Method of Characteristics for Pure Advection By Gilberto E. Urroz, September 2004

Transcription

1 Mehod of Charaerss for Pre Adveon By Glbero E Urroz Sepember 004 Noe: The followng noes are based on lass noes for he lass COMPUTATIONAL HYDAULICS as agh by Dr Forres Holly n he Sprng Semeser 985 a he Unversy of Iowa Pre adveon The paral dfferenal eqaon of pre-adveon s 0 s he adveon veloy and represens he qany beng adveed eg a onenraon of onamnan n a one-dmensonal flow Ths paral dfferenal eqaon an be wren also as an ordnary dfferenal eqaon namely along he haraers rve d d d d Mehod of haraerss The followng ompaonal grd llsraes he solon of he ODE d/d 0 along he haraers rve d/d We are neresed n fndng he vale of along he haraers rve ha sars a poson ξ a me level Inegrang d/d 0 along he raeory llsraed n he fgre prodes smply: 0 ξ C To deermne he vale of ξ e he vale of a he foo of he raeory we an negrae he eqaon d/d from pon ξ o pon e ξ d d Inegraon along he haraers rve prodes he followng resl: ξ d T For non-onsan veloy he negral n he prevos eqaon an be appromaed by

2 d ξ The epresson for ξ sng a lnear nerpolaon along s: ξ ξ eplang he las resl no he prevos eqaon and ha resl n rn no eqaon T prodes ξ T For he smple ase n whh onsan he raeory s a sragh lne: r d ξ r s he Coran nmber for he ompaonal grd The followng fgre llsraes hree possble sragh lnes represenng he haraers rve for onsan For lne r > and he foo of he lne falls osde of he - ell For lne r and he haraers lne s he ell s dagonal Fnally for r< he foo of he haraers lne falls whn he - ell Lnear nerpolaon for onenraon Earler on we fond ha he solon o he governng eqaon namely d/d 0 along he haraers rve s smply ξ To deermne he onenraon a he foo of he raeory ξ we an se a lnear nerpolaon of he onenraons a pons - and a me level namely: r ξ Ths smple lnear nerpolaon approah works fne for r sne no nerpolaon error s nvolved In sh ase ξ - However he me sep sze s onsraned by and If we ads so ha r> ξ s allaed by erapolang osde of he - ell whh

3 ypally prodes an nsable solon On he oher hand f we ads so ha r< lnear nerpolaon wll damp non-lnear omponens of he solon In general lnear nerpolaon s no he bes approah o esmae he onenraon a he foo of he haraers rve Hgher-order nerpolaon polynomals an mprove he performane of he solon sheme Ths approah s appled ne n he Holly-Pressmann sheme The Holly-Pressmann sheme Ths mehod s based on onsrng a hrd-order nerpolang polynomal o allae ξ based on he nformaon from wo pons - and The polynomal s wren as r A A r A 3 r A 4 r 3 HP r ξ and he A s are onsan vales The meanng of he parameer r s llsraed n he fgre below showng he haraers lne n a ypal ell of he ompaonal grd Noe: for a onsan vale of he haraers rve s a sragh lne and he parameer r s also he Coran nmber of he ompaonal grd: r / In order o evalae he onsans of he b polynomal n eqaon HP we wold have o provde for vales of r However sne we have only wo referene pons s known namely - and we need o nlde also he dervave vales / a hose referene pons Noe ha an be wren n erms of r as follows: / d/drdr/d r -/ Sne n he one of he allaon ell ξ he resl dr/d dr/dξ -/ follows from eqaon The eqaon o se for dervaves s herefore r - r/ -A A 3 r 3A 4 r / HP The fgre above llsraes he fa ha he vales - - are known These vales orrespond o r for - and r 0 for Ths replang he vales of and r for pons - and no eqaons HP and HP prode a sysem of lnear eqaons whose solon s gven by 3

4 A A - A A eplang hese onsans no eqaon HP and olleng erms allows s o re-wre eqaon HP as r a - a a 3 - a 4 HP3 a r 3-r a - r 3-r a a 3 r -r a 4 - r-r Also eqaon HP now beomes r b - b b 3 - b 4 HP4 b 6r-r/ b 6rr-/ b b 3 r3r- b 4 r-3r- Noe ha hs solon assmes ha ξ s sh ha - < ξ < as n lne 3 n he fgre below I may happen however ha he foo of he raeory o pon falls n a ell o he lef of [ - ] as llsraed by lnes and n he fgre below The solons for r and r provded by eqaons HP3 and HP4 respevely an be appled o any ell regardless of wheher ξ s whn he range [ - ] or n a ell o he lef of hs range The fgre below llsraes he saon for any nerval [ L ] NOTE: for lne n he fgre above - L -3 e 4

5 Ths regardless of whh ell he vale of falls no he nerpolang polynomals an be wren as: r a L a a 3 L a 4 HP5 a r 3-r a - r 3-r a a 3 r -r a 4 - r-r Also eqaon HP now beomes r b L b b 3 L b 4 HP6 b 6r-r/ b 6rr-/ b b 3 r3r- b 4 r-3r- In hese epressons - L Adveng / Whle he Holly-Pressmann sheme provdes a more sable approah o he problem of pre adveon of he onenraon also reqres he adveon of he dervaves / An adveon eqaon for an be obaned by akng he dervave of he adveon eqaon wh respe o so ha we an wre Ths resl s eqvalen o solvng d d along he raeory d d Inegraon of he ordnary dfferenal eqaon for along he raeory resls n ξ d The vale of ξ r s allaed wh eqaon HP4 whle he negral n he rgh-hand sde s appromaed by he rapezodal rle: d ξ ξ 5

6 eplang hs resl n he prevos eqaon and solvng for resls n: ξ ξ C Noe ha f onsan / 0 and ξ For non-onsan he vales of / a pons ξ and an be esmaed wh fne dfferenes and nerpolaon sne he veloy feld s spposed o be known The followng formlaons an be sed o allae he dervaves / shown n eqaon C The dervave a pon an be deermned as DU To esmae he dervave / a pon ξ we need o ake no onsderaon he fa ha he foo of he raeory ξ may fall n he nerval [ - ] or n any of he nervals o he lef as llsraed n he followng fgre Ths one he nerval ξ s loaed has been denfed we an esmae he dervave a ha pon as ξ DU ξ ξ In hs formlaon sands for he nde of he rgh-sde lm of he nerval ξ s loaed For eample referrng o he fgre mmedaely above for haraers lne - Smlarly for lne - and for lne 3 If he vales of he veloy are only known a he nodes of he allaon grd wll be neessary o nerpolae ξ from he known node vales say and L Here and L sand for he ndes of he rgh-sde and lef-sde lms respevely of he nerval ξ s loaed Ths n he fgre mmedaely above for lnes and 3 L -3 - and - respevely Smple lnear nerpolaon of along reveals ha ξ ξ L L herefore he dervave / a pon ξ an be allaed as: 6

7 7 L L L ξ DU3 Consder he ase n whh he haraers rve sars a pon 3 or larger b hs he frs ell along he as a a pon sh ha < < Ths ase s llsraed n he fgre below To deermne he vale he haraers rve hs he lne we negrae he haraers rve d/d beween pons and The resl s: d Lnear nerpolaon of he veloy wh me a loaon ndaes ha eplang hs epresson for n he eqaon for and performng some algebra manplaon prodes a qadra eqaon n : M - M M 3 0 ETA M M and 3 M There are wo possble solons for from he qadra eqaon shown above The solon of neres s he vale of for whh < < If he ase desrbed above arses n a solon wll be neessary o allae he followng dervave o perform he adveon of /:

8 8 P P pon P orresponds o and In he fgre above s he foo of he haraers raeory leadng o pon The vale P an be obaned by lnear nerpolaon wh respe o namely P Wh he epressons for P and fond above he dervave / a pon s now wren as α α DU4 α The eqaon for allang for hs parlar saon s: C3 Noe: he dervave / a pon an be deermned as n eqaon DU Inal ondons The nal ondons for he onenraon and s dervave n are smply vales of and These vales shold be known If only he vales of are gven he vales of / an be obaned sng fne-dfferenes eg for 3 n and n n n These resls apply o a solon doman wh n sb-nervals of lengh L/n L s he lengh of he reah of neres along he dreon The grd pons n are allaed wh - for n

9 Bondary ondons Only he psream bondary ondon s needed The smples ase s when vales of boh and / I are provded Alernavely only he vales of are provded Ths las ase s dsssed laer n hs seon Ne we analyze a possble saon assmng ha boh and are gven The possbly ess ha a he psream bondary he parameer r s larger han Ths saon s llsraed by he fgre below The sse n hs ase s o be able o deermne he vale he haraers rve hs he lne and hen fnd he vales and The followng fgre s sed o develop an algorhm for deermnng he vale of Alhogh he grd shown apples o any vale of he approah s o be sed elsvely n he psream bondary e for To deermne he vale he haraers rve hs he lne - we an negrae he haraers rve d/d beween pons and The resl s: d Lnear nerpolaon of he veloy wh me a loaon - ndaes ha eplang hs epresson for n he eqaon for and performng some algebra manplaon prodes a qadra eqaon n namely: K - K K 3 0 K ETA 9

10 0 K and 3 K There are wo possble solons for from he qadra eqaon shown above The solon of neres s he vale of for whh < < If he ase r> arses n he psream bondary wll be neessary o allae he followng dervave o perform he adveon of /: P P pon P orresponds o and In he fgre above s he foo of he haraers raeory leadng o pon The vale P an be obaned by lnear nerpolaon wh respe o namely P Wh he epressons for P and fond above he dervave / a pon s now wren as α α DU4 α was defned earler The eqaon for allang for hs parlar saon s eqaon C3 wh The dervave / a pon needed n C3 s deermned as n eqaon DU wh eall ha hs approah s only needed f ξ< e f ξ falls o he lef of he pper-bondary ell herefore s only sed wh If only he vales of are provded s possble o allae he vales of as follows: for 3 n and n n n Assmng ha none of he haraers lnes ever h he as e here s no need o deermne pon as he foo of one or more haraers lnes hen he proedre s sraghforward wh he nmeral solon adveng he vales of and n me

11 If he ase ors ha a haraers hs he as as llsraed n he fgre below wll be neessary o deermne he vale of nmerally We only need o do hs wh he frs haraers ha hs he as sne we wll be sng a hrd order polynomal n o nerpolae vales of and of for and < < Any oher haraers ha sars a he as wll have vales of and nerpolaed from he known b polynomal Le he rve sarng a and endng a pon as shown above be he frs haraers rve ha hs he as for he perod < < The vales of and are known and he vale of has been allaed wh As a frs appromaon o we se a lnear nerpolaon on namely Then we an esmae as C3 The nerpolang polynomals for a are gven by r α - α α 3 - α 4 HP7 α s 3-s α - s 3-s α α 3 s -s α 4 - s-r and r β - b b 3 - b 4 HP8 β 6s-s/ β 6ss-/ b β 3 s3s- β 4 s-3s- In boh eqaons HP7 and HP8 he vale of s represens

12 s α S To mprove he allaon of an erave proess an be mplemened by whh a new esmae of h s allaed wh eqaon HP7 hen a new esmae of s allaed wh eqaon C3 The eraon onnes nl wo onseve vales of are whn a margn of error ε e nl k - k < ε Implemenaon of nmeral solon The followng s an olne of he algorhm for he nmeral solon of he pre-adveon eqaon lzng he Holly-Pressmann sheme: Apply nal ondons o load vales of and for all vales of For every me sep say 3 m and for every nner pon say 3 n: Compe raeory from a me level bak o ξ a me level by sng eqaon T 3 Consr he nerpolang polynomals for r and r as defned by eqaons HP5 and HP6 Basally yo need o allae he oeffens n he polynomals namely a a b b 4 Callae r wh eqaon deermne he vales of he nde and L by deermnng he nerval [ - ] ξ s loaed e and L - f - < ξ < and evalae r and r wh he oeffens allaed n sep 3 5 Compe ξ and from eqaon C The dervaves / reqred n eqaon C an be allaed sng eqaons DU and one of eqaons DU DU3 or DU4 [NOTE: eqaon DU4 s only needed n he psream bondary ell f r> and he deermnaon of a from eqaon ETA s reqred see sep 6 below] 6 Apply psream bondary ondon o load and Wre he ode o aon for all possble ases as olned n he seon on he bondary ondons Eerse Modelng he pre adveon of a Gassan onamnan dsrbon Consder a Gassan dsrbon of onamnan gven by he eqaon 0 0 ep o σ wh σ 64 m Use hs eqaon o provde he nal and bondary ondons for a preadveon smlaon n a hannel ha s 0-km long L 0 km f he smlaon lass for ma 60 mnes The waer n he hannel s movng wh a onsan veloy 05 m/s A me 0 he rve s enered a 0 0 nal ondons and for >0 he laggng half of he dsrbon eners he model as he bondary ondon a Wre a Malab fnon o mplemen he nmeral solon of pre adveon sng he Holly-Pressmann algorhm b Use a srp o perform o avae he fnon and perform he pre-adveon smlaon

13 wh 00 m and 00 s 00 s 300 s 400 s and 800 s Le he srp also plo solon rves vs for he dfferen vales of a 800 s 600 s 400 s and so on nl reahng ma Also plo he ea solon e he adveon of he Gassan rve a a onsan speed for he mes ndaed above n he same graphs epea one of he allaons n b b hs me se 0 a me 0 e 0 0 Dsss he effe of hs nonsseny n he resls 3