Radiation Diffusion: An Overview of Physical and Numerical Concepts

Transcription

1 UCRL-PROC-95 Raao Dffuso: Ovevew of Physal a Numeal Coeps F. R. Gaza Jauay 8 5 Ope ssues Uesag Coe Collapse Supeova Seale W Ue Saes Jue hough Jue

2 Dslame hs oume was pepae as a aou of wok sposoe by a agey of he Ue Saes Goveme. Nehe he Ue Saes Goveme o he Uvesy of Calfoa o ay of he employees makes ay waay expess o mple o assumes ay legal lably o esposbly fo he auay ompleeess o usefuless of ay fomao appaaus pou o poess slose o epeses ha s use woul o fge pvaely owe ghs. Refeee hee o ay spef ommeal pou poess o seve by ae ame aemak maufaue o ohewse oes o eessaly osue o mply s eoseme eommeao o favog by he Ue Saes Goveme o he Uvesy of Calfoa. he vews a opos of auhos expesse hee o o eessaly sae o efle hose of he Ue Saes Goveme o he Uvesy of Calfoa a shall o be use fo avesg o pou eoseme puposes.

3 RDON DFFUSON: N OVERVEW OF PHYSCL ND NUMERCL CONCEPS FRNK GRZN Lawee Lvemoe Naoal Laboaoy Lvemoe C 955 US ovevew of he physal a mahemaal fouaos of aao aspo s gve. Emphass s plae o how he ffuso appoxmao a s aspo oeos ase. ovevew of he umeal halg of aao ffuso ouple o mae s also gve. Dsussos ee o paal empeaue a gey mehos wh ommes oeg fully mpl mehos. ao fe ffeee fe eleme a Pe epeseaos of he v-ga opeao s also susse. he So Wha? Queso: Why Raao aspo Maes Phoos be hey he ao opal. X-ay o gamma-ay poo of he aao speum leave he mak aoss he fab of he uvese a mulue of ways. he smalles asoomal sales aao aspo s ual o uesag he amosphees of plaes. he lages sales he uvese s bahe a afeglow of s bh alle he osm bakgou aao. s s well kow whole fels asoomy ae evoe o he suy of ffee poos of he eleomage speum. Phoos a a as a sgaue of some asoomal eve. ao beause of he esy a empeaues eouee may asophysal applaos phoos a effe he moveme of a gas o flu a he moveme of he gas o flu a u affe he behavo of he phoos. Raao pessue hages s speal shape ue o movg flus a PV wok oe o he aao fel ae all mpoa examples of he eao of mae a phoos. Wh he ave of lage sale ompug he omplex sysem of equaos volvg aao aspo a flu yams oul be solve. Cuely wh he ouo of paallel ompug s ow possble o moel D asophysal pheomea suh as supeovae wh upeeee auay a wh he luso of omplex physs. all of hese smulaos aao aspo emas a exg bu hallegg obsale. mul-physs oes es o omae CPU me. hs s easy o see whe oe oses ha D

4 yam applaos he soluo of he phoo aspo poblem volves solvg a seve mesoal olzma equao. hs equao s geeal hghly o-lea a o-loal. ao s ouplg o he flu mofes boh he flu equaos a he usual hyoyam equaos. hs eao bewee aao a flus efes he fel of aao hyoyams. s a vas fel wh exelle efeees by Mhalas a Mhalas Pomag owes a Wlso a Caso 5. eses beg vey goo gues o aao hyoyams hey ae exelle soues fo he fel of aao aspo geeal. he halleges of solvg he aspo poblem have le eseahes o solvg a smple poblem. may applaos he physs allows oe o solve he ffuso appoxmao o he full aspo poblem. y gog o he ffuso lm of he aspo equao he umbes of egees of feeom D ae eue fom seve o fve fo mul-goup a seve o fou fo Plaka. hs appoxmao s by fa he mos use appoxmao o he aspo equao. fa makes he aao poblem so aable ha he ffuso appoxmao s use egmes whee oly a aspo espo s val. he use of aspo oee ffuso suh as flux lmes helps exe he applably of ffuso. he beef s of ouse he heape os of ffuso ove aspo bu a he pe of eue auay. hs pape s evoe o sussg he geeal famewok of aao aspo a paula how ffuso ases fom. ao a evew of he umeal eames of he ffuso opeao a how he ouple aao maeal equaos ae hale s gve. Due o he lak of spae he subje of aao hyoyams s o gve. eese eaes ae uge o ea he above lse esoues. Howeve he wok pesee hee shoul always be hough of he lage oex of a mul-physs oe. ao subjes o ovee hee lue opaes a saeg. Deale sussos of hese ops a also be fou he above lse efeees.. Revew of Raao aspo Coeps.. Classal a Quaum Popees of he Raao Fel he lassal mafesao of he aao fel s base o he wave popees of lgh. lassal espo of aao s osse wh he popees of polazao ffao a efao. Ufouaely he fa ha

5 a lassal aao pes a Raylegh-Jeas Law fo he emsso spea volao of he expemeal aa meas ha he wave espo of lgh s o omplee. s s well kow he quaum mehaal espo of aao esbes a wealh of obseve pheomea. he phooele effe whee phoos whh ae mpgg o a meal sufae elease eleos above a ea heshol fequey s a well kow example ha eae Ese he Nobel Pze. Compo saeg whee he fequey of ha X-ay phoos s shfe owwas ue o he omg phoos saeg off of saoay eleos a asfeg some of he eegy a momeum o he eleos s aohe example. he obseve emsso spea fom aoms ae a lass example whee he quaum mehaal espo of aao explae obsevaos ha wee pevously uexplae. he quaum mehaal eame of phoos a he espo of he blakboy speum s a famous a sgula suess ha heale he begg of a ew age. Fally he beauy beh he quaum mehaal espo of aao ulmae wh he ufao by Da of he pale a wave espos.... he olzma Despo of Raave asfe he saa espo of aave asfe ess o esbg he aao fel as a phoo gas movg wh he spee of lgh a eag wh a meum va absopo emsso a saeg. Fo smply he effes of maeal moo ae goe hee as ae he effes of efao ffao a speso. he aao fel s assume o oss of po pales phoos. ssoae wh eah phoo s a fequey eegy h a momeum h /. ay me sx vaables D ae eque o spefy he poso of he phoo phase spae. hee ae hee poso vaables a hee momeum vaables. he hee momeum vaables ae we ems of he spee of lgh a he phoo eo. Usg he gas aalogy a phoo sbuo fuo f s efe suh ha f s he umbe of phoos a me a poso oae he ffeeal eleme sol agle eleme wh fequey avelg he eo subeg a. he leaue he phoo sbuo fuo s

6 aely use. sea he spef esy o agula flux s he quay mos use a he obje efee o hs pape whe he aao fel s meoe. s efe by f h geeal o f shoul have a aoal ex esbg he paula polazao sae. hee ae geeal fou ompoes of he spef esy alle Sokes paamees eessay o esbe a polaze beam of aao. Oe of hese ompoes s he aao spef esy efe above. he ohe hee ompoes ome fom he plae of polazao a he ellpy of he beam. hs pape s assume ha he aao s ehe upolaze o he polazao saes have bee aveage ove. Fo eals o he aave asfe equao fo polaze aao please osul ehe Chaasekha 6 o Pomag. Usg he spef esy as he fuameal quay of ees a umbe of physally eleva objes a be efe. f h P f h M f h F f h u he quaes u F M a P ae espevely he eegy esy aave flux momeum esy a pessue eso of he aao fel. wg ow he equao of aave asfe a peagog appoah s usually ake. y usg he aalogy of a lassal gas a sem-lassal equao a be we ow base solely o osevao of phoos. ems ue o sk a soue ems a saeg ae lue a sem-pheomeologal fasho.

7 5 j S S h h he gh ha se of equao [] epeses he eao of he aao fel wh mae hough hee bas mehasms; emsso absopo saeg. he quay j epeses he emsso of phoos fom he maeal a s alle he emssvy. he quay s he absopo oeffe a has mesos /m. s elae o he opay κ a he esy ρ by he smple elao κ ρ. he quay S s he saeg keel. ypally wll epese he poess of Compo saeg. Fo smply he es of hs pape saeg s goe. he aave asfe equao s sem-lassal aue. he ke equao he phoos ae eae lke ay ohe gas. he quaum mehaal effes ome hough he absopo emsso a saeg ems. Eah of hese hee poesses esbes a a mo-physal level he quaum mehaal eao of mae a aao. he ex seo a smple example of he mo-physal ogs of he emsso a absopo mehasms wll be gve alog wh osequees of he mae fel beg hemoyam equlbum.... he Ese Coeffes a he Plak Dsbuo hs seo a evao of he Plak sbuo wll be gve. he evao pesee hee base o he book by Pomag eluaes he aue of loal hemoyam equlbum LE a o-le NLE a yels sgh o he sk a soue ems of he aave asfe equao. Cose a aom wh a umbe of bou saes. Cose wo levels wh eeges E a E m a sasal weghs g a g m. he pobably pe me ha a aom sae m expose o aao of fequey wll absob a phoo fom he aao fel s gve by. P m m m 5

8 6 h E E 6 m m m s oe of he Ese oeffes. s a osa of popooaly epeseg he aso ae a whh he pesee of he aao fel ues a upwa aso eegy. m s he spef esy of he aao fel whh we wll efe a mome a s he sol agle subee by he phoo. eses beg absobe he aom a also em a phoo. he pobably pe me ha a aom sae wll em a phoo of fequey s gve by. P [ ] m m m m 7 he em popooal o m s he las Ese oeffe. epeses he aso ae a whh he aom uegoes spoaeous emsso. ha s hee s a fe pobably ha a aom a sae wll em a phoo a uego a owwa aso eegy he absee of a aao fel. he seo em popooal o m epeses he effe of smulae emsso. s he aso ae a whh he aao fel ues a aom o uego a owwa aso eegy. ha s he pesee of a aao fel self wll ehae he emsso poess. hs smulae emsso s a osequee of he quaum sass obeye by bosos. he a oeffes ae elae. Gve he Hamloa of a aom a aao fel me epee peubao heoy a be use o ompue he aso aes fo absopo a emsso. he fuameal elaoshp a be eve m / / h m 8 hs po he susso ohg has bee assume abou he absopo a emsso poesses. he agumes ae ompleely geeal a ae applable o boh sysems a ou of hemoyam equlbum. f howeve hemoyam equlbum s assume he aoal esuls egag he Ese oeffes a he assoae aao fel a be eve. omplee hemoyam equlbum he pple of eale balae hols. ha s hee exss a eale balae bewee all absopo a emsso poesses. Mahemaally hs meas he pobably fo emsso exaly equals he pobably fo absopo. m

9 N [ ] N m m m m m m 9 N efes o he umbe esy of aoms sae. Solvg fo he aao fel yels m / m m [ N / N ] m Se omplee hemoyam equlbum has bee assume he populaos ae sbue aog o he olzma sbuo heefoe he Plak sbuo s obae N N m g m h m k e m g m h / h m k / e m m m m... Loal hemoyam Equlbum LE a Khoff s law 7 he smple example above poves he po ha solvg he aave asfe equao whe he mae ems absobs a saes aao s omplex. hs s beause geeal eale kowlege of he aom populaos a ozao saes makg up he maeal mus be kow f he absopves a emssves ae o be alulae. he oep of LE s a smplfyg assumpo of he mae ha gealy eues he omplexy of yg o solve he aave asfe poblem whe aao-mae eaos ae mpoa. he esseal po esablshg LE ay gve maeal s ha he popees of he mae ae omae by aom ollsos whh esablsh hemoyam equlbum loally a a spae-me po a he aao fel oes o esoy hs equlbum. heefoe he ma ffeee bewee omplee hemoyam equlbum a LE s ha LE oes o eque he aao fel o be Plaka. he mplaos ae ha a a gve spae-me po oly he aom omposo a wo hemoyam quaes esy a empeaue ee be spefe. he LE assumpo of ouse assumes vey spef feaues of he saes of he aoms a moleules makg up he

10 8 maeal. he fs s ke equlbum ha s he eleo a o sbuos obey a Maxwella. he seo feaue s exao equlbum ha s he populao esy of he exe saes of evey spees mus obey a olzma sbuo. h ozao equlbum ha s he pale eses fo euals eleos a os obey a olzma lke sbuo volvg ozao poeals. hs s he so-alle Saha equao 8. Fouh s he Khoff-Plak elao. hs s a amazg elao ha eues he emssvy o a pou of he absopo oeffe a a Plak fuo whose empeaue s haaes of he loal maeal empeaue. Smply pu j. heefoe he maeal-aao eao eues o a suy of he absopo mehasms a plasma whee ke exao a ozao equlbum hols. Whe oes LE hol? oe fo aom ollsos o omae ove aave poesses s lea he plasma mus be ese. Gem 7 has osue a eo base o he ao of aave o ollsoal aes a hyoge-lke aom whee he plasma s opally h eough ha phoos oe eme ae o e-absobe. he oo fo LE of aom level s. 5 Z * 7 7 / Z ev *.6eV / ρ << * he effeve ozao s eoe by Z. Gem s eo lealy shows ha fo hgh esy a/o ho plasmas LE s a goo appoxmao. hs pape LE wll always be assume uless ohewse oe. So fa he susso has fouse o he aao fel. mos applaos wh opally hk mae he aao s absobe eme a he maeal empeaue hages. Usg he assumpo of LE evey pee of mae as lke a blakboy aao emg phoos wh a Plaka speum haaes of he empeaue of he maeal. heefoe gog saeg usg Khoff s law fo he emssvy a wg a eegy balae elao fo he maeal we have [ ρc V ] [ ] 5

11 9 wg ow he lae equao s assume ha he opaes ae epee of agle a all ouo effes ae eglgble.... he Equlbum Raao Fel he soop a homogeeous sbuo obeye by a phoo gas omplee hemoyam equlbum a empeaue s he Plak fuo h h / k e 6 hs shoul o be ofuse wh LE. he Plak fuo a be eve a vaey of ways. We have see oe suh meho seo... Physally s a osequee of he phoos uegog hee bas poesses absopo smulae emsso spoaeous emsso a he fa ha he phoos ae bosos. he soop aue of he equlbum aao fel meas ha he eegy esy aave flux momeum flux a pessue eso quaes a be easly evaluae. Subsug equao o equao yels 5 8π k u 5h F M a 7 P a 5.7Mba [ kev ] s expee fo a soop gas he aave flux a momeum esy ae zeo a he pessue eso s agoal. he oep of a empeaue fo he phoo gas s uquely efe hee. ha s fo a equlbum sbuo of phoos he empeaue s efe by he Plak sbuo. Oe oul hk of he elaoshp bewee eegy esy a empeaue as a efg elao. Howeve fo o-plaka sbuos hs s o ue. Seveal efos of empeaue ae eouee mul-goup ffuso a aspo 8. he above elaos fo eegy esy a pessue gve a sog ao of he powe of he aao fel o move mae. kev he

12 pessue exee by he phoos o mae s 5.7 Mba. ao hs pessue aply ses wh empeaue. hs fa meas ha fo hgh empeaues he aao fel a make a subsaal o oma obuo o he oveall pessue a flu...5. ssumpos of a Ke heoy a he Mo-physal Fouaos of Raave asfe he spef fom of he aave asfe equao whose umeal soluo s he fous of hs pape ess o a umbe of assumpos. he fs se of hese assumpos s o hee he ke heoy of aave asfe bu ahe a smplfao. s meoe pevously polazao effes ae goe. ao efao a speso effes ae also goe. aave asfe equao opoag hese effes a be ealy osue Pomag bu fo smply hey ae goe hs pape. he seo lass of assumpos s hee he sem-lassal ke appoah aope hee. Se phoos ae eae lassally he wave behavo s goe. heefoe he possbly of efeee bewee ffee phoos s goe. Hee he spea of he phoo wave pake s assume small o he esoluo we ae eese. Se he phoos ae eae lke a lassal gas ohe quaum pheomea suh as phoo umbe fluuao ae goe. Fally all ollso absopo a emsso poesses ou saeously. he fuameal espo of phoos a he eao wh mae s base o quaum eleoyams QED. s possble o ave a a fully selfosse espo of aave asfe fom a fuameal espo? he aswe s yes. lhough a eale espo of he evao woul ake us o fa a fel suffe o say ha a umbe of auhos have eve ke equaos fo he phoos sag wh he Hamloa of QED Gelas a O 9 Cao Degl`oe Gaza. example of he poeue volves usg ehe he esy max fomalsm o he Hesebeg equaos of moo o osu yam equaos fo he umbe opeaos of he plasma a phoo egees of feeom. quaum mehaal sbuo fuo smla o he Wge sbuo a be osue fom pous of phoo opeaos. he Wge sbuo s a opeao ha s a quaum mehaal geealzao of he spef esy. he quaum aave asfe equao omes fom he yam evoluo of hs Wge fuo. he plasma egees of feeom ee aually o he quaum aave asfe equao beause

13 of he eao ems pese he QED Hamloa. he lassal spef esy s alulae as he quaum aveage of he Wge fuo. Of ouse he quaum aave asfe equao s moe geeal as allows oe o ompue fluuaos of he spef esy opeao. Classally hese fluuaos ae eglgble. Examples of whee suh a appoah has pove useful ae evg aave asfe equaos wh polazao fom fs pples quaum ops a NLE plasmas.. he Dffuso ppoxmao he aave asfe equao [] beg geeal a sx mesoal olea ego-ffeeal equao s o ouve o lose fom soluos. fa s a ue hallege he asophyss amosphe physs hgh eegy esy physs a ulea egeeg ommues o solve umeally. heefoe boowg a page fom he ke heoy ommuy a meho ommoly use o smplfy he aave asfe equao s o osu momes of he equao. he fs ype of momes ha a be osue of he spef esy ae ave a by mulplyg equao [] by vaous powes of a egag boh ses of he equao ove all sol agle. Defg φ Π 8 hese quaes epese he zeo fs a seo oe momes of he spef esy. Physally hey ae he eegy esy flux a pessue eso pe fequey. We wll efe o as he speum. Usg hese efos he aave asfe equao a be we as a se of ouple paal ffeeal equaos

14 [ ] "Fouh Oe Mome"... Π Π φ φ π φ 9 hs se of mome equaos exhb he lass losue poblem. ha s zeo oe momes ae ouple o fs oe momes; fs oe momes ae ouple o seo oe momes a fum. he eal queso all appoxmaos o he aave asfe equao s omg up wh a suable losue sheme. hee elae ops ae exame hee; vaable Ego faos ffuso a he elegaphes equao... Vaable Ego Faos Vaable Ego faos s less a losue sheme ha a way of easg he ouple mome equaos. Defe a eso quay alle he Ego fao by χ Π Physally hs quay epeses he mea of he eso ove all eos weghe by he spef esy. Subsug equao [] o equao [9] yels [ ] φ χ φ π φ

15 lhough he above se of equaos s seemgly lose s fa o ue o he pesee of he Ego fao. Howeve by wg he ouple mome equaos s hs fasho he Ego fao poves a egee of feeom o haaeze he aao fel. Fo example s possble o solve a aspo poblem evey o yles o so a osu he Ego fao hs Ego fao woul he be use he soluo of he ouple mome equaos a woul be upae whe he aspo solve was pefome. hs s esseally wha s oe ZEUS-D... Mul-Goup Dffuso Fo a soop aao fel he Ego fao smplfes oseably o χ f s assume ha eaos wll mae omae he flow ha s φ φ << he he flux equao beomes smply a Fk s ype law φ a he eegy esy equao beomes he famla ffuso equao [ ] π hs equao alog wh he maeal empeaue eegy balae equao [ ] [ ] C V ρ 5

16 fom he bass fo he mul-goup ffuso equaos. shoul be emake ha he above aalyss s equvale o assumg ha he spef esy s ealy soop φ π π 6 o see how hs ases physally assume LE a goe saeg a eaage he ems he aave asfe equao suh ha Pefom a expaso o fs oe / o oba 7 Noe ha a expaso Cosug he flux ely fom equao [8] yels Fk s Law! ha s 8 / s equvale o a expaso. π φ 9 heefoe Fk s Law ases ely fom a vese mea fee pah expaso o equvalely a expaso soopy. he mul-goup ffuso appoxmao woks well whee he aao fel s ealy soop ye he meum s aspae eough ha phoo mea fee pahs ae log a heefoe he aao empeaue s o eeme loally bu ahe s eeme by soues ha ae fa away. he mul-goup ffuso appoxmao s fequely apple asophysal a eal ofeme fuso CF applaos. Fo example fo a CF apsule bahe aao he mul-goup appoxmao woks vey well peg apsule pefomae. s lea fom he fuoal fom of he Plak fuo ha he above equaos foms a se of ouple o-lea ego-ffeeal equaos. s hs fa ha makes he soluo a hallege. s he aspo poblem he

17 5 pesee of a maeal empeaue ouples all goups ogehe a s hs fa ha makes he soluo a hallege. hs pape wll suss seveal mehos fo he soluo lug paal empeaues gey mehos suh as Lu-Wlso a soue eao a full max mehos... he Plaka Dffuso Equao aoal smplfao ases f s assume ha he phoo speum s Plaka. hs wll ou opally hk mea whee he aao fel s eeme by may absopos a e-emssos. hs ase he aao fel aply beomes Plaka a a empeaue o eessaly ha of he maeal. ssumg he aao speum s Plaka wh a haaes empeaue R.e. R equao [] a be egae ove fequey o yel [ ] R R P P R R R R Whee he Plak a Rossela aveage opaes ae efe by π π R P a a he maeal eegy balae equao s jus [ ] [ ] R R P P V a C ρ he Plak opay P ses he me sale fo loal eegy exhage bewee mae a aao base o emsso a absopo. he Rossela opay R eemes aveage aspo popees of he aao flow.

18 6.. Some Obsevaos Regag Dffuso s lea ha by gog o he ffuso appoxmao he equaos of aao aspo have bee fuameally hage. Fom a mahemaal sapo he aave asfe equao s a hypebol fs oe equao equg fo bouay oos ha he al spef esy be spefe alog wh fo example a omg value of he spef esy a a bouay. he ffuso equao howeve s a paabol equao eeg wo bouay oos o be spefe alog wh a al oo. Fom a physal sapo he ffuso appoxmao s aausal. o see hs ose he mul-goup ffuso equao wh o sk o soue ems hs equao esbes a sgal popagag wh a veloy V whh s obvously volao of he spee of lgh eso. s lea ha applyg he ffuso appoxmao o aspae mea whee a sgal a popagae a o ea he spee of lgh s volao of he appoxmaos use evg equao [] so s o oo supsg ha he aausal aue has eae s ugly hea. Howeve all s o los. Flux lmes whh wll be susse ex save he ay a allow oe o apply he ffuso equao o applaos whee omally oly a aspo espo base o he aave asfe equao woul o. efoe movg o a queso aually ases whh s; wha has bee los by howg ou he me evave of he flux em? wha happes f s esoe? Cose fo smply he ouple se of mome equaos a vauum φ φ akg he me evave of he fs equao a usg he seo equao o lose he se yels

19 7 5 hs s he wave equao ha esbes gog a ougog waves movg wh a veloy of /. Hee ausaly has bee esoe; howeve he fe lgh spee s oo small by a fao of.577. he ouple mome equaos he pesee of mae yel he so-alle elegaphe s equao aspo Coee Dffuso: Flux Lmes a vauum he aave asfe equao pes ha he eegy esy popagaes wh a veloy a he flux s gve by s he sol agle subee by he ay. hs s he maxmum flux allowe physally se epeses phoos movg umpee. Dffuso pes a vey ffee behavo. s show above he flux fo he ffuso equao he absee of sk a soue ems s φ. Whee φ. Howeve s lea ha fo vey sog gaes a/o whe he ffusve flux a be geae ha he maxmum flux allowe. Jm Wlso fs popose lmg he ffusve flux o oe fo he fa ha ffuso pes fase ha lgh flow spees aspae mea. Defe a ew flux wh a geealze ffuso oeffe whh epolaes bewee he ffusve flux opally hk mea a he aspo flux opally h mea.

20 8 fo log phoo mea fee pahs fo sho phoo mea fee pahs φ Λ Λ Λ 6 he leaue 6 oas a vaey of hoes fo he flux lme. s a ase Levemoe a Pomag 7 have eve he flux lme a goous fasho by pefomg a Chapma-Eskog lke expaso of he aspo equao. he esul s o quoe hee bu eese eaes ae uge o ea he papes. Lase flux lme Maxmum flux lme Wlso sum flux lme MX Λ Λ Λ 7 Fgue shows a ompaso of a vaey of hoes lug he Levemoe- Pomag esul R s ploe o he hozoal axs a Λ s ploe o he veal axs. slgh vaao o he sum flux lme appeas Lu a Wlso a ha s hey lue a fao of / m fo of he absopo opay. Fo ha phoos hee s o effe bu fo sof phoos he fee seamg lm s eovee.

21 9 Fgue. Compasos of seveal flux lme hoes ake fom Olso ue a Hall JQSR V Fgue shows he Levemoe-Pomag esul log ash-sho ash as eve fom he aspo equao. Clealy he Wlso sum flux lme es o es he flux oo muh whle he MX flux lme oes o es he flux eough. eesgly he Lase flux lme wh oes a vey goo job of mahg he aspo soluo.. Numeal Mehos fo Dffuso Complex applaos a he aue of he ouple aao mae equaos meas ha aaly soluos ae almos mpossble o f. sea a umeal appoah s sough. he sezao of he mul-goup o Plaka ffuso equaos meas ealg wh seveal ssues. he fs s me sezao. s assume ha he ouple aao a maeal equaos ae o be solve mplly. Expl me ffeeg yels a Coua sably oo whh seveely ess he me sep a heefoe s o wely use. hee ae uooally sable expl shemes fo ffuso ha have bee exploe 8. Howeve he auay of hese shemes beyo seveal Coua me seps s poo. hs pape a mos applaos oes he smple bakwa Eule ffeee s use. ha s

22 whee - 8 spe of hghe oe me ffeeg shemes suh as Cak-Nholso he smple fs oe bakwa ffeeg me s wely use ue o s obus behavo he lm of lage me seps. hs s ue o s o-osllaoy behavo a s ably o eove he sa soluo as. he seo ssue s how o seze he ffuso opeao. Spae lmaos peve a leghy susso of hs mpoa op; howeve a omplee suvey of mehos suh as fe ffeee fe eleme a Pe wll be susse. he susso wll be ese o qualaeal meshes D. Geealzaos o D ae saghfowa. he h ssue s he eame of aao-mae ouplg. Eve he smples ase of Plaka ffuso hs poblem ases as he equaos goveg aao ffuso ae ouple olealy o he maeal empeaue. mul-goup ffuso hs mafess self as he maeal empeaue beg ouple o all phoo fequees.. Spaal Dsezao... Fe Dffeee he ype of spaal sezao pefome o he ffuso opeao s maely oee wh he mesh ype. he smples mesh ofe eouee s he sa Eulea ohogoal ype. Fe ffeee mehos ae fequely apple hs ase yelg a see epeseao of he ffuso o v-ga opeao. Cose fo example he D zoe J/ -/ J J / J J-/

23 ssume he epee vaable R o s efe a he zoe ee. Cose he ffuso opeao as he gae of he flux a egae ove he ell volume of he j zoe. he followg esul s obae. V Whee j j ab V Φ V a Φ ab V j j Φ / j Φ / j / j Φ / j j / Φ j / j / Φ j / 9 ae he sufae aea a flux espevely a he a b fae. he flux a he fae s jus a fuo of he ffeee of he zoe eee quaes. Fo example Φ Φ / j j / D / j D j / x j / j y j j / j j Reaagg ems yels V j V j j V j V Φ α [ ] α [ ] / j α [ ] α [ ] j / j j j j / j j/ j j j j Whee α / j / j x D / j / j a α j / j / x D j / j /

24 he me epee ffuso equao s smples fom whou sk o soue ems s almos eay o be solve. he las ssue s he me samp assoae wh he ffuso oeffes. geeal hese quaes epe o he mae empeaue whh epes o me. he smples appoah s o evaluae all ffuso oeffes.e. opaes a he ol me sep value. hs meas ha he see epeseao of he ffuso equao s sem-mpl.. ha s he soluo a he avae me sep a be we ˆ Mˆ Φ Φˆ Φ he ffuso max M s symme a posve efe. ao s a M-max meag ha ay gve agoal eleme s he egave sum of he oespog off-agoal elemes. hs has mpoa osequees ha mples ha he soluo veo Φˆ s posve. he suue of M s -agoal wh sub a supe agoals epeseave of he 5-po sel equao []. he me lag of he ffuso oeffe meas ha some ype of me sep ool mus be efoe. hs s o esue auay. Fally he sem-mpl equao fo Φˆ a be mae mpl smply by wappg equao [] a eao loop. s of ouse eque ha he ffuso oeffe be oually upae wh eah eao. Mehos ha o hs wll be susse a he e.... Fe Eleme a Pe Opeaos Whe he mesh s o alge wh he ooae sysem whh a ou fo example aao-hyoyam oes usg Lagaga o LE bay Lagaga Eulea mehos omplaos a ase whe osug a see epeseao of he ffuso opeao. paula pesevg seo oe auay a posve-efeess o o-ufom gs oues o be a hallege. hs seo wo elae mehos fe eleme a Pe whh have fou we use oes usg omplex zog wll be susse. he susso wll be base o vaaoal mehos a s base lagely o upublshe oes by R. po 9. o beg he me seze ffuso equao wh o sk o soue ems a be eve fom a vaao of he ao

25 Θ Φ V D Φ Φ Φ Chagg oao slghly oe o mmze es he ao see fom yels Θ a C Φ a a a b a Κ Φ a Φ a a ab Whee a eoes oe umbe.e. a a whee C a a V a a a Φ b Φ a V 5 Noe ha Κ ab Θ δφ δφ δ 6 he K max epeses he spaal sezao of he ffuso opeao. he hoe of hs opeao eemes he see fom of he ffuso opeao. Cose he ouous fom of he ao. he fe eleme meho assumes hee exss a se of bass fuos suh ha a b Φ x y ϕ N x y 7 a Subsug hs epeseao o he v-ga obuo o he ao yels Θ Κ Dvga ab a a b V D N a a K ab ϕ ϕ a N b b 8 po s applao of fe elemes eques wo aoal gees. Oe s he fa ha fe elemes lke o have he ukows lve a he oes wheeas has bee assume hee ha he ukows lve a he zoe ees. po efes a ual mesh whose oes lve a he zoe ees of he egula

26 mesh. ao sea of usg qualaeal bass fuos he spls he qualaeal o agles a uses bass fuos assoae wh hem. Se hee ae wo ways D of splg a qualaeal o agles he K max s osue as he esul of aveagg he wo splgs. Fo a agula bass fuo a -z ylal geomey Κ ab πds s a a s s b b 9 Whee sa s a veo efe o he oe oppose oe a. f he ohe wo oes ae eoe by b a whee b a o o equal a he sa les omal o he b leg a s mague s equal o he legh of b. Defg θ ab as he agle oppose oes a a b he K max beomes fo a sgle agula bass fuo Κ Κ ab ab π D o θ ab D o θ ab - z geomey plaa geomey 5 Wheeas he fe eleme esbe above eles o agula bass fuos he Pe opeao eles o b-lea qualaeal elemes wh wo po quaaue. sgle po quaaue wh b-lea qualaeal elemes wll o wok ue o he fa ha fo squae o hombo meshes ouplgs bewee eghbog zoes ous oly hough oe ouplgs a o hough faes. Hee he mesh a evelop he so-alle hekeboa sably. Cose he mesh logal ooaes. Cose a spef zoe whose oes ae efe wh he so-paame ooaes ξ η ± ±. he b-lea bass fuos ae smply N N ξ η N ξ η ξ η N ξ η 5

27 5 he v-ga obuo o he ao omes abou by subsug equao [5] o equao [8]. he esul s Θ Dvga ξηd π g J ξξ ϕ ξ g ηη ϕ η g ξη ϕ ϕ ξ η he g s ae he me eso a epeses he ooae asfomao bewee he logal ooaes a physal ooaes. Fo example ξη x x y y ξ η ξ η g 5 he Pe epeseao fo he ffuso opeao omes fom a wo-po quaaue appoxmao o he egal he above equao fo Θ. Dvga hs s oe by akg he ual mesh qualaeal a eomposg o fou equal aea seos. heefoe he K max beomes g g g πd ξξ πd g πd g πd g ηη ηη ξη ξξ ξη πd g πd ξξ πd g πd g ηη g g g ηη ξξ ξη ξη πd g πd g πd ξξ πd g ξη ξξ g g g ηη ηη ξη πd g πd g πd g πd ξξ ξξ ηη ξη g g g ηη ξη... Loal Suppo Opeaos Reely Jm Moel 5 a o-wokes have evelope a pomsg D a D ffuso sezao sheme base o loal suppo opeaos. Lke he mehos susse above yels a spase max epeseao fo he vga opeao. he meho has efe avaages. s seo oe auae o soe meshes goously eas maeal soues a has a symme posve efe max. he savaage of he meho s ha eques fae ee as well as ell ee ukows. hee s some subley egag mplemeg flux lmes o he loal suppo sheme. Howeve Dav Mlle 6 has oe hs suessfully

28 6.. Mul-goup a Plaka Dffuso o summaze he soluo of he maeal empeaue a speum ees o be fou fom he followg ouple equaos [ ρ ] C V [ π ] [ ] 5 5 he hee ffuly s solvg hs se of equaos s ha eve hough he equao obeye by he speum looks lke s epee of fequey goup ouplg he maeal equao eques kowlege of he speum a emsso a all fequees. Hee he emsso fuo also eques kowlege of he speum a all fequees. heefoe he equao fo he speum s heely o-lea ue o a effeve goup o goup ouplg. he paal empeaue a gey mehos ae examples of ehques whh have fou suess ealg wh hs poblem. hese ae susse ex. Plaka ffuso a be hough of as a subse of mul-goup ffuso. Physally of ouse Plaka ffuso ases fom assumg he speum s Plaka a he egag ove all fequey. Numeally we wll see ha he paal empeaue sheme a be meely hough of as a speal ase. all of hese mehos hea apaes a opaes ae me lagge so eve hough he soluo mehos ae o fully mpl. ao ue o hs fa all of he mehos osee wll eque some so of me sep ool o he maeal a/o aao empeaue. s we wll see he paal empeaue meho eques a vey spef ype of me sep ool. ao he mehos hee shoul be hough of he we oex of a mul-physs oe. aoally physs pakages ae opeao spl. Whehe hs s oe fs oe o seo oe me hs fa aloe lms he auay eve ases whee he aao pakage s solve fully mplly. Examples of fully mpl mehos wll be susse seo... he mpl me ffeeg a he spaal sezao mea ha he ouple aao-maeal equaos wll fom a sysem of equaos. geeal hese sysems wll be o-lea. Due o a vaey of ehques o be susse ex hs o-lea se a be appoxmae by a sysem of lea equaos. heefoe lea solves play a mpoa ole he mehos pesee hee.

29 7 s ue o avaes lea solves a he peooes ha have mae aao ffuso poblems moe omplex geomees a D possble. hs s ue o small pa o saleable boh he poblem sze a paallel sese algohms.... Mul-goup Dffuso: Paal empeaue he ouple se of equaos s ffeee mplly assumg ha he hea apaes a opaes ae evaluae a he ol me sep. gog fo he mome ssues elae o spaal sezao a seg all eal a exeal soues o zeo we oba π 55 π ρ V C 56 he summao s ake ove he goup ex a us fom zeo o g N. Noe ha he Plaka ffuso sheme s jus a speal ase of whee. Now assume hee exs a paal maeal empeaue obuo fo eah goup. ha s [ ] N g a whee τ τ τ τ [ ] V C π τ τ ρ 57 We ow appoxmae he emsso o Plak fuo as f wee omg fom eah goup. ha s [ ] τ τ τ τ τ 58 he followg lea sysem esuls [ ] V C τ π ρ τ π τ τ 59

30 8 Upo appoxmag he emsso o Plak fuo as befoe he mul-goup phoo equao beomes [ ] τ τ τ π τ π 6 Subsug equao [59] o equao [6] yels a lea mpl soluo fo. hs may be solve va a vaey of peooe lea solves suh as mul-g peooe ojugae gae. wo of auo oeg paal empeaues s oe. he meho s sable a obus. Howeve a suffe auay uless he paal empeaue swgs fo eah goup fom yle o yle ae lme va a me sep ool. Plaka ffuso se hee s jus oe goup he empeaue swg s jus he hage mae empeaue fom yle o yle.... Mul-goup Dffuso: eave Gey Mehos hee exss aohe lass of mehos ha aemp o solve he maeaao ouplg poblem va a eave poeue. hese mehos eque a gey ffuso aeleao oe o spee-up ovegee. We wll suss he fully mpl meho of Lu-Wlso a s vaa ue o Eppley. We wll he suss a sem-mpl meho ue o Moel Lase a Maze a s fully mpl exeso ue o Gaza. he Lu-Wlso meho eplaes he se of mul-goup ffuso equaos wh he se [ ] π 6 [ ] π ρ V C 6

31 9 he ex s a eao ex whee oespos o he ol me sep values. Noe ha upo ovegee a he ouple mul-goup equaos ae eovee. Defe he gey oeffes Σ Σ α π π 6 s popooal o a effeve aao empeaue ase o he fouh powe. α s speum aveage opay. he mae equao we make he appoxmao α 6 he lae appoxmao s exa upo ovegee. Usg he gey oeffes he mae equao beomes [ ] Σ Σ V V C C ρ ρ α 65 hs equao a be solve pove we kow. hs quay s obae by summg he mul-goup phoo equao ove fequey. hee ae wo ways of og hs epeg o how he ffuso oeffes ae aveage. Cose he ffuso obuo o he mul-goup phoo equao. lso we assume a smple D fe ffeee epeseao of he v-ga opeao. We a fom a equao fo by summg he phoo mul-goup equao ove fequey goup.

32 . D D / / Lu a Wlso appoxmae hs expesso as follows γ D / γ / he equao fo γ / [ ] [ ] [ ] D / [ ] [ ] [ ] / [ ] [ ] beomes Σ Σ [ ] α [ ] γ / [ ] hs s he so-alle gey equao. he above equao s a -agoal equao fo whh a be solve va bak-subsuo mehos. wo o hee mesos he above equao s a max whh a be solve va peooe ojugae gae mehos. solvg he ouple mul-goup equaos he followg seps mus be pefome. Evaluae he gey oeffes [6] usg he las avalable speum. Solve he gey equao fo. Compue he upae empeaue. Kowg ompue he upae speum 5. s he mae empeaue ovege < δ? 6. f o epea seps -5 usg he laes speum o ompue he gey oeffes.

33 7. f yes values fo mae empeaue a speum ae aepe. 8. he meho ue o Lu a Wlso seems o wok well mos ases. oes have oe awbak howeve. he gey aveage ffuso oeffes ae o guaaee o be posve. hs a ause havo fo max solves. Eppley s meho es o umve hs poblem by efg ew gey aveage ffuso oeffes. Eppley s sheme sea of oe ype of gey ffuso oeffe he efes wo. Namely / / / / D D δ δ 69 / / / / D D 7 he gey equao s ow slghly mofe fom befoe [ ] [ ] / / / / Σ Σ δ δ α

34 Eppley s meho poues posve efe gey ffuso oeffes. Howeve he os s a o-symme max fo he gey equao. pae he obusess of Eppley s vaa of Lu a Wlso seems o wok well. Mehos suh as GMRES ae sasfaoy fo solvg he o-symme gey equao. Mehos ha aemp o solve he gey equao by splg he max o symme a asymme obuos a lug he asymme pas he oveall eao loop o o seem o be obus. lhough hey have he avaage of allowg he oe o use symme max solve mehos he meho oasoally fals o ovege. Eve hough has o bee explly meoe as suh he soluo of he gey equao he Lu-Wlso a Eppley vaa aeleaes he eave poess. Whou he gey soluo sep he umbe of eaos a gow o he housas egmes of he poblem whee aao a mae ae ghly ouple. hs s ealy ue he ex meho whee a gey aeleao s explly oue. he sag po of he soue eao meho of Moel Lase a Maze s o expa he Plak o emsso fuo evaluae a he upae empeaue abou s value of he pevous me seps empeaue. ha s [ ] 7 hs s pefome boh he phoo a he maeal eegy balae equaos. y og hs expaso he ouple se of mul-goup equaos s effevely leaze. Subsug he above equao o he maeal eegy balae equao a solvg fo ρc yels [ π ] 7 V Subsug hs expesso o he phoo equao yels Q Whee a ew se of gey oeffes ases ηχ 7

35 η χ Q ρc π V π πηχ 75 hs s a leaze fom fo he phoo mul-goup equao. hs po he goup o goup ouplg sll exss. he soue eao poess volves eplag he above by Q ηχ 76 heefoe whe he above equao oveges o he leaze fom gve by equao [7]. Noe ha wheeas wh he Lu-Wlso meho we ovege o mae empeaue hee we ovege o he speum. pae a ovegee ea base o ovegg he speum ove all goups s o eessay. seems o be suffe o ovege o he aao empeaue o he aao eegy esy. he beef s a smalle umbe of eaos wh vey lle loss of auay. heefoe he poeue s. Evaluae he gey oeffes usg he ol me sep values fo empeaue a speum. Solve he phoo equao [76] va eao. Oe s ovege ompue he maeal empeaue Noe ha he gey oeffes ae evaluae oe a he sa of he yle as oppose o Lu-Wlso whee hey ae evaluae a eah eao.. ao hee he maeal empeaue s evaluae oe he speum s ovege

36 wheeas Lu-Wlso s upae evey eao. pae he soue eao meho woks well exep he umbe of eaos o ovege he phoo equao ses seeply opally hk egmes. oe o oe hs efey he eao poess s aeleae va a gey equao. he pape by Moel Lase a Maze beses he speum hey also efe a quay whh s he ffeee of he speum bewee he exa soluo a he laes guess fom eae. We eoe hs quay as δ 77 ems of hs vaable he mul-goup phoo equao beomes δ δ ηχ [ δ ] 78 hs meho he gey equao s eve by assumg ha he mul-goup speum s gve by he equlbum speum. he equlbum speum u s gve by he soluo o equao [7] whee he gae em vashes. ha s δ χ Γ eq Whee he gey oeffe s efe by Γ χ Subsug hs expesso o equao [78] a egag ove goups yels 79 8

37 5 Η η η D D 8 he gey oeffes ae gve by Η χ D 8 Η Γ Γ Η χ χ D 8 he mul-goup soluo poeue s smla o he uaeleae poeue. Evaluae he gey oeffes usg he ol me sep fomao. Upae he speum usg he phoo mul-goup equao. Solve he gey equao fo he egae speum. Coe he speum by ag he oeo em χ δ Γ eq o he esul fom sep 5. Repea seps - ul ovegee s eahe woul seem ha hee aga he ovegee eo mus be base o ovegg he speum all goups. pae howeve us ou o be suffe o ovege o he aao empeaue o he aao eegy esy. he umbe of eaos a be subsaally eue wh lle loss of auay. moe mpoa ssue s he gey equao. Ufouaely he gey

38 6 equao lke Eppley oes o have a symme max. hs s ue o he D ~ em. Whou ay jusfao oppg hs em oes o seem o ham he aeleao poess sgfaly. hs s fequely oe pae. Howeve hs ssue ees o be looke a lose. hee exss a vaa of he Moel Lase a Maze meho ue o Gaza ha solves he o-lea ouple mul-goup equaos. he sag po s o expa he emsso o Plak fuo o aou he ol me sep bu ahe he ol eae. ha s [ ] 8 We go hough he same algeba as pevously oe howeve a he e we e up wh a slghly ffee equao fo he speum. Q ηχ ~ 8 he gey oeffes ae efe by ~ V V C Q C ρ πηχ χ π ρ π η 8 heefoe he Q em s slghly mofe. he poeue fo solvg he olea mul-goup vaa follows he same seps as he lea meho. Howeve he ovegee eo s ow base o < δ

39 7 he elave mes of solvg he o-lea vaa of Moel Lase a Maze has o bee vesgae. Fuhe wok ees o be oe.... Commes o Full Max Mehos Wh he ave of ease memoy a he pogess peooe lea solves he ea fo fully mpl mehos has sae lookg lke a aave aleave he mehos susse above. We gve hee some efeee o a samplg of he eseah hs aea. Re Koll a Olso use Newo-Kylov mehos alog wh mul-g peoog o D a D oe empeaue Plaka ffuso. hey showe ha he fully mpl meho gave ease auay ove he usual sem-mpl meho whee opaes a hea apaes wee lagge. Smulaeously bulg o eale wok usg he ODE egao mehos of xelo Dubos a Rhoes ow Chag Gaza a Woowa apple hese ehques o D mul-goup ffuso. Mousseau Koll a Re exee he eale wok o wo empeaue Plaka ffuso. Reely ow Shumake a Woowa osee fully mpl vesus sem-mpl mehos whee abulae opaes ae use a exeal soues omg fom hemoulea fuso ae lue. hs lae ssue s mpoa as he soue em he aao equaos omg fom fuso soue ems s a vey sog fuo of empeaue a hee plaes a osa of me seps a auay. hee olusos wee ha a fully mpl meho a aheve moe auae soluos ha sem-mpl mehos a a os ompaable o sem-mpl mehos. ao he meho s sales vey well o paallel mahes. kowlegmes he auho wshes o hak oy Mezzaappa fo hs k vao o we a pape fo hs volume. Refeees. F. Gaza a G. Olso aspo Mehos: Coqueg he Seve Dmesoal Moua SCaLeS epo UCRL-JC-5.. D. Mhalas a. Mhalas Fouaos of Raao Hyoyams Oxfo Uvesy Pess 98.. G. Pomag he Equaos of Raao Hyoyams Pegamo Pess 97.. R. owes a J. Wlso Numeal Moelg pple Physs a sophyss Joes a ale Publshes 99.

40 8 5. J. Caso Raao Hyoyams Cambge Uvesy Pess 6. S. Chaasekha Raave asfe Dove Publaos H. Gem Pples of Plasma Speosopy Cambge Uvesy Pess M. oulos P. Fauhas a E. Pfee hemal Plasmas Pleum Pess R. Gelas a R. O. Phys C. J. Cao he asfe of Speal Le Raao Cambge Uvesy Pess Degl oe. F. Gaza JQSR J. Soe M. Noma a D. Mhalas sophys. J. Suppl S. ze a J. Meye-e-Veh he Physs of eal Cofeme Fuso Oxfo See Publaos. 5. elegaphe 6. G. Olso L. ue a M. Hall JQSR D. Levemoe a G. Pomag sophys. J E. Lve a. Glasse J. Comp. Phys ; Rhaso Feell a Log J. Comp. Phys ; F. Gaza J. Comp. Phys R. po pvae oes LLNL.. G. Pe J. Comp. Phys C. alw P. ow R. Falgou F. Gaza a J. Joes J. Comp. Phys C. Lu a J. Wlso Numeal sophyss Joes a ale Publshes 985; K. Eppley pvae oes 98; J. Moel E. Lase a M Maze JQSR 985;. Wslow J. Comp. Phys ; F. Gaza pvae oes.. D. Koll W. Re a G. Olso JQSR ; W. Re D. Koll a G. Olso J. Comp. Phys ; V. Mousseau D. Koll W. Re J. Comp. Phys xelo P. Dubos a C. Rhoes J. Comp. Phys ; P. bow. Chag F. Gaza C. Woowa eave Mehos Sef Compuao V 999; P. ow a C. Woowa SM J. S. Compu. 99 ; P. ow D. Shumake a C. Woowa Fully mpl Soluo of Lage-Sale No-Equlbum Raao Dffuso wh Hgh oe me egao pep. 5. J. Moel R. Robes a M. Shashkov J. Comp. Phys ; J. Moel M. Hall a M. Shashkov J. Comp. Phys D. Mlle pvae oes LLNL.

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