analysis of the sor iteration for the 9-point laplacian

7/30/2019 Analysis of the Sor Iteration for the 9-Point Laplacian

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Analysis of the SOR Iteration for the 9-Point Laplacian

Author(s): Loyce M. Adams, Randall J. Leveque and David M. YoungReviewed work(s):Source: SIAM Journal on Numerical Analysis, Vol. 25, No. 5 (Oct., 1988), pp. 1156-1180Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2157663 .

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SIAM J. NUMER. ANAL. ? 1988Societyfor ndustrial nd Applied MathematicsVol. 25, No. 5, October 1988 012

ANALYSIS OF THE SOR ITERATION FOR THE 9-POINT LAPLACIAN*LOYCE M. ADAMSt, RANDALL J. LEVEQUEt, AND DAVID M. YOUNG?

Thispaper s dedicated o Werner . Rheinboldt.Abstract. The SOR iteration for solving linear systems of equations depends upon an overrelaxationfactor t. A theoryfordetermining t was given by Young ("Iterative methods for solving partial differentialequations of elliptic types," Trans. Amer. Math. Soc., 76(1954), pp. 92-111) forconsistentlyordered matrices.Here we determine the optimal t for the 9-point stencil forthe model problem of Laplace's equation on asquare. We consider several orderings of the equations, including the natural rowwise and multicolororderings,all of which lead to nonconsistently ordered matrices, and find wo equivalence classes of orderingswith differentconvergence behavior and optimal wo's. We compare our results for the natural rowwiseordering to those of Garabedian ("Estimation of the relaxation factor for small mesh size," Math. Comp.,10 (1956), pp. 183-185) and explain why both results are, in a sense, correct, even though they differ.Wealso analyze a pseudo-SOR method for the model problem and show that it is not as effective s the SORmethods. Finally, we compare thepoint SOR methods to known results for ine SOR methods forthisproblem.Key words. SOR, consistently ordered, multicolor orderingAMS(MOS) subject classification. 65

1. Introduction.he SOR method successiveoverrelaxation)s a standard tera-tive methodfor olving inearsystems fequations,particularlyarge sparsesystemsarising rom artial ifferentialquations.Convergence f hemethod sgreatlyffectedbythechoice of overrelaxationarametero.A standardmodelproblem or nalyzingSOR isthe ystemfequations rising rom finite ifferenceiscretizationfLaplace'sequation on a rectanglewithzero boundarydata. The solutionof thisproblem sidentically ero; hence,the terates f SOR also representhe error t each step.Theconvergence ropertiesf SOR for hemodelproblem lso apply oPoisson'sequationwith eneralDirichlet oundary ata,sincetheerrorswill till atisfyhehomogeneousequation.The 5-point pproximationo the Laplacian is(1.1) Uj-l,k + Uj+l,k + Uj,k-1 + Uj,k+l-4Ujk =O , j, k= 1,2, ,N

Ujk=0, j=O,N ork=O,N,whereUjk approximates he solutionU(Xj, Yk) withxj = jh,Yk= kh, nd h= 1/N.Thisgivesa linearsystem f (N- 1)2 equations n (N- 1)2 unknowns. he exactform fthe matrix quation, nd theform hat heSOR iterationakes,dependson the orderin whichthe unknowns jk are arranged n the vectorof unknowns.Two standard

* Received by the editors October 28, 1986; accepted for publication (in revised form) July 14, 1987.This research was supported in part by NASA contract NAS1-18107 and AFOSR contract 85-0189 whilethe authors were in residence at the Institute forComputer Applications in Science and Engineering (ICASE)at NASA Langley Research Center.t University of Washington, Seattle, Washington 98195. The research of this author was supported inpart by U.S. Air Force Officeof Scientific Research grant 86-0154.t University of Washington, Seattle, Washington 98195. The research of this author was supported inpart by National Science Foundation grant DMS-8601363.? University of Texas, Austin, Texas 78713. The research of this author was supported in part byDepartment of Energy grant DE-A505-81ER10954, National Science Foundation grant MCS-821473, andU.S. Air Force Office of Scientific Research grant 85-0052.

1156


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9-POINT SOR ITERATION 1157orderings re thenaturalrowwise NR) ordering nd the Red-Black RB) ordering,in which he grid s colored n a checkerboard ashion nd the Red points re orderedbefore the Black points (by rows within ach color). For the model problem, heoptimalco and rate of convergence re the same for both of these orderings. hismodelproblemwas analyzed byFrankel 1950] and also by Young [1954], who gavea moregeneral heory fSOR for wide class of matrix quations nwhich he matrixis "consistently rdered."Another tandardmodel problem s the 9-point pproximation o the Laplacian:

4uj-,,k + 4Uj+l,k + 4Uj,k-1 + 4Uj,k+l + Ujl,k-1 + Ujl,k+l + Uj+l,k-1 + Uj+l,k+l - 2OUjk = 0,(1.2) j, k= 1,2,* N

Ujk=O, j=O,N ork=0,N.Again there are various waysto orderthe unknowns, ut none of these leads to aconsistently rderedmatrix nd so the theory f Young does not apply. Multicolororderings, imilar o the RB orderingmentioned bove but usually involving ourcolorsfor he 9-point tencil, re of particular nterest orparallel processing pplica-tions.RecentlyAdamsandJordan1986] have studied hisproblemna moregeneralcontext nd identified 2 distinct our-color rderings. hese can be grouped ntoequivalenceclasses that are known o have thesame convergence ehavior.For themodelproblem onsideredhere, heir heory educestheseto six classes of orderingsthatcould potentially ave differentonvergence ates, lthough he actual rate wasnotdetermined or ny class.In thispaper,we analyze four f these ix classes and show that hesefour lassescan be reduced to two. One class is shownto have thesame convergence ehavior sthe natural owwise rdering,uttheother lass is shown obe distinct ith differentoptimalcoand asymptotic onvergence ate. The eigenvectorsf the iterationmatrixaredeterminedor hree eparateorderingsndthecorrespondingigenvalues whichdetermine heconvergence ate)are found nterms fthe rootsofquartic quations.The optimalcoforsmall h is givenby an asymptotic xpansionabout h= 0, and isverified umerically.In ? 2 we use a separationof variables techniqueto determine he eigenvaluesand eigenvectorsor heNR ordering. he resulting uartic quation s used to derivetheexpansionfor heoptimalco.Our results or hisordering iffer rom hosegivenbyGarabedian 1956]. We explain whyboth results re, n a sense,correct.In ? 3 we discussthe variousmulticolor rderings. he maintechniquewe use isa change of variablesfromn,the iterationnumber, o v,the "data flowtime,"asdefined y Adams and Jordan 1986]. The factthatthis change of variablescan beused to simplifynd relate various SOR methods was observedby LeVeque andTrefethen1986]. Theypresent simpleFourier nalysis fSOR onthe5-point tenciland show theequivalenceof NR and RB usinga changeof variablesmotivated yGarabedian 1956] that s equivalent o thedata flow imes.In ? 4 we analyzea pseudo-SOR methodbased on a Red-Blackcoloring or he9-point tencil.Although his methodhas attractive eatures orparallel computers,we show that t is unsatisfactory,eingan order ofmagnitude lowerthan thetrueSOR methodswithoptimalco.Finally,n? 5,webrieflyiscuss ineSOR methods ndcompare heir onvergencerateswith hepointSOR methodsdiscussed n thispaper. We summarize esults orthe5-point nd 9-pointpointand line SOR methods n Fig. 5.2.


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1158 L. M. ADAMS, R. J. LEVEQUE, AND D. M. YOUNG2. Analysisof the natural rowwise rdering. or the 9-pointstencil withNRordering,he SOR method akes the form

n?1 n n?1 n~~~~~~~+UUnk = ( 1- &))Uj'k +-(U jk-l1 + UJ-1,k+ UJ?1k?k Ujk+1)J 5 j,~~~(2.1)2 (U.1k+l ? UYJ-l,k-1 UY+l,k+l + Uj+l,k-1)

We assumethat his teration as a solutionof theform(2.2) ujk AW(Xj, Yk)-Then A is an eigenvalueof the iterationmatrix, nd the vector withcomponentsW(xj, Yk), j, k= 1,- - , N- I is thecorrespondingigenvector.utting2.2) into 2.1),cancelling commonfactor fAn,and dropping hesubscripts n x and y givesAw(x,y) =Akc(w(x- h, y)?+}w(x- h,y- h)?+w(x, y- h)?+ w(x+ h,y- h))(2.3) ?cw)(0w(x+h,y)?+ 1w(x-h,y+h)+lw(x,y+h)+?1 w(x+h,y+h))

+(1-co) W(X, y).The eigenfunction (x,y) mustbe zeroon theboundary f theunit quareinviewofthegivenboundary onditions.We use separation f variables nd letw(x,y) = X(x) Y(y). We also seta - A /2Substitutinghis nto 2.3) and dividing yX(x) Y(y) gives

a2+Ct-1 1a 2 (y - h)+ Y(y +h) 1 a2Y(y - h)+ Y(y + h)c 5 \ Y(y) /20 \Y(y)(2.4) (X(x- )+X(x+ h) 1 _a2X(x_ -h)_+_X(x_+ h)X(x) 5k X(x)Now let(2.5) Y(y) = a y/hsin 7qyto get

a?2+C-1 2a X(x - h) X(x + h)(2.6) =-COS 7h?+(,2.6) - 5 X(x) X(x)where

12a 1a(2.7) a=a +-COS 71h, 2 =- ?cos -qh.5 10 5 10We note that D and (12 are independent fx and y and let(2.8) X(x) = ((1/(2)x/2h sin gx.Then

(lX(x - h) 2X(X + h)2 1/2(/2 COS h,and usingthis n (2.6) gives(2.9) cx- cos ih 2I1/2I /2 osgh.(1) 5


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9-POINT SOR ITERATION 1159Squaring 2.9), using 2.7), and rearrangingerms ivesa quartic quationfora:

a -[5cos COSh + 25os2 (h cos 7h]a(2.10) [2(1 - w) - 42 cos2 qh ?2I cos2fh(cos2 jh+4)]a'

+ [4CO I -_C) COS qh A2w02 COS2 (h cos qh]a+ (1I_ )2 = 0.An eigenmodeof the terationmatrix as the form

Ui'k= A'X(Xj) Y(Yk)(2.11) uJAXx)(k= a 2n+k [(,D,/(D2)1/2]j singxj sin qYk-In orderfor heboundary onditions o be satisfied, and -rmustbe integermultiplesof 1r.The eigenvalue s A= a2, where is a root of (2.10). Note thatwe must hoosethecorrectquarerootof 11/12 in 2.11) (recallthatwesquared 2.9) toobtain 2.10),introducing dditional olutions).The correct ignfor p1/2I1/2 is determinedytherequirementhat 2.9) be satisfied,nd thisgives hecorrect ignfor D1/2/I /2 as well.The frequencies g and -reach range over the values iT,2 , ** * (N - 1) rI/2).This givesa set of (N- 1)2/4 pairs offrequencies. orrespondingo each pair (g,7)there re fourrootsof thequartic 2.10). In generalthese roots are distinct,nd sowe obtain N - 1)2 eigenvalues nd eigenvectors,hecorrect umber.Whentherootsare notdistinct, rincipal ectorswillbe obtained ndthe number feigenvectors illbe less than (N - 1)2. Recall thatforthe5-pointdiscretization, principalvector sassociatedwith he optimalco, see, e.g., Young[1971]). We makeno attempt ere todeterminehe principalvectors r thevalues ofcoforwhich hey ccur.However, ycontinuity, e do obtainall oftheeigenvalues.The frequencies (N- 1)/2+ 1)Tr, , N- 1)IT,whichone might xpectto beincluded s well,giverepeats f theeigenvectorslreadyfound.Replacingg byNIT fleaves (2.10) unchangedwhilereplacing qby NIT -?) simplynegates hecoefficientsof a and a3. In either ase thesquaresof theroots are unchanged.The eigenmodes(2.11) are also unchangedbythesefrequencyeflections.The convergence ate of the method forfixedco) is determined ythespectralradius of the terationmatrix,which s

p=max A(, 71).To determine heoptimalcw, e need to minimize over co.To help characterize he roots of (2.10), we firstolved thequarticnumericallyforvariousvalues of theparameters. or example, he solid lines n Fig. 2.1 (the +'swillbe explained ater)show themagnitude fthe fourrootsplotted s a functionfco when == q = XT forh= 1/10,1/100, nd 1/1000. When co is small there re fourreal roots.As co ncreases, wo of therootsbecomecomplexconjugates.The optimalco occurs when these complexroots ntersect he largestreal root. As co increasesfurther,wo more roots become complexconjugatesand near c = 2 thereare twocomplexconjugate pairs. The same behavior was observed for smallervalues of husingvariousvalues of A and 71.We now show thattheseobservations re correct y determiningheoptimalcand correspondingpectralradius for mall h.We let cohave the form(2.12) co=2-klh+ O(h2)as h- 0. At each value of h,f and q range over T, 2I, **, (N - 1)IT/2where N = 1/h,and so cos (h and cos qh areselectedfrom setofpoints n [0, 1] thatbecomes dense


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1160 L. M. ADAMS, R. J. LEVEQUE, AND D. M. YOUNGas h- 0. At each h,the optimalco is the one that minimizes hemaximum oot of(2.10) over all allowablechoices of4 and 77.We willsimplify he problemby makingthereplacement

cos (h p, cos hv-qin (2.10), and proceed o determinehemaximum ootoftheresulting uartic quationoverall p and q in [0, 1] forfixedkl. Wewill find hat hemaximum ootoccurswhenp = q = 1.Wewill how hat hismplieshathe owest requencies 7= 7r eterminethe spectral adiusofthe terationmatrix s h-* . Makingthisreplacementn (2.10)yieldsa'- [_)q+ 2w2p2q] 3 [2(1-c -)-5?2q2+5 O2p2(q2+4)]a2(2.13) ? [ -w(1o-w)q -Aw2p2q]a + (-Ico)2 0.

We wish to analyzethe behaviorof the rootsof thisequationas h-*0 withcooftheform2.12) and p and q fixed.To determinehe imiting ehaviorwe set h = 0, co=2in (2.13) to obtain(2.14) a4_ [q2qp2] 3 [2+ q2 5p2(q2 +4)]a2 _ q qp2] +1= .In this imit ll the rootsof (2.14) lie on theunitcircle ince theymusthave modulusno greater han one and product qual to (1 - c)2 = 1. Setting = e"' and multiplyingby e-2O reduces 2.14) to a quadraticequationforcos 0

COS2 0 _ (8q8 qp2) cos + 2 2 2(q 4) 0This equation has roots(2.15a) cos =y(8+ 8 p 2) P q2+2 p2q2 +?4and(2.15b) COS2 4=(5? +5 ) q2 ?p2q?4PThe correspondingootsof 2.14) are e?ioland e i02.From 2.15a), -1 < cos 61 1 forall p, q E [0, 1], and hence (2.14) alwayshas at least one pair of nonrealroots. Theright-handide of (2.15b) is a strictlyncreasing onnegative unction fp and q forp, qE [0, 1] and gives cos 02= 1 only forp = q = 1. Thus forp = q= 1, there re tworeal rootsat a = 1,butfor ll other hoicesofp and q (2.14) has fournonrealroots.We nowwishto determinehe behaviorofthe rootsof (2.13) as h- 0 with to fthe form2.12) for ome fixedk, (wewilloptimize verk1 ater). f p, q) # 1,1),thenfor ufficientlymallh,theperturbed quation 2.10) also has fournonrealroots,bycontinuity.his might lso be true forp = q = 1 and will be true fork, sufficientlysmall,as we willsee. For largerk1,two rootsare real and thiscase willbe discussedseparately elow.

We assume fornow that 2.13) has twocomplexpairsofroots, xpandthem boutthe imiting alues,and look forrootsof theform(2.16) e1iol(l - f31h) 0(h2), eii02(1 -f32h) 0(h2),where/3 and /2 could be complexbut will in factbe real. From 2.13), we see thattheproductof theroots s (1- co)2. Using (2.12) and (2.16) and equatingthe 0(h)termsyields(2.17) l1 f2= k,.


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9-POINT SOR ITERATION 1161Similarly, he sum of the rootsmustbe (4/5)wcoq(2/25)w(02p2qnd againequating he0(h) termsyields(2.18) 131cos 01+?82 COS 2=k1(q? q +p25 )Ifp#0 thencos 01$ cos 02 and we can solve (2.17) and (2.18) for 3 and 132.Doingthis and using 2.15a) and (2.15b) we obtain(2.19a) (3~=2 51(13/5)q2 +(4/25)p2q2 +4)and(2.19b) (2 51(13/5)q2 + (4/25)p2q2+)By continuity,hese expressions re also valid in the case p = 0.Since p and q are nonnegative,l3 - 2 and the maximum ootof the quartic(2.13) (forp, q,k,fixed, ssuming ll roots re nonreal)has magnitude - /31h 0(h2)as h-O0. Maximizing hisvalue over p and q in [0, 1], we find hat themaximumoccurs forp = q = 1 where(2.20) 131 26k1Moreover, f we replace p and q by cos (h and cos r1hwith g and Y7 ixed s h * 0(low frequencies)we obtain/3, (11/26)k, 0(h) from 2.19a) and consequentlyhevalue of I31n (2.20) is valid for hediscrete roblem s well.We concludethat f wevary o s in (2.12) whenh- 0, and iftheresulting uartic(2.13) has nonreal roots then the spectral radius of the iterationmatrixwill be1-f3,h + 0(h2) with/3, ivenby 2.20). But we must tillconsider hepossibilityhattwo oftherootsof (2.13) are real. Recall that hiscan onlyoccur whenp = q= 1 andthat nthiscase the imiting uartic 2.14) has a doubleroot t a = 1. Setting = q = 1in (2.13) we find hat = 1 is in fact rootfor ll co.Recall, however, hat npracticep = q = 1 cannot occur forh 0 and themaximum alue thatp and q can actuallytake s cos Irh 1-_172h2+ 0(h4). Consequently, e must eturno theoriginal uartic(2.10), set cosh1h -2h2?0(h4), cos-h=1- 2 h ?0(h4),and look for realroot of the form(2.21) a = 1- clh- c2h2+ 0(h3).We taketerms utto the0(h2) term ecause now the0(h) termswillcancel denticallyand we obtainan expressionforcl only by equating he coefficientsf h2.Also,wemust xtend heexpansion n (2.12) as

co 2- klh k2h2?+ (h3)although he coefficients2and k2willdropout.Usingtheseexpansions n (2.10) andequatingcoefficientsf h2gives

13C12- 15k1C1 9((2+? 2) = 0.So,(2.22) cl= -5k, 21 225 2 468(2 + 2).


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1162 L. M. ADAMS, R. J. LEVEQUE, AND D. M. YOUNGWe see thatfor givenchoice ofg and 71, 2.10) has realrootsonly fcl is real,thatis, onlyfor(2.23) kl_ 225The largestreal root is obtainedby takingthe minussign in (2.22) and choosing= 7 = Tr.This gives(2.24) cl=2k -V225k 1- 9361X2which s real fork,> 2.04XI.We must till determinehecomplexroots n the case where here re two realroots.Take these to be of theform '0(1 -f(3h) O(h2). From 2.10), theproduct fall fourrootsmustbe (1 - c()2. Usingthis form fthecomplexrootsand (2.21) withcl givenby 2.22) for hetwo realroots, quating oefficientsfh intheproductyields

-2f3 30k,= -2k1.SO,13= 26kl-Note that thisagreeswith he value of 3,ffoundpreviouslyn (2.20) forthelargestroot n the case of fourcomplexroots.This indicates hat t is thepair of complexrootswith mallermagnitudehat plits nto real roots s the critical alueofk1givenby (2.23)Yispassed. This agreeswithwhat s observed n Fig. 2.1.We now determineheoptimalco.We need onlyminimize

max 1-13h, 1- clh)where3,f nd cl aregivenby (2.20) and (2.24), respectively.ince3,f s an increasingfunction fk, and cl is decreasing unction fkl, the minimum ccurswhere3,3= lwhichgivesk, IT /9236=z~2.116X.

Consequently,cl 13126kj 0.895-T

andtheoptimal o ndcorrespondingpectral adius,p = a 2= 1-2clh, have thevalues(2.25) wOopt2- 2.116TrhO(h2), Popt 1 1.791Trh O(h2)as h 0.For comparison, hecorrespondingalues for he5-pointmodelproblem re

t)opt 2- 2ITh, p1o - 2ITh.Noticethat or he9-point tencil,he pectral adius s slightlyarger, iving omewhatslower onvergencehanfor he5-point tencil, lthough hetwo arevery lose. Moreimportant,oth are 1- 0(h) as h- 0, giving symptoticallyhesame order f conver-gence.By contrastheJacobiand Gauss-Seidel methods, nd also thepseudo-SORmethod nalyzed n ? 4, have spectral adii 1- 0(h2) as h- 0.It is very nterestingo compare heseresultswith symptoticesults btainedbyGarabedian 1956], especiallysince theydo not agree and yetboth are, in a sense,correct.Garabedian'sanalysis s based on viewing he SOR iteration2.1) as a finitedifference ethodfor time-dependentDE. Expanding nTaylorseries hows thatthisdifferencequation s consistentwith hePDE(2.26) 5Cu,+2u, ?+3uyt 3uXx3uyy, u = 0 ontheboundary


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9-POINT SOR ITERATION 1163where C and co are relatedby

2(2.27) i +ChIf we fix C > 0 and choose coaccording o (2.27) foreach h > 0, then 0< cK< 2, andso themethod 2.1) is stable. Since it is consistentwiththe linearequation (2.26),iterates k withn= T/h must onverge o solutionsu(xj, Yk, T) of 2.26) as h- 0 (bythe Lax Equivalence Theorem) if we choose u3?k y discretizing ixed nitial datau(x, y,0). Consequently, tudyinghe decay of solutions o (2.26) gives nformationabout the rateof convergence f SOR.By introducinghe changeof variables

x y3 2'

(2.26) is transformedo5Cus 3us = 3uxx3uyy.

Separationof variables hows that igenmodes f this PDE havethe formu(x, y, s) = e-Ps sinx sin qy,

whereg and -r are integermultiples f XTand p =p( , 7) is a rootof the quadraticequation(2.28) -2 - 5Cp+3( r2?72)0Transformingack to the original imevariablegives(2.29) u(x, y, t) = e-P(t+x/3+Y/2) in gx sin 7y.Note that n a timestep of lengthh, this solutiondecays by a factor -Re(P)h. Theeigenmodewith lowestdecay s obtainedbytakingg= q = iX and thenegative quareroot n solving 2.28), giving(2.30) Pmin f3(5C -25C -261 ).

Ifwe obtain nitialdata for OR by discretizinghe correspondingigenfunction,Ujk = U(Xj, Yk, 0) from2.29), itfollows byconvergence) hat hedecayfactor or heSOR iterationmusthave theform(2.31) JAmaxl 1-Re (Pmin)h 0(h2)as h- 0. Sincetaking ther igenfunctionss initialdatagivesfaster ecay,one is ledto theconclusionthat n order o obtain the fastest ossible convergence, e shouldmaximizeRe (Pmin)nd henceminimizeJAmaXj.ecall that the value of C is stillatourdisposal.We canminimizeRe (Pmin)ysettinghe radicalto zero n (2.30), giving

C = 26 v1.02 T, Pmin 6T2/ 132.35XT.5By (2.27) and (2.31), we obtain thefollowing redictions orthe optimalcoand thecorrespondingecay rate, s in Garabedian 1956]:

2l?Ch~~--2(1Ch)=22.04iTh,.) o*pt = 1 -C h 1 2.34h,(2.32) -mnA*t -z /- Pmin h = I - 2. 5 oTh.


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1164 L. M. ADAMS, R. J. LEVEQUE, AND D. M. YOUNGThesevalues do not agree with hevalues (2.25) foundbycomputingheeigenvaluesof the terationmatrix. he reason s thefollowing.While 2.31) does indeedgiveacorrect xpression or he argest igenvalueof the terationmatrix orrespondingoa decay factorforthe PDE, there are other, purious,eigenvalues of the discreteproblem hathave largermagnitude orconear2 and hence determinehespectralradiusp. This is seenclearly nFig.2.1,wherewe haveplotted e-Phl for hetworootsofthequadratic 2.28) (as + 's) alongwithJA,hemagnitudes fthe ctualeigenvaluesobtainedbysolving hequartic 2.10) (as the solid lines). One pair of discrete igen-values closely matches e-Phl for small h,while the other complex conjugate)pairdoes not. In fact,Garabedian's resultsmayalso be obtainedfrom ur approachbychoosingk, in (2.24) to maximize l, therebygnoringhe effect f the other oot.For each fixedh,we can choose initialdata (namely, s a spuriouseigenvector)so that onvergences slowand determinedythespectral adius.On the otherhand,these purious igenvectors ecomehighly scillatorynd do notapproacha limit sh-O0. In particular, hey do not approach the eigenmodesof the PDE as h -*0.Consequently,fwe obtainourinitialdata bydiscretizing fixed unction fx and yat each h (as is more realistic n practice),we would expectto see vanishingly mallcomponents f thesespuriouseigenvectorss h- 0. Forpracticalpurposes, hen, hevalues (2.32) obtainedbyGarabedianmaybe moremeaningfulnd usefulthanthe"true" values (2.25).This is demonstratedn Fig. 2.2, wherewe showthedecay of 11u2 forvariousinitial data. For initial data obtained by discretizing he smooth data u(x,y) =(x2-x)(y2-y), the observeddecayis initiallymuchcloserto jAmaXIn,as predicted y(2.31), thanto pn. However, as the iteration ontinueswe would expectto see theeffectfthespurious igenvectorsake over. With heNR ordering n smooth nitialdata,thishas notyet ppearedover henumber f terationssed. Withdifferentnitialdata,uij= 1at all interioroints so that here s a discontinuityttheboundarywhereu = 0, and hence morehighfrequencies re present), hisdivergence oes occur andtheasymptoticlope appearstoagreewith hespectral adius.This effects even morevisible nFig.3.11 where he sameexperimentsreperformedor differentrdering.In that ase, even with mooth nitialdata,theasymptoticonvergence ate s clearlygivenbythe spectral adiusalthough own to an error evelof10-6 orso Garabedian'sestimate s valid.To verify hat the eigenvector orresponding o the spectralradius is highlyoscillatory, e note thatby (2.5) and (2.8) an eigenvector as the form(2.33) Wjk= ak [(P1/2) 1/2]jsinfxjinN'Yk-Inserting complexeigenvaluea = e'0(1-,8h)+ 0(h2) into 2.33) gives

Wjk= eiOk(1 -oh)k[(Q1/ I2) /2]j sin fxjsin 7Yk-Since 0 #0, this functionwillbe oscillatoryn k andj and nonconvergents h- 0.Bycontrast,heeigenvectororrespondingo the otherpairof rootsconverges o theeigenfunctions2.29) ofthePDE as h- 0. For thesevectors urprevious rgumentsshowthatAhas theform f 2.31) andhencea, thesquarerootofA, anbe expressedas(2.34) a = 1-4fph 0(h 2).Expanding O1/P2 usingthisvalue of a in (2.7) shows that(2.35) O1/P2= 1 -j3ph+ 0(h2).


11/26

9-POINT SOR ITERATION 1165

Vlovil

VII

Cil

o In o ~~~In o e IUI o d7 ojj 0

H o o o o (, ~ ~ o Vl

e1111 VI

,~VII

t~~~~~~~~~~ ,_

+ x

+ Ii

m tD v __o_ _ _ _ _ _ _N


12/26

1166 L. M. ADAMS, R. J. LEVEQUE, AND D. M. YOUNG

10-9

0 20 40 60 80 100 120 140 1601 t era t io n

FIG. 2.2. Convergence istory2-normferror ersus terationumber) ortheNR ordering ith = 0.05and w= 1.86. Three hoicesof initialdata are compared: a) An eigenvectororrespondingo thespectralradius. b) An eigenvectororrespondingo the argestnonspurious igenvalue.c) uk= (J- x)(yk - Yk).(d) u=o1.Putting 2.34) and (2.35) into 2.11) gives

n (1-1ph + O(h2))2n+k(l -_ph+ O(h2))j sinOhsinrkh= (1 -ph + 0(h2))n+j3+k,2 sinOhsin rkh.As h - 0 witht= nh, =jh and y= khfixed, his pproachesthe eigenmode 2.29) ofthePDE.(See note dded nproof.)

3. Multicolor OR. In this sectionwe consider he SOR method pplied to the9-pointmodelproblemwith everalalternativerderings fthe unknowns jk. Theseorderingsre determined ylabelingthegridpointswithfourdifferentolors Red,Black, Green,and Orange) and thenordering he pointsby firstisting ll thepointsof one color,then a second color,and so on. The overallordering f grid points sdetermined ytwofactors: a) the manner n whichthe gridpoints are labeled (thecoloring fthegrid)and (b) theorder nwhichthe colors are taken the ordering fthecolors).Four-coloringsre of nterest or he 9-point tencil ecausewithfour olors t spossibletodecouplethegrid, nthe sense that he resulting OR formula orupdatinga grid pointof any givencolor involvesneighboring rid points,all ofwhichhavedifferentolors than the centerpoint. This is advantageous n parallel processingapplications inceall gridpointsof the same color can be updatedsimultaneously.For the 5-point tencil,two colors suffice o decouple the grid using the RBcheckerboard attern iscussed n ? 1. Recently eVeque and Trefethen 1986] havepresentedn easyway oanalyze he5-pointmodel problem sing changeofvariablesfrom he teration umber,n,to the earliest ime,v,that he unknown t a gridpointcan beupdated ssuming neupdaterequires netime nit. hisvariablevcorrespondsto the"data flow imes"discussed nAdamsandJordan 1986] and closelyresemblesthechangeof variablesused by Garabedian 1956] to analyzethe PDE. This changeofvariables llows the use of Fourier nalysisto determine heconvergence ateandoptimalCt. t also givesa straightforwardroofof theequivalenceoftheNR and RBorderings.


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9-POINT SOR ITERATION 1167Here weuse this ameapproachto analyzethefour-colorrderings or he9-pointstencil. eforentroducinghese rderings, ebrieflyeview he nalysis or he5-pointNR and RB orderings o introduce otation nd motivate ur 9-point nalysis.For the5-pointmodel problem 1.1) withthe NR ordering,he SOR iterationtakesthe form

(3-1) u n?1= _0Un "(n 1+ n?l n Un(3 1 UJk = ( 1-()) UJk 4 j-l,k+ Ujk- + U,k+1 + U+1,k)-The stencil orupdating gridpointon iteration + 1usingtheNR orderings givenin Fig.3.1. To assist ndetermininghechangeofvariables, he earliest imes t whichan unknown an be updatedon thefirst wo terations singthestencil n Fig.3.1 arelistedbelow each node in Fig. 3.2. These times define heiteration ariable P. Eachnode in Fig. 3.2 is updated at timev+ 1 by the stencil hownin Fig. 3.3, and thecorrespondingOR iterations(3.2) u k1 = (1W)uj k ?-(U> 1,k + Ujk-1 + Ujk+1 + UJ?1,k)-Figure 3.2 showsthat the timesalong lines + k are constant nd that terationsnvariablen occurevery wo timeunits.Hence,theproper hangeofvariablesbetween(3.1) and (3.2) is(3.3) v=2n+j+k-2.The advantage f thischangeof variables s that heeigenmodes f 3.2) areeasyto determine. heyare simplyFouriermodesof the form(3.4) Ujk = g" sinfxj in qYk-These gridfunctionsatisfyheboundary onditions rovided hat4and iq are ntegermultiples f iT,and substitutingnto 3.2) givesthefollowing quationforg:(3.5) g2 = (1 - G)+ ?g (cos eh+ cos h).2

nn+1 n n

n+ ~n nn+IFIG. 3.1. NR stencilnvariablen.4,6 5,7 6,8 7,93,5 4,6 5,7 6,82,4 3,5 4,6 5,7

1,3 2,4 3,5 4,6FIG. 3.2. Times ortwo terationsf NR.

vc vN-1 iv

FIG. 3.3. NR stencil n variable '.


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1168 L. M. ADAMS, R. J. LEVEQUE, AND D. M. YOUNGFor each f and 7q,hisequation has twosolutions, iving wo eigenmodes.We obtainall modes by letting 5 and q range over the values , q = iX, 21i, * * *, (N - 1)1J-,or atotalof 2(N - 1)2 modes.This is thecorrect umber ince 3.2) requires wo previouslevels of data to calculateuJklBythe hange fvariables3.3),aneigenvalueAof 3.1) isg2 and the orrespondingeigenvector as components j+k sinfxj in qYk- Wenow appear to have twice s manyeigenmodes s required or 3.1), but here ach mode s repeated incereplacing (, 7)by N1J- ~, Nir- D) simply egates heroots f 3.5). SinceA g2 this eflectioneavestheseeigenvaluesunchanged.The eigenvectors also unchanged ince

(_g)j?k sin NT -()xj sin NT -)Yk = gj?k sinfxj in vYk-Equation (3.5) givesthefamousrelationship etween n eigenvalueA of SOR and aneigenvalueA = 2(cos (h + cos qh) oftheJacobi teration:A+ Cto-1 12(3.6) = A /2yGto

Equation (3.6) can be used to determineheoptimalvalue ofw as a function f A(see, e.g., Young [1971]) and thisderivationwillnotbe repeatedhere.A similar nalysis anbe carried utfor heRed/Blackordering fthegridpoints.In the variablen there re two stencils, ne for he Red nodes and one for heBlackones,as given n Fig. 3.4. The correspondingteration sR: Ujk =(1-)U+(Uik+1+U,k1+UJ-l,k+Uj+I,k)4,(3.7) B: u`k~=(1-o)u~',.I+ u(k+ n? 1k)jk = - jk + 4 j,k+1 j,k-1 J-1,kUj

Theearliest imes orrespondingo (3.7) aregiven nFig.3.5. In thedataflow ariable,P,both R and B nodes havethe same stencil, amely, hestencil f Fig.3.3. Equation(3.2) gives the update formula or ll nodes. Hence, (3.5) and (3.6) also hold for heRB ordering. igure3.5 shows that he times longall lineswith + k even are equal,n n+1

n n n n+1 n n+1n n+1

Red BlackFIG. 3.4. Red/Blackstencil nvariablen.

B R B R2,4 1,3 2,4 1,3R B R B1,3 2,4 1,3 2,4B R B R2,4 1,3 2,4 1,3R B R B1,3 2,4 1,3 2,4

FIG. 3.5. Times ortwo terationsfRB.


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9-POINT SOR ITERATION 1169and similarly or + k odd, andthat terationsnvariablen occur every wo timeunitsin P.Therefore, hechange ofvariables s

v=2n+[(j+k) mod 2]-1,and theeigenvectoromponents f this teration orrespondingo , i1 are{sinxj inX'Yk,

Wik-A 1/2sinexj in Ykfor heR and B nodes,respectively.hisagreeswith heresult iven nYoung [1954].We now turn o the 9-pointstencil,withorderings ased on fourcolors. Theresults f Adams and Jordan1986] applied to this modelproblem how that he 72distinct our-colorrderings an be grouped ntosix equivalence classeswithregardto convergence ehavior see Fig. 3.6). Representativerderings rom ach of theseclasses are:Ordering 1: Thegrids colored s inFig.3.6(a)with rderingIB/GIO.Ordering 2: Thegrids colored s inFig.3.6(b)with rderingIB/GIO.Ordering 3: Thegrids colored s inFig.3.6(b)with rderingIBO/IG.Ordering 4: Thegrids colored s inFig.3.6(b)with rderingIG/BIO.Ordering 5: Thegrids colored s inFig.3.6(a)with rderingIB/OIG.Ordering 6: Thegrids colored s inFig.3.6(a)with rderingIG/BIO.

G O R B G O G OR B G O R B R BG O R B G O G OR B G O R B R B

FIG. 3.6(a) FIG. 3.6(b)We will show thatOrderings#1, #2, and #4 are in factequivalent o theNRordering iscussed n ? 2. Ordering#3is different,owever, nd givesslightlylowerconvergence ased on thespectral adiusand slightlyaster onvergence ased on theeigenvalue hatdominates he teration or mooth nitialdata. We havenot beenableto analyzeeither rdering#5 orordering#6.Ordering#1. Figure3.7 shows theupdate timesfor his ordering,whichdefine

the data flowvariableP.In thisvariable, ach node has the same stencilwithupdate formulak = (1-) (U +Uj+kl+kUJ Uk+1 k-)(3.8)

+20 J-1k-1 + UJ-1l,k+1 Uj+1,k-1 +U1,k+)-G 0 R B

3,7 4,8 1,5 2,6R B G 01,5 2,6 3,7 4,8G 0 R B3,7 4,8 1,5 2,6R B G 01,5 2,6 3,7 4,8

FIG. 3.7. Times ortwo terations ithOrdering#1.


16/26

1170 L. M. ADAMS, R. J. LEVEQUE, AND D. M. YOUNGThe change ofvariables s givenby(3.9) v =4n+c-3,wherec=0, 1,2,3 for heR,B,G, and 0 nodes of Fig. 3.7, respectively.

To see that hisordering s equivalent o theNR ordering,we only need to lookat the updatetimesfor he NR ordering, hown n Fig. 3.8.

7,11 8,12 9,13 10,145,9 6,10 7,11 8,123,7 4,8 5,9 6,101,5 2,6 3,7 4,8

FIG. 3.8. Times or the 9-pointNR ordering.If we define hevariablevby these imes,we again obtain teration3.8) with hechange of variables

(3.10) v= 4n +2k+j-6.Since a change of variables gives (3.8) in eithercase, the eigenvaluesand henceconvergence ehaviorwillbe identical. f g is an eigenvalueof (3.8), then A= g4 iSan eigenvalueof bothNR and Ordering#1.The eigenvectors, owever,will not be the same, since a differenthange ofvariables s used in each case. The eigenvectorsor he NR orderingwere determinedin ? 2. Using theseand the above changesof variables allows us to determine heeigenvectors orOrdering#1. In analyzing 3.8) we viewit as applying o all meshjoints j, k) at each level of v,althoughnourapplications t s appliedonly opointsof a single olor neachstep. Notethatwe couldapply ttoall pointswithoutffectingtheresults, utthe workrequiredwould be increasedby a factor f 4.) Since (3.8)requiresfour evels of priordata to determine 'k+4,an eigenvectorf (3.8) consistsof4(N- 1)2 values,

v =V=v2'whereV' = { Vj)k} or, k = 1, 2, * , N - 1. Ifg is thecorrespondingigenvalue, hen

-vl- -v0-V3gL].This indicates hatVV+" gV' foreach v, and hence

V= [vo] E R4(N-1)2


17/26

9-POINT SOR ITERATION 1171If we now let V? be the vector onsisting nlyof the values V3?korwhich j, k) is aredpoint, nd similarly orV?, VGand VO, hen n eigenvectorftheoriginal terationin then variable withOrdering#1) has theform

VR(3.11) V()=|2 _ R(_22 1

g3 vOwith igenvalueA= g4.On the otherhand, bythechangeof variables 3.10), an eigenvector or he NRorderingn the n variable has theform

V(NR) = g2k+j VOagain with igenvalueA= g4. Equation 2.11) gives he igenvector or heNR ordering,

V5kR) = (Dk[((F/(2)l/2]j sinexj sin7)Yk,where DI and (2 aregivenby (2.7) and a = A /2= g2. Using thiswe obtain

Vj9k = g 2k-j V5NR)= g-j[((l/(2) 1/2]j sinexj sin iYk-

The eigenvectors are now determined by (3.12) with q, j = , 2 , *- *, (N- 1)(X-/2).As before, he frequencies , iq = ((N - 1)/2+ 1) , * , (N - 1)ir give repeats of theeigenvalue-eigenvectorairs alreadyfound.Ordering#2. For Ordering#2, the associated earliesttimes for the firstwoiterations re given nFig. 3.9. An inspection fFig. 3.9 shows hat heR and G nodeshave the same stencil n the variableP,with hefollowing pdateformula:

R,G: uk (= 1- )Uk+ (UJ1 k+Uj+l1k+Uj-2l+Ujk+l)(3.12)

to~ivi3 + U>3 k? 3? + Uiik1)3+20 (Uj-1 k-1+Uj-l,k+l + j+l,k-l +j+l k+l)-20Likewise, heB and 0 nodes have the same stencilwithupdate formula:

B, 0: uk = ( 1-)uk ?(U k+ UJ+ k+ Ujk-1I + Uj,k+l(3.13)+2 (ulk-1 + UJ-l ,k+l + UJ+1,k-1 + Ujv+l k+l )-

G 0 G 03,7 4,8 3,7 4,8R B R B1,5 2,6 1,5 2,6G 0 G 03,7 4,8 3,7 4,8R B R B

1,5 2,6 1,5 2,6FIG. 3.9. RI B/GIO times orOrdering#2.


18/26

1172 L. M. ADAMS, R. J. LEVEQUE, AND D. M. YOUNGThe change of variables fromn to v is again given by (3.9), wherec = 0, 1,2 and 3for heR, B, G and 0 equations,respectively.Now, the equations n (3.12) and (3.13) have thesymmetryeededto verifyhat

jk= sinexj sin qyk.Thus the methods n (3.12) and (3.13) have the same value forVjokut differentmplificationactors,ay g1and g2. A single tep of the fullmethod,in variable n, consists f four weeps,two with 3.12) and two with 3.13) and hencehas amplification actorA = g92-

In order o determine ,wefind = g9g2 byconsidering wo weepsofthemethod,Red and Black, say.We substitutehe following aluesUJkk=e'(Xj+ yk

z,+1UJk = g2 UJk,v+2(3.14) UJk =glg2u k,

UJk =glg2uJkzv+4 = 92 2 vUjk =gl2ujk,U+5 2g3 UUjk glg2 Ujkinto 3.12) to get

(3.15) 2 2(-c) 22csh+ggco h+1~oseh cos qih).31) gg 19-2 ) + - (292 COS eh+ 2g,92 COS 77h 9192 CO hcsrh)5Next,we take a step with 3.13) to update theBlack nodes, and obtainu5. Putting(3.14) into 3.13) for v+5 we get(3.16) 9192 = -w)+ (2 r2coseh+2g,g2C cos ehcOS-7h).5Equations 3.15) and (3.16) can be equated and gl and g2 eliminated ogivea quarticequation fora. Surprisingly,hisquartic s again (2.10), thequarticobtained n ouranalysisoftheNR ordering y separation fvariables.This shows that hisorderingis equivalent o theNR ordering,nd hence s also equivalent o Ordering#1.The eigenvectorsn thevariablen can be seen from3.9) and (3.14) to be

v(#2) 9g2VB j R(N-1)2g2 VO

where V? = sin exj sin 'yk, e, Y7 rr, IT,, * ,(N - 1)(Xr/2) with eigenvalue A= g2g2Again, we findthateigenvalue-eigenvectorairs are repeatedfor the frequencies((N -1)/2 +1)T, - *, N-1)ir.Ordering#3. For thisordering,heearliest imesfor hefirst woiterations regiven n Fig. 3.10. The R and 0 nodes have the same stencil n thevariablev withthefollowing pdateformula:

R, O:jk 5 jk+ jk- j+ + j-1,(3.17)+?- (UJf,+k-1 + Uj-l,k+ll j+,k-1+ Uj?+l,k?l)-20+ 2 +


19/26

9-POINT SOR ITERATION 1173G 0 G 04,8 3,7 4,8 3,7R B R B

1,5 2,6 1,5 2,6G 0 G 04,8 3,7 4,8 3,7R B R B1,5 2,6 1,5 2,6

FIG. 3.10. R/B/O/G times orOrdering#3.Likewise, heB and G nodes are updated by theformula

B,G: uto4=(1-w)uk? (U l?+ u l?U k+ uJ-1k)5,(3.18)+ 2(U-l,k- + Uj-i,k+l + U)'lk-1 j+l,k+l)-

Substituting3.14) into 3.17) and (3.18) yields(3.19) ghg+(92w)+S2eh)3.19) g,2g2- ( - (s) + - (919 o rhCOS (h) + - g1,92 COS rqh os eh5 5and(3.20) gg2 =(1-w)+ (gl cos rh 192cos eh) + 9192 COS ehcosrqh,5 ~~~~~~~5respectively. s before,we equate (3.19) and (3.20), eliminate l and g2, and getthefollowing uartic n a = 9192

a 4 - (5CI COS -qhcos eh + 4w2 COS rqh os eh) a 3(3.21) +(25 cos2 qhCOS2eh+ 2(w -_1) - 24 2O 2 qh- 2 cos2 h) a2

+(3w(1-w) COS qh oseh-_22 cosehcos qh)a+(w-1)2=0.The changeof variables fromn to v is again givenby (3.9) wherec = 0, 1,2, and 3correspondo theR, B, 0, and G equations, espectively.ollowing he ameargumentsas before,we find hat heeigenvectorsn variablen forOrdering#3 aregivenbyI- 1?V(#3) 92 V B )2

glg2 VOwhere V3. sin exj sin rqYk, , q = -r, i-, * * , (N - 1)(iT/2) and the eigenvalue is A=912. Again, we find that eigenvalue-eigenvectorairs are repeated for e, i =((N- 1)/2+ 1), * * *, N- 1)7r. his canbe seenfor heeigenvalues rom3.21) sincereplacing4,7q)by N7T e,N7T 'q) leaves thequartic nchanged ndreplacing4, q)byeitherN7T e, q) or (4,N7T 'q) negates hecoefficientfthe a and a3 terms.The quartic n (3.21) does notagreewith hequartic n (2.10), and the rootsdonotagree ngeneral.Consequently,heoptimal oandcorrespondingonvergenceateare differentor hisorderinghan for he other rderingsonsidered o far.


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1174 L. M. ADAMS, R. J. LEVEQUE, AND D. M. YOUNGNumerical esults how that heroots f 3.21) have thesamequalitative ehavioras shown n Fig. 2.1 withe= q = X giving heslowestdecay.The optimalto s alsoseen to occurwhere the largestreal rootand the complex rootof largestmodulusintersect orthe frequencies = q = 7. We confirm heseobservations y performingan identical nalysis o thatgiven n ? 2 for he rowwise rdering.We summarize urresults elow.Thecomplexrootof 3.21) withargestmodulusfor mall hoccurswhen Tand is

(3.22) eil(1 -_3k1h) O(h2)where01=cos-=(-CO ).The largest eal root of (3.21) for mall h also occurs when 7T= = X and is(3.23) 1- c1h O(h2)where(3.24) cl=5k1-8A25kj-96&7.The optimalw occurs when themodulus of the root n (3.22) equals the modulus ofthe root n (3.23). This value of to s(3.25) to =42- i7h+(h2) : 2 -2.1387rh.The corresponding alue of thespectral adius s(3.26) Popt= 2c,h + O(h2) 1-1.604irh.This RBOG SOR iterationwas programmed nd the resultsof (3.25) and (3.26)confirmed.y comparing 2.25) and (3.26), we see thatdifferentvaluationorderingsfor hesame coloring fthegridpointscan lead to differentsymptotic onvergencerates,based on the spectral adius.The eigenvector ssociated withthe spectralradius is highly scillatory s themesh s refinedrecallthatthis was trueforthe rowwise rdering lso). An analysissimilar o Garabedian'scanbe performedofind heconvergence ehaviorfor moothinitialdata. To do this,we choose k1to maximize 1 n (3.24) to get

k,= g:= 1.95959-rand

2c1 Vr 2.44949Xg.The corresponding alues ofp*ptand to*tare(3.27) pt = 1 2.44949 h, t)o*t 2 - 1.95959 h.The values in (3.27) show thatfor mooth nitialdata thisorderings preferredverthe rowwiseordering nd Orderings#1 and #2. Note thatthe values in (3.25) and(3.26), based on thetruespectralradius, ead to theoppositeconclusion.However,theresults f 3.25) and (3.26) are valid fornonsmooth nitialdata. Figure3.11showsthedecayof 11u12forvarious nitialdata. For initialdata obtainedby discretizinghesmoothdata u(x, y) = (x2-x)(y2-y), the observeddecay is initiallymuch closer tothatpredicted y (3.27), thanto pn.Ordering#4.This orderingeads to a quarticequivalent o (2.10) withe and i1interchanged. ence,for square gridwith tepsizeh inboththex andy directions,tooptnd Popt or this ordering are also given by (2.25).


21/26

9-POINT SOR ITERATION 1175

lo-,

1W2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~d

0 20 40 60 80 100 120 1 40 1 60lt er a t l on

FIG. 3.11. Convergence istory2-norm f error ersus teration umber) or Ordering#3 with = 0.05and w 1.86. Three hoicesof initialdata are compared: a) An eigenvectororrespondingo the pectralradius. b) An eigenvectororrespondingo the argestnonspuriousigenvalue. c) Ujk = (x-x)(y -Yk).(d) uo=1.Anyof thetwenty-fourossibleorderings ssociatedwithFig.3.6(b) canbe easilyproved to have the same eigenvalues s Ordering#2 or Ordering#3 that we haveanalyzed here (Adams and Jordan1986]). We have found two equivalence classesthat haracterizehe symptoticonvergence atebehavior nd therelaxation arametercl. Orderings#1,#2, and #4belongto the first lass and Ordering#3belongs o the

second one. In addition, here ould be twomore classes correspondingoOrderings#5 and #6thatwe have not been able to analyze.4. A 9-point seudo-SORmethod.We nowconsider pseudo-SOR methodwiththe stencil n Fig.4.1, and iteration

n?1 _ ) n to n 1 n?1 Un UUjk ( 1- )Uk +-(UJk-l + UJ1,k+ UJ+1,k + jk+?1)5(4.1)+? (UJ1,k+1U- + k j+l,k+l + U?+l,k-1)

which differsrom 2.1) in the last two terms.This methodcan be analyzed by thetechniquesof ? 3. The earliest imesfortwo iterations orrespondingo Fig. 4.1 areequal to those ofFig.3.2. That s,the terationxpressed nterms f the variablev isUJk = (1- w)Uk +-(UJ7k-1 + UJ-l,k+ UJ?+,k+ Uj,k+l)(4.2)

+ (U.-1 + Ulk-1 + Uj+,k+?1 + +k-1),with hechangeof variablesgivenby (3.3). It is interestingo note that he times nFig.3.5 andthe terationn (4.2) are also obtainedfor n RB ordering f thegridwith

n n n-In+1 n nn+1 n+1 n

FIG. 4.1. NR modifiedtencil nvariablen.


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1176 L. M. ADAMS, R. J. LEVEQUE, AND D. M. YOUNGtheRed and Blackstencils hown nFig.4.2. Since two colors do not decouple a griddiscretizedwith he 9-point tencil, t s tempting o use old informationor he BlacktoBlackcoupling s shown nFig.4.2to obtain method uitable or arallel omputers.This modification as consideredby Kuo, Levy, and Musicus [1986] fora 9-pointstencil rising rom discretization fa PDE with cross-derivativeerm. heyshowthe convergence ateof SOR for heir roblem o be 1- 0(h) inthe regionwhere helowest frequency ominates.We show by analyzing 4.2) that the use of only twocolors s not sufficientor he 9-point tencil rising rom he Laplacian. In particular,we will show that he method onvergeswhenever < l


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9-POINT SOR ITERATION 1177As h- 0, (4.7) showsthat he method s divergent orcl 3. Also (4.6) shows that hevalue of JgJ2s maximizedwhen cos eh= -cos -jh.This occurs when = N -e (lowfrequencyn one direction nd highfrequencyn the other) nd this maximums(4.8) JgJax=(1 + 5 COS2 h)-1.As h O gmax.jcv_-11.Second, assume the radical in (4.4) is positive.Then, the maximum alue of goccurs when4:= ,= X and is

g2max = 2 Cos2 h+c1 + cos2 1Th25 ~~~~~~5(4.9) + 5 cos 7rh c2 Cos2 1Th (1 - v)+ 5 os2Th.5 25 5It is interestingo notethat he cothatminimizes4.9) occurswhenthe radical s zero,

2and as h-*0, co 5/2.That is, the method s divergentor hecothatminimizesmaxof (4.9) for the lowestfrequency.Recall that the cvthat is optimalfor SOR forconsistentlyrderedmatrices orresponds o the owest requency. or thepseudo-SORmethod,the optimal cv occurs wherethe modulus of the eigenvaluesof the twofrequencies 7 N - e and 77 4:= IT are equal. This co is determined y equating 4.8)and (4.9) to get(4.10) 6c Ccos Ih cos2 rh c - 1) -4((o-1)2=0.As h-*0,co- 5/3,so we look for solution o (4.10) oftheform(4.11) co(h)=5+c,h+c2h2+.Substituting4.11) into 4.10) and equating ermsyields

C1 = 0, C2 =-IT ,and thecorrespondingalues of theoptimal o and spectral adius of (4.1) are(4.12) opt3-902h2I Popt= -3ir2h2.Comparing 4.12) toCOS2 ih 1- &2h2,the pectral adiusofthepseudo-Gauss-Seidelmethod, howsthat as h-*0, thismethodwithoptimal cl is onlythree imes fasterthanwithcv 1.This is notnearly s good as the trueSOR methodsdiscussed n ? 3,where hedecayfactor s 1 0(h) for heoptimalc.

5. Comparison fpoint nd linemethods. et thesystem x= b be blocked asD, -U12 . Ulm [Ul [bb]

-L21 D2 ... U2m u2 b2-L32 ..L.=[.L-Lm-Lm2 ... Dm [UmJ L bm

The line-Jacobimethod s defined s(5.1) D1u7n+= Liju7+ j3U nu+biji


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1178 L. M. ADAMS, R. J. LEVEQUE, AND D. M. YOUNGand the ine-SOR method s

Diu"'=wZc) E Liju + cl) E U1ju7+wbi,+ (1-wt))Diu7jiwhereforboth methodsui correspondso thenodesinrow i of thegrid.The spectralradius of theJacobimethodfor he5-point nd 9-point tencils an easily be foundby separation of variables, since sinfxj sinT)Yk is an eigenvector f iteration5.1).These results re given n Fig. 5.1.The spectralradiusof the ine-SORmethodforthe5-point nd 9-point tencilscan now be foundusingYoung's theory orblock-consistentlyrderedmatrices. hatis, f/us thespectral adius of ine-Jacobi,henwo0ptndpoptforine-SOR aregivenby(5.2) ()opt 2 2 Popt=Wi)optl.These SOR results re summarizedn Fig. 5.2, whereGarabedian's resultsforthe9-pointpoint methods re given n parentheses. igure5.2 showsthat,based on theactual spectralradius,the line methods onvergefaster han the pointmethodsforboth he5 and9 point tencils. he9-point ine SOR method as the ameconvergencerate as the5-point ine SOR methodfor small h; whereasthe 5-point OR methodwith consistent rdering an be expected o converge lightly aster hananyofthe9-point ointSOR methods hatwe analyzed.However, heconvergence atewillbeobserved n practicewithsmooth nitialdata forthe9-pointpointmethodsmaybecloserto Garabedian'sresults. hat s,we can still xpectthe 9-point inemethods oconverge lightly aster han thepointmethods, utnow,the 9-pointpointmethodswillconverge asterhan he 5-point ointmethod. his atter act s encouraging incethe9-pointdiscretizations more accurate han the5-point ne.

Method Spectralradius5-point oint cos irh 1- 1 r2h25-pointine cos 1rh 1 2h22- cos rrh9-pointpoint 4cos 7h+ cos2 1Th

1 _ 372h2

9-point ine 5 COS rh OS2 h 1 2h21 s COS rhFIG. 5.1. Spectral adiusofpoint nd lineJacobimethods.

Method Ordering SpectralradiusPop,popt)5-point oint Rowwise orRed/Black 1-2-rh5-point ine Rowwise orRed/Black 1- 2vxiTh9-pointpoint Ordering#1 1- 1.79-rh1 - 2.355h)9-pointpoint Ordering#2 1- 1.7917-h1 - 2.35-h)9-pointpoint Ordering#3 1 1.6OiTh1 -2.4517rh)9-pointpoint Rowwise 1 1.79irh1 -2.3517rh)9-point ine Rowwise orRed/Black 1- 2f iTh

FIG. 5.2. Spectral adius orpoint nd lineSOR methods.


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9-POINT SOR ITERATION 11796. Conclusions.The SOR methodwith several orderings nd a pseudo-SORmethodhave been analyzedfor he9-point aplacian. TheFourier nalysis echniquesproposedby LeVeque and Trefethen1986] and separationof variables techniqueswere used to determine he eigenvalues nd theeigenvectorsfthesemethods.We examined heSOR method or he9-point aplacianusing henatural owwiseordering nd severalmulticolor rderings. or all theseorderings, e gave a quarticequationforthesquare root of theeigenvalues s a function f thefrequenciesndt. The optimal was foundfor he rowwise rderingnd Ordering#3 by asymptoti-callysolving quartic quation for he ntersectionfthe argest in modulus) ofthecomplexand real roots.Ourresultswere confirmed yperforminghe SOR iterationwith n initialguesscorrespondingo theeigenvectorssociatedwith hespectral adius.The observed ateofconvergencematched hatpredicted y thetheoryo five ecimalplaces. The SOR

iterationwas also performed yusinga smooth nitialguess,obtainedby discretizing(x - x2)(y y2) forvariousstepsizes,h. In these cases, theobserved onvergence atemoreclosely resembled hatpredicted y Garabedian.The results lso show thatdifferentrderings f thesame coloring an lead todifferentpectral adii:R/B/G/IOand R/B/O/G for hecoloringnFig. 3.6(b) havespectral adii of 1- 1.7917rhnd 1- 1.6017rh,espectively.or smooth nitialdata,thesetwoorderingslso led to differentffectivepectral adiiobserved npractice, amely,1-2.3517rhnd 1- 2.4517rh,espectively.his informationan be useful n selectingcoloring, n ordering, nd appropriate nitial data to use withmulticolor OR onparallelcomputers Adams and Ortega 1982]).An analysisof the pseudo-SOR method howed that heoptimalt occurswhenthehighand low frequencies ross and that hecorrespondingpectral adius s only1-31T2h2. his is inferior o both the5-point nd 9-point OR methodswe analyzed.In addition,for mallh,thepseudomethod nlyconverges or0 w 3.The 5-point nd 9-point nd line SOR methodswerecomparedfor themodelproblem or mallh.The line methods onverge lightlyasterhan hepointmethods.The 9-pointineand5-point inemethods ave the ame asymptoticate fconvergence,butthe5-pointpointmethodwith consistent rderingwas 1.12 timesfaster, asedon thespectral adius, nd 1.23 times lower,based on Garabedian'sarguments,hanthe best9-point ointmethod hatwe analyzed. Hence,for smooth nitialguess, he9-pointpointmethods an be expected o be moreaccurate nd converge aster hanthe5-point ointmethod.

Acknowledgments.he authors hankLloydN. Trefethenorprovidinghe nitialstimulusfor this workand for severalvaluable subsequentconversations.We alsothankElizabethOng forprogrammingheSOR methodwithour orderings.Note added in proof. n unpublishedworkcarriedout in 1951, the third istedauthorofthispaperused the methodof separation fvariables o derive he quarticequation 2.13) for heeigenvalues fthe SOR methodfor he9-point quationwiththe rowwise rdering. ecause ofthe ack ofsufficientomputational aculties, ewasunableto usethis quationtodetermineheoptimum alue ofwOndthecorrespondingconvergence ate of the SOR method.The work ay dormant ntil ts resurrectionnthesummer f 1986.RichardVarga and JohnBuonibrought o our attentionhat an analysis of theSOR methodfor he 9-point quation with he rowwise rdering ad been carried ut


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1180 L. M. ADAMS, R. J. LEVEQUE, AND D. M. YOUNGby A. I. van de Voorenand A. C. Vliegenthart1967]. This analysis ncludes caseswhere hemesh izesAxandAy nthex and y directions,espectively, aybe different.For the case whereAx= Ay,the authorsobtain thesame asymptotic esults orthespectral adiusas givenby 2.25). In this ection,wehave carried heseresults urtherby finding he eigenvectors s well. These eigenvectorswere used in our numericalexperimentss starting atatoverify2.25). Theeigenvectors ere lso usedtoexplainthediscrepancy etweenGarabedian's results nd thosegiven n (2.25).

REFERENCESL. M. ADAMS AND J. M. ORTEGA [1982], A multi-colorOR methodfor arallelcomputation,roc. of the1982 Conference n Parallel Processing,EEE Catalog N. 82CH1794-7,August,pp. 53-56.L. M. ADAMS AND H. F.JORDAN [1986], s SOR Color-blind?,IAM J. ci. Statist. omput., , pp.490-506.G. FORSYTHE AND W. WASOW [1960], Finite-Differenceethods orPartialDifferentialquations,JohnWiley,New York, p. 266.S. FRANKEL [1950], Convergenceatesof terativereatmentsfpartialdifferentialquations,Math.Comp.,4, pp. 65-75.P.GARABEDIAN [1956], Estimation f he elaxationfactorforsmallmeshize,Math.Comp., 10,pp. 183-185.C. Kuo, B. LEVY, AND B. MUSIcUS [1986], A local relaxationmethod or solving ellipticPDEs onmesh-connectedrrays, IAM J. Sci. Statist. omput., ccepted.R. LEVEQUE AND L. N. TREFETHEN [1986], Fourier nalysisoftheSOR iteration,CASE Report No.86-93, ICASE-NASA LangleyResearchCenter,Hampton,VA,to appear in IMA J. Num. Anal.A. I. VAN DE VOOREN AND A. C. VLIEGENTHART [1967], The9-point ifferenceormula or Laplace'sequation,J. Engr.Math., 1, pp.187-202.D. M. YOUNG [1954],Iterativemethodsor olving artialdifferentialquations f elliptic ype, rans.Amer.Math. Soc., 76, pp. 92-111.[1971], Iterativeolution f LargeLinearSystems, cademicPress,New York.

analysis of the sor iteration for the 9-point laplacian

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