(_ -_----_===__

AIAA 2001-2570

Textbook Multigrid Efficiency for

Computational Fluid DynamicsSimulationsAchi Brandt

The Weizmann /nstitute of Science, Rehovot 76100,Israel

Boris Diskin

�CASE�NASA Langley Research Center,

Hampton, VA 23681

James L. Thomas

NASA Langley Research Center, Hampton, VA 23681

15th AIAA Computational Fluid DynamicsConference

June 11-14, 2001/Anaheim, CA

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics

1801 Alexander Bell Drive, Suite 500, Reston, VA 20191-4344

https://ntrs.nasa.gov/search.jsp?R=20010062528 2018-08-27T06:21:51+00:00Z

Textbook Multigrid Efficiency for

Computational Fluid Dynamics Simulations

Achi Bran(it *

Th, c W_',izmann I'n._titute of Science, Rehovot 76100. I.sra, d

Boris Diskin t

ICASE/NASA Lauqlcy Re,search. Center, Hampton, I'_4 23681

James L. Thomas*

NASA Langley Re,search (:enter, Hampton. I,'_4 2,7681

Introduction

Consideral)le progress over the past thirty years has

been Inade in the dew,lopment of large-scale comlmta-

tional fluid dynamics (CFD) solvers for the Euler andNavier-Stokes equations. Computations are used rou-

tinely to design the cruise shapes of transtmrt aircraft

through comph,x-geometry simulations invoh'ing the

sohltion of 25-100 million equations; in this arena, the

numl)er of wind-tmmel tests for a new design has been

substantially reduced. However, simulations of the en-

tire flighl envelope of the vehich,, including maxinmm

lift, lmffet onset, flutter, and control effectiveness, have

not. t)eell as successful in eliminating the reliance on

witM-tmmel testing. Thes(, simulations involve un-

steady flows with more separation and stronger shock

waves than at cruist,. The main reasons liiniting fur-

ther ilm)atls of CFD into the design process are: (1)

the reliability of turbulence models and (2) the time

and expense of the munerical simulation. Because of

the prohibitive resohgion r('quiremonts of direct siin-

ulations at high Revnohls mmlb(,rs, transition and

turbuhm(:e modeling is ('Xl)('('t('(l t() r(,main an issue

ff)F tit(' Ile_tl" t(Wttl. I Th¢' f(t('{Is" ()ic t_lis p;t[lof _t(t(|t'()sstas "

the latter prol)h,m l)y att(,ml)ting to attain ol)timal

"Director, The (:arl F. (;auss Minerva C(!nler for Scientific

('.omputation al The Vqeizmann hlstitut.e of St:ten(e, Rehovot76100, Israel, (email: achi©wisdom.weizmann.ac,iX)..

tSlalf Scientist, I(?.\SI-, (email: bdiskin©icase.edu). This

research was sllpl)ortt'd |)y the National Aerollalltit:s alld SpaceAdministration under NASA (_ontracl N(). NAS1-97045 whih'

the author was in residence al. the Institute for (:oiltpuler Ap-plications ii1 Sciente and Engineering (1CASF.), NASA l.angley

t_esearch ¢ ?erlter, Itamf)tot_, Virginia 23681-219Y.; (Mmputational Modeling and Simttlation I/ranch, Mail Stop

128. NASA Langh,y Research (:enter. tlamptotL Virginia 23681

(emaiI: j. 1. thoraaslIarc, nasa. govt.('Ol)?,'righl @ 2001 by the American Institute of Aeronautics and

Asif['Oll3,111ics. hie. No copyright is assorted in the [hilled States

under Title 17, 1J.S. ('ode. The U,S. (;overnnlent has a royally-

free li('(mse to exorcise all rights u,,(t('r the COl)yrigh! claimed herein

tkII" (_ox'ernlll('llta] |)Ul'[)os(?s, All [)lh(,r rights are reserve(l by the

('opyrighi owner.

t

efliciencies in solving the governing equations. Tyl)i-

tally era'rent CFD (:odes based on the use of multigrid

at'eehwation tet'hniques and nmltistage liunge-Kutta

time-stel)l)ing schem(,s are able to COllverge lift and

drag valu('s fbr cruise configurations within at)proxi-

mately 1000 residual evaluations. More ('Oml)lexity in

the geonletry or physics generally requires many more

residual evaluations to converge, and sometimes con-

vergelwe (:aIlllOt I)e attained. An ol)timally conv(,rgem

method ix defined'-' r, as having iextt)ook multigrid efli-

cien('y (TME). meaning th(' solutions to th(, governing

system of e(luations are att ahmd in a computational

work which is a small (less than 10) multit)h' of the

operation count in the discretized system of equations

(residual evahlations). Thus, there is a I)otential gain

of more than two or(h,rs of magnitu(h, in oi)eration

('Otlllt reduction if TME could b(, attained.

In this pap(w, a distriimted relaxation al)proach to

achieving TME fi)r Reynolds-averaged Navier-Stokes

(R.ANS) equations is discussed along with the fomt(la-

tions that h)rm the basis of this al)l)roach. Because tit(,

governing equations are a set of eOul)led nonlinear ('on-

servation equations with dis('ontinuities (shocks, slip

lines, etc.) and singularities (flow- or grid-induced),

the difficulties are many. The TME mmhodology in-

sists that each of the (tifficulties should Im isolated,

analyzed, and solved systematically using a carefully

constructed series of ino(hd l)roblents. An ilnportmlt

aspect of the distributed relaxation approach is a SKI)-

arate treatment of each of the factors (elliptic and

hyperbolic) (:onstituting the system of I)artial differ-

ential equations. Another distinguishing aspect of the

approach is that these fa('tors me t realed directly for

steady-stale flows rather than through pseudo-time

marching methods; tim('-dep(,ndent fl()w soh'ers can 1)(,

('oilstru('ted within this al)l)roach an(1 in prin(:iple are

simpler to develo t) than stea(ly-stat(, soh,ers. An ('x-

tensive list of envisioned difliculli(,s in attaining TME

o1,' 15

AMERICAN [NSTITV'I'E OF AERONA1-TI('S AND ASTI/ONAIVII('S [)APEI/ 2001-2570

for CFD simulations, ahmg with possible solutions, are

discussed elsewhere, c _' This paper also smnmarizes re-

cent progress towards ttw at tainment of TME ill basle

CFD sinmlations.

Foundations for Textbook Multigrid

EfficiencyTilt' basic framew()rk for TME solvers is full multi-

grid (FMG) algorittmls. '-''a'<+ " Ill FMG algorithms.

tilt, solution process is started on a very coarse grid

where tilt, eotnlmtational cost of solution is negligibh,.

The coarse-grid solution is then interpolated to the

next fine grid to fl)rm an initial at)proximation. Few

nmltigrid full approxinlation scheme (FASt cycles, or

possibly just one, are performed next to obtain an im-

proved fine-grid solution at)proximation. Then, the

process proceeds to finer grids until the solution on

the target finest grid is achieved.

In solution of highly nonlinear problenls, a good ini-

tial guess is important. A general way to obtain such

all initial guess is ])y continuation, in which the so-

lution to the target t)rotflem is approached through

the solutions of a sequence of paranwterized problems.

Usually the prolflem starting tile continuatioil process

is easy to solve, and ditfieulty gradually increases with

control parameter at)l)roaching the target value: this

COlltitluatioli process earl often be integrated into all

FMG solver. For examl)le, with viscosity as the control

parameter, at the coarse grids 1here artificial viscosity

can be used, then gradually be taken out as the alge-

r|tirol proceeds to finer levels. Such FMG continuation

is often natural because larger numerical viscosity is

introduced on coarse grids, even without ainfing at

continuation.

A version named A-FMG algorithm provides tile

device needed fl)r Ol)timal a(tal)tive local refinement.

Effi('ient multigrid solvers based on this apt)roach have

}teen demonstarted, m

The objective of FMG algorithnls (and TME meth-

ods in particular) is fast convergence to the solution of

the differential equatitms, not necessarily fast asymp-

totic residual ('onvergene(,. The natural solution toler-

ance is the diseretization error defined as the difference

between the exact solutions of (tiserete and differential

l)roblems. Thus, tile quality of a solution approxinla-

lion on a gix'en grid can be measured l)y the relative

magnitude t)f algebraic errors in eonq)arison with the

dis('r(,tization error h,vel. The algebraic error is defined

as the differ(uw(' between the exact and at)proximate

solutions of the t[iscrt,te problenl. ()n any grid in all

FMG alger|tirol, we ext)ect the algebraic errors after

fl,w multigri(1 ('),('lt,s to t)e always less than tile dis-

cretization error.

()n the oth(,r hand, a fast residual convergence is

et)nsittered as an itnl)t)rtant monitoring tool. Ill ttlally

l)ra('tit'al eases, it is t)ossil)le to (tevelo 1) a st)lver ex-

hil)iting fast residual eonverg(,nee rates without (:onl-

pronfising TME. Note however that sometinms the

quality of the target-grid solution can be much im-

l)roved by doul)le disert'tization methods applying for

relaxation a (tiflbrent scheme than that used ill cal-

culating residuals transferred to the coarse grid; zero

target-grid residuals nfight not be tilt, aim in this east,.

Standard multigrid methods eflieient for elliptic

prot)lenls separate tilt' treatnmnt of oscillatory and

Stllooth error (:OlllponelltS. Th(' folIll('r are elficient]y

reduced in single-grid iterations (relaxation): th(, lat-

ter ar(, well al)l)rt)xilnat(,d on coarse grids and. hen('e,

elinfinated through th(, coarse-grid correction. The dif-

ficulties assoeiated with extending TME for solution of

the RANS equations relat(, to the fact that thes(, equa-

tions are a SVS)("III ()f (:ou])led lloIlliIwar etlttatiolls that

is not, even for subsonic Maeh nunfl)ers, fully ell|It-

tic, bill contain hyt)erbolic l)artitions. Tile etfieieney

of classical multigrid ntethods s(werely degrades tbr

nonellil)ti(' t)roblelns becaus(, some Slnooth characteris-

tic components cannot be adequately al)proxinlated on

coarse grids, lj la The characteristic conq)onents art,

nmeh smoother ill the characteristic directions than in

other (tirections. To ])e effit'i(mt, a nmltigrid solver for

nonellit)tic problems has to adequately address three

tyl)eS t)f errors: (1) high-fl'equeney error ('tmq)onents,

(2) unifi)rmly smooth error (:omponents. (3) ('harat-ter-

istie error eonll)onents.

If the target discretization is strongly h-elliptic

(or .semi-h-elliptic) ()he can design a h)t:al (or bh)ek-

wise) relaxation t)roeedure efficiently reducing all high-

frequency error ('Otll|)Oll(?tlt, s. By. (tefillttlOll,-"" " ') :¢' 8, I 1 _.l

discrete scalar (not necessarily elliptic) operator L[u]

possesses a good m(,asur(, of h-ellil)ticity, if the abs()-

lute vahle of its sylllbol tL(0)l = [e-i(°J)L[(,i(°'J)]l iswell separated frtml zero for all high-frequency Fourier

modes. Here j = (j,., Ju, J=) at'(, the grid indexes and

# = (o_.,o.,o=),o <_ 10.,.I,10.1,10=l_< ,_ _,,, nornlalized

Fourier fl'equeneies. High-fl'equeney Fourier nlo(les are

the modes satisg'ing max(J0, I, [0,[, 10_-I) > ._. For sys-

tems, the measure of h-ellipticity is defined as the

llleasure of the deternfinant operator.

Coarse-grid correction is usually efficient for uni-

forniIy smooth error components. All effective reduc-

tion of characteristic error COIllpOllelltS call ])e achieved

either 1)3' designing a prol)er relaxation s('henw reduc-

ing n()t only high-frequency but smooth error eonl-

ponents as wetl (which (:all be done in many non-

unifornfly-elliptic cases by downstreanl ordering of r(,-

laxation Stel)S I t. tel or t)y adjusting ('oarse-gri([ opera-

tors for a ]letter cllaracteristi('-('Oml)onent al)l)roxinm-

tion.

Multigrid nlethods efficiently reducing all tilt, three

afl)renlentioned tYl)es t)f errt}r have been develol)ed for

scalar n(meIIipti(' el)craters, la _? Sinfilar elTit'ieney fi)r

solution of the RANS systenl of differ(,ntial equations

can l)e achieve(t t)y ('xl}loiting the syst(un faett}rizabil-

itv. T}zetorizability is a t)rol)erty of th(' syst(,nt det(u-

2 OF 15

AMERICAN tNST1TI'TE ()I-' AERONAITI('S AND ASTRONAIVl'I('S PAPEtt 200t 2570

mimmt to be factored as a product of simph' scalar

factors representing tile (elliptic and hyperbolic) par-

titions contributing to the target differential system.

For the sllt)soIlic c()mpressibh, Euler equations, the

first TME solvers exploiting factorizability of the sys-

tem have been developed by Ta'asan.l_ 2o New canon-

ical 'variables have l)een introduced, and in these vari-

ables, the Euler system of equations has a blot:k upper

triangular form with the main diagonal blocks con-

sisting of the basic components of the systenl. '-'! The

main disadvantage of this formulation is that it is not

easily generalized to viscOllS and unsteady problems,

especially in three dimensions.

Another approach toward achieving TME for so-

hi(ion of the Euler and incompressible Xavier-Stokes

eqlmtions'-,'- >2_ is based on the pressure-equation for-

inulation which effectively separates el[il)tic and hy-

perbolic factors of the system. This formulation

has been extended to generalized coordinates. '':,'2_i

(;rid-indepel)dent convergence rates bare ])eel_ demon-

strafed for inviscid flow around airfoils and for viscous

incomt)ressible flow past a finite flat plate and a non-

lifting airfoil. This approach has met difficulties in

g('neralizing to vis('ous ('oInpressible flows.

A more general approach is the distributed re.laxatio'n

meth(id._, a, l'_,.2v 2,) Th(, details of the method are pre-

sented in the next section. An important fi,ature of

this atlt)roa('h is that the sel)aration of different fa('-

tots (:onstituting the system determinant oc(:urs only

in ('omlmting ut)tlates at the relaxation stage of an

(litter multigrid s()lver; th(" original COUlfled (conser-

vative) equations at'( ) always used to cottlpIIte resi¢]-

uals. This feature allows a lot of Keedom in relax-

ation schenm design since different schenws may be

at)plied to different flow regions. The distributed re-

laxation approach reduc(,s the problem of relaxing a

complicated system of discretized coupled differential

equations t() relaxation of scalar factors constituting

th(, systenJ determinant. Note that relaxation schemes

for the scalar factors may include separate multigrid

soh,ers. Usually, distributed relaxation can be ap-

plied throughout the entire domain having th(' fifll

effect away fl'on! discontimtities (shocks, slip lines) in

the regular (smoothly varying) flow field. Sore(, local

relaxation sweel)s should l)e applied in these special

regions after (an(l perhaps also before) the distributed

relaxation pass to l'edu('(' r(,siduals. The general ruh,

for eHieient adat)tive relaxation is 1o allIily additional

relaxation sweeps wherever local residuals significantly

exceed the av(,rage level characterizing the n(qghl)oring

r('gular flow fi('Id.

Tit(' distril)uted relaxation schenle design for the

IIANS systenl of equations can lie significantly simpli-

fied if the targt,t diseretization is also factorizable, i.e..

the dis('ret(, system det('rltlillant (:all I)e reprosellted as

a l)ro(hlct of discrete scalar factors, each of them al)-

proximating a corresl)on(ling factor of t|te deterlniltant

of the differential IIANS equations. In fact, since dis-

tributed relaxation is apl)lied only for solution updates

in a relaxation sweep, the factorizalfility property is

only required for th(' principal lmem'ization el)cra-

ter. The l)rincil)al lin(,arization of a .s'calru" ('(tllati()n

contains the lin(,arization terms that make a maj(/r

contribution t(t the residual per a unit change. The

princil)al terms thus generally d(,peiM on the scale, or

mesh size. of interest. For examph', th(' dis('retized

highest derivative terms at'(' 1)rincit)a] on grids with

small enough mesh size. D)r a discretiz('d sy.stem of

differential eqllatiotls, the l)rincil)al t.erlllS are (hi)st,

that contribut(, to the principal terll).S of th(' sysIOlll

determinant. If rite prin('it)al linearizat loll is discretely

factorizable and efficient relaxation schemes for the

corresponding discrete scalar factors are available, dis-

tributed relaxation efficiently reduces high-flequency

and characteristic erx'or components as well.

For nonfactorizalfle (but h-elliptic) discretizations of

the RANS equations, the general schenu, for relaxation

updates should include tw(i different passes: (I) Di-

rect relaxation of the target dist:rete st:hem(' that is

efficient fin' high-fl'equency error reduction attd (2) dis-

tributed relaxation based Oll reasonable discretizations

(if the scalar factors of the differential system dettami-

nant (these dis('retizations are not derived froln the

target discrete system) tlt)at eliminates ehara('t(,risti('

error (;Olllt)Ollellt s.

Distributed Relaxation

Th(, system of time-delien(h,nt COml)ressible Navit'r-

Stokes equati(ins ('an b(, written as

0,q + R(Q) = 0. (1)

where the conserved varial)les at'(' Q

(ou, p'v, pw, p, pE)'r representing tlt(, m(mmntum

vector, density, and total energy p('r unit vohmw.

and R(Q) is a spatial divergence of a ve(:tor fun('tion

representing convection and viscous and ]mat transfer

effects. In general, the silnplest form of the differential

equatiollS corresponds to ttotl(:OllSel'vative equations

expressed in primitive w_riables, here taken as t.he set

composed of velocity, pressm'e, and internal energy,

q = (u,v,w,p,e) T. For a perfect gas, t h(, primitive

and conservative variables are cotlnpeted through the

following relations

p = (:_ - 1)pc,

,( )e = E-_ _[-'+ +w'-'

c _ = ;,p/p,

where c is the sl)eed of sound and is the ratio of

st)ecifi(' heats.

Tit(, tinm-del)endent llOn('onservative e(lllatiolls at'(,

fOUIld readily by transforming th(, time-del)endent con-

servativ(, (_qllat.iolls.

3 OF 15

:\MEtlI('AN [NSTITI'TE OF AERONAIITI(_S ANI) ASTRONAUT[('S PAPF;tl 2001 2570

;'q R] o,oq [0,Q + =£a ROtq + t)Q = O,

where _ is the Jaeobian matrix of the t ransfinina-0Q

tion. For steady-state equations, the tittle derivativ(,

is dropped. In an iterative t)rocedure, the correction

6q ___qn+l _ q,, where r) is an iteration counter, can

be ('()reputed Kern the equation

i) q

LSq - 0Q R, (2)

where L is a lin(,ar ot)erator containing the viscous and

inviseid terms of the nonconservative operator that

are priIwipal at the st'ale h. Thus. a good ('orreetion

is exl)ected away fl'om discontimfities in tit(, regular

(smoothly varying) flow field.

Usually. the princit)le tinearization operator L is de-

rived under the assumt)titm of solution smoothness

that requires the magnitudes of all solution differences

t(, be sn|alh,r than the sohltion magnitude. Th(, at)-

proximation inherent in this princit)al linearization ran

1,e ilhlstrated t)3" using the nonlinear conve('tion e(lua-

ti(m.

the entire system associated with L. |n relaxing s('alar

factors, ttw ('hanges imr(Mu('e(l in the "'ghost" vari-

ables 5w (tit(' variables dw ar(' "ghost" because they

need not [)e exl)lieitly used in ('()mtmtations) during

relaxation are distrii)uted, with the t)att(!rn of distri-

l)ution matrix M, t() the p)'imitive variables. To obtain

the optimal (textbook) etIi('ien('y, relaxation of each

factor shouht in('ort)orate the essential part of an (,f-

ficient multigrid solver for its ('orrest)onding ot)erator:

sometimes this essent ial part is .just the relaxation l)art

of that solver, sometimes this may even ])e an entire

sel)arate multigrid soh,er al)plied at some t)rot)er sub-

(h)mains.

hleompressible Navier-Stokes Equations

The stead.v-state in(:()mt)ressibh, Navier-Stokes

e(lllatiolls (:all ])e writtolt as

Q,,u + VP = O.

V'u=O.

where u = (u, c. ,,)1 is tit(' veh)('ity w,ct()r an(t Q,, =

u. V - v__ is a convection-(tiffusion operator. (Q = Q0

denotes the t)articular case with zero (v = 0) t)hysi('al

(liffusion.) Under the solution smoothness assuml)tion

the t)rincipal linearizati(m ()l)erat()r is given I)y

.\'(u) - ,,0t/,, = f .

A flfll linearization for a ('orreetion ($u results in

0N

0----75,, = d,,O]lu + uOf, l&, = f - N(u).

Tit(' prineitml linearization of this correction equati(m

at scale h is

u0_t<iu = f - N(,),

where the t(,rm duOJ_u can be neglected as h --_ 0 as-

suming that @u is bounded. This at)proximation is

als() termed Pi(:ar(t iteration, which is exact for the lin-

ear ease. Note that on coarser grids, the term &,0_!u

may not be so small. The FMG algorithm plays a very

important role in preventing fine-grid initial approxi-

mations with large high-frequency algebrai(' errors vi-

olating the smoothness assumt)ti()n.

While significantly siml)lified by retaining only prin-

('ipal terms, the system (2) is still a set of eout)led

e(tuations containing elliptic and hyperbolic compo-

nents. Therefi)re, ('()lh,('tive Gauss-Seidel relaxation of

L is not often effective, and fa('torizal)ility of L must

l)e exl)h)ite(l. The (tistritmte(t relaxation method re-

t)lares 6q in (9) 1)v Mdw, s() that the resulting matrix

L M t)e('omes lower triangular. The diagonal elements

of L M are comt)osed idealh of tit(, S(,l)aral)le factors

of the matrix L determinant. These factors me scalar

(lifferential ol)erators of first or secon(t order, so their

eifi('ient relaxation is a mu('h simt)ler task than relaxing

i

Q,, 0 () 0,.

0 Q,. o O.

o o Q,, O:0,. 0. O: 0

(a)

.)

(let L = -QT, ._k, (4)

where the coefficients (u,v, w) in Q,, are COml)uted

fl'om the previous solution apt)roxinmtion and fixed

during ea(:h (listributed relaxation stet). An approl)ri-

ate matrix M is

M __

1 0 0 -(9,,.

0 1 0 -0_ I

0 0 1 -0:

0 0 0 Q.

(5)

yiehting the lower triangular ()t)erator

LM =

Q,, (} 0 0

0 Q,, 0 0

0 () (2,, 0

0,. 0,, 0: -5

(6)

Euler Equations

The (:onservation term for the Euler equations is

given by (1) with

R(Q) = 0,.F(Q) + 0_G(Q) + 0:H(Q), (7)

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AMEItI('AN INSTITUTE OF AERONAI_TICS AND ASTRONAIVt'I('S PAPER 2001 2570

F(Q) =

PU_ + P

flll t

ptt It'

p,

p,E + .p

, G(Q)=

flt_t_

pv _ +p

lit!lit

t)v

pvE + rp

( /0

0

-- II

-- I/

In the regions of smoothly varying sohttion, the l)rin-

cipal linearization of the nonconservative ol)erator is

Q 0 0 l 070,0 (2 0 !0_ 0

L = 0 0 q _0- 0 (8)

pc'-'O._ pc'-'O:, pc20: Q 0

:2 , _0.dO _0,, o Q

The determinant of the matrix op(u'ator L is

a,,t L = O:_[q-' - ,:'-',] , (9)

where .5 is the Laplace Ol)erator. and Q'2 _ c'-'.X repr(,-

sents the full-l)ot('ntial o])erator.

A t)ossible distri|)utiou matrix M is given t) 3"

M z

and

(20

LM = ()

p(.2 G_.,.

Compressible Navier-Stokes Equations

The conservative compressibh' Navi(u'-Stokes e(tua-

tions are formulated in the form (1) with R(Q) defined

in (7).

F(Q) =

p,'2+ p _ 2pO.,.u- A(V .u)

l,,r- I,(0.,.,'+0_,,)

p..'- tl(O,..' + O: .)

pup_lE + ¢11_- Au(V • u) - pvl -- _'0,.(

G(Q) =

p.e- p(O,.r + O,,u)

i,,"-'+ v- "2,i,,, - X(v.-)t'""' - t'(011"' + 0._,,)

m,prE + _'1' - Av(V " u) - pr., - _0_

H(Q) =

I,U.' - tz(O.,.w + 0:.) )

Iw.. - t,(Ouw + 0: +,)

pw e+p-2pO:w- A(V-u)

pwpwE + wp - Aw(V • u) - pr._ - h'O:_

W|I(W0

r, = 2.0.. + c(0,.c + 0,j.) + W(0rW + 0:.),

7'2 = 2C0VV + U(0rr + O,,u) 4- W(O?,O' + 0:c).r:_ = 2w0:w + "(0.rW + 0:u) + c(Ot,., + 0:_,).

p and A are viscosity (:oel-fi('ients, and t," is the coeffi-

ci(,nt of heat conductivity.

The ('orrest)on(ting nonconservativ(, formulation is

given t)y

1 0 0 --10 0p x

[) 1 0 -._O u 0/,

0 0 1 -iO- 0p

0 0 0 q 00 0 0 0 1

(10)

0 0 0

Q 0 0

0 q 0

"_0 "_0. "_ ._X2, Y -r - p h

0

0

0 . (11)0

(2

(u. - - - +o.,,,)+ ,,.(u v) ,,.• ._ - 7(0._:. + O_,:'r) + =

\

P('2(V" u) + (u- V)1'+ (2, - 1)(-_A( + 4)) = (),

4(v. u) + (u. v)v- _( + _+ = 0.

(:I)_ p (2(Oq,rU) 2 + 2(O_r) :_+ 2(O=u,) 2

+(O,,,v + Ov_)" + (0.,,., + O: ,,)_ + (0_,.' + O: v)")

+A(0.,.. + 0v'_' + 0:.') _-

Assullling_ ,q()lllti()ll SlllOOthll(_Ss all([ (-OllStalll viscos-

ity and heat condu(:tion coetficienls, the t)rin('it)al lin-

earization ot)erator L. keet)ing th(, terms t)rint:it)al onboth t,h(, vis(:ous an(t invis(:i([ s(:ales, is given |)y

5 OF 15

.-\MEItI('AN INSTITIrTE OF AERONAUT[('S AND ASTItONAIrTI('S PAPEII 2001 2570

L _-

i

- ; 0_,:

I pc':O, pc'-'O.

L 4°I

p !'= 770_ 0

pc-'O: Q (1 - _)K

dO: o (2

(12)

" h-(i+l,) _.,de( L : Q_, _"-_X2:, + Q(-c'-'A+ _ -)p-

+Qj, (131where, nondinmnsionalizi|tg 1)y density and sound

speed and applying Stokes hypothesis for the bulk

viscosity term. tile coefficients l)eeonm p/p =

.,'_I_/(p Re). n = 3l_:_/(Re Pr). and i = A + I, = p/a,

M,x is tilt, hee stream Math munber, anti Re and Pr

are Reynohts anti Prandtl numhers respectively. In-

stead of devising a suitable relaxation method for the

t'omplicated scalar factor in tilt, brackets of (131, one

can opt to a (tistrilmted relaxatio|! t)artially decoul)ling

the linear system associated with operator L (121. In

l)ar|i('ular, the distrilmtion matrix

1 0 0 - lO.,. 0P

() 1 (1 - _[)_j 0

M = 0 0 1 -±0 0 (14)p :

ion, io:,,\o= oP

0 0 0 0 1

results ill

Q_ I) 0 o o 10 Q, 0 0 0

LM= 0 0" Qe 0 0 , (15)

"PO.r PO_ PO'. QQ_c"A (1 - 7)_A

where "P - tu"-' + iQ. Tilt' last two equations re-

main tout)led, requiring a bh)ck 2-115"-2 matrix soh|tion.

This dislritmted relaxation scheme is still much less

ext)ensive than direct relaxation of matrix L requiring

solution fl)r a block 5-t)y-5 matrix.

Relaxation of Scalar Factors

Etfit'ient'y of the distrilmted relaxation schemes out-

lined in the t)revious section is determined by tile ef-

fi('ien('y of tilt, relaxation (solution) schen|es for scalar

factors appearil|g at the main diagonal of the nlatrices

LM.

For unif()rnfly ('lilt)tit' operators su('h as Lat)la-

('tan. (liffusi(m-thmtina|ed (:onvection-diffusion opera-

tor. anti subsoni(' full-f)otential ot)erator many efli-

dent relaxation tedmiques are available (set, text-

t)ooks:_'<s"). For such ot)erators, an important re-

laxation re(tuirement is efficient reduction of high-

fl'equency errors. All tilt' smooth coinponents art, well

aI)proxinmte(t on ('oarse grids buih by standa|'d (full)

('oarsening: therefor(,. Ill(, coarse-grid (:(wre('tion is (,f-

fit:tent ill rt,duction of smooth errors.

For nonelliptic and weakly ellit)ti(' factors, e.g.. con-

ve(:tiOll. (:OllV(,(:tiOll-dOllliilat(_d (x)llve('ti()ll-(]if[usioll,

transonic an(l supersonic fifll-t)otential operators.

(smooth) chara('teristic conq)onents cann()t t)e al)l)rox-

imate(t with standar(l multigri(t metho(ts.11 t:'. n_.:m

Several al)t)roat'hes aimed at curing tilt,

('hara('tt'ristit'-comt)()ne|lt l)rt)l)tem have heen stu(lie(l

in the literatur< These ai)t)roach(,s fall into two

(:ategories: (1) devvlolmmnt of a suitable relaxation

s('henw (singh,-gri(1 m(,tho(t) to (qiminat(, not only

high-fie(luen(:y error COlllt)Ollellls ])llt the (:haI'a(:t(,ris-

tic error conq)ol|ents as well; (2) devising an a(ljuste(t

('oarse-grid operator to at)l)roximate well tit(, fine-grid

('harat:teristi(: (,rl()l (:()ml)Oll(q|ts.

Single-Grid Methods

Downstream marching

For hyl)erl)olie problems, the Mint)lest first-category

meth(M is downstream marching. If tile correspond-

ing discretization is a stal)le ut)wind diseretization an(t

the charat'teristi(' fieht (h)(,s not re('irculate, then down-

stream marching is a very efficient solver that yMtls

all ac('urat(, solution to a nonlinear hyI)erboli(' ettua-

tion ill .just a few sweeps (a single (h)wnstrea|n swee 11

provi(tes the exa(:t solution to a linearize(l t)rol)lem).

The (loWllSti'ealll marching t(whnitiue was successfully

apt)lied in solving many CFD prot)lems associated with

non-recirt:ulating flows (see. e.g../_' u',,25,',r,2u). How-

('vt'r. if a discretization ol)erator is not fifth ut)wind

(e.g.. is onh" upwin(t t)iase(l), straightforward down-

streant marching is unstal)h,. For the st'homes that

cannot be (tire('tly mart:he(t, there are two possil)le al-

ternatives (also of marching tyl)e): defect-c()rrection

and t)redictor-('()rre('tor methods.

Defect CorT_ction

Let us consider a defect correction method for a dis-

cretize(t hyperl)olic equation

Lhu,_,i'_, = fil.i._,, (16)

with st)ecitied inflow boundary conditions u0.i:.

Lc.t [lil,i._, t)(_ the current soh|tion al)l)roximation.

Then th(, iml)roved al)l)r()xinmtion _il,i2 is ('alt'ulate(l

hy defect-correction st'benin in the tbllowing two stet)s:

1. The correction ¢,,_.i_ is ('alt:ulate(t l)v soh, ing ol)er-

ator L_} with a right-hand side ret)resented t13" the

residual of (16) (:onqmte(t for the current at)t)roxi-

mation fii_.i._,. Tit(' inflow boundary ('onditions for

r are initialized with the zero values.

t' ' _ (17)Ld_i, ,i,: = fi, ,i,, -- Lhhil ,i.,.

2. The current apt)roximation is corrected as

[lit,i=, : (til,iu -'[- Ui] ,ie" (18)

(i OF 15

AMERI('AN [NSTITU]E OF AI<HONAUTICS AND ASTRONAUTI('S PAPER _00[ 25_'0

The operator L,') is called the driver operator. It is

chosen to be easily solvable and usually less accu-

rate than the target olmrator Lt'; the latter can be

very general. If the iteration c(mverges, steps (17)

and (18) can be repeated until the desired accuracy

is reached. Usually the eflit'iency of (lefet't-corre{:ti()Ii

methods is {lllite satisfactory. 27"2s'31 3:_ ovell though

in principle the cotwergence rate of a defiwt-correction

method for nonelliptic operator is normally mesh-size

(tel)endent.:_.t a6 as ext)lained below.

In several papers (e.g.,a2':_7), authors stu(tying the

defect-correction method for nonellit)ti{' 1}rol}lems oI}-

serve(t a Slow ('onvergeltce or eVell a divergen('e iit s()ille

conmton error norms for the initial iterati{ms and good

asymt)toti{' convergence rates afterward. This behav-

ior is different from that observed in solving ellii)ti('

I)roblems by the defect-con'ection method, where the

a symi}totic convergence rate is the sl{m'est one. This

nonelliptic feature is ext)lained by some properties as-

so{'iated with the {'ross-chara('teristic interaction (e.g.,

dissipation and/or dispersion) in the operators in-

volved in the defect-{:orrecti{m iterations. SI}e('ifically,

this cross-characteristic intera('tion defines th(' pene-

tration distance (also ternmd "survival distance"l'-')

()f a chara('teristic component. The t)enetration dis-

t.an{'e is the distance fl'om the inflow l)oundary wilhin

which tile discrete solution of the homogeneous pr()l)-

lent reasonably al)t)roximates the continuous (me (i.{'.,

tit(' discretization error is substantially smaller than

tit(, solution).

The penetration distan('e of a characteristic con>

t}(ment is roughly tnoi}(n'tional to _, l(_'h)-{, where

q is the highest order of differentiation in tit(, hy-

t}erbolic (}l)erat{}r ltnder considerations, p is the

discrete-ol}erator approximation order, w is the cross-

chara{:teristic frequency of the characteristic (:(mq)o-

nent, and h is the mesh size. Tit(' ratio of penetration

distances of the operators L/' and LI} is an imt)ortant

factor for {tet.ermining the mmd)er of deflect-correction

sweeps required to r('{hwe the algebraic error to the

discretization-error level or to reach the asymptoti{:

(:Oltvergence regiille.

When the ot}erators L I' and L_ have the same

al)l)roxiination order (p = r), efli('iency of the

deDct-correction method is (}t)timal and mesh-size h-

independent If however the (}t)erators L _' and L_} have

different approximation orders (p and r, respectively.

p > r), then efficiency of the defect-correction method

is h-det)endent; i.e., the maximal mmtl)er (}f sweel}s

which might t)e required to reduce the algel)raic er-

ror to the (liscretization-error level (or to rea{:h the

asymptotic convergenc{_ rates) is larger on fine grids

than on coarse grids. This is I)ecause one has to iterate

LI} as many times as needed to attain accuracy Up to

the L _' t)enetration distance. The worst (largest) ratio

of l}enetration {list, an(:es is ot}tained t"(}1 characteristi('

{:{)report{rots f{}r which tit(, I)enetration distance in the

7 ov 15

target /*-order a{'curate diseretizati(m al)I}roaches tit(,

del}th R of tit(' {'(}mlmtational (l(mtai|t in the {:harac-

teristic (tire('tion. It follows that the re(tuired munl}er/222

of iterations is l}rot){}rtional t(} T

PTvdictof Corrector

One l)()tentially efficient but yet unexl)l{}ited

met h(}(l to overcome gri{l-del)en{lent convergence ext)e-

rien{:ed in defe('t-('orrection iterations is the t}redictor-

corre('tor te(qmi{tue. A (tetaile{I l{}ok int(} the defect-

correction iterati(m reveals that the ('omt}|itational

work distrilmti(m is unl}alan('ed: (1) Driver ot)erator

iterations at lo{'ati(ms I}eyond the t)enetration distan('e

do not improve the solution appr(_ximation. (2) h] suc-

(:essive iterations, the solution approximati(m near the

inflow t}oundary I}(w{}mes much more accurate than intlte interior; the ('omt)utational efforts st)ent in this re-

gions coul(l l}e more t}roftal)ly invested at tim accuracy

frontier.

Tile predictor-corrector metho(l has been exten-

sivel.v use{| for ordinary an{l time-dependent differen-

tia] e(luati{}ns; :_s':m however, at)i)li{'ations for stea(ly-

state nonelliptic t}roblenls have been very linfite(l. In

predi('t{)r-corre('tor schemes, the final ul}date (}f the s{}-

htt, ion at a t}articular point is eomtmted from the local

sohttion of the target ot}erat{}r. The sohttion vahles

at downstreain p(}ints inchMed in the target-ol)erat(}r

stencil are pr(,di('ted front the solution of the driver

(t)redict{}r) (}t}erat{}r. In order to (lefin{_ a family of

l)re(Ii(:tor-(:orre(:tor schellles, oIle (:all divi(t(, the COIll-

putational dolnain into several time-like layers; the

frst layer {-ontains all tit(, grid t)(}ints adja('ent t{} the

inflow boundary. Each next layer is composed of tit(,

grid points that contritmte to the stencils of target ol}-

erators defined at the points of the previous layer and

(1o n(}t t)el(}ng to any of the previ(}us layers.

N{}w, a family of predictor-corre('tor s{:hemes f{}r

solving the (:orr{'ction equation

Lhvi_ .i.., = R_', .& = fi, ,_., - L/'u& .i.2, (19)

where L/' is tit(, target operator, fi&.i__ is the current

solution at}l}r{}ximation with residual RI' ,i,_,, and Pi,.,_

is the desire{I (:orrection flmction, can t}e {lefined as

PCo: The solution of (19) is apt)roxiinated t)y s(}lving

h, h (2{})Ld_ &,i2 = I{&,i2.

This scheme is identical with the {lefect-{'orrecti{m

scheIlI(,.

PCt.: Recurs|re definition of the derived t}redietor-

corrector s('helnes (recurs|on with rest)e('t to k)

can t)e (lone as follows: Assume the (j- 1)-th layer

have aheady been finally ul)dated in the current

sweep. Then. to cal('ulate new values at t.h{, next

j-th layer one has to l}erform the following three

ste])s:

AMERICAN INSTITUTE OF AERONAUTI('S AND ASTRONAI'TI('S PAPEt/ 2001 ')570

1. To predict values at the j-th layer with

PC_._ I scheme.

2. To Inedict values at the (j + 1)-th layer with

P('_ - 1 scheme.

3. To update value,_ at tlt(, j-th layer by directl,v

relaxing (19).

The sehenms for k = 0, 1.2 have been tested for a

linearize(l supersonic full-potential olterator. 17

Multigrid methods

Recirculation

Downstream mar(:hilkg methods are not viattle for

pi'ol)lenls with closed characteristics. Alternative dis-

('retization and solution te('hniques should 1)e (:onsid-

ere(t. The discretization issue becomes especially im-

l)ortant for flows with streamlines that d() not start

and end at l)oundaries, but ('onstitute closed curves. In

su('h cases, (,yen a very small viscosity t)lays mi |roper-

tan! roh' in determining the flow through(rut the do-

main. The solution in the linnt (if vanishing vis('ositv

det)en(ls very strongly on h()w the vis('osity coefficients

tend to zero. The l)rot)agation of information flom the

boundary into the domain is governe(l by tit(, viscous

terms n(t matter how small they may be. It has been

sh()wn m for both the scalar convection-diffusi(m itro|)-

h,m and the in('Omltressible Navier-Stokes equations

that varying ('ross-streant mmlerical viscosity (('aused

usually by varying angles between the stream and the

grid lines: e.g., in standard upwind att([ upwind bi-

ase(t schemes) may prevent ('onvergen('e to a t)hysically

realizable solution. In tit(, most general case. it can

1)e shown that even isotropic viscosity is n(tt suffi-

('lent for convergen('e to a physical solution, and one

must actually sl)e('ify a 'un_fo)-m vis(:()sit3,. However,

fi)r the hom(tgelmous l)i'oblents there at(, several indi-

('ations 40"41 (thollgh no t)roof) that isotropy suffices.

To obtain a discretization scheme that exhibits the

at)t)ropriate 1)hysical-like behavior for vanishing vis(:os-

it3. (me must ('ither add suffi(:i('nt explicit isotrot)i('

vis('osity that will doIninate the anisotrot)ic mmmri-

('al vis('(tsity of the ('onveetion operator, or else derive

a discrete conve('tion operator with numerical viscos-

ity satisfying the con(lit|on of isotrot)y. An upwind

isotr(tt)ic-visc()sity dis('retization has been derived. '11

()ne general approach t(t th(, algebra|(' solution

()f nonellit)tic equations with ch)sed characte)'istics is

to apply a multigri([ method with an (tverweighted

upwin(l-biased residual restriction. Efli('ient nmhigrid

solvers fi)r re('irculating convection equation have al-

ready t)een demonstrate(1. 41,42 This al)t)roa('h is well

(:ombined with the distributed relaxation method for

th(, RANS equations, because within a distributed

relaxation sweel) a mull|grid solver with optimal over-

weighting ('an t)e al)plie(t to a sel)arate scalar nonellip-

tic e(tuatioll with closed ('hara('t(,risti('s.

Another solution at)preach is to apply some gen-

eI'al techniques to altproxinmte in(tireetly Sln(toth ('har-

acteristic ('Omltonents. Among hell)tiff t e('hniques

are reconfi)ination ()f iterants. ('v('h_s with high in-

dexes, and implicit alternative-directi()n relaxation.

Re(:oml)ination of iterants (solution approxinmfi()ns

(m diflbr(,nt stages of a multigrid alg()ritlml) at ('ach

grid level elinfinates several (numlter of iterants re|-

tilts ()tie) erFor (!Olllt)Oltent.'-i, those, ln()Y(, speci_ic;-tHv.

that are most l)rondnent ill tlt(' residual flmction.

Making increasingly many ('oarse-grid iterations p('r

eacll fine-grid iteration, e3(:l('s with high in(lexes

Solve tile (:hal'a(:teristic-contl)onellt t)roblenl (Ill (X)al',',ieF

grids. Implicit alternative-direction relaxation simu-

lates downstream marching in the regions with open

('hara(:teristies an(l efficiently transfers information in

the regions with charaeteristit:s closely aligned with

the grid. Theoretically, ea('h of these meth(tds can-

not completely resolve the probl(,n_ of poor coarse-grid

('orrection t() the fine-gri(l smooth chara(:teristi(' err(it

('(nnl)onents. Th(, l)rol)lem ah'eady manifests itself in

two-h,vel algorithms with any tyi)e ()f local relaxation.

()n fine grids the numt)(,r of l)rol)h'mati(" error ('ompo-

nents may increase, and many ('ycles may be needed to

colh'ct the necessary number of the fine-gri(l it erants Io

exelu(le all the tr()ul)ling error (:omt)onents. However,

it has lteen shown ext)erinlentally |:' that a (:ombination

of implicit alternative-direction defe('t-eorrecti()ll type

relaxation, re('ombinati(tn of iterants on all the levels,

and W-cych's ('an result in a relatively (,tfieient multi-

grid solver for recireulating flow pr()bl(,nis (m l)ractical

gri(l._.

Full-Potential Operator

Tit(' full-t)otential operator is a variable tyl)e ot)er-

ator. an(t its solution requires (liflbrent pr()cedures in

subsonic, transonic, an(l sut)ersonic regions. In d(,ep

subsonic regions, the fidl-potential operator is mfi-

fornfly ellit)tic and therefore standard nmltigrid meth-

ods yield optinml effi(:ien('y. When the Math num-

})eF aI)l)rO_-lches unity, the ol)erator I)eeomes in(Teas-

ingly anisotr()pi(' and, 1)eeause smooth characteristic

error components (:annot be at)l)roximated a(tequately

on coarse grids, classical nmltigrid m(,tho(ts severely

degrade. In the dee t) supersonic regions, th(, full-

potential operator is unifornfly hyperbolic with the

stream direction serving as the time-like dire(:tion. In

this region, an efficient solver can t)e obtained with a

downstream marching method. However. downstream

marching becomes problemati(' |i)r the Math nmnber

dropping towards unity, because the Courant numl)er

associated with this ntethod becomes large. Thus. a

s])eeial procedure is re(luir(,(t to provide an efficient

solution for transonic regions. A t)()ssib]( , l(tca] pro-

cedl.lrel3" 15, 17. :¢0 is ))_t,'-;od on pieeewise senlicoal'bening

and some rules for adding (tissipati(tn at the e()arse-

gri(1 levels.

8 OF 15

.:\MERI('AN INSTI'IUI'E OF AERONAI'TI('S _NI) ,@4TRON _UT[("-; ])APER 2001 2570

V

V V

u e,p u c,p u

V V

Fig. 1 Staggered arrangement of primitive vari-ables for Navier-Stokes discretization.

A similar technique can be used to construcl an

efficient marching-fl'ee nmltigrid solver for convection-

(hmlinated equations. This method 16 employs a co]-

oretl relaxation schenm and is very att.racliv(, tbr mas-

sive paralM c(nnputing. A highly para.lM nmltigrid

solver for the supersonic full-potential ot)erator may

lie obtained 1)v methods similar to the wave/ray nmlti-

grid./4

Discrete EquationsTraditional Factorizable Schemes

As mentioned in tilt, introduction, factorizability tff

a target discrete schelne significantly simt)lifies the

(tistributed relaxati(m design. The main obstacle ill

this case to efftqellt solution is that the discretiza-

tions o|)tained for the s(:alar factors in the dis(:rete

deternfinant are not always convenient. The search for

new factorizabl(, discretization schemes (see Summary

of Recent Progress below) is chiefly motivated 1)3' the

need t.o derive discrete schemes with the resulting dis-

t:retizatitms of scalar factors satisfying some desired

l)rt)l)erties (e.g., correct alignment with tile physical

anist)tr(q)ies, ('onq)aetness, availability of an efficient

relaxation s(:heme, etc.).

Staggered-Grid Discretization flu Navier-Stokes

Equations.

The staggered-gritl discretization dating back t() the

mid 60's 44 4(i is one of the frst factorizable discretiza-

tions for incoInpressible flow equations. Comt)ress-

i|)le flow discretizations with a staggered arrangement

of variabh,s hart, also been studied. ;_'_r'_r A usual

l)lacement (if prinfitive variables ill tWO dimensions is

depicted in Figure 1. With this staggering, (a) the

off-diagonal first derivatives in (3), (8), and (12) can

be aI)t)roximated as short central differences: (b) the

see(rod derivatives in (12) ('an t)e COlnp(isitions of cor-

responding central first derivatives; (c) the conw,ction-

diffusitm operators, O,,, can t)e al)l)roxinmted by any

proper tliscretizations Q_',. For discrete fat:torizal)ility,

!) OF

it is inlt)ortant to have the sanle discretization i_r ea('h

of th(' (),,-ol)eratt)rs in the lliOlllelltlllll e(luatit)ns: the

conv('eti(m-diffusi(m operators in other c(ttlations call

1)e different. The convet:tion terms in the momentum

and energy t,(luations are usually upwind or upwind-

t)iased; for simplicity, below we assume that all rims(,

terms art, the same. With such differencing, the dis-

crete schenl(,s mimic the factorizatlility l)rot)erty of the

differential equations, and the discret(, system deter-

minants can Im factore(l as ttet L h = (O h)e2x_' (incom-l)ressil)le Navi,,r-Stokes) or detL h = (Oh):_(QhOl, -

c_-s h) (c()nqir(,ssibh _ Euler), where_S t' ill thre(' dimen-

sions is th(, seven-point h-Lat)laciml, Qh is an upwind

or ut)wind biased discretization of the convection op-

erators ill tll(' illomentillti ail(] ellergy (,(tuatiolls, Oh

is a conv(,(:tion-operator discretization for the pressure

term in the fourth equation of (8), hem:t' QhO,, _ c'_,_S_,

is a tlis(:rt,tt, al)i)roxinmtion t() tilt, ful]-t)otentia] op-

eratol. The ttiscrete determinant ('onqmted for th('

('ompressil)le Navier-Stokes e(luati()ns is sinfilar to the

(tifl'ert,ntial deternfinant (13).

The discrete distritmtion nlatri('es fl)lh)w directly

from tilt' continuous matrices (5). (10), and (14). Th('

short central differen('es are used for th(, alll)r()xinla -

tion of all the off-diagonal first derivatives: the con-

vection parts in th(, Q-operators art' tile saint' as those

ill the IllOllielltUlli (,quations. Tilt' resulting l)rottucts

LhM h art, similar to those for tilt' continuous l)rob-

lems with th(' main diagonals (:Oml)OSt'd of the factors

of the discrete deternfinants.

Distrit)uted-relaxation solvers have been sut'cess-

fully at)l)lie(l to the staggered-grid (liscretization

s('hemes for subsonic c(mq)ressibh ,_r and incompress-

il)h0e.2s flow 1)rol)lems.

In COml)uting the Euler system of equations, the

nlain disadvantages of the stagger('d-gri(t sch(,me relate

to the discrete stencil of the flfll-t)otential olierator.

For subsonic flow l)roblems, tile downwind differencing

apt)lied for the (_t_ term resuh.s in a fu]l-t)otential-

operator sten(:il that is sonwwhat wide (t)ecause of

the Q"@ term) and l)oorly aligns with the t)hysical

(cross-strean|) anist)trot)ies in at)proa('hing the tran-

sonic regime. For SUl)ersonic flow, where the protfleni

is purely hyperbolic, tile stencil is not flflly Ul)wind

(even if the (_h term is upwint[ differen('ed) implying

lilt)r(' involve(1 ttl_trt:hillg S('lleIlleS.

l/ecently, a new at)l)roach to trail(ling dis(:retization

seheilles that alh)ws any desired differencing for tilt'

full-potential factor of the systenl (tetern_inant with-

out conlpromising the scheme factorizahility has been

discovered. This apliroach is (tis('ussed subsequently

ill apl)lication to central collo(:ated-grid discretizati(ms

(see alsotU), I)ut it apt)lies to staggered grids as well.

15

AMERICAN INSTITUTE OF AEI/ONAIr'II('S AND ASTllONAI'TICS PAPF, R 2001 2_570

Collocated-Grid Discrctizations fin" the EulerEquations.

Anoth(,r examph, of a faetorizM)h, schelne is a

colh)cated-grid st'hem(, with the second-order central

differencing for the off-diagonal first derivatives in (3),(8). and (12). The convection operators in the mo-

mentum and energy equations art, again upwind orupwind biased: the differencing of the conve('tion term

applied to the pressur(, may alternate fl'om down-

wind (downwind-I)iased) in subsonic n|ode t() upwind(upwind-t)iase(t) in sut)ers(mi(" mode.

A typical ditfi('ulty associated with this type of

s('henws is a poor measure of h-ellipticity in the dis-

('ret(, approximati()n fi)r tit(' flfll-t)otential factor of the

system (teterminmlt. To be lnore specifi(', let us (lefine

the collocat(,(t-grid dis(Tetization L h of the matrix op-erator (8) as

()l,

0

L h = 0

p_.'2i)i_t_

c'-' _')2b

0 0 ±02''p .l'

()h 0 _-021'P g

0 (2_' -10.-`hp -

'"-'i) 2h "='O'-'l' 0

{}

0

0

0

Qh

(21)

wh('re tit(' (liscrete derivatives. /)2t, D2h O'_,h in all off-

diagonal terms are the wide (with mesh spa('ing 2h)

second-order-at'curate ('entrai-differencing at)proxinm-

tions. All the diagonal terms, Qh ex/.et)t (_h ill thefi)urlh equation, are (tiscretized with the same second-

order-accurat(, upwind (or upwind-biased) diseretiza-

lion scheme. In the subsoni(' regime (Iu[ "2= 02 + _'_,+w'-' < ("'), the Qh-term is discretized with a second-

()rder-ac('urate downwind (or dowmvind-biased) dis-

cretizati(m. Tlw (h'terminant of the matrix operatorL h is given I)y

I0 0 -where A 2h is a wide (tis(:retization of the Laplace

Ol)erator. Tit(, fllll-I)otential-ot)erator at)l)roximation

aI)l)earing in tile brackets has two major drawbacks:

(1) For slow velocities (lul << c), the discrete oper-

ator is dttnfinate(t by the non-h-elliptic wide Lal)la-clan. and effieienev of any local relaxation severely

degrades. (2) For near-sonic regimes (the .Math nun>t)er M = ]u[/c _ 1), the discrete operator stencil does

not refle('t tit(, physical mfisotropies of the differential

full-t_otential operator: the (tiscrete ol)erator exhibits a

very Stl'Olt g coupling ill tile streamwise (tirt,ction, while

tit(, differential t)t)erator has strong COul)ling only the('ross-streanl directions.

S[,veral approaches tt) cure the lack (if h-ellit)ti(:ity

(mainly in al)t)li('ations tt) inc(mq)ressible-flow equa-

tions) have been proposed in the literature (e.g.,4u's°).

Some ()f tit(, approaches are associated with introduc-

ti()n ()f additional terms increasing the measure of

h-ellil)ticity in the systent of equations, others pro-

pose averaging (filtering) stmrious oscillations. Tile

l)roblem of wrong anisotrolfies in the fldl-l)ott,ntial-

operator has not l)een sufficiently investigat(,(l, littwo dinlensions, it is I)ossible to c()nstru(:t a discretiza-

lion that satisfies the following prol)erties: (1) At low

Math nuntbers, the discretization is dominated 1)y the

stan(tard (with mesh sl)acing h) b-ellipti(' Laplacian.(2) For the transoni(' Math nunlbers, the dis('retiza-

tion tends to t h(' optimal (tiscretizati_)nla' J7. :m for the

soni(:-flow flfil-l)otential operator. (3) For supersonic

Math numbers, the discretizati(m becomes upwin(l

(upwind-biased) and can be solved by marching. The

t)rol)lem of ('onstrueting a goo(l high-order discretiza-

lion for the transoni(' full-p(_tential ol)erator in three

(limensions is still open.

Non-Factorizable schemes

Tilt' nta.jority of discrete schemes in cllrrent USe. es-

t)eeially for eompressibh, flow but also more recently

for ineoml)ressible flow. art, based upon a flux-splittingat)prt)ach. The basis of this approach is tit(, solution of

tilt, Rienlmm problem (i.e., the time evolution of two

regions of flow initially separated by a (liat)hragm) ap-plied on a dimension by dimension basis. This method-

ology has ena/)led the robust treallnenl of flows with

strong shocks anti cOnll)lex geonletries. However. tilt'

derived schemes are not discretely factorizable, ex(:et)lin one dimension.

These (lis(:rete etluations have always I)een solved us-

ing collective relaxation (or t)seudo-tinm-stel)l)ing) inmultidimensional multigrid algorithnts. A t)etter effi-

('ieney should t)e realizable with a relaxation st:heine

that efli(:iently reduces both tilt, high-fre(tuen('y andchara(:teristi(' error components. Such a scheme should

cond)ine two different rt,laxation methods: (1) A local

relaxation scheme treating directly the conservation

equations and re(tu('ing the high-fl'equency error con>

I)onents. (2) For redu(:tion of charaeteristi(" error con>

t)onents, a defect-correction (or l)redictor-correct()r)

method with a faetorizable driver (predictor). This

aI)proach has not been tried as yet.

Boundary Conditions and

Discontinuities

Boundaries and discontinuities introduce some ad-

ditional complexity in distrit)utetl relaxation. The

deternfinant of L M is usually higher order than the(h,terminant of L. Thus. as a set of new variat)les.

5w woultt generally need additional boundary condi-tions. In relaxation, t)ecause tit(, ghost vm'iables can t)e

added in the external lmrt of the (tonlain. it is usually

I)ossible to determine suitable boundary (:onditions for

5w that satisfy tile original boundary conditions for

the primitive variables. Examples are available 2s for

inconq)ressibl(, flow with enterint4 anti no-slip t)ound-

aries. However. to t:onstruct such (,xtra 1)OllndHry

lO OF 15

AMERI('AN INSTITUTE OF AEIIONAIV|'I('N _ND ASTRONAUTI('S PAPEII 21101 2570

conditions may be difficult and/or time-consunfing in

general. In addition, enfi)rcing these t)oundary con-

ditions causes the relaxation equations to be coupled

near the boundaries, not decoupled as they are in the

interior of th¢, ¢lomain.

Thus, near boundaries and discontimfities, the gen-

era] aptnoach :_ :' is to relax the governing equations

directly in t.erms of primitive variables. Sew,ral extra.

sweeps of robust (bill possibl,v slowly converging) re-

laxation, sltch as t)lock-Newton-I(acmarcz relaxatit)n,

can lie made in these special regions after (and t)erhaps

also before) the distributed rebtxation pass to redm'('

residuals to tilt' average hwel characterizing the regu-

lar flow fiehl. The additional sweeps will not seriously

affect the overal] comph,xity because thv numl)(,r of

l)oundarv and/or discontimfity points is usually small

in ('Oral)arisen with Kit(' number of interior p()ints. An

example of TME using local relaxation at shocks at,t

l)oundaries and distrilmted relaxati(nt over thv rest ()f

the domain has })(,ell demonstrated. ='_)

Analysis

As mentioned previously, it is important in attain-

ing optimal eflMency to understand all the diHieulties

that present themselves in ai)plication. Analysis meth-

ods are quite helpful in this regard, amt tile main tools

are discussed below. In iterative methods solving el-

liptic l)rol/lems, the main mechanisnt of convergence is

damping of error components. In solution of hyper-

bolic scalar equations, there is another very imi)ortant

eotwerg(,tl(:t' mechanisnu the downstream evohltiott of

tit(' error eonq)onents. In the I)resence of this addi-

tional mechanism, the accuracy first achieved near the

inflow I)oundary and is then propagated into the inte-

rior of the domain.

The recognition of this additional convergence mech-

anism urges modifi(:ations in the standard analysis

developed for elliptic t)roblems. Basically, one can

distinguish flmr types of analysis applied to nonel-

liptic problems: (1) standard linear-algebraic inatrix

analysis, (2) nlodified zero-mode-exclusion flfll-space

Fourier mode analysis, (3) half-space analysis of the

first, differential approximations (FDA)) l, a I. a2 and (4)

discrete half-st)ace analysis. Briefly, the first differen-

tim approximation {also called modified equation) to

a difference operator on a grid with mesh size h is the

Tayh)r expansion of this difference operator in terms

of h truncated to the first terms including the least

ItOllzero power (if h. The quality of an analysis applied

to nonellil)tic l)roblems is determined l)y how well th(,

analysis handles the characteristic components.

Matrix analysis

The most general and precise analysis in(,thods art,

the linear-alget>ra matrix analysis methods applied to

the corresl)onding linearized problem. This analysis

c(msiders th(, differen('e operators without assmnt)-

it(ms about the solution and l)oun(lary conditions. It

can be applied to variabh,-coel'Hcient i)robh,ms as well.

This analysis was finm(l very useful for analyzing one-

(lim(,nsional l)roblents. However, the enormous conqm-

rational complexity of this analysis makes it n(lt vial)le

for multidinmnsional prol)lems. Although, tile analysis

(:Oml)lexity can lie reduced considerat)ly 1)y assuming

D)urier modes in two of the three spatial directions.

Modified full-space Fourier mode analysis

The modified fllll-space Fourier ltlOde analysis is a

modification of the standard fltll-sl)ace Fourier nlo(h'

analysis excluding from )It(' consid(,ration all th(' zero

modes (tit(, modes with vanishing symbols). :) It is the

simplest and most popular type of analysis (e.g., see

appli(:ations in :c->'a:s). This atm lysis estimates only the

amplification (damping) factor. Its inherent disadvan-

tage is the inal)ility to take the influen('e of the inflow

l)oundary into account. This exl)lains its failure in

descril)ing tlt(, dOWllstreatll error (,volution. However,

the modified full-space analysis can also be useful for

analyzing the effect of forcing terms.

FDA half-space analysis

The FDA half-space attalysis is a relatively simple

and efficient tool for atmlyzing the effe('t of lit(, inflow

])oundary. Examples of at)pli(:ations of this analysis

at(, available._ t. t',_,r,l,52 The first differt,ntial approxi-

mations are ('()nsidered on a half-st)ace in('luding cut

I)y an inflow })oundary. Tilt' boundary co/lditions are

rel)resented t)y one Fourier mode at a time, Th(' FDA

analysis l)r()vides a good qualitative descriI)tion (if the

downstr(,am error evolution. This analysis fill'uses

()it characteristic COml)t)nents and, therefore, c(msid-

ers homogeneous t)roblems. Note that a conll)ination

of the FDA analysis with the/nodified full-st)ace anal-

),sis can provide a goo(t insight for nonhomogeneous

l)roblents as well. Tile disadvantages of this analy-

sis are the inability to l)rovi(h, quantitative estimates.

to attalyze the effect of different t)oundary (:ondition

discretizations, and to address the asymt)toti(" t'onver-

gell('e rate.

Discrete half-space analysis

Tilt, (tiserete half-space analysis :m" :_r,,3(_considers tile

discretizations in their exact form rather than their

differential apt)roximation, while the I)oun(tary data

are represented I)3" a Fourier component. This analy-

sis translates the original multidimensional t)rol)leln

into a one-dimensional discrete t)roblem, where the

freqtlell(:ies of the hOlllidarv Follrier Cotlll)OllelltS art,

considered as I)arameters. To regularize the half-spa('e

t)rol)lem, tit(, sohltion is not allowed to grow faster than

a l)olynonfial flm('tion. This tool is very at:curate; it

(:an lie used to ext)lain in detail nlany llhenomena ob-

served in solving nonelliptic equations and l)r()vides a

close prediction of tile actual solution behavior.

The one-dinmnsional solution obtained in the dis-

I1 OF 15

AMERI{'AN INSTITUTE OF AERONAI'TI('_q AND ASTFIONAUTI(_q PAPFR '200] 2570

('rete half-space analysis has two different represema-

titre forms: (1) away fronl the boundary, the sotuti(m

is defined as a linear combination of a finite nmnt)er of

analyti('al components; this region is called the anal qt-

ical representation region: (2) in the region adjacent to

tilt, inflow boundary, tim solution is defined pointwise:

this ztme is refiu'red to as the pointwise representa-

tion region.. \Vith each fiwther iteration described

I)v tilt, analysis, the t)ointwise r(,l)resentation region

t)('llOll'atos l)y a finite IlUllll)or of mesh sizt,s illto tilt'

interior. By using these representations, the cOnll)U-

ta!iomd eomI)lexity of tilt, analysis t)eeomes much less

than that associated wilh the (me-dimensional matrix

analysis. In the asymptotic regime, when the point-

wise rel)resentation zone covers all the (lomain, this

analysis 1)et'onws a dis(:rete one-dimensional matrix

analysis of the multidimensional prot)tem.

The discrete half-spat't, analysis provides a quantita-

tive description of the al)l)roximate solution: it t)redicts

tilt' eonvergen('e rale for each iteration and the asymp-

totic eonw,rgenee rate. It can In, easily adjusted to

analyze the global effect of any local diseretization of

the inflow boundary conditions. This adjustment can

Im done just by widening the initial pointwise repi'e-

sentation region at the inflow l_(mndary. If necessary,

the analysis can take into a('('ount the influen('e of

tilt' discretized outflow bountlarv conditions as well.

Generally, this discrete half-st)at'e analysis treats (:om-

t)h'tel.v I)oth me('hanisms of convergence, dmnping and

downstream evolution of (,rr()rs, associated with nonel-

Ill)tit' t)roblenl solvers.

Summary of Recent Progress

The first TME solver at)l)lying the distributed re-

laxation approach for st)lution of a free-stream in-

c(>mpressil)le Navier-Stokes equations has bt,en dem<m-

strafed long ago. l_ Recently, TME has been achieved

for high-Rcvnohls-nulnber inconq)ressible wake flow

and the boundary layer flow associated with a fi-

nite fiat t)late. "-'s An initial extension of this work

to compressibh, (substmic) viscous flow has also been

('omph'ted. '27 In all these calculations, a staggered ar-

rangement of varial)h,s on Cartesian grids has been

used. With distrilmted relaxation, the system of equa-

tions has been decomposed (i.e., factored) everywhere,

excet)l near t)OUlldaries whore the ettuatiolls relllailled

('out)led. Two-dimensional FMG solvers with just one

multigrid cych, per grid level and a total computa-

tional work equivalent to about l0 target-grid residual

evaluations converged the drag to the discretization

a('('ura('v.

Re('ently, a new mullidimensional factorizal)le

scheme for the Euh'r e<luations has been developed _:_

for Cartesian ('oordinates and extended through gener-

alized coordinates to external lifting flows around air-

fi)ils with t)oth sul)critit'al and super('ritical fi'eestream

Math munt)ers. '54'5F' This scheme is the first flux-

differenee-st)litting discI'etization faetorizable in mul-

tiple dimensions. The starting point for the scheme

is the first-order diseretization of the flux-difference

splitting st:heine of Roe. Correction terms are added in

the fi)rm of mixed derivatives to make the scheme both

second-order accurate and discretoly fa('torizable. The

resulting scheme is second-order accurate and compa(:t

in comi)m'ison to other scheme. Discrete fa('torizat)il-

itv is a('hieved t 6 using some non-standard wide ap-

proximations tbr sI)atial derivatives to ensur(, that the

identities

O r,rOqy ---- O ryOrg ,

O:,..,.Oy = O.,.!lO.,. ,

OuuO r = O.ruOu

are satisfied on the discrete level. The determinant

of the resulting scheme is composed of an upwind

differenced convection factor and an h-elliptic approx-

imation for the fifll-l)otential factor. The distributed

relaxation is possible by using a left and right distri-

bution matrix, although this has not 1)een applied as

vet.

In immerical tests performed for this scheme, th(,

nmltigrid solver emph)yed ahernating-direetion col-

lective Gauss-Seidel relaxation. The alternating-

(tire('tion relaxation is imcessarv sin(x, the fllll-potential

factor is not separately treated. Computations for sul)-

sonic and transoni(: channel flows with essentially grid-

independent convergent'e rates have 1)een l)resented. 54

Grid-indet)endent convergen('e rates have also been

attained for a flow with stagnation t)oints, at' The

subsonic-flow convergence rates observed in multigrid

V-cycles were quite fast (about (1.3 per cycle) and only

slightly grid dependent. The rates somewhat deteri-

orate in transonic/sUl)ersonic (:omtmtations. Further

deveh)pments of this scheme are l)resented in two 1)a-

pers at this confi, rent'e. 7,7''a(_ The scheme at)l)ties at h)w

Ma(:h numl)ers although it has yet to I)e extende(l to

vist:ous ttows.

Another at)proach to Imihting factoriza])le st'heroes

with suitable discretizations for scalar fa('tors has

been exl)lored in paI)ers of the second and third au-

th()rs.4S.57 The al)l)roach is t)ased Oll a collocated-grid

scheme with a mechanism that allows one to iml)rove

the h-ellipti('ity measure by obtaining any desired dis-

cretizations for the fifll-potential factor of the system

determinant without compromising the discrete factor

izability. Also. the distrilmtion matrices fi)llow directly

fi'om the dist:rete forms for M presented earlier. The

same at)proach t:an I)e al)plied for incomi)ressible-flow

t)roblems and to staggered-grid discretizations as well.

The starting t)(fint is the discretization (21). The

way t)roposed to improve the discrete flill-potential op-

erator is to change the diseretization of (_)h to Qh +Ah.

Then tilt, discret(, flfll-1)otential Ol)erat()r is changed to

Q_'A I' + Qt'C,f' _ c='_"1,.

12 OF 15

AMERI('AN ]NSTITI'TE OF AERONAUTI('S AN[) ASTRONAUTI_"S })APE[I 2001 2570

( )-lwhere A l' = Qh _nh _h = y, _ (?hc)h _

(:_,_h), and .7"I' is a desired approxinmtion for the flfll-

potential factor. In smooth regions, .;I I' is second-order

small (proportional to D'-'), hence the ow_rall s{,cond-

order discretization accuracy is not compromised. The

()'operator &)h is a non/oca] operator and its im, ro-

{ttlctioIl call ])e elCt'ected through a new auxiliary vari-

able t :'t' and a new discrete equation Qhch = "1"}#'1}'.

Thus. the corrected discrete approximation to (8) is

{Iefitt(,d as

L h z

Qh {I 0 0 ±0 2j' 0p d"

{} (p#' {} 0 1De#, {}-- --U

0 {} Q_' {} ¢0 _' {}

{} {} {} 0" '-'_" {}') "_'}h "} q'}D ") ,_'}h Ohpc- o.7. pc- o_ pc- o;" 1 (}

/-'0 '-_' /"<7-'* KO-' t' {} {I (p_'"r .t' ",, --IJ r ~

(28

The corresl}onding distril}ution matrix, M t'. for dis-

tril}ute{1 relaxation is defin{,d as

M h =

t _)2111 {} {} {} -7 ._'(} i {} (} - ±0 _h

p Y

{} {} 1 {} -10'_ -'#_p

0 (} {} 1 'D #'

0 0 0 {} Qh

0 {} {} (} 0

0

{}

(}

{} '

0

1

(24)

so that the resulting matrix L h M h becomes lower tri-

L h M h =

angular as

(4}h 0 (}

(I (2 I' 0

1} {} CY'

0 {} (1';' _'2_,'}/I "} '_')/i

pc-o.7 pc" c,_ pc'-' O_h

/->0" "Z-a '-t' <'-]-OP'

{} 0 (}

{} {} {}

{} {} {}

CY' {i {1

1 2_'I_ 0

0 -<<"A ''h Q_'_p

• (25)

The schenm as defined above is valid f{}r ilOll{'Oll-

servative flows. A version to be used for {tistributed

relaxation of conservative equations has also 1}een de-

signet1. 5r

Concluding Remarks

A general multigrid approach to attain TME soh,ers

for large-scale CFD al)plieations has been outlin{,(l.

This ai)pr(}aeh focuses on fast convergence to tile sohl-

lion of the differential equations, not necessarily pro-

riding fast asylnt)toti{_ {:OllVel'geli{'e rates. The consid-

ered llieasllre of the nmtho{1 et'licien{2v is C(}IIVel'gOll{'e

to the ditferential solution, i.e.. fast reducti(}u of alge-

t}raic errors t}eh}w tit{' discretization error level on each

Iliesh.

Beeaus{, the governing equations are a eout}]e{l set of

nonlinear conservation equations with dis{'ontinuities

l 3 01.'

an{1 singularities, attain|neni of full eflicien{'y re{tuires

that each of three error COlilt}OlieiltS l)e addresse{t: (1)

high-frequen{'y OH'Ors (2) unif{n'iuly sliiooth errors alid

(3) chara{'t{TiSii{' erl'ors. These err{its at'{, re{hiee{1

thr{}ugh a combi|lati{}n (}f distributed relaxation and

h}cal relaxation at each grid. The relaxati{}ns are f{}I-

]{}wed by all F._S COFi{wii{}ll fI'OllI a {:{}._ll'SOl' grid Wll{Te

t.h{' initial coarse-grid residuals are found from the

fine-grid residuals {}f the c{mservative equath}ns with

(conservative) full-weighting restri{:tioll.

The distributed relaxation procediire is designed to

reduce errors in tile smoothly varying regions of the

dtmiain and relies on the factorizability tirotierty of

the governing {tifferential equations to is{}late anti treat

optinlally different factors arising ill the {teterniinant {if

Eli{, differential otmrator. {)t}timal relaxation of sonte

t}artieular factors niay itself involve a set}arate inner

nulltigrid {'ych' (}ver a liniited sub{toniain.

Local relaxation is a procedure designe{| to re{bite

large residuals of the conservative equations at dis-

e{mtinuities/singularities/boundaries and is apl}lied in

diese regions as well as in general where tit{' resi<ht-

als are largest. It. emIll{}ys a robust st:heine (e.g., sortie

bh}ek relaxation) apt}lied l}efor{, and after a distrihuted

relaxation sweei/.

The l)artieular factors arising iii distribut{,d relax-

ati{}n of the Euler and Navivr-Stokes e{luations have

l}{,{,n discussed fl'oni the stall{ttioini of the differential

all{J the dis{Tete eqllatiolis. Methods for relaxing alid

analyzing these factors within tile multigrid {:ontext

have l}een l)resented and evaluated. Recent l}r(}gress in

(hwelopment textl}ook-effi{'ient nntltigrid solvers base{l

on the distributed r{,laxation at}t}r{lach has been sum-

marized.

References

] Huliinstein, I_., Rtunsey, (!. L., Salas, M. D.. all{] Tiionias.

.1. L., "Turbulence nlodeling workshop," ICASE lnlerinl Ilepori

37, NASA {_tt-20{}1-21{}811, March 20{}1.

2Brall{]l, A., "Gulch, to muttigrid developln{,ni," Multi.qrid

Methods, edit{!d I}y \V. Hackl)uscil and 1_. Trott{mherg, l_{,{'l, ure

Notes ill Math. 960, Spritiger-Vvrlag, Berlin, 1982.

3Brandt. A.,"MulligridTechniques: 1981 (;uidewithapl)li-

cations to fluid dynamics," L¢cturr Notes for the (?omputational

Fluid Dynamic,_., Lecture Series at the Von-Karman Institute

Sor Fluid Dynamics, The Weizmann lnslilule of Science, He-

hovot, Israel, 1981, 1St3N-3-88,157-08l-1. (;Ml)-Studien Nr. 85,

Available from {;MI)-AI\V, Postfach 1316, I)-53731, S1. Au-

gustin I, (;erniany. ?{is{} available fr(nn S{,crelary. I){'liarlnlenl

{if Iklatli{unali{:s. Universily of (2ihira{]o al J){,nver, {:{) 8{}'20-t-

5300.

4Braudi, A., "Barrh,rs tl) ac}lh,ving lextt}ook mullJgr]d ef-

fi{:[eil{:y ill {TD," ICASE hiterim f{epor! 32, NASA (IH-1998-

207617, April 1998, Updaied version available as Gauss ('PIllPl •

Report \VI/(;(_- 10 al The _VeizIliann [liSlil.ule of S('ieli('e, |{{!-

hov{}l. ]srael, December 1998.

5 Brandt, :%.. "tle{:{!nt (leveloptn{mls in mulligri{t el]ichni{'y hi

conlputaliona[ fluid dynanli{:s_" A1}pen{tix (_ {}[" IVlUltlgT"id |}y U.

']'rott{'nb{!rg, {L _V. Oosterh,{, ancl ..%. Sch/iler, A{'a<h!nii(' Press.

l,ondon, 2{}00, Pl}. 57.3 58!}.

6Briggs, \V. L., Xl{'Cornlick, S. F., and I'2.Hens{}n. V., Multi-

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15

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