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>\ ASA1CR-?002-2 1 1 762[CASE Report Igo. 2002-28
Py Nonconforming Finite Element Metho l: for theSolution of Radiation Transport Problems
Ka h Seok KangICASE, Hampton, Virginia
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NASAfCR-2002^211762tE Report No. 2002-28
Pl Nonconforming Finite Element Method for theSolution of Radiation Transport Problems
Kab Seok KangICASE, Hampton, Virginia
ICASENASA Langley Research CenterHampton, Virginia
Operated by Universities Space Research Association
Ci Prcrued fix LxVk-)- Rocarch C merender C nnu u^ n AS 1.9'bdf•
August 2002
4vw'.-Ar trr $t fr'Y-.•n^
%A-SA Ccr :ir (cf "cSrs" WirnrY.rr 4('ASIA %st ,ar1 Trrlm, ^I 1^Fw-.at . n per. u.*rflS% i21 S•,aedard Drvc 12V i,r Ro,a R"a..H"% -T %ID1t q "±.1)IU 4t•.g6rtk VA -2 171I WO Alf 4, i9fi1-,)j W46 54
P 1qON4CO%7O1tt%IT%,G M. r".. CLrW..NT METHOD FOR THE SOLUTION OFRADIATION TRANSPORT PROBLEMS
Pq
KAU SE rik KANG•
A vilraet. The ettaulati-m if radiation trantpnn in the optically ihiclt Aux-hmlced diff-aiirm rxrirrw hatbeen DUP rn m+dtimuerWcomplex geometr y. the radiation trans;i in systrns mu +t nkrn be mired Cri unstructured grids. In this paper.sr invowigatr the, behavior and the befwfitt of the unstructured T . noatonforming finitt element metbud.nhict.'; ,,roven to be l ndble , w-pi P"iw or, related transport pto kips. in s:!vin5 unstrr-tl y mitiilcitr>anlinezr radiataxt ebffission pr-,!erru uuig Newton and Picard hn-wiWinn rnribnel+.
Key words. itnnronforminit finite elements, radiation !rannwrt. blexact Nesion lieu-aadation multigridprey onditiming
Sub*1 ekwWation. Applied and Numer.cal Mathematics
1. Introduction. RAItation transpire in astrophywdl phenomena and inertial rrmfinorrrnt fisirrn isofun rnodek-d using a diffui,.un appruaimslwun 112 17 It. 2fi. 21. 22. 24J. 11,1" the r"(P.)n field ii not Inthermodyni:mic ryutlibrium with ttir mare-rittl a rn-rpib-d set of time dependent diffusion equations is used to.kiarnbe tnergry tra..-rPert. Tiesp oTiatir.ry are iti0Jy ranlitiear atvf exhib it multiple tine and space staler
lmltlicit init.yraiion methods are desired to o+trr-x:ie time step r+strwixisN:Ar _.rmira fimt p-element mert►ods ha-.e prvvm (iexibl! , anol effective u ,ri.-xnpr+wiw Aidd firm
problems turh w Suit, and N:vier Strir+ eri iiatione [i0. 111 In the I ii nortcon'cmr.ingrrwthexf. the der wrs of freedom lie rri mrdpointt of edges Thetefore. the number of connemiorr of degreesrf fre+doen .itk wart r0wroi at trait four (four at interior edger and two at boundary edgn+l whirf- is thraaae s'anbt. rf connections of degrees of free;rm. in structured finite dtfierence methuds in amtraw,is the iii confonni::g metbrA tlr a ,.t-nber of erm-Iertimt rl degree} of free rknn is At Imut Lnrt vrrefd at
bousdazy pair-f it and depends tb+ triangulat y-iri and pcxitirxi of {rein Tlw ra,mt.- of ronnertixis ofdMrr-ex r-f fre«irxt. Are,rmn-er the neirr w cf nrxirerr, mrrwa of irr norated rratri--es and pla y s an tssrntial
role hi pertrirmanre of parallel irnpierrientvions because of the cnmro tnkatiuri required art ktnwl upr•ratirxitlike metri-e-wrtor -nultipbeatwn f'i norcorticrming methxis sererart rnmrirtF that hA ye a rim+tan+ snuff
ri mb,s 4 mincery -verses for each row anti the.xfore ha':r snore AAvWASe+ in parallel ImplerrientPt.,r and
Twin rr mvo.Beraase mm.v rrr,li-war elkptir pimbierry ire :eil solved by uafr,rrruny, fin.te elrTnr •nt rrie,th , irix next•
ranfrxnung metlictds are std! rant"surd pirobierns Habit= ttuncose'--mirw me t hestis may I rwpl ye festwes
of soiutr t:, r-( nobler-r ar prt" n s nit wv it rrtrrrr. nerd in rrvt4irming methr4t In !hie research, a nonron-
(arrwstg nwthriqu tbrmn u- restive vet. sharp !han ger of en*rxies on het e rngenerms 4,anabis The results
are +tire aiwAm to the eolutioeu d the finite v-Aurne nw ilxi with an edge-brw'd flu% I' -ter ;191
To salve wmiir.nr protierrr, one tuualiy rrrploye Im•arizatko vebwgtt" %fait± M-MirlerF use Pxard
and W 4W.rti'i lnfths.ci+ u, h-var.ee Picard'- nvth-r1 .. ratty rt, inderstatA and imple-mritt, b',s! ronyerg?s
I WASE. t/ai Stop MC. SAGA l w.(Uy Rmewj. kends'. N%m;rt-. VA [)t4.-J,V9 email: tikI1070iraw.e,dti I-
rWarrt wet pa^elal'r rip2,.xerd y)' rpr Nat/xs A+nnyt.!f-a err, i;,&- Un..wttatru eider NASA Nq NAe -
9 ,1A saOt t it• ai. • I-.r we it. m ywtr v Ir'+4T N t!A 7.4rri" R~&rra Cmit, Ka.nytm VA /Yia:•J.vv Tate work ww
L4 party! r Rppor6+d tri IwWr „ t^ amwig h.ucu ,n 'KU5EF,.
slowly. Newton a sart)tod hat a seccad-(order conv"graare rata iAr rryiimvii the Jarn5tan of the originalsaulinear rystam In many nrwilinear problerns an inexact \rwim method works "D. with )eaa storageOW operatim count expanse 11(. to this paper. we study the hehavfra of their three methods on a modelradiatioin transport problem.
herai N' the ity iripm gmKated from mine Iinmrizatirm of the nattlinear probkn, it usually ncas .manrrme.wo wtw prsr•ondhionod GUIRES (21' As a preeonditioner, we consider multigrid. Multigrid reprrtrnts annapmunt advance in &!4orittmic efficiency for the sdution d loge yroblcm y (2. S. 4. 14. 19, 25)
To use mult'grtd, we med to define intergrid tramfer operata s betmeee t onmifortning R nitedi txnneutspaces Due to the ran-ntwadxi s- d nrinrrx fr,mring rpaotr. there ns no riat Ural intergrtd t: anefpr operator. Inprevkow studies of the n nc nitorrrtmag muitignd method. tier :nc rare ,ahiv of twin adjarmt e! srnents is used toget the miteerpdated valur sr. a ntrir Nary-onforming rnJtigr:d wi,h this interg: lei trnntder operator is a Roodsolver fur linear invemw and vmr.e nonlinear iysti . mt w•:th tn"th tu,nlinrar rref}irtontf (1. S, 6, 9. 13, 161.
Rosi'r'vw th" "rrMd tmoff- oprrattw duce nr,t twet.&rve pneitirity e4fmrirtintis, which is an etseneial part d;r;narwn transport part,;-ms iur • .rtse eoe o! and temperat wo are always posAive. Iherefrxr some rrxd:neai-prohimis with durontirnmv rnellkientit bmirv! rr ttrirrtt;s nn and rapidly changing soluti•xis.tike the radiatiewi traropnrt yr44m.., rains nr tit n irdrghd tmnsfei r4erator'xe.7true the ctmirse :ctrlaaprmiwwim otAained fir n- the fine appnnumttion dries not uthfv rolutiort c,,; 6s, wk; c.,ew caurimgenerate the coarse k •:el sywtnr : s x scivv the marse level proble=9 ( 15). lo overr•omr thew diffu• , Jt;es. weease a new mind simple krorgr4. 9 transfer o;4.-tator that pri.-n" posntn ity and w4ves the a wv m•ntiorrdprob{an However mufrn •nd wtth this imergr:d transler operate, is slower than with the prey -Cris operator
Tbnv§ore. we use the Wmple irdergrid transfer operate to derive tiara: level srrt••mt and the avPraRe valueUK@-grid operator to solve the linear systetcr
Tt,e rest r,f the paper is rrgan . red as b ili, In on-ti-min 2. rr de"be a mrndel rarl :ar. ,,n transportAM its P, nnnr . xnf,nrrn .r,R dwr :^tixwi ,wn Tr rert.i.rn 3. wY consider a disrretez abort ie dine. derive thehnear- :aatwm. by Piewd add rrr•hr . and describe the inexad Nimton meriwd It,, section 4 wr
Aeorribe C`d1iE.S and the timconforming riming.-id r.ee mdttloner Numeric :J vsperimentr
We given in teCtica N.
2. Rwilatiaa tranapewt model stun P turar ,infomiing disc rrtiration. I'rider the as-oumptkx, 'A
an `rpts, -ally thick rr,rnln l:n i5kv ,ri 'rNa, fair path .f pEn, or: r, a first pr.rripl-tt slat emtfit :if rad'at:on transprrt
reduna to the ra'Itatif n diffisiin hmit A ^artirtilar id ea lir.vi dir .enr`.nnlew (owns of rt^ih a nyrtem. knrm-r,
a1 thv 'Ir model ran be sir,ur% rte
8E c. , D.VE T" E). (2.2.1;At
Mc in, —, 7, = o,IT'_F.... (22.2)
Witt.
IT F.Sa e
^^EI' Aral Dr To - OCT, -3)rm
Here. E'z.f) rn4eserrs the ptvtrnt eno!cy. Tfs.t, if the mat t -rim ] teruperature a s i+ ate eyarity and
a Is the watertal cranrfurtivity Iii the n-xi cq,ulibrimr car•. Ow rr ,r,l Flea: r Nlrre ?*fins or the tight-ham
side ate mumcro and Sinew. the transfer 1 rortf, &tw, .^n the radnarow, fold aid material temprtature
Add_"al wA, lirit•arttirs ar e R , v rrata! }n the panne liar tram of the difFutnn ereflkierala. Wh.ct: are hmncu.r,.
d the• F_ and T fnel,ls In parts uw tin erwin l,ffjww r . Ifriont, Dr!T, EI oretalnr the terra CtI whisk
r4 rc
Fir.. I. Dimis.st of mulel proWerr,
rt+lr!n to -1r xrar}j, rt c (E. T,k hen if icier, u u aniIleui WNW of erisurin5 physir illy " 'anirrgful c ncrrypropsiption speeds fie. no facia- than the speed d light). the atomk number r is a material res ►ff,riant,sad rhlat It may tK NgM vamble it is only a fur.maur 4 positicxt ( i.c. z = J(,, y) to two dimensional.
The tt•a model problem + cunsidmd in this study er• iakon frrrrm 110) and depicted in Figure 1. We*-nfirfer a trait #rpuue dorrimn of similar nuterial with aurmic rrsmber z = 1 and a unit square dom. iii of
two dissimilar material. nhm the outer region corAsins material sith an atornir nunitxr of z = I and theinner region ?1J3 < r < 2,'3. 1;3 < Y < 2,+3i roothim, mat.rirl k iiii an wnnur wimbv. of z - 10. The topand brAtt in, wa!Is are in► ilarrd and inlet and out lot houn rlariee are sperifird using mixed ( Robin) boundaryrrarditkins. ar shcnr, in the figure For convenience. we rep mrn rno boon sin • r = 0 0 c y < 1 and r e 1.
0 < y < 1 by Cs. and citherwise by 1 1. Ilan lire boundary condition of the pruWem is
an r,
of r..hfi—r,.
on bfibn
ahr•r• n A 11W nra rw,twa: d nc: n.nl trr+ rrr of the urarndar•
Lquatkd f2 2 11 and (2 2 2! (+Tm a ss-s r en, c r caupWA norilin ear partia: define nt:al eariatvrr, pt shich
roust !o ducretinw , , spruce and ur g e In 'his v trey wr mrr,(Ier a #,seer+izan,m to #pace and will consider
is disrretiunon is t;rrw in Jh next anion.The saritdxW krir. of 12.2 11 and 1.2 2.2; can be writter. a follnvs Fim T T) c (H' 111111'' b T'
surf ther
-
/8Et.d^ • / D•C£ Cwf
1 I F.,.dn
I M r, 2
1 (Jrosl4- E w,-/ 2gudn =4.
I. r.
rj isle - / D,7T CAr - f as I T, ` - Elva! - 4. (L 1.5
r. t.
(-e a7 (I., V" a (H I In)' V'd fr.- an t E 'Q fmax
!Ate dlrretiar fl by using a tnar , gulr vd containing ,.dtsa, fiIwrwn in Pigi-r 2 The rnel i s 4errra ted by
cunnecting I ti►r mrd {rAws of the •Agra of tie• tr ,angleti fn,in the rr,usrrxr dirret iWicin T; . whkh rrn•TJns
Ft0 2 L fametrafe r of A' fr oth
adg►t atarl ronfr^t to the mato-W interface brnsndarhea in 94 A a way that — tnanRle odroo cram thutv w,dwy Let h t and 'T• , it Tr. k-r : w 1. .. , J. be ip .n whirr T, a a partition ri 0 into trundrs and htis the murmum d:araew of the eknwnts c.f 7 -) -
Defmr !hc Pt -ncrtrrmfmmtng fitsit^ ► lematr: Frmr- n
{v E L ?+A) : r;K is hn•ar fo' all K E T,
is crxttimx.ut as C.v widpmr im 4 in • ior •dp.o).
Then tke noaeonfora.Lna fin i te ekmmt dirr!tirs±wn of (2 2 . 1) awl (2 2 6) can *w %-ritten „s . Find
tE..Too E Wr (U. tt.AXj,3 such tha,
f)F..u1r^ D.U & .T XEa Cadt + 1F► ^ufn
8t -."2 12261Ia^(Tt ► "Tti ' - F_a r - f. 2g,.dtt - 9.
fn ,
1,t ,4r • 1 U,'E,.T, rT, vady ' o.(Tio , T. - F.,-4. = t1. 1 2'1
&x all (a. t ; E V" arvl fur aA t r I, tm aihis afxnT tquatians r.r Ire-i^.ra Oho , nu/ra+hrM in /y_n, w ► Wr a thrt• grint q,:adrature ruir un earf,
tr"gk m Tr Bersusr , Lt• fxarcta w4mos Ito ,frl7ee+ n( (her-rfom are defined and the quatirrtur • prints of
tr,ar,& art. Ox sarrw we tar+ Pauly crrnp,,te ifs ito pirat irm 'm" A trhang5 end
fr U!r,l^tiI Mr*tsdr = —3w
for W basis hw-s at ♦, of V and K E T; Aiw, beta sv Cu t- A pw",ar ennositni rm each triana:e in ft
fn if 6 E V. w+ rnteputr Cu' fr+y-1 u: P. ,•a.0
1. Tirrw intepal" and nonlinear iteration. In tla; aediUi, wr: cotMidef n dwretiration in time
aril three WM11-sar he r atrons ii . \awtun Picard o-vi ,rwYart %ewl ,as itrr„rwr.
4
ilr trrrw dewrivitres aK dwrre4trrc) w Mtt urdn Gwkawel dtlMrre" shh himping of Ju nrAn matrix,iwlr119 v, an I PIWIf /rt"w wh>:rl, raqurrer the r,lufinn of a dmlhrar prob:em as each time step. Thisappmech is Arst-order wwate in tone. and is c.hom mnvl y for ronveniencr. since the principal 0b)-diveis the study of the solution o( th., nonlinear system Higher cider tempos! dixretirations are dc irwmttntedto be tsvrth while in (IN,
To siNve the nonlinear problem ^2 2.6) and i"2 2.7) we consider the Pic ard lincariisuorr the-tboei and theN euton I:rwar tr.at kic rnethod. In both ►netlands. or cowl to solve linear systetrs to w corrections at earhnOl MOAt ttelattnn ate;..
The fully nnOnt Plrnrd itrwariratmn nwhr.d eeparmeo the operators into :inear parts and non:i.-warparts and a i noWirwat parts are e\aluatal at the previous nonlinear iteration lewl. k - 1. This reeult y tothe "losing s0-stem of equari-trip
I_
/
F —F— udr+ l D;' - 'C^'s •Gtdz- 1 E: ► udo
u 3t a r, 2 (3.3.1)ww.• t 1i7A'' t sT'' - ri;.A''i'1tdz - / 2yude - 0.
n t.
fa It • r -:T2T" Tvd,
u '► ' n
f ews -. (!r. t)ss_r4.a)fdr
6,r 49 's,a) t V j. 8eraum ; 33 lr and ' 33.2 are Imear sy%fe ru in ;E^'',TA s ', w ran easily atln,larethen Jacobian
To Kei the ocievrtums (6E dT) in the Pfear:l llrthcd at l,wr•I k, we tud.e the `nllowmix haPu sys!ems
dFiudr+ D;' 'Cd£ Culx- dEudo
It, Jt ^i ^r- 2 (33.3'._loa;wr((T,°'"I)'dT-d£udz-F;a
^ T'.dl+1u D;' - "VJT Ccdi
1 o; `((T;.a t ) s tiT - dF.,dz a pro s'tla;.
G,f all (a.0 t t'j •here
Fes. ^„ r
1 t• ^.'s 3 ;
-I I E^'ud-1 (?.'r,Ts.a-E^')wls+I 2yu,Lr
r, 2 o r.
^, ' ` ' s "!
TI . Jt 1 r1;..r•T^.• C'aer
n (35Rr
u
For the fud; urplirk Newton lirwariration tw •thod it s suss rl,st m . rt (,,fj,p!w;kte4 to rion,pette the
Jaoblan a! apyroxlenate WAtiewr, ;trims ':o Ken it,, Jw„lean ur have to r;Jrulars the der , vatftes of the
nsten, sub r..sf,M t tr, ;0, t',) 6,r &]l I W 4 funrtfr,n . rn T j x L j.
As tlr rm* cl tig-molatimm with respect w JA, 9 rj. to Rot this rn►n,rtintte (4E, ffP In %Mlcm tMothnrt r keel k. tae '''leer tM jnM)wing hnetar 4atrma
I!—'W&+ f
n:^ "'VE reds + / D; /EVE,",' ' Cud'n it
! n n
I ^ n;T 'aTCE;''-' - char + f ^MudoI !r. (3.3.7)
u
E,.-t1 - 3-^^ --
T,.. t 1 6T d:u + s"' - ^dEudr = f'"'^"'lu;./
f Jt t dr -
rD"' CAT Cudt / 17i ; -'w'er'e -'
nct dX
+ o"/ E,..-, \
1 - 3 ^ - 6Tcds -^ n^ .k 'eF., pdr :,j1 n;338;
F-T a-' (v),n nfrx an (r, r) E N a'}rro
r,' • _ {ce; '^ lD"' aV_r •+—
D".OrL' eta
r'—iT 1'
D;f=R6T;•^,h
whore oer'•
*± ran tx rani; r. el ,.rt.d ,n, .v,.} -istripjas in Tj.
Aher ItneArttattcm rp have to tdw the hnea! aluernt
1F. f ' l^• dT) I-A-t 1 (3.3.9;
T"" 1
i,c event /teP where V r a larrthtar, whirh N rtrn&',Mit d by Plemd 't Wthod of Newlon't methc.dIn a.:h►r nwh wi . ntwi kr ►'base»" tit, r oneml .M tttep : IKttlth (J ,.he►e
(T.- ') (^_- ) ' (AT)0
In th,r, n ady, se rootrr.l Cho ate, ienirth h! 'Impl y halving o soul thr :mdual of ttr uprlatcrl W"Iltu.A ulr.r than tho pro-.1nui residual In this control, we roatetiarrr tall to st, a W'JPt r rip k•uRth. +r; we nap at r.fixed step length and ;,+Tft,tm I he ne:t nonlinear Ivratirm It the num v n r 4 fa-' „rem e n nr dr a fixed of - t>rr.tber: we go tc Bert tino type t»' using t!r !ro t appti-smatim, akwh hay the analieat oonhr3ar r ►ndual
k t.MAR ►. 3 1 The Nevvon ntO&-1 Ae.. IM1,Mprotr nilt a •er-md order contrrVenre for nonimear p,,4-lern' an! the Pteard rfrtl.ol 160L only a fir ,i or&, rmm ,•rWv;ri Howetrr th e "Pultrnp hnrnr prvUr too the
l-trard 7KtA4 r1 fn, rr daily ordt ed b,ott tlw t of the .letrtnn mr thvf Genus, the Ptro tnt-tAod b-tit Oht
conaft hott ter'b L drMrs or4 tr. r'f F)I
T-, tm•rrr»v Ove P&sonn if it,e Npv"rt. rnet ittd . we ran use M u"art N'ew'tr,tn trw , ittd [A W-son
the Nnyvm neratrm k 'fa:" tom come o rgenco (Ir the rntdua! is 1a141 • 1 ilrrr IF ne, roaann err Bran the
bear trrom acrwatdy. However. when the New-t,t. I+rtat tnr, is 'r'ryr" , flit rr4 ,dual is "small - i the
crKtrergrnrr rate of Nertart ' s raethod to tt51 th vol l,ki tit, the arrurary of th e linear en!uttrm Tj adlusr Ud,
aa,gtnt. 1 WO-k drMir In the- [:near 04%c 'r.H a rrm ye • rKwo k,l►rar rvi t'e offplc y an tromwt \rat, n n.ott'rtrl
Ln a it emit Nvirtm appraarA, thr rxrwerp" mkk-ia kr tl,e linear su1WWI* proprwtiawal to tho remualio • Ire ftDa hear tterstim in e7uatK,rr firm tw, 1s
•here +,t : 1.0 r 10- 2 r the vary used in this stud, unlem othrrwL%e rmotpo, Wo n rtr e that (13, shcmt howW adWiveb (elect -n to recover esvrnptuttcall y full earn(& ordrr mnvvTgenre.
4. PGMRPS and utaltigrid preconditioning. In this w-tion xo rnplain PGNIRFS, which ie A
crimbination cf a Krylov • barted linear iterativ7 method. and multillrid. which is well known as a pvcmduiprrcoeiditionw, as wad i as a erruahk ttolvrr rven in unweeleratrd kxm for n.any pro44vn4.
GORES '131 is a cash limoun enlvr kx non-Pensitian probie-ns In practice. G%IRFS an tw restartedafter T setups. where in is arrtn o firer; integer parameter, to eav p sGxags by acrilvittnr r ren►ra:l) Ws rapidconv"Spece.
Wo, &►scribe the -mitart,1 PGNRI;S far Thong
Arz - A
(4.4.1)
witb prmnr,.ht„omn6 uk"Ir. Rr
PGM!"..94m) Algorithm 4.1.( 1) Scar C%v~ to and n ,ml,ute r„ = RJ rb - A iral. i - irt'12 and vt - ral3
(2 i It-rate : Pw a 1.. . m dr.Compute it = Rr A i r r
Frn , - l... dn:
, - to.,tie)
b = 4 - ), rV,
Fiat”,C,omp,.,• Ar. i - {^' p and t r.i = r^^,N•rLnddrr
(J) rr,rrr, th.. approximation wutton:netn7e V;. -a (t.,• ,W..*.
R,. — 0I h<t-i.i lc7 <.,
and pit r = to + t„y_, %I-it i,- rn,r.,m,rn ilr; - if-1,11. y
4 4) RJ^rart:Cr,mpute e* . Ri (i1- Air.,), if sattsfi-d thn n,,p6W crimpute rp 5... 3 - fir, I anri t is ? .w i.9 and Ku it, "71
AmAdt itt-s tion mUrtrurtr an O"1104"etla. barb r,( it, ,, 1,, of pt""ClI,twt ,erl Kty !rn .uhspdce
Span((, Bi A i ro .(B;Ar'.-.r.)
It uses a modthod Gram - Sek-At ;(oxen. in *r6i, h tlr r,w vrrtn• ,o he nrthOKc.nalired is obmnrrl frm,
the prmrA p N st trR M the pr,.cesb Ali rcvidual vvrr rs ar,d they ( nrrrmr tl.ar are rov .pu, ed by tla alr eithn.
owr prprmd to the precirtw lrmed rroat als mode. tr, s A,4 -- Ajs,s}, iw pad a( the mional Impt^et+wt-duased) teMeW d - Ajs.. In addtttaa, there is no Pain arreas to oho" anpw radltkoiwd restdsals, un.wthry we txur.pestr+d eaylieith So %*- rnunttor thew prrrm4 . tinrtM rewdtsAlr aj stop PGMRM Iteralkin tosohe pwear prublrm
hut. we caat+idet 'Muhigrid PrerrtnditionrT Bi.To define a muhigrid method, we need w define I nt"grid transfer uprratot. etw~ nrrtrnuf 'rrtning
finite tberr.@m sWwa. Due to the of rwmmnfnr ing tep ar. ,. there to not a natural IrWgridtrans&r operator In ptro yloas etudtes of nooconfcxtrtrg multigrid rrv4hcN1! I ; 6 91. aye-aRc Val to tt( two&d)amr r dement!t are used to set the tdw 0( a wj-& A nonrorrfrt • tit step, mvitiorid mrthrwl with this intw&ndeta *r ngerstor is a good sown t f it brwsr "onns and tome noolkwar systems that have sw,,o)th twmhnearcorgrimts
70 on tiar cow" Irv!! appffuzi na r 111. ►ar sy Nero, fri 0 J 0). so tried roars! .e%vl appraxinst tune of^w t r a t ) and J A - . T^ +) If the ajKw',una+e 4r wn (E.' a '.T, t ) vvkt rapidly in space.
tyr tttt vMW MAO-V 4v► I Rp( rrrr'rlitewont If ! r a "' . T; s ' 1 may have roVtive tralun Arwerr-,vt, re, fared to he prritrw for • iw ct,mput%t ; on of D'-* EltSwr we can wx Rue s- r thr rrarrta burl s)stemtnr the may bartwo nearly nr,gubv, m4king it hard to anhrt the cream 1r%A pro rlerrn
To averru-ne tlw dAw.rltwo. wo tsse a rrw and timple Irr rr-rVid tranrirr nperatar caArd tiw rrevolismo
h&AAd iyvemc d trandor operator, w•htrh preserves onl y pterrwise trmstant fumt.rms 11;) It is well known
that. to pr. a pod canwrgeare (actor to mul-igrid d"thrns aampit! tra-tsfer c4wrarms sit7uM preserve
hiRttrr orbs functiotw 1 19 Tt.PWwe th► mulupid rnilsorl . nth thu tnte!grtA trAnefe- cTwator converges
skrsdy ct,mpared to avecarte valteo tuergrid 4peratoor w raw. Itrwar ,.stw,v Hnwrver presrrvatton of pos-,u.hy Of Widal twiues of the firWit to naval Sc, wr IV* the rovoftupe-harr►d mtowgrid tra rofer opetatur
w crbetata the . r„orer, 1~4 9)wrnr And the average val+se : nterend oprstor to interpolate the tsdurr,s. betuvwn byrls IroartN t0 Ry e anr! l+ rr to roars•' ••hat roirtnR the hase w systems In Picard ru(.rhud to New trtn's
root hnA
Lot A Wj ; s ftj)'. J kr the d, srrm umstw, njw-atrc on k,+I and 1, '1 ! - t , r -' (L^,r
1 . 1 J br ttv rrsarse tr, fir:.+ tntrrrnd tram rprrsttjr ASK, w* deftr.e thrr Arrw-usroarsr ira trod
rrar.der npnraar P, -I !ii) t -• W -t )r by
w it a t, , art. E W) Ifs F r" '
Ftsully. Irt R, : rI r n (/; ' (,t t J is t!a iufir ww,rAhmr #,,^Atom. Ire pr :rrwte the
ad)aht of R; wnth t"tO r t to the moor pr edtsr • and h•firr
r R; 7 rrM
HT t eti en
Tt.1kw,:r .F (Sr, the srshiCrid operator B.. (1',) 2 -0 (L t r is rle!Sn►A re,urst. .ly r (0,MP
Mrltitrid Al rerithm 4.2 1r 1 1 J % J tad p br a P—OW rareRrf Set B, - A t ' Ar.vru • that
R. t hit, irrn (lvfit wA ar.i dArw B y Ia p '. I,)' by
l 4tis•0vAgc.•(a, r"rwr' frw I- I.. .,mat h.
a3 rn r' , R&.M1.:tp_Air' t
(11 Diifir.- 1"0, w T**It t 1,4e, wbt- r fu i w 1. . .p is d#Otrwl Iry
(41 Utfint V for I r W) + 1. 2m )1 by
111 _ 0-1 - Rt1+ taU;1 lp _ A, ll- t ^,.
(11 Set H' t or yowl
1s) %1WtlPd algontha, 1 1 to )) it k the r.11n.o.r I pre- lord IteTAhrn- and can van'ar A fum -IM of ) If p n 1. we have a l'-cock multtRrvl alcurithm if p = 2, w, have it W ycla mutt rldalgt,tstha, Otlw ve -tsiuro of n.0 togrid algorithm. n i • t.rmt pm a pnr'- *r ewnihirtl itttativt can be an.%Ivtrdwrr.ilarl. A vxns 'tlt 1' rvrie rt I-.g,wJ algnrlthr1 u tF,af (ar whirA the numDet of sr oothing ml) I L•xzentnto-,WWWWtally u ) Arrr"LWA ( , p s I and in - If i;.
RIU,aK 4 1. Otte eon ,am thr muWgnd dtonMvn to io1.Y the ty*terns at a frre•otandtnq arratrttrtwedod Cntaur. one am$ I -eyrk and 14 -vie mwlrrgnd alpn ►hmr fn t o.ls end ant ones V -f r1r andverwW, 1'-.yrk rsuiss9od rnrin gd sA prrror.Arttow-n o, Xrytnr type rr - • h(.1n +u,A a- Pr G. Mraasr..hen .4,u "wrin , p seat r J,find, tj.e V (yelr 'wwl o wrtd cytrwo-r R, is v..ymmo t-r pnwrite drfrtrtr operator en
(Vt )s t bw tAe if' ryrlr n..'rtp-,d ore--tt w tq not to p""slly jJj Va►p re"wrr errs alotr that rnn • rjWOvnof SA' - ryrlr atulfW%1 for U., wnvnrsfnr"Ifig and ronfm-mrng •:aces and t'•cyek onaht9nd fur the coafurnungerK ere pail prrtnstuhtton • rr jl, f. S. 6, 9, 14, 16 ll r In this priblem t oe nor V eyrie mult:9nd rnrthnd
e. a prnoaklsower of G'•1RE5
S. Algorit hm pariotrwatacw mod rwtults. in this w"sun we stud y tir'erforsawwN d U'rPirud. and saga , • %owusn rnehr4o on P dR,coofrrmlrg fins, r elm.rf:t n,t!vx1 nn Iwo mr.Ani prottlemswith thr twill drRrrrrwr beviomm tow pt ykIrrno tr,ng hnntnrrrrits In the Iwo exarrrp :r. we up The ,areatr.anr"t ms. /anti) 122, truth. 19296 rdXrf and 6497 vertim Bresuw nodes ut (in rrtrip,tnt. ifedges in a 1, nrrscoufr ,rining roothod. tw ttumber o' of freedatn of this pr4k Tii to 34M
few problem 1. w r rsojer a hur ,ogntr-rss rme .rid wnh alrm.y n.unbrr r = I And n - G 01 t,n t:e
wtr;k doa.asr. The u .ival w,a•1nu,rst we F' UO r io1 anti r - (r it 3, tar probinn is run aut toume t M 30 Mori rtrtrJtr , t-wr (Lrtarewtr tt,1►ra,t+ wsthin a Itmr *tr( a A.}inwi as ' Vub )Nt <_ 10 A 10' 6.r^rnbltrn I Rr ells) wt*} sEVrral tare *trlA n( 0 nftl 0 002. 0 0r1.,. aryl 0 01.
In Fxu -o S. wo p:or the rowun of ternprtaturt T r f = 1 0 . 20 J 0 Table 1 affpam itnrar "we
ro.varnrneats asvl rvmllw- to aitim rhos) d dope- lt tar mmtrva: cor,v#f ,-rrrr trttt v ;. ;t ,A N.wttat
rurehod Picud awtbod and Jim A ct Semou row!v d at torte t a 1 UFigure 7 elbow* tJua trmtrran r/. tertpnal , r p"4 atatr Itarallrl to Ihr :nls f hn^tnela: v aryl reptoAwo- on
an :mmttrurtuev ,d grA it., y-t t;r ptv.a of tit r.nr 4.rr nst-s,Al rat Twt.I. I ants FiKurr / Area that Nrrt , m s
%U*N- ,d it vs,v 4&-s w -ompwvd to r.eard s tswOvA ar.A sptptli; afore dfu;rnt ex,n.;wrtd eft tl.. ur tort
\.vets* mot" m torn,* of nexiUr at lonsf:nrto per urw stop
I(raact Nrww-+ W-wis rrw-r rrwJtrwa:.imaine" Ir rnwpanvn u, Ntwttn'* nwtlxrl. bu t has tM hest
;v Arc rums , overall brtatty. • Ino rnrtt. d rmvlo the rrnkF-w rr:r.Ihtr of lirrar tVratvAu A6 "ImUr 1 Afro 0
I" . tv rn;mtrrr of !peas Irnattnr;a n, oath non: ,trar sttratkn of tie I'Kard xwo hod it amalk: than t hat of
' loo \oohs) N104 'sod 'Moo ►risen that tho iinsty f-(.er. PirzrA s mNhcA are m— t twili rtl"A 1h4n
the hrou ovet ma from Newt 7n rr m hr,d
In Tank 2. we- report the arrurari , As a funs " f „n'r ot.yt b, tie 1' on (.f ito &.fitter, Shirt.
u dr6tad r Iv - w.._ s inter, .t.n, r tttAirrrl ant•,/ A torn. step 0 OKI Thin result 4'to 0 that the
L s trrw u, arise- » f ri , ord.+
0
(at I V U)
I I : '.O
{r I I - I-)
Fv. I Cnawvur of Tempe •o?wv, of Jim( lem 1
For "obim 2 M r• rr osdvr an ntre :ai wi , h atale'r ntiniho f ! a I 1 Inm4p !hr I1` is all•)
go) t1 Outr.l0 at uF, f1rife i W. .hailed W , I1ry111:n►4T (00%"KM. • ..Lreran \1 1 1,1+1 a um.• Mr1j
1.1` tO rFtUa ) b < 1(, 0 I3- 4 tit tr.l'o•^ IM a.n 'I lit , ,-in 14mo-
(rI FIR'J" G •e ON thr nrwr.y• rd letnye.or.lre Tat r s I G T n 30. 111 S.(t. Taile 1 c,anpafrt limar
..T requerwrie a noma rar hr,Mi mt, of d number u/ fa br it. In rrAwo t M r,m"i *not* I rAvran- t[Ure+
9 10 dt rmstraw flit LorAnw rgr.mopiv tahrrt it .i.h0 firwT-i Pk aM awl vwx&rf %. wl-n M.1hv.11.
at twon I.I6 f LL. ands w4hAe r w(r prapagrro. vap•ratu». ta;041r rha► R1 o.a; the fr.m . aryl 1.1 y. r In0 I..., 01101P the tUrl
/bl/w•rw wmte wa nm- As tr. ,rre IMr pAmwo ti.r tmi.poiaw tu0 sn," es. `IRurt h. tI 1(t ahoy
t1,at tG0r0 are many rmp hair!, rarlttaio to Rr the adutlxl r,/ dw ". Jirv< protAm-1 wbon Itv .dut .n
dw4r.9 r 4pt41iy ft + 13 1 - Y ; tul thr• .r W1 1`10.4 4,r tarp 10nph •tA t1`.! . tv n tFr o,l.,tr.s. a saw"h
(I 0 41,, Jr u,1` d tb► 1 hfe0 mt.-O-.
Tatits I AlgrWk-% pWoowwn•• as 7 tvnrtr" of trnr st'f for ►rtW,m ! ora trmt lion t,4 nJ.!!1
UrdwW. df A cf of r a • r.f • rot s are r/ ave 0 linwm4n• oaelm Lnew hn Sf ;wmr:r.
0 txll 3UIx1 6226 2 1 49851 166 b GNet-c•
i 0 002 1500 41:6 27 3erl7u 26.1; 91511" t" 0 005 6M2120 , 35 2k1-47 46.9 133
0.01 30u 1334 14 4 11996 73 7 165
- -- 0 001
1
3cr3 - 24935 ^e a_ ._
1e1ii7 `GO 4 -3Tirarf 0.002 WO 15319 103 126152 44.1 E.2
I Sfmbcd O w, I Gui 77e4 133 7116 " 1 16 9 9 04'r l 1f^1 53T1 1 77 •16 191 IS-1.0 _F.7
Inr>rart 0 '00 315-0 af.1A ' 2 A 246;' 9.5 3 3Newva 0'x12 Mat 4531 30 15626k 1n.6 3.5UKkd .t "ores Grit 2154 3 P 94. 16.6 4 4
0 )1 i 301 14.9) 4.0 .- - - T761 25.9 54
TAN. t 2 L t -error of t 3 -0
time -r.lift Vier-or,L Oil u 1x1'Yr4
U fxr2 4 01: !?6G Ox U ^14'lflj(1 1), O.,A6-6
Fiviro 6 fWrws thr ttr owbr.ion of the rvm,;nf,-mtng finite Plfmaro mrth-x) is tr: nn.iwtr t, fit,s3krvA J iruw volume arthod with edg-based flux inn:tm 126,.
In the aspect of parfnrmanco, the trhavior rA prohlem 2 is u-niAr to prublern I with tho a:re lttbm thatproC4err.: dooi " ru;t::re stop lengtf. rcvtrcl
To egtunate the ar-urs-v ae a :tlrrctkm ,d time suy • irf, sn rr pnn the L' a-: ',r Yt , hr sautirir: in Tallo4 homed xt an arrurstr urlutior, wtth df = 00001) Trnf rr•tetive L' error o(simulatkn+ with df : G fOnD.OVJ. 0 O rnrnpAre.! I the P-Offor of dr - G.G41 is p!oted m a furicti-,r, of erne in Figtuf 7. M-v result•fbem- that the L ; -errce in t;rry it first o-*t a• the tx;imur, g t f sin. ilar )n u'wi t ^ 3f, but rraAtiallideteriorates 77ds dnerixatko rr.ay be introduced by thf: tvx,iwar rvn ,.PfKP , ro firm atthin a limo sir,,
because the ar rumulat y f. 4 tiw mnhnesir a.n,^-gPrtt errr ,r w 11 ck: -r.matf other errors . spa ,-* ar.A rirr:f
d.w"Imitr,a emu ar trrnr soya irrr,ws I t wr uu• a fir &r nr:ntirw.ir rnnvftsrnfr role-anrr, thin err ranArlav thi+ drrf+rra • io" ti, kiew" t:rr
a. C,oririnsioris. u'f w l'.xd unit -x4y Implicit norb is r ar rariiatlrw. drf ui vo. prui ,lnna to a-1 urMtrur-
r,;rM P, -Vr fnforrrirW timiv e;ernerr rnrslv^i. Pt norx nfoming hniv eiern- m na"Im As rrr, Jvr• vfr y +harp
4angu a enrrgks jr.' th ► ivtetoRrtlrwas •mean. strnitarly t , rr, i:rs of riw Frwe vr,lumo method with asedgo . bwed Ou: ; wniter Tha irwsart \ tzton rnr• irMf Lai • 110 Iw•. t I.afframan ra -'Horan and PrfcrxditionM
GkIRES with rvxvorJortntnR muhlfrwl prrYOhdlab oi Pr tr, salvo li w'ar pr :star arks sell In P top:rlin
harnir, f mutrgrwl the rrwrrhrma - baa, l irwx7id tfm ,sfaf cpfravxs are useful tf . sr-v.- radratw'r rranq,rrr
c:
19 a 'c o e dt = G rA!
co° dt = ()()I ' t o0°s
'a
c
^ `Oo
r I 1 3 a 5 6 itetaticws(a^
°°^errc: ► .oc
cIl" t 'oe,
t ,,oc
moo
\
''`0O
oc^c
C2
C 111 15 26 15 :terwicnsrb.
O1
r^
G 1 Z 3 5 ^. . twatwro(r^
Fu. 4 C'on • nt • nr+ plot on ymNerr. J of tan., r = 10 la y ^'^rl •n .1/-IA,rd. /sJ Ptrar4 ','-IArd It/ lviacr -V-uton MclUi
pr rpble>rne btkraare th? pownit; grtw vir.R Ixoperty n twded
Ach y wbdsrnrnu. 7be Avttor % rwld like to thank U E. Keym ,./ Old Dt nomun UO o"011 Y few ha
VA IAMP Advice in the prepsraticA o! tid y peper
12
RF FEREV'E 5
(I; D k;P. ALtt AsC R VIII '►±H J,1Lhyrsd +tvthods fo- t17r,r,,mjQrmn4 firs+, ^l,ment mehrob SI.%Mlunwr Anal 19'19% , pp 9Yl WK
(Zj J BIN Amsar 1tlrsitiVA M-thuds. Pttrr.an ltxwlrtn. 1.19313, J BFLAMT%t[. J PA c rtAk , r.n J Xtr, n, rmetgat of ntJhgr:d tiporulmt u-!th nor.-netted spacer
or wcm-m&-rind gbsdrattr fo^t,s Math Cump 56:1991,. pp 1-31.Nl A BRAT .VUltlgnl tuh Mb-at3m" to ftuad dynsmtes- 1994 guide. in VKI Lccturr
Sr:ns. 'Nar 19*4, 1 76 ;,;j(Sl S An FnttF.R .a rt ,yhm^' wr rPti t ty ^id •retAod jr- P, non^onformmg finttr el,mmu. Math. Comp
52" 19A'i',. pp. t I ,
f6' S BRila\rlt rrmr-rar . -e of nonanforrtung milhgrid me pthods tnthott fill eMphc rrgulo •"r4. Math.Crxnp. pp. 21t-53,
(7j P X. BROWN AI;D C. S ^1'o:,'rA:tRD. Precondthontnp Qt-+Wears fo- Ftidly Impliva Rudtntwn NfJu-
non nvh Mote-►si Energy Trwu;r. SIAM Jouma: on Seirtr :flc Cmpulmg, 23 .,2001;. pp 409-115.[A1 X C CAI A2SL D. E KLtLS. Nrr-.hn ,-.rly Prewridtttoned Inezact Krul'un .I1qun8MA. STAM J. Sri.
CcAnput. 24r 2 ,Xr7 t pp 153- 2{Xt[V Z. CKE►y 71w sraslyets of anforInd trans!rr optrotors and Msl(egrsd method, fcr nonrora/r+ntnq frut,
elements. ETNA StIWS; pp -, b-96101 S H Cuou. Anadyr , a and con-try%nce of s oaro:srnr med,od fr th, y.nr-;h.rd Stakes Prot7km, s:ath-
C%ay x(1999,. pp % 164.III M CV :t •7rre ,Y7 P -A RAV!ART 1'b,.4 • m,n2 and rwn ,ryn;o--nt,tfl 4r. •t rLr•rtcrtt m<tAod tortolr^ny
the statweas•y .^t ak^s egsatseme I R 1,M(, R- 3 -1973), pp 3-3'75[12) Lf D 'Avirr,. A nmthnror Et e,^r:wn hydrndynam
v.. J. (ju t. Spam. and RM Tray . 644 19951. pp. 655 669(T3: S C E:srt?TA- AAD H F WAL KF jt. Gk'.tlty mnt-,rrn-r tnrzae Anr.on ffwfh 14 91ANI J Op•i•
aJaation 40M). pp 393 422.(14: W. JJACKSUSH Mvlttjnd .'dtttode snd .4pyltrettov S;xinSr. r Va:tlag. & rltn G.rrnazr 19+55.15 K c- KA.-mu C'r+olaect•Yarrd wa-rg-sad trcna/^r ap^ra ror en PI r,"rnn/rmtrsg mul!tgrtd mrthcd.
ICASE R"rt No 2TYXI-15 %ASA/CR 2102 211G!a5 2'er_I:K f, S IAA%:. AAt' S. Y LEE Nis mtny,rd tre.ufr vpnalr an mt.$t^rrd mtthod fr f 1 "twn ^r,n!ormtn^
fir,stt ettrr, ,ni wwtAui AY Aw,: Ma • ir rrmum wA f omp"ztttion 100 . 1999;. pp 139 111)17 D- A KxOtL W J Rlnra Avn G. 1. OISOY. An tfftp !nt nonttr,^s- solubn mrtAad f t novyot
)berms, r2dtflt,o,t dltffssion .1 Qty-rt Srwr and RAd. Trano.. 67:1999" pp. 15 2ri
;18: D A K%r,t t W J RiDFIt ♦ kr, G L C;lfr.M Vonbnear wtrergen:r ate ttrar 6 sad t me sirp
coM►ed m art.• 7udtl- turn rsda.te7n 1%fu L J (rune Sear and Rat! Trans 704 TV t pr,. 25- 30
JIS D. J %JAVRtoLts N:lttgrsd techntgw1 fo- a rtetrsetvrtd meshes ,n %-Kl T-Ort..r► S^rwr VNI - LS 1495-02. %tV 1995.
(26', D. J NI At R : ► US An Aseetorwn r of Lane• l'rrsuA Yon-:na,r M+,k,gnd ,t!•t.'.ods for Crnet►vtt.rw!
Steal, Spirt-s ICASE Rt;rrxt t„ tJOI I: JOWILA] Of fnmtr1'4ttcs1a1 PLN-Otrs. 175 :.12 p1 . Y,2
lrr,(21 1, V A SlouSsfAr D A K%01 I. acvl 3S' J. Ral f- Phyerre 5,26 1 1 pn^unfit :,<ntnq And the
.tYttrtnn-Rrglor. m,rhLi/',- nor: ^raltln-im r ydta •wn d,Juston Joums. AG,a,J11- 1-noll Ph),ire.
:1
1 ft xa,', . pp 743 ?6;,
(n: W J Arntra. Ti A K %c,T t .,n G. L Ot frm A .4frk,q-vf Vetiton Kridot Me-1lu4 for Mull-matonit
F.qurh4naw. Rahehen II.fJwv.a Jrnrnal olydimrrastiovil M -i ire. 151199EI:. pp 164-19:
(23 Y SAAG .4-4G ILI it 5041ATZ, GURFS A Generahaed Mmmot Reatival Alporahm for'0ohmq
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