms ar rlacd wih h ollowing iciious ons (s also igur ): n n P 4 / R 4 / m 5 / m ( ) ( ) m m ( ) ( ) whr P in E. () is h wd rimr and uncion () in E. () and () is givn rscivly by: m m () () ( ) () 5 / 5

10 Cosanza Aricò ( ) (4) Whn h local war dh dros blow h minimum valu, bcaus o h rducion o h nw rsisanc rms wih rsc o h ral ons, h war dh dros uicly o h zro valu a h * im. A his oin h ODEs sysm ingraion is l and h small war volum nring h cll during h rmaining - * im is sord in h sam cll o mainain h mass balanc. m () ( ) ( ) m m Figur. Funcion aroimaion or small war dhs. Soluion o h diusiv corrcion sysm Diusiv corrcion sysm () is urhr simliid and linarizd, using a ully imlici ini dirnc discrizaion. Th rsuling sysm is: ( [ ] / / / / T ),,, N (5,a) 4 4 g R g n g R gn / / / (5,b) whr h cross scion im drivaiv discrizaion has bn rlacd in E. (,a) wih a uncion o h oal had im drivaiv discrizaion. E. (5,b) can b wrin in h ollowing orm: dis dis / (6) whr / 4 / / R gn g dis. Mrging E. (6) in E. (5,a) w g: 6

11 Th Schm or Shallow War Euaions T / ( ds ds ) ( ds ds / / /,,,, ds, dis, ds ),,, N (7,a), dis (7,b). Using h oal had gradin discrizaion givn in E. (9), E. (7) ar changd ino a sysm o linar uaions, wih a symmric, osiiv dind and wll-condiiond mari. Ar sysm (7) is solvd in h unnowns, h low ras can b comud by mans o E. (6).. Boundary condiions Boundary condiions assumd in convciv rdicion and diusiv corrcion ss ar rord in abl. h n h c ar rscivly h boundary (nown) war dh and h criical war dh corrsonding o h low ra b in h las cll o h domain. Th Froud numbr o h downsram, laving lu is comud using h domain low ra and war dh valus, h Froud numbr o h usram lu is comud using h rnal low ra and war dh valus. In h convciv rdicion sysm boh h incoming volum and momnum lus ar assignd a h usram cll. I h incoming lu is sub-criical, h momnum lu is comud as uncion o h war dh in h usram cll comud a h nd o h rvious im s. In h cas o surcriical downsram low, soluion o h diusiv roblm (7) is obaind by sing zro diusiv lu o h downsram boundary (cas n., 5 and 6 o abl ). This is uivaln o s: / b b (8,a) In h cass and 4 o abl, a war dh corrsonding o h nown valu is assignd: h b hn (8,b) In h hird cas o abl, a criical low ra corrsonding o h comud war dh is inally assignd: b gh (8,c) / b b cas n. Flu Fr. numbr h n /h c Boundary condiions laving No bound. cond. laving h n laving < < Criical war dh 4 laving < h n 5 incoming n h n 6 incoming < n Tabl. Boundary condiions EXTENSION OF TE PROCEDURE TO TE COMPLETE D DYNAMIC SALLOW WATER EQUATIONS In h mos gnral cas o D or D low, h vlociy dircion and orinaion is no nown. I h low ild has a scalar onial, li in h cas o diusiv modls (Noo and Tucciarlli, ), h 7

12 Cosanza Aricò low dircion is, a im, oosi o h gradin o h onial comud a h sam im and i is ossibl o ordr and solv h lmns according o h hir onial dcrasing valu. In h mos gnral cas, i is sill ossibl o nd h rocdur adding a urhr convciv corrcion s and using an auiliary scalar uncion, calld aroimad onial, urhr dind. Call,b h lu bwn cll and on o h wo conncd clls (b - or b ), an osiiv or ngaiv i h lu is rscivly laving or nring cll and v, b h vlociy o h corrsonding aricls, ha is:,b ( b ), v,b, i ( b ) > h,b ( b), b Obsrv ha, according o h rvious rul, condiion Assum ha a scalar valu ϕ and ( b ) > ( b) (9,a) b v,b ohrwis (9,b) hb, b b, b always holds., calld aroimad onial, is assignd o ach comuaional cll a h bginning o ach im lvl. Th chosn volum and momnum lu comuaion ruls can guaran h isnc o a onial uncion ha is ac a h bginning o h im s. This aroimad onial can b ound by sing arbirarily is valu ϕ in h irs cll and by assigning o h ohr clls h ollowing valus: ϕ ϕ i <, ϕ ϕ i or,,n (), Evn i h low ras ar consisn wih h onial gradin a h bginning o h im s, hy can chang sign along h soluion o h convciv rdicion sysm (8). For his rason, h soluion o a nw convciv corrcion sysm is now ruird. Sysm ()-() can b solvd by mans o h sunial soluion o h ollowing hr PDEs sysms: ) convciv rdicion sysm: ma ma (,sign( gradϕ ) (,sign( gradϕ ) g ) convciv corrcion sysm: min min ) diusiv corrcion sysm: (,sign( gradϕ ), (,sign( gradϕ ) gn R 4 / () () 8

13 Th Schm or Shallow War Euaions M g / g / S n 4 / R g / n 4 / R / In E. (), as in h cas o mono-orind low, w hav assumd h dirnc bwn h gradin o h momnum lu and h gradin o h man valu o h sam rms comud in h convciv rdicion and convciv corrcion ss o b ngligibl. In E. ()-() h argumn o h uncion sign( ) is minus h roduc o h gradin ims h low ra and h uncion is ual o or - i h argumn sign is rscivly osiiv or ngaiv, is h ind o h nown im lvl and variabls wih ind ⅔ ar assumd nown in sysm () rom h rvious soluion o sysms ()-(). Obsrv ha sourc rms ar nglcd in sysm (), sinc h roaional comonns o h lus ar assumd o b small wih rsc o h irroaional ons. Sysms () is ual o sysm (8) i h valu o uncion sign( ) is ual o. In h ohr cas, sysm () dgnras in h ODEs sysm: d d d, g 4 / d gn R I h valu o uncion sign( ) is ual o sysm () dgnras in h sady-sa condiion: d d, d d Ar ingraion in sac and im, sysms ()-() can b wrin as: ) convciv rdicion sysm: (4) (5) () / ( ),d, d (6,a) / ( ),v, d, v, d ( ) whr,b,b ) convciv corrcion sysm: ma min (,, ), i ϕ ϕ b (, ), i ϕ < ϕ, b B U d (6,b) (6,c) / / c c ( ),d, d / / c c ( ),v,d, v, d (7,a) (7,b) 9

14 Cosanza Aricò whr c,b c,b min ma (,,b ), i ϕ ϕ b (, ), i ϕ < ϕ,b b (7,c) ) diusiv corrcion sysm: / c c ( ) d d ( ),,,,,, (8,a) gn (8,b) R / ( ) g( M, M, ) d g S d B 4 / whr,, c, ar h man valus o h volum lu bwn clls and, as comud in h convcion rdicion and corrcion ss. Obsrv ha in E. (6) sricly ngaiv (nring) lus ar always coming rom clls wih highr onial and sricly osiiv (laving) lus ar always moving o clls wih lowr onial and vicvrsa in h convciv corrcion sysm (7). This allows o solv irs h convciv rdicion s moving rom clls wih highr o clls wih lowr onial and h convciv corrcion s moving rom clls wih lowr o clls wih highr onial. Th sum o sysms (6), (7) and (8) is an aroimaion o h ingral o h original sysm ()- (), and his aroimaion is as good as h dirnc bwn h lus and h sourc rms in h corrcion sysms is ihr small or im-indndn.. Soluion o h convciv rdicion and convciv corrcion sysms A s ordr saial aroimaion o h unnown variabls (h) and is assumd insid ach cll, as in h soluion o h convciv rdicion roblm shown in scion. According o his aroimaion, as wll as o h sady-sa assumion o h saial oal had gradin, ach sysm (6)-(7) can b viwd as an ODEs sysm. Convciv rdicion sysm (6) can b wrin as: d d d d whr d d () δ () ( δ ) φ ( δ ) φ δ,,,, (9,a) () v () ( δ ) ξ δ () v () ( δ ) ξ ( U ) δ,,,,,, B ( ) Convciv corrcion sysm (7) can b wrin as: δ sign (4) c c c c c c () δ () ( δ ) φ ( δ ) φ c c δ,,,, (4,a) (9,b) d d c c c c c c () v () ( δ ) ξ δ () v () ( δ ) ξ c c,,,,,, δ (4,b) 4

15 Th Schm or Shallow War Euaions whr ( ) c c sign δ (4) Sysms (9) can b solvd sunially in ach comuaional cll, as rviously sn or h cas o mono-orind low, moving rom h highr o h lowr aroimad onial. Ar soluion o all sysms (9), sysms (4) can b solvd sunially moving vicvrsa rom h lowr o h highr aroimad onials. O cours i an ac scalar onial iss and i is nown, h convciv corrcion sysm rducs o h idniy: ( ) ( ) / / h h, (4) / / In h convciv rdicion sysm, h izomric gradin o cll a h inrac wih cll b downsram along h low dircion a h bginning o ach im s can b comud as in E. (9). Th olynomial im aroimaion o h laving volum and momnum lus ar h nring valus or h n lmn in boh convciv rdicion and corrcion sysms. Ths olynomials can b simad as uncion o h (h) and valus comud during h running im s, as alrady sn or h cas o mono-orind low.. Soluion o h diusiv corrcion sysm Diusiv corrcion sysm () is urhr simliid and linarizd as rviously laind or h cas o mono-orind low, o g h ollowing ully imlici discrizaion: /, /,,, / / T,,, N (44,a) 4 4 g R gn g R gn / / / (44,b) Th rror du o h linarizaion o E. () gos o zro along wih h siz o h im s, as wll as h dirnc bwn h rdicd and h corrcd lus and war lvls. Th man valu o h nown war lu o E. (,a), as wll as o h sourc rm o E. (,b) hav bn aroimad in E. (44) a im lvl ⅔, ha rrsns h bs nown im lvl simaion or h vcor variabls U. E. (44,b) can b wrin in h orm: dis dis / / (45) whr / 4 / / R gn g dis. Mrging E. (45) in E. (9) and hs ons in E. (44,a), w g: 4

16 Cosanza Aricò whr ds, b / dis T ( ) ( / ds ds ds ds / ) /,, /,, T, b i ; ds, b dis b b, b < i (47),,,N Ar discrizaion o h oal had gradin as givn by E. (9), E. (46) orm a linar algbraic sysm, wih a mari ha is symmric, osiiv dind and vry wll condiiond. Th low ra unnowns, a im lvl, can b comud by mans o E. (45).. Boundary condiions Boundary condiions ar h sam summarizd in scion.. An aroimad onial gradin consisn wih h lu sign a im is assumd a h wo boundaris o h domain; his imlis ha zro nring volum and momnum lus ar assignd a boh boundary clls or h soluion o convciv corrcion sysms (7). Boundary condiions o h diusiv roblm (4) can b obaind by E. (8), changing E. (8,a) and (8,c) rscivly wih: (46) b / b, b gh / b b (48) 4 D NUMERICAL TESTS 4. Dam-bra wih ini downsram war dh In his s h channl is ricionlss, horizonal, ininily larg and m long. Bor is locad a m, wih a war dh ual o m in h usram ar o h channl and a war dh ual o m in h downsram on. Numrical rsuls hav bn comard wih h analyical ons obaind by Sor (957), in rms o war dhs and low vlociis. Numrical soluion is comud a im 9.9 s, using a cll siz m and a im s siz. s. Th maimum valu o h CFL numbr is aroimaly.5. A comarison o h rsuls obaind using dirn lici numrical modls can b ound in Zoou and Robrs (999). Th Auhors comard h L norms o rrors o war dh and low vlociy dind as: L h, nu,, N, h,, N h h,, N v,, N v L, v (49) v whr sub-inds and nu indica rscivly h ac soluion and h numrical rsuls. Th modl rovids L,h. - and L,v 7.6 -, whr h bs norm rovidd by h s saial aroimaion ordr mhods is.4 - or h war dhs and or h vlociis. Th wors raning in h vlociy simaion can b laind by h us o h low ra insad o h low vlociy as unnown, as don in som o h ohr mhods. In igur 4 h comarison bwn analyical and numrical rsuls is shown, whr s. s sands or., nu, 4

17 Th Schm or Shallow War Euaions I w assum h comuaional cos o b roorional o h numbr o lmns ims h im ss numbr, i is ossibl o mainain h sam comuaional cos by halving h lmn siz and doubling h im s siz. In his cas, vn wih a maimum CFL numbr ual o., a br rormanc is obaind or h norm o boh h had and h vlociy rror (L,h and L,v ). Also or his cas, comarison wih h analyical soluion is shown in igur 4. Obsrv also ha or boh comuaional mshs no nroy glichs aar in h numrical soluions. Ths disconinuiis lagu svral numrical soluions, also obaind wih numrical mhods wih saial aroimaion ordr grar han on (Zoou and Robrs, ). [m] [m] ac soluion s. s m s. s 5 m v [m/s] ac soluion s. s m s. s 5 m [m] Figur 4. Dam-bra wih ini downsram war dh. War lvls and low vlociis 4. Sady low ovr a bum wih hydraulic um A sady-sa ranscriical low ovr a bum, wih a smooh ransiion ollowd by a hydraulic um is simulad. Th channl is ininily larg, horizonal, ricionlss, 5 m long. Boundary condiions ar givn by: h (, ).8 m (, ). m / s Iniial condiion is givn by a consan war lvl ovr h channl, ual o. m, and bd roil is givn by: ( ) (5)..5 i 8 z ( ) (5) ohrwis Th s has bn carrid on using.5 m and. s, wih a maimum CFL numbr ual o.6. In igur 5 h comud war lvl and low ras ar comard wih h corrsonding analyical soluion ar iraions (T s). Obsrv ha h hydraulic um is caurd in w clls and ha no oscillaions occur in h war surac and, mos imoran, in h low ra roil along h bum. Incrasing h comuaional im, low ra valu nds asymoically o h saionary valu.8 cm/s in all h domain. For h sam s, w also rovid in igur 6 a comarison o h convrgnc hisory o h roosd schm wih rsc o h SGM mhod by Zhou al. () and o h wll-balancd high ordr WENO schm by Vuovic and Soa (). Th global rlaiv rror R is dind as (s Zhou al., ): R i,n n i n i n i (5) 4

18 Cosanza Aricò whr n i is h war dh in h i h cll comud a im lvl n h and ind n- indicas h corrsonding war dh comud a h rvious im lvl. Dsi h saial aroimaion ordr, schm convrgs much asr han h ohr wo highr ordr schms. Obsrv, in igur 6, h rlaiv low rrors obaind in h sam s by h algorihm. In his n i cas in E. (5) is h low ra comud in h cll i., z [m] [m] bd lvl ac soluion [cm/s] low ra ac soluion [m] Figur 5. War surac and low ra roils ovr h bum or ranscriical low ar s R (h ).E.E-.E-.E-.E-4.E-5.E-6 SGM WENO R ( ).E.E-.E-.E-5.E-7.E-9.E-.E-.E n. iraions.e n. iraions Figur 6. Global rlaiv rror o war dhs and low ras or s LVu s s roblm (998) In h LVu s roblm (LVu, 998) a non-saionary low rovidd by wo dam-bras roagas in oosi dircions; whil h usram moving ron asss ovr a horizonal rivr bd, h downsram moving ron roagas ovr a bum. Bd roil is givn by: z ( ) ( cos( π (.5) ) ).5 i.4.6 ohrwis Channl is assumd ricionlss and ininily wid. Th iniial condiions ar: v (, ) m/s h, (, ) h z( ) z( ) i.. ohrwis whr h is h high o h uls ual o. m. In igur 7 comud war lvls and low ras ar shown a im. s, using. m and (5) (54) 44

19 Th Schm or Shallow War Euaions. s. Th maimum CFL numbr is.55. Obsrv h disconinuiis in boh war lvl and low ra roils and also h corrsonding rrors wih rsc o h rrnc soluion, comud using a valu o h CFL numbr ual o.58 and a numbr o lmns ims highr. Th rror dnds on h arial rlcion, du o h bum, o h downsram ravlling wav in h usram dircion and on h corrsonding raid invrsion o h low dircion. Th low invrsion is no ollowd, in h momnum uaion o h convciv rdicion s, by h oal had gradin invrsion. This rror is wll corrcd by h diusiv corrcion s, unlss h diusiv corrcion s dos no rduc h local chang o h unnown variabls. This is avoidd i h convciv chang can cross svral lmns, du o a CFL numbr much largr han on. Obsrv, in acs, ha rining h msh using a numbr o lmns ims highr oscillaions disaar. Similar disconinuiis in h numrical soluions hav bn obsrvd also by LVu (998) and by ohr Auhors (Vuovic S. and Soa, ; Zic al., 4). [m] r. soluion [m] [cm/s].4. r. soluion [m] Figur 7. War lvl and low ra or LVu s s roblm a. s 4.4 Erimnal dam-bra in a rismaic channl Som rimnal wors carrid ou by h U. S. Army Cors o Enginrs (USACE, 96) hav bn usd o comar h modl rsuls wih rimnal daa. Th rimns hav bn rormd in a m long,. wid rcangular channl lind wih lasic-coad lywood. Th boom slo is.5 %, h dam has bn lacd in h middl o h channl ( 6 m), and h war dh immdialy bor h dam is ual o.5 m. Two sris o rimns hav bn carrid ou using boh smooh and rough suracs, wih n Manning ricion coicin ual o.9 s/m / and.5 s/m / rscivly. Iniial condiions ar givn by:. 5 i 6m h, v i > 6m ( ) m/s and ar shown in igur 8. A h downsram nd o h channl zro war dh is assumd as boundary condiion. In h modl run.765 m and.5 s aramrs hav bn usd. In igurs 9,a-,b h war dhs a abscissas.5 m, 6 m and 85.4 m ar lod vrsus im agains h corrsonding masurd valus. Numrical rsuls givn by a nd ordr MUSCL schm (Zhang al., ) ar also shown. Th maimum CFL numbr is aroimaly.874 or n.9 s/m / and.8 or n.5 s/m /. In igur h masurd and comud war lvls a im s or h n Manning coicin ual o.9 s/m / ar shown. Comarison wih rimnal daa is good and h maching is similar o h on obaind by h nd ordr aroimaion schm. (55) 45

20 Cosanza Aricò dam h m 6 m m Figur 8. Iniial condiion or h rimnal dam bra s.. h [m].5..5 m masurd MUSCL h [m] m masurd MUSCL [s] [s] Figur 9,a. War dhs a.5 m, n.5 s/m / Figur 9,b. War dhs a 6 m, n.5 s/m / h [m] m masurd MUSCL h [m] m masurd MUSCL [s] [s] Figur,a. War dhs a.5 m, n.9 s/m / Figur,b. War dhs a 85.4 m, n.9 s/m / 46

21 Th Schm or Shallow War Euaions.7.6 [m].5.4. masurd [m] Figur. War lvl a s, n.9 s/m / 5 TE APPROAC IN TE D CASE 5. Ingral orm o h SV uaions Th D SV uaions can b wrin as: h uh vh y (56) uh ( u h) ( uvh) y ( uh) ( vh) h z n u gh gh h 7 / (57) vh y ( v h) ( uh) ( vh) h z n v uvh gh gh y y h 7 / whr and y ar h saial coordinas, u and v ar h and y vlociy comonns, is h im, g is h graviaional acclraion, h is h war dh, z is h ground lvl and n is h Manning ricion coicin. Th unnowns in sysm (56)-(58) ar h war dh h and h wo low ras comonns r uni widh uh and vh. Assum a riangular msh and a irs ordr aroimaion o h variabls h, uh and vh insid ach lmn. Ingraion in sac o E. (56), (57) and (58), as wll as alicaion o h Grn s horm rovids: (58) h F, (59), uh M, R (6), 47

22 Cosanza Aricò whr is h lmn ara, vh y y M, R (6), F, is h volum lu hrough h h sid o lmn, and ar M, y M, h and y comonns o h momnum lus along h h sid o lmn and h sourc rms y R, R ar dind as: ( uh) ( uh) ( vh) n R g h (6), 7 / h R y ( vh) ( uh) ( vh) n g h 7 / y h whr and h ar rscivly h lmn war lvl and war dh. (6) 5. Fracional im s dcomosiion φ Assum a scalar valu, calld aroimad onial and urhr sciid, o b nown a h bginning o ach im s in ach comuaional lmn. Th volum and momnum lus hrough ach sid ar dind as uncion o h lu o h sciic low ra comonns o boh h lmns and sharing h givn sid. Th volum lu hrough h h sid o lmn is ual o: FL, ( uh) ( y y ) ( vh) ( ) (64) whr is h nod o lmn ollowing nod in counr-cloc wis dircion. Th volum lu is dind as: F FL,, FL, > FL, > FL, m i and (65,a) F, FL,m ohrwis (65,b) y, F, FL, M, F,u M F, v, i (66,a) y M, F,u M, F, v, ohrwis (66,b) whr is h ind o h lmn sharing is m h sid wih h h sid o lmn. Obsrv ha condiion F holds or all h inrnal sids. I is h osiiv (laving) lu o a boundary F,, m F, sid, h condiion F, FL, holds. Ar ingraion in im, rdicion s o sysm (59)-(6) is dind as: h / h F, d (67), 48

23 Th Schm or Shallow War Euaions / ( uh) ( uh) M, d R d (68), whr: / ( vh) ( vh) y y M,d R d (69), ( uh) ( uh) ( vh) n R g h (7,a) 7 / h and ( vh) ( uh) ( vh) y n R g h (7,b) 7 / y h y F ma(, F ), M F u, M F v i φ φ (7,a),,,,,,, (, F ) y F min, M F u, M F v i φ < φ (7,b),,, Th irs corrcion sysm is dind as: c, h / h / ( uh) ( uh) / ( vh) ( vh) (, F ), / / /,, c F, d (7), c M, d (7),, M cy, d c c cy c F ma M F u, M F v i φ < φ (75,a) c, (, F ),,, c c cy c F min M F u, M F v i φ φ (75,b), Th scond corrcion sysm is dind as: h h /,, F,,, d,, c ( F, ), F, (74) (76) 49

24 Cosanza Aricò whr F,, c / ( uh) ( uh) R d R (87) / ( vh) ( vh) y y y R d R (78) F,, R, R ar h man (in im) lu and sourc rm valus, as comud in h rdicion and in h irs corrcion s. In E. (77)-(78) w nglc h dirnc bwn h man (in im) convciv inria rms and h sam man valus comud rom h soluion o h rdicion lus h irs corrcion sysms. This is uivaln o assum, in h scond corrcion sysm: M,d M,,,, y y M,d M,,,, M M c, cy, (79,a) (79,b) This imlis ha convciv lus ar missing in E (86)-(88) and w call h rdicion sysm convciv rdicion sysm and h irs and h scond corrcion sysms rscivly convciv and diusiv corrcion sysms. Iniial condiions o h convciv corrcion sysm ar h inal valus o h convciv rdicion sysm and iniial valus o h diusiv corrcion sysm ar h inal valus o h convciv corrcion sysm. Saial izomric gradin is assumd consan in im in all h sysms. Obsrv ha summing all h rms o E. (67)-(88) h ingrals o E. (59)-(6) ar ormally ound again. 5. Soluion o h convciv rdicion and h convciv corrcion sysms A irs ordr saial aroimaion is alid o h unnown variabls h, uh and vh. Obsrv ha, according o h lu diniions givn in E. (7) and (75), h lu ingrals rom lmn o lmn in h convciv rdicion s ar only uncion o h lmn unnowns i φ φ and ar only uncion o h lmn unnowns i φ < φ. In h convciv corrcion s h oosi holds. This allows o solv ach sysm as a sunc o small ODEs sysms, on or ach comuaional cll, ar ordring h clls according o h hir scalar onial. Th rioriy is givn o h clls wih highr onial in h convciv rdicion s and o h clls wih lowr onial in h convciv corrcion s. In h rdicion cas, h ODEs sysm is: d ( uh) ( ) d, dh F, () (8) d, ( M () R () ), (8) 5

25 Th Schm or Shallow War Euaions d ( vh) ( ) d, y y ( M () R () ), (8) Call ou (in) h ind o any sid shard wih any lmn such ha φ in h convciv rdicion s ( φ < φ in h convciv corrcion s). Ar h singl sysm o lmn is solvd along h im s, h man valus o h lus laving hrough ach sid ou hav o b simad. Th man valu o h laving volum lu F, ou can b simad by ariioning h oal man laving lu, obaind rom h mass balanc in h lmn and ual o: ou F,ou in F,in h / h Obsrv ha h man valus o h nring lus F, in ar nown rom h soluion o h lmns wih highr scalar onial. Pariion can b don adoing or ach lmn sid a wigh ual h arihmic man bwn h iniial and h inal lu valus, ha is: F, () F, ( ) () F ( ), ( ) F,ou,ou ou ou F whr is also h ind o any sid shard wih an lmn wih lowr onial. Givn h iniial, h man and h inal valus, a arabolic volum lu im aroimaion in ach sid is inally comud and usd o sima h nring lus in h conncd lmns wih lowr aroimad onial. Consrvaion o h man valus can b asily rovd o guaran h local and global mass consrvaion. Th man laving momnum lus ingraion, hy can b comud as: M, y M M, () F, ( ) () F ( ), ( ) F,ou,ou ou ou F F,ou φ (8) (84),, can also b simad in a similar way. Ar Du o h non-linariy o h momnum balanc uaions, h man valu o h oal laving momnum lu a h r.h.s. o E. (85) has o b numrically simad rom h singl im valus. In h codd algorihm, h laving momnum lus in hr Gauss oins, slcd in h im inrval hav bn comud using a C inrolaion o h soluion valus roducd by h Rung-Kua mhod adod or h soluion o h ODEs sysm. A arabolic momnum lu simaion is inally carrid ou or ach lmn sid using h iniial, h inal and h man valu. A similar rocdur is carrid ou or h soluion o h convciv corrcion sysm, wrin as: d, M,ou (85) dh c F, () (86) d ( uh) ( ) d, () c M, (87) 5

26 Cosanza Aricò d ( vh) ( ) d, () y M, (88) Obsrv ha h sourc rms hav bn oally allocad in h convciv rdicion s. This simliis h soluion o h roblm and is comuaionally inciv du o h small siz o h corrciv lus, sough ar wih a good choic o h aroimad onial uncion. 5.4 Th aroimad scalar onial Fracional s mhodology rovids accura rsuls only i h soluion o h convciv rdicion s is clos o h corrcd on; ohrwis, h comuaion o h saial gradins a dirn im lvls can srongly ac h soluion. In our cas, o minimiz h variabl chang in h convciv corrcion s, i is imoran o choos an aroimad onial wih a gradin lu oosi in sign, as much as ossibl, o h war lu along h lmn sids. I an ac onial iss and his condiion is always aaind, convciv corrcion sysm vanishs in h ollowing idniis: h / / / / / / ( uh) ( uh), ( vh) ( vh) h, (89) W s o minimiz, a h nown im lvl, h ollowing uncional: (,( uh) ( y y ) ( vh) ( ) F min (9), whr is again h ind o any sid shard by an lmn such ha: φ φ (9) Th bs s o scalar onials, ha minimizs uncion (9) subc o consrains (9), is h soluion o a global minimum sarch, comuaionally vry nsiv. Eclln rsuls can also b ound mor asily by rorming a local sarch, i a good saring oin is chosn. This oin is h onial s corrsonding o numrically simad gradins as oosi as ossibl o h vlociy gradins ims h war dh. In a D sac, i can b viwd as h s o onials ha ar as clos as ossibl o h lans maching h onial valu a h riangl cnr, wih a gradin oosi o h vlociy gradin ims h war dh (s igur ). Th uncional comonn corrsonding o h sid o ach lmn is wighd wih h absolu valu o h lu and o all h lmn comonns a naly rm rlad o h chang wih rsc o h rvious valu is addd, o inally g: min N ( F' ) [ ( φ φ ( )( uh) ( y y )( vh) ) FL, α( φ φ ) ] whr, y ar h coordinas o h cnr o lmn, and α is a small osiiv numbr. Th roosd uncional is conv and h minimum can b ound by sing o zro h arial drivaivs wih rsc o all h lmn aroimad onial. Th rsuling linar sysm is sars, symmric, osiiv dini and wll condiiond. Th naly rm in h suar bracs is aimd o avoid in h linar sysm zro diagonal rms whn h sciic low ra comonns a im ar ual o zro. Incrasing h naly rm, a onial disribuion mor similar o h rvious on comud a im is obaind. To obain a comromis bwn comuaional icincy and h aroimad onial ualiy, h coicin α o h naly rm can b normalizd wih rsc o h diagonal rm o h corrsonding original uaion, wih a lowr boundary givn by h machin rcision. Ar h minimum o F is ound, h aroimad onials can b urhr rind wih a local sarch, iraivly adusing h lmn onial valus unil a local minimum o h uncional F is ound. (9) 5

27 Th Schm or Shallow War Euaions φ ( uh) ( ) ( vh) ( y y φ φ ) m Figur. Linar aroimaion o h aroimad onial around a cnral valu 5.5 Comuaion o h gradins or h convciv sysms War lvl gradin is usually simad, in h cas o riangular msh and irs ordr aroimaion, by alying Grn s horm a h D ingral o h war lvl ovr h lmn (s or aml Anasasiou and Chan, 997; Pui al., 99). This simaion is accura only i rgular mshs ar usd. In h cas o unsrucurd, auomaically gnrad mshs, i is acd by larg rrors and hrognous gradins ar obaind also in h cas o a linar variaion o h war lvl insid h domain. A mos robus chniu has bn dvlod, similar o h algorihm roosd by ubbard (999). Th gradin is comud as a wighd avrag bwn on or wo vcors, ach on corrsonding o a vr o h riangl. Each h vcor is comud assuming a linar variaion o h war lvl insid a nw riangl, dind by h cnr o lmn and h cnrs o h wo surrounding lmns sharing h h nod. Zro wigh is givn o h h vcor corrsonding o a ngaiv scalar roducs bwn h lmn vlociy and h dircion o h lin conncing h cnr o lmn wih is h vr. Each vcor is h gradin o h linar uncion maching h war lvls in h cnr o lmn and in h cnr o lmns, m sharing vr wih riangl (s igur ). Th inal gradin is givn by: ma (, n n ) uv (, n n ) uv (9) ma whr n n uv is h scalar roduc bwn h cnr-vr and h vlociy dircions. I lmn has an dg on h imrvious boundary, is vlociy is aralll o h boundary and on o h hr condiions ruird o comu h gradin corrsonding o h downsram vr is missing. This condiion is rlacd by h ualiy o h cd oal izomric gradin dircion and h imrvious dg dircion. 5

28 Cosanza Aricò (u, v) B C O A Figur. Comuaion o h saial war lvl gradin 5.6 Boundary condiions or h convciv rdicion sysm A h bginning o ach im ss, lmns wih on boundary sid ar slcd. Th Froud numbr r, o h lu r uni lngh is comud as: r FL,, / L, h g FL, i, b b ( y y ) h v ( ) b b h u r, / L h, g i FL, < whr L, is h lngh o h h sid o lmn. Boundary condiions o h rdicion roblm (67)- (69) dnd on h lu sign and on h Froud numbr r,. On o h ollowing cass can b slcd: ) Flu hrough h boundary sid is nring h domain. In his cas h ollowing ualiis hold: b BF BM BF, BM,, b F BF, M BM, M BM, h, b u b BF h b,,, b b ( y y ) h v ( ),, BF b u u, M y, BF, v b ( y y ) h v ( ), u, BM y, BF, v y,, y, (95) r i, (96) i r, > (97) whr h, u and v ar h assignd boundary war dhs and vlociy comonns; ) Flu hrough h boundary sid is laving h domain. In his cas war dh gradin is no nown and convciv rdicion sysm is rlacd, or lmn, wih h hyohsis o nglcing h low ra and momnum local changs in h sam lmns. This is uivaln o say ha h laving volum and momnum lus ar ual o h nring ons. From h rvious hyohsis, w obain: (94) 54

29 Th Schm or Shallow War Euaions M ( ),in / h h, u / in in F,in (), v / wih h sam manings sciid or E. (65)-(66). ) Flu hrough h boundary sid is zro (imrvious boundary). In his cas h ollowing condiions hold: in in M F y,in,in ( ) () (98) F, (99), n, whr is h war lvl gradin and n is h uni vcor normal o h imrvious sid., W assum h scalar onial immdialy ousid h boundary sids o b consisn wih h lu sign. This imlis h volum and momnum lus o b zro in h convciv corrcion s, ha is: F c, c cy, M, M,, () 5.7 Soluion o h convciv ss wih small war dh Insabiliis occur in h soluion o h convciv ss whn iniial war dh valus ar ngaiv, and convrgnc is vry slow whn osiiv, bu vry small valus ar comud in h ail o h roagaing wav. Small ngaiv war dhs can b comud by h diusiv sysm in h rcding im s, whn h rsul o h convciv ss is a vry small osiiv valu. Th ollowing rocdur is ollowd in ordr o avoid hs inconvnincs and o mainain h local mass consrvaion rory. A vry small minimum war dh h min is chosn ( m) and boh h momnum lu and h ricion rms ar comud, or h < h min, as h irs ordr ansion around h min, ha is: R R M y, ( uh) ( uh) ( vh) ( vh) ( uh) ( vh) ( uh) ( uh) ( vh) n 7 n g h ( h ) 7 / hmin / hmin hmin ( vh) ( uh) ( vh) n 7 n g h ( h ) 7 / hmin / y hmin hmin ( uh) ( h h ) ( uh) ( vh) min ( h h ) ( vh) min () y F, min F,, M, F, min F, () h h h h min This chang allows a as soluion o h roblm or small, bu osiiv war dh valus. Whn a ail occurs, soluion convrgs anyway o ngaiv valus. To avoid his, a conrol is s in h ODEs solvr in ordr o rurn h im valu corrsonding o a zro war dh valu. Whn h solvr sos bor h nd o h im s, h rmaining war volum nring h cll is l in h sam cll assuming zro laving lu ar h rurn im. Whn h iniial war dh h (or min in h convciv corrcion s) is ngaiv, wo ossibiliis occur. Call Vol h volum corrsonding o h cll ara and o h oosi o h ngaiv iniial war dh. I h nring volum V in F, in is smallr han Vol, incras hn Vol, s h h / h, assum a zro laving lu, and mov o h n cll. I Vin is grar, rduc h man nring volum and momnum lus according o h Vol / in 55

30 Cosanza Aricò valu and solv h ODEs sysm. 5.8 Soluion o h diusiv corrcion sysm Diusiv corrcion sysm (76)-(78) is simliid by changing h ingral man valus wih h ully imlici im discrizaion and linarizing h sourc rms. This rovids a diusiv c, ha gos o zro along wih h siz o h sourc rms and h diusiv lus. This siz is roorional o h im s, as can b obsrvd in E. (77)-(78). According o hs hyohss, diusiv corrcion sysm (76)-(78) is uivaln o h ollowing ully imlici im discrizaion: / ( uh) ( vh) y () / ( uh) ( uh) g n ( uh) ( uh) ( vh) gh / gh / ( uh) ( uh) ( vh) 7 / / ( h ) g n / 7 / ( h ) / / (4) / ( vh) ( vh) g n ( vh) ( uh) ( vh) gh / ha is: y gh / ( vh) ( uh) ( vh) 7 / / ( h ) y g n / 7 / ( h ) / / ( uh) lm cos ( uh) / (5) (6) whr: y / ( vh) lm cos ( vh) (7) y / / g( h ) 7 / / ( h ) g n ( uh) ( vh) (8,a) lm / and cos y lm, cos lm (8,b) y whr h variabls wih ind ⅔ ar nown rom h soluion o h convciv corrcion sysm. 56

31 Th Schm or Shallow War Euaions Mrging E. (6)-(8) in E. (), h ollowing rlaionshi is obaind: / whr diusiv lus ar discrizd as:, F δ F (9),, lm L, cos F, y ( y y ) cos ( ), lm L, cos y ( y y ) cos ( ),, i () FL,, i < () FL, δ i h h and h nods o lmn ar shard by h lmns and (inrnal sid), δ i no (boundary sid). Ar comuaion o h nw war lvls b comud wih E. (6)-(8) using h nw convciv gradins ( laind in scion Boundary condiions o h diusiv corrcion sysm I r, >, zro diusiv lu is assignd, ha is F,. I r,,, h low ras r uni widh can, ) comud as y wo ossibiliis is. I h assignd boundary war dh is smallr han h criical dh corrsonding o h sciic low ra, ha is: h ( uh) ( vh) / b () g a oal lu corrsonding o h criical dh is assignd o h lmn boundary sid, ha is: F g / / ( h ) L F /,,, () I inualiy () dos no hold, boundary war dh is assignd o lmn, along wih h corrsonding Dirichl condiion and diusiv lu is comud a osriori by mans o E. (9). 6 D NUMERICAL TESTS In h ollowing, rsuls o h modl ar comard wih rimnal daa o wo laboraory ss and wih rsuls obaind or h simulaion o h sam rimns by ohr auhors. In all h simulaions, h comuaional msh is unsrucurd and auomaically dsignd by h ARGUS ONE msh gnraor (Argus oldings). Th war dh and vlociy comonns in a givn oin wih, y coordinas hav bn assumd ual o h h, u and v valus o h lmn conaining h givn oin, wihou any urhr inrolaion. 57