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INSTYTUT BADAŃ JĄDROWYCH
ИНСТИТУТ ЯДЕРНЫХ ИССЛЕДОВАНИЙ
INSTITUTE OF NUCLEAR RESEARCH
REPORTNcH W/CYFRONET/PM/A
TWO-SWEEP ITERATIVE METHODSFOR SOLVING LARGE LINEAR SYSTEMS ANDTHEIR APPLICATION TO THE NUMERICAL
SOLUTION OF MULTHGROUPMULTI- DIMENSIONAL
NEUTRON DIFFUSION EQUATION
N01447
/CYFRONET/PM/A
Z.WOŹNSCKI
WARSZAWA
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ИНФОРМАЦИОННЫЙ ЦЕНТР ПО ЯДЕРНОЙ ЭНЕРГИИпри Учреждении по Атомной З а е р г ш
Дворец Культуры и НаукиВаршава, ПОЛЬША
Available from:
NUCLEAR ENERGY INFORMATION CENTERATOMIC ENERGY OFFICE
Palace of Culture a.id ScienceWarsaw, POLAND
Drukuje i rozprowadza:
OŚRODEK INFORMACJI O ENERGII JĄDROWEJUrzędu Energii Atomowej
Warszawa, Paląc Kultury i Nauki
Wydaje Instytut Badań Jądrowych
Nakład i 8 7 e gz. , Objętość ark. wyd.6,38, A r k . druk. 9,0 , Datazłożenia maszynopisu przez autora 20. VII.73 г. f Oddano do druku
6.XII.73 , Druk ukończono w w styczniu 1974 г.. SP-09/250/66,Zam„ 320/73
INSTITUTE OF NUCLEAR RESK.lftCH
TWO-StfSBP -ITERATIVE METHODS POR 30LV1NG LARGB
LINEAR SYSTEMS AND THEIR APPLICATION
TO ТНБ NiMEBICAL SOLUTION OP MULTI-GROUF
MULTI-DIMENSIONAL NEUTRON DIFFUSION EQUATION
DVi'UPHZlSBIEGCWE METODY ITEKACWNE ROZWI^ZlftlfANIA
DUŻYCH UIOADOV/ LINIOWYCH I ICH ZASTOSOWANIE DO
NUMERYCZNEGO ROZWIĄZANIA WIELOGHUPCVYCH
OH ШПИЛЯ DYFU2JI NBUTUONW
ЕВУПРОХОДЕМа ИТЕРАЦИОННА МЕТОДЫ ДЛЯБОЛШ-К ЛИНЕЙНЫХ СИСТМ И ИХ ПРИГлЕНЕНИЕ
К ЧИСЛОВОМУ РЕШВ1Ю МНОГ0ГРУШ10ШХ,УРАВРШИЙ ЛДФФУЗИИ НЕЙТРОНОВ
Zbigniew Woźnicki
Dootoi'al Dissei*tation
i i
Abstract
The two-sweep iterative methods described in this
paper for .«solving large syeteaa of linear equations
are a generalisation of the procedure "forward elimi-
nation - backward substitution" used up to BOW only
as the direct method /the Gaussian elimination/ for
solving systems of linear equations with tridiagonal
matrices« The theoretical foundations of the two-sweep
iterative methods are described on the basis of the
theory of non-negative matrices«, The numerical results
presented in the paper show that the application of the
two-swoop methods to the numerical solution of the
smlti-uimonsional neutron diffusion equations provide»?
a considerable acceleration of the convergence in com-
parison to other iterative methods», ,?,м_Ч"; I
Opisana w raporc ie dwupraebiegowe isetody i t c r a c y j a eroswiąajrwania dużych układów гб^пай l in io^ycb są uogól-nieniem procedury "postępowaniu proste «- postępo^raaie,odwrotne" 'stosowanej dotychczas tylko jako bezpośredniasseioda / e l i m i n a c j a Gaussa/ $© roswiasywaaia «składów
o sac ierzaeb tr«5j diagonalny oh „ Teorety—przefeiegowfeix n«etou iterscyjnyoJi opi«>
sańo w oparciM o te&ttę Шас^аггу ai^w^^fflnjrch* l>rs©astap»wioń© w r a p o r c i e wyniki nuraexyo- is wslsaaująs ź© sast©»
po«»! saacsiae przyspieszenie abieżnośoi w
r6%?s.aniu do innych metod
I l l
Аннотация
Описаны двухпроводные итерационные методы решениябольших систем линейных уравнений. Зти методы ЯРЛКГОТСЯ
обобщением процедуры "прогонки", до сих пор применяемойтолько в качестве непосредственного метода vэлиминацияно Гаусу) для решения токовых систем, в матрицах у кото-рых три диагонали!. Теоретические основы двузпроходныхметодов изложены в "связи с теорией неотрицательных ма-триц. Приставлены чиолеыные результаты указывают,, чтоприменение двухпроводных методов для решения многоди-менсиокнш. уравнений диффузии нейтронов даёт заметноеускорение сходимости в сравнении с другими итерационнымиметодами.
iv
CONTENTS
Abstract ii
Is Introduction 1
II* Theoretical Background G
111„Two-Sweep Iterative Methods 22
IV* Application of the Overrelaxation Processes 65
V e Model Probleia Analysis 78
VI„ Application of the Two-Sweep Methods to theSolution of Ыхь Neutron Diffusion Equationsand Numerical Results 90
VIT*Conolusions iOS
Acknowledgements iiO
Tables из
Figures 119
Keferenoes 135
I. INTRODUCTION
The numerical solution of the energy^dependent
multidimensional diffusion equation for a steady-state
nuclear reactor problem is usually accomplished in
several steps« In step number one, the multigroup approxi-
mation ia developedо In step number two, the multigroup
problem is solved by the souro© iteration method; and
the solution of the multigroup problems is reduced to that
of a sequence of one»group problemsa In step number three,
one-group equations are represented ia a finite difference
approximation. Finally0 ia step number four, th© resulting
equations are more or less effectively solved by various
iterative methods«,
The mtaltigroup diffusion theory saodal is mathema-
tically equivalent to an eigenvalue problem for a system
of coupled elliptic partial differential equations of the
second order of t&© form
V
where the notation is standard as defined by Hsbettlerand Mart ino [i] s and the reisoval cross section 2E °can be expressed as followss
S q -bL. Q-> e
2
Th"ise equations are supplemented by group-dependentlogarithmic boundary conditions at the outer surfaceof the reactor
9 Эф 1 3 , S л (")Эп
where к is the outwardly directed normal at the surface!
The basic method of solution of the equations (i)
is th® scixre© iteration procedure in whioh th© group |
equations «sre successively /frora the i*>st to the G -th I
group/ solved t*ith the assumptioa that the source term }л
in any group is determined for each source iteration |
prior to ih© calculation of fluxes a f
Th® numerical solution of this eigenvalue problem |
for realistic heterogeneous reactors is usually accom- |
piished by applying finite difference approximation raethod|
In this way the numerical solution of the continuous jf
problem C&M b& reduced to the solution of an eigenvalue |
problem for a system of homogeneous algebraic equations 1й
representing the discrete fou of the multigroup diffusion|
theory equationse In matrix notation those equations |
оen b© written as follows! |
кI
The order I of the square matrices u end С is |equal to th® ntraber of @rc©rgy<*>sp@ce mesh points3 у Isa column vector reprsseatisag the /approximate/ neutronfins at the I energy-space rassh poiiats and к is th©eigenvalue /fc-effectlve/ of the ргоЫеш» The saatrix 3describes th© diffusion, removalv and scattering processsee, and the matrix С represents tbe fission processes
IS
In the case of tlie discrete form ( l i t ) of tfoefew-group diffusion equations BirHioff and Vargaproved in 1957 mider the assumption of transitivityth© existence of a unique positive flux vector ф anda corresponding single positive eigenvalue к largerthan the absolute value of any other eigenvalue of theргоЫешо Next, in 1968, Froehlicb [з] prov©<I theseimportant existence and positivity properties undervery weals assumptions of connectedness t covering essentia-lly all problems of practical interest« 'Шиз, w® ae©that the theoretical foundation for the discrete foraof t.lse raultigroup diffusion equations is quite satisfac-tory at present*
The application of the зогзгее iteration method tothe system of equations Cii i) allows i t bo be reducedto a system of noRhomogeraeous equatloss of & lovm? order.
In tlłis paper we ere interested in <rae»g two*g andproblems for uniforai or
е Approsiiaating tbs ooe«»e two«5
three-dimensional diffusion theory equations (i") iaeacto energy group, for insfcase©,, by tte sa©st widelyused central difference selseEses, wa obtain, respeetively B
3°>polBt$ 5-poiats and l^potnt finitein tlis forsa of a system of !£si©ar
ш С.
equal to the шшЬэз1 ®f spas© aasla poiiail-a0 шй Ш®©istri©@ ©f Д «©emr in 0©ш©
diagonals; ф is a column veot.or of order П repre-
senting the approximate neutron flux in a given group,
aad С is а column vector of order fl containing the
fission sad scattering source terms for a given group•
For a derivation of the difference equations see
Chapter 6 of Heference 4<*
Th© nest problem is to solve equations (iv) bearing
i a mind that typical tw©*»d±mensi©nal problems involve
froia iOOOO to 2OO0O mesh points ia ©ach energy groupe
and that for three-diroensional problems th© number of
mesh points varies from 30000 to 100000» There exist
raany methods of solving SUCH large systems of linear
equations*, The successive overrelaxatien method oloasiy
related to the Gauss-Seidel Iterative method is most
widely used. There are maay variations of this method;
the point metbods the I in© method, the milt Alin© method y
the simultaneous group method @tc« At present, the flux
synthesis methods for solving the multigFotip diffusion
equations are bojng developedp as fox* езеавзр!©, th©
coars@-mesh variational diffvsioa theory techniques [s] „
Tii© purpose of this paper is to giv© theoretical
foundations sad to describe practical application of
two=»sw©ep iterative saethods used to solv® systeiss of
linear equations e Che use of the nass© Htwo»sw8©pH for
these iterative Methods is Justified bv their practical
realization which is & gemerslisatioia of the widely
used sweep teehaique® n forward elimiaation-bacterard
substitution" ia the usual Gaussian elimination for
solving eyatents ot liaear eupafcioas with tridi®goa,al
/e@© Chapter i of Sefforeace 6/e
The simeriosl eolutiosa of large eystejas of 11к©аг
equations arising from discrete approximation of th©
niblt-i-diaension l neutron diffusion theory equations
in eaoh energy group by the two»svisep oeihods presented
In this pspar is more efficient than that with the
Gums^Seideł iterative iaetbodw Moreovert the applica-
tion of th© successive overrelaxation process provides
a further aeooloration of the solution ooavergence»
Tho choice of a sero starting vector ф /approximate
neutron flux/ for all ener^ groups ia always allO; ad,
and leads to tlie acceleration of the $Ш1Уег^даг$ in
к -effective and noutroa fliuc Ф ^ n ^ ® ^irat stago
of th© iterative рго«з©азв
In reaotor problems ^her© certain energy»3£>aci3
points may ba strongly еощиой со some of Uieix*
s, aad we^cly coupled to the 'remaining onea,
the conventional iterative methods result isa slow
oonvergenoe or ©von somi—convergeao©* Typical situations
of this bind are encountered in problems with highly
non-milfona raeah spaoings, heterogeneous sa -rial
properties, or in cell calculations ith a very flas
mosh spaoing and/or weak enevgv group coispliage Is'the "
cas© of two—sweep methods w© observe ths opposite
situation, and the more шш-uniform is the mesh spacing
ia a given problemt the faster is th© convergence of
the solution obtained Ъу the two«sweep methodss This
phenomenon for a typical ttaro-dimonsional reactor proo?.ein
will b© preaeated in Chapter VI e This Isatur© иакзз
the two»sweep methods undoubtedly v®vy attraotlv© in
tho study of reaotor problews with ©oaplicsted geometry,
t?ith 1ао1ешг gaps, thin trails, ©tioe
diagonals; ф As а оо!шш vector of order a repre-
sent ing Ш© approximate neutron flux in a given group,
aacl С is a eolumn vector of order П containing the
fission aad scattering source terms for s given group«.
Por a derivation of the difference equations see
Chapter 6 of Reference 4*
She nest problem is to solve equations (iv) bearing
in mind that typical twoe-dimeasion&l problems involve
froia 10000 to 20000 mesh points in each energy group,
and that for three-dimensional pr obi eras the number of
mesh points varies from 30000 to i00000e flier© exist
шопу methods of solving such large systems of linear
equations» The successive overrelaxation method closely
related to the Gauss-Seidei iterative rasthod is most
widely used* there are sany variations of this anatbod;
the point jsothoć, the line method, the raultiline method,
the simultaneous group method etc. At present, the flu."
synthesis methods for solving the multigroup diffusion
equations ar© *seing developed,, as for esaapla, tho
coarse«»iaeeSi varlationel diffusion theory techniques [б] 0
*Ш® purpose of this paper is to giv© theoretical
foundations and to describe practical application of
two-sweep iterative methods used to solve systems of
linear equations» The use of the name ntT?o=»sweep!1 for
these iterative methods is justified by their practical
realization w&icii is a generalisation of the widely
used gws©p techniques ^forward •»•*. 'jalnatlon^baokward
substitution" in th© usual Gaussian elimination for
solving systems of linear equations with tridiagonal
matrices /ее© Chapter i ef Eeforemce Q/-,
mra©ri©al solution of large systems of linear
eqaations arising Srom discrete approximation of the
multi-disKmsional seutroa diffusion theory equations
in eaełi energy group by the two—sweep methods presented
in this papox* is mor© efficient than that with the
Gauss-Seidel iterative method» Moreover, the applioa»»
tioa of tho successive overrelasation process provides
a further aoe©l©ration of the solution convergence •»
The choice of © aero stilting vector ф /approxiraat®
neutron flux/ for all energy groups is always ellov.ad,
and loo-is to the acceleration of ЬЫ> o,?mver «3no§ in
к «effective and aoutron fliix f in Ш © iirnt stag©
of th© iterative process.
In reaotor probleas wher© certain eiies*ey-space
raash points may Ъо strongly coupled to some of their
liOtghbourSj end weakly coupled to the remaining ones,
the oonventiORal iterative methods result in slow
oonvergonoe or oven somi-coiavorgQnc©» Typical situatioaa
of this kind are encountered in problems with highly
non-uniform mosh spaoings, heterogeneous m .rial
properties, or in cell calculations with a very fisse
mosh spacing and/or weak enevgy group coupling 0 In'th< '
case of two-sweep methods we observe th© opposite
situation, and the вюге non-uniform is the mesh spacing
in a given $>roblemj the faster is th© ooHvergeace of
the solution obtained by t"h© two»sw©ep methods e This
pheaosjonon for a typical two«dimensioaal reactor problem
will be presented in Chapter VI a This feature makes
th© two«*sweep methods undoubtedly very attractive ia
the study of reactor problems with eosiplicsted
i«.ee with holes, gapa9 thisa walls9 ©t©«,
TI-«' two-sweep methods are described is this paper
on the basis of the theory of non«-negativ© taatrices
developed by Perron and Probeniws and the form of
presentation is such as in Yarga*s comprehensive шопо=>
graph [4} • 1» Chapter II the basic definitions and
theorems are given as presented in the first thre®
chapters of Reference 4 « Proofв of theorems omitted |
her© оаа tee found in this reference* In Chapter III |
the theoretical description of the two-sweep iterative f
methods is given together with the practical implements?» |
tios of these methods* Chapter IV discusses the successiv© §
overrolaacatioa process as applied to the t o«-sweep
iterative metbode^ Ш® next tw$ chapters are isainly
concerned with the extensive analysis of numerical
results* The final discussion and conclusions are given
in Chapter VXI*
IX« ТгВ01Ш?1вМ> BACKGROUND
Now the taattseraatical formulati^Q of our problem
* Let А«(О-цр ъ© ass
lat Ш@г© be sought th® eoluttloB of Ш® folicmi^g
systets of liEieasr equations
which will be writ ten £a raatris aotati©sa as
0,
«here С i s @ given real ео!шш veetoi'o Us® ©oliatioav©©tor © @si©ts cad i s emi^a® if ®sd ooly i f А 1ш
nonsingular, and this solution vector is given explic-
itly by-A
f-A С (£>We sssurae throughout thai the Matrix A is n on singular,
, moreover,, that i t s diagonal entries U, t are all
nonzero numbers«
Several definitions and theorems which will be
used in further considerations are given below* We shall
assume that all matrioes and vectors are real»
Definition,!
Let A—fait]) and B=(bij") be two ПХГ matrices»
then, A ^ B ( ^ B ) if a^bi,j(^bi,^ for aii
14] 4t r .If 0 is the null matrix and
\ть sfQ' that A is a non~negatiye ^positive/ aatrisa
/The notation A ^ B means that at least for on© pair
of indices М ^ ^ Ц ц / .
Since oolunm vectors are OKi matrices , via
shall use the terms non^negative and japsitivjs vector
accordingly throughout»
Definition.^
Рог П^.2. f an n x n matrix ^B ia reducible
if there exists an ПХП permutation matrix P such
that
PBP о (3)
where ^4Д is an fx r submatrix, bg,^ is an (n-r*) к
x(n-r) submatrix, and Ь .г is an rx(n-r') submatrix9
with 4^Г<^П a If no such permutation matrix exists,,
then В is irreducible» If В ia a 4x4 matrix,
then В is irreducible if its single entry is nonzero,
and reducible otherwise„ A permutation matrix is a
square matris which in each row and each column lias
one entry equal to unity aad all the others equal to
Definition 3
An n*n rostris ц ) is diagonally, dominant
if
for allщ
i- H e AD. П Х П matrix Л is etrictly
^ ^ ^ dominant. if strict inequality in (4) is
valid for all 4 ^ Ц n . Similarly, A is
dominant,, if A is irreducible and diag-
onally dominant, witb. strict ineqiaality ia (4) forat least om© i o
Let В=(Ьц) fee em siatrix with eigenvalue®
Таеп
" I te)9 (В)t l i S spQfitral radius of th© matrix В »
5
Ъ&Ь s&O tea аза irreducible ПхП E atr i s» and let кЪ© the mtssTber of eigenvalues of D of modulus Q(B).If k—4 9 th©ss В i s primitive. If к > 4 P then В
index
Definition 6
Let D ее an Плп saatriSe Then D is convergent /tothe null matrix/ if the sequence of matrices B,B^B^....converges to the null matrix 0, and is divergentotherwise *
Theorera и i
If & is ш nxn matrixs thea В is convergent if andif
Theoremм 2 /Varga/
Let B ^ O b© ©ss П>4П EatriZo s
i« Б has © aois-oegativ© real eigenvalue equal to i t sspeetral radiua 0 Moreover, this eigenvalue is posi-»
uisl©ss В i s reducible in the s©aa@ of Definition 2t
10 §
and PSP is a strictly upper triangular matrix in I
which all entries on anil below the main diagonal are §equal to zero and the spectral radius is equal tozero*2. To ^ & ) there corresponds an eigenvector X$sQ»3. <?(P>) does not decrease when any entry of В is
increasedeFor irreducible taetrices we have the following versionof this theorem.
ITheorem 3a /Perron-Frobenius/ |
•4
Let B > 0 be an irreducible nxn matrix* Then, |
i. В has a positive real eigenvalue equal to its f
spectral radius» «
2» To C>CB) there corresponds an eigenvector X^>0«
3* Ś£(D) increases when any entry of В increases,
4» (8>)is a simple eigenvalue of B.
Theorem 4
Let ЪА and В г be two nxn matrices with B,>B 2 >0Then,
(б)
If B | and Bg, are i r reducib le matrices, then s t r i c tinequality in Сб) is valifi.
Lcsnma 1
If B^Cbi.p^O is an irreducible ПХП matris9 then
either
ii
for all
or
mm
fheoraą S
If В is an nxn matrix with p(B)< 1 , then I-Б is
nonsingular, and
the series on tbs right-hand side converging* Conversely,
if the series on the right-hand side converges9 then
All iterative methods of solving the equation (i)
considered in this paper can be described by a general
scheme• Let us express a nonsingular nxn matrix A
of (i) in the form
where H and W are also ПХП matrices» If H is
nonsingular, we say that this expression representsot t Ł t e raatri3: A , and associated with
this splitting is an iterative method
12
which we crrtts equivalently as
vrhere the subscript J denotes ths iteration numberSJUU a g\iess is made lor the init ial vector ф .W© shall call the matrix M M the ij^ration^ raat^lxassociated with this method.
With the iterative method so defined we can associa-te error vectors £ ^ defined by
Ш is the unique vector solution of (i) , and
from the definition OA the iterative method C±2) , we
can express the error vectors Z * aa
As we know /Theorem i/s the error vectors £ J ov a given
iterative method tend to sero veotors for all £ with
i-*, сю t it and only if the spectral radius €>C * N^
of tho iteration matrix M M is less tlum uaity. The
smaller the spectral radius Oi the iteration matrix9the
faster is the converse nee of a given iterative method,.
We restrict further considerations to the iterative
methods, which correspond to a regular splitting of
the matrix A in the equation (i^ , as defined below*
Efefjnition?
Por nxn real matrices A>M, and MsA«H-t4is a regular э д И Ш п | oj the matrix A , it H isnoasingul&r with H"4^0 and N ^ G «,
Theoirecs__6 /Varga/
If Д ^ М - N is a regular splitting of th© matrix Mand A >0 , thos
r < i (45)
Thus, tna ssatrix M N is convergent, and the iterative
method of (il) converges for any initial vector ф .
In diffusion reactor calculations the matrices A in
the equation (i) consist of 3, 5, or 7 /for one-,
two-, and three-dimensional calculations, respectively/
diagonals of non-zero entries located symmetrically with
respect о the main diagonals The entries of the aain
diagonal are positive, whereas the non-zero off «-diagonal
entries occuring in each row of A are negative» The
symmetrical location of negative entries 'frith respect
to the main diagonal secures the irreducioility of the
matrix A ; physically, this means that a neutron can
get from any point to another point in th© space aaesha
Moreover, the removal cross section !E* in the equa-
tion Ci) is non-negative in the whole reactor and this
implies the irreducible diagonal dominance of the
matrix A e
A / 4.=(Дц) ia the
equation (i) as a sum of the diagonal matrix К and
strictly lower and upper triangular matrices L aa«l U>
respectively. Thus,
кц a for iчО for»
•ац for iО for iО
-аи for К]
Об)
where
We shall assume ia further considerations that the
matrix A defined by G.^ is irreducibly diagonally
dominant, that is
for all
irith strict iaequality in (i6s) for at least oae I e
We now r oint out that an ПДП matrix A defined
by (iGj and atisfyibg the assumption (д.6а) is noa«=»
singular and the matrix Д is nonne^ativea
From the definition (i6) v& get that tne diagonal
matrix К is nonsingular aad К ^ 0 e №©n we can write
the matris A as
~4 -vMatrix К (Li-U) i в nonnegative and. irreducible andfroa the aasumptioc. (l6a) i t follows that
Moreover, from Lesma i we see
On the basis of Theorem 5 we conclude that the matrix
r-K~4i-+l0 is nonsingular and[l-K4(LHJ)]" 0 , whiohconsequently gives
Tłius, wo Have proved the following theorem.
l /Birkhoff and Уш-ga/
Let A be an ПХП matrix defined in (l&) and satisfy-
ing the assumption Cl6a) u Then the matrix A is non-
singular and the matrix A~ iś nonnegative»
Moreover, Varga pointed out that if А=(о.ц) is a real,
irreducibly diagonally dominant ПХП matrix with Q.i
for all L fcj aad йЦ>0 for all ''4Ц n then A/see Reference 4, p e85/ e
Tv7o basie point iterative methods ere described below»
A* The Jacobi Point Zteratiye^Metbiod,
This itsrativo method is also called the Richardson
method or the method of simultaneous displacements»
Por an ПКП matrix A of it) defined as in & б )
and satisfying th© assumption C±6&) , w® can аэзшю the
16
following regular splitting of A
A=M 3 ~N 3 • H 3 = S R>O Nj»UU>0 (49)
Substituting M3 and Щ lnt© (ii) , we have
and since R i s a aonsingular matrix ш& К«et 4
We shall ©all i&is i tera t ive aethod the Jagobli terat iv» method, and we shal l ca l l the matrix
the Jacob^ f^olnt matrix: associated with the matrix ЛAocording to (ix) we see that ^ C B ) < ^
of the result of Theorem 6 e W© shall дав© сСВ") as abasie quantity with which we sha l l оошрате spectralradi i of [email protected] sastriess obtained, fro@ other methods
Be fhe Gauas^Seidel_PoibŁ I te ra t ive
This i t e r a t i v e method io also oslled tit®ot suooeseive diaplaoenosats ®rs 1& oertalB oases, Ш®ЕЛеЪшшш methods
assume © different regular s p l i t t i n g of Ш©<SofiB@<l as above
Substituting H Q and NQ. into (ii) , we get
U\ Frora the definition of A vre see that К L is a r»on»
j -negative strictly lower triangular matrix **or which
* t ^ 0 and it follows frum Theorem 5 that ,
\ (|-1Ś"'LT^.O * Thus, we see that the matrix M Q is пои-
| singular
сад -writ© (34") oquivaletstly ea
I Wo shall call this iterative raethod the Cteggs-SeideJL
iterative method, and we shall call the matrix
I tlso Gauss~Seidel point matrix associated with the matrix
I This iterative BK thod lias the computational advantage
I that it does Bot require the simislfcaneous storing of
I components for isioth vectors ф ^ ' and ф У in th©
i course of computation as does th® Jacob! point iterative
| Iho following theorem gives as a comparison of
| these two iterative methodse
II1 IMCggSLlL /S^eis аай Eoseab©irg/
•5I
I Lot th© Jaoobi matrix !Б^К (L+U) be a mra-nogativoI ПХИ matrix with aero diagonal entries9 aad let ŁĄ besi
Ii%4
ta
the Gaues-Seidel matrix of (2б) • Then, one and only
en® of Ш© following mutually exclusive relations is
2.
3.
B snu the GauSS»SQidol matrix
are ©ithor both convergente or both ó3hrarg©Bto
W© shall show the proof of this theorem for the oas©
that the matrix В ^ 0 is Irreducible, iae* the case
c ( B ^ = O is oxcludod. The assumption that X> is irredu-
cible is valid in our овз© of the discrete diffusion
equations* For the oie», two»», aad three-dimensional
discrete diffusion equations the Jacob! raatris З К
with aero diagonal entries h&& respectively only two,
four, and six non-negative diagonals located symmetrically
below and above the main diagonal»
W© introduo© HOW the following definitionв
Let E aad г be respectively strictly lower and
upper triaagwlar nxf» aatrieese Then, r n ^ *3 ^ ' ^ " * ^ - '
and nicOSęttłoCFh бС^О. ^Prom the definitions of the functions fTBtoLI ana iHetri- з
i t is obvious that ш(О)=п(О) — О» ш(4)гап(4)г=г^СЕ4"Р^ 1
19
/ 1 oC>O. (a?)
LejamajB
Lat B=ss£i>F be a non«negative irreducible n»0
matris and E and F be respectively strictly
lower sad uppor triangular П*П matrices» Then,
the functions m(oC) aod rv(dC) are both strictly
increasing for c£^O.
Proof» For o C > 0 the non-negative matrix M(oC)—oCE^F
is also irreducible, and we see from the P©rroa«=Frobsnius
Theorem 3 tnat oa increasing oC the spectral radius
of M(oC) also increases. The same argument shows that
n(aO is also a strictly increasing function, which
completes Ш б proof of this Lemma»
For pCf£)>Q , we represent this result graphi-
cally in Figure io We can now return to th© proof of
Theorem 8 e Defining the matrices E = K L and p=R U,
w© have
Since the matrix <м is non-negative, there exists a
non-zero vector X ^ Q suoh that
where %
It can be deduced from the non»n@g&tive irreducible
character of b = E * F that %Q, is In fact po9itiveBand
X is a positive vectore Thus, we ©an vr*ite
equivalently as
20
Since
(XgE+F) anare non«aegative irreducible matrices, then using
Definition S, we have
r a nd n(VA<^i- Си»
c l ^ then n(f)=M,and it follows from themonotone increasing behaviour of n(eC) in Lemma 2
that Х л ^ М • The converse statement is also true4
This proves poiat 3 of the theorem*
Kow, we вввиюе that 0 < f (S>)< e Sinoe f )
and п(*/Л£^)~^^Ье monotone behaviour of n(cC) again
shows us that 7 Д & > 4 , or CKX<j<4 о But аз m (oC)
is also strictly increasing and Ш (1)== C!B>) t it
follows that O^.X^fC^V^CBK'J /point 2 of the theorem/,,
Similarly, if f(!B>)>4 , then Ttg^f (^)> ? CS)> 4/point 4 of th© theorem/«. Thus, we have proved the ^
theorem of Stoin and Rosenberg for the case tvhen the
matris 5>^0 is irreducible A«e« С (S)>0/- ^The Stein-Rosenberg theorem tells us that when
0<C€(Sl ^ d s the Geuss-Seidel point Iterative method
is more rapidly convergent them the Jacob! point itera»
tive method«
Before leaving this sectioB, ws point out that
statement 3 of the Stein-Eoaenbarg Theorem S easi also
be obtained on the basis of the theory of regular split»
tings* VI© hav© th© following-
21
Th_eqr;ejain9 /Varga/
Let A^Hą-Mf^igpN^ b© two regular splittingsof A , where A ~ 4 > 0 . If Н а > М 4 > 0 , then
Por tli© matrix A=R-L i l defined as in, (i6) .
and satisfying the assumption (i6a) „ where the non-aegaiivo matrix jifr5^ *(L+U/ is irreducibl©8 ®э
define
It is obvious that these ar© both regular splittingsof A , and A~>0. But as Mj^ N^^1: О 1 vt® oonclud©directly from Theorem 9 that
к"4 [ L + U ] ]
22
III» TWO-SWEEP ITERATIVE ЩЛИС©5
The idea of solving large systems of linear
equations by two-sweep iterative methods has been
developed originally by the author and applied in
19GS in his first two-group two-dimensional diffusion
corte ИЕ>/А»11а* This code which has been used in many
critical calculations for reactor design, has furnished
up to now quickly convergent solutionsa The algorithm
of this code has been reported for the first time at
the КРУ Seminar on Numerical Solution of Multi-Diraen«=
sional Diffusion Equation hold in Warsaw, March 1969,
and after the modification of the overrelaxation pro»
cess it has been reported at the Seminar of Reactor
Physics Calculations, Budapest, 20-24 Octobers1969s
ami reprinted in Reference il« Later, the algorithm
of two—sweep method used in the code "EWA-IXR was
modified in order to provide a further improvement
of the convergence and applied in 1971 in a new codeяДОА-11-Аяо More detailed description of both codes
is given ia Chapter Via
The concept of two-sweep9 iterative methods con-
sists in © specific partitioning of the square matrix
Д in the equation \^lj , different from those used
in the iterative methods described in Chapter lie Xn
both Jacobl and Geuss<°Seidel Iterative methods, the
ПХП matris Д is expressed as the difference of two
ПкП matrices9 each containing part of the entries
of Д о In tho two-sweep methods the matrix A is
expressed as a difference of products of certain
23
triangular matrices. Such a partitioning of A allows
us to apply tile well łmown numerical procedure of
two—sweep : "forward eliminatiogg&iibkward substitution"
which is used ia the Gaussian elimination method for
solviag linear equations» It is obvious that in this
case the splitting of the matrix is also regular.
In this chapter the two—sweep iterative methods
Ш А and AGA are described o The names EWA and AGA used
far both methods stera from the names of the correspond»
ing codese
A» The EWA Twof-Sweep Iterative Method
The nonsingular ПХП matrix A in the equation
defined by (±б) and satisfying the assumption (l6a)
be expressed as follows
(32)
where Fjf^&QQy^is an П П diagonal matrix such that
for a l l 4 4 i ^ n .
We sesume that the matrices Kf=cll<łC}Ckj jJ and
satisfy the following conditioa
K > P e ^ O (35)where кц>рьд ^ 0 for all" 44 U ft.Then
is also a noa-aegativ© diagonal matrix and d-ц ^>0 forall 445-4^ ^ so tbat Up ^ О , and we can write
24
step is to express tfcte matrix Dg—L""U
as a difference of products of triangular matrices,
i,e«P
v»e express the noa-negative matrix ł-Bg U «ts the suraof two non—negative matrices
L D V U — V E + T E > ($7)where Kc dia ćl-Dg U; is a dittbc-nal raatrix whose entries
are the diagonal entries of LQf: U and the matrix Tg
has zero entries on the main diagonal and its off-
-diagonai entries are those of LDg U.
Substituting (36) into (35) and talcing into
consideration (37) , тге obtain
^ V R E + P E . (38)
If we choose the diagonal raatrix Pg in (32) such that
then vm get
A=(r-LD"E)DECT-Diu)-TE = N£-NE , fto)
where
' ) D
25
Since L and U are strictly lower and upper trian-
gular non-negative matrices respectively, and D E ^ Q
then on the basis of Theorem 5 we conclude that the
E?a£rix N g is nonsingular eaad that !4g О » Tlius,
the equations (40) aiid (4i) represent the regular
splitting of A , and the iterative method associated
with this splitting can be written as
and the iteration matrix associated with this methodis
The iterative method (42i is easily realized in
practice by applying th© two=-sweep procedure whioh
eliminates the prooess of calculation of inverse
triangular matrices. Let us multiply (42/ on tlić left
by (I-DE* U ) and shift В Е 1 ) Ф ^ + 4 ^ to the right--hand sldej we obtain
3)enoting
aad multiplying again this expression on the left by
(I- LB £ ) t we have finally
26
Ргок the fact that all the entries in the first
of i;he striotly lower triangular matrix LDg
are equal to zero we see that the first equation in
(44) can be solved explicitly forj^ ? provided
ф ^ ' and О are known.» Successive components of
В ' ' can be calculated recursively for increasing
indices» Similarly, the last row of the striotly upper
triangular matrix Bg U contains only zero entries
and this allows us to solve explicitly the second
equation la (44) for (j>Ci+4' , provided j&CJ*^ is
known. Successive components of ф J can be calcu»
lated recursively for decreasing indiceso
Thus we see that in this method each iteration
consists of two sweeps г j[oryfar<i elimination for
reourslv© calculation of the components of 6 •! '
^nd b^c^ard substitution in recursive calculation of
the components of фЧ"™'. For this raason w© propos©
to call this method the ЩА two^sweem .iterative method,
and th© fflstris бд of (43) , the Щ matrig associated
with the mętna A . Thue8 the equations (44) repre-
sent a practical fora of the iterative method (42) for
Ф ОЛ*
In our considerations we 5iav© assumed that thenon-negative ssatris L Bg U has non«»z®ro satriaa Kotonly on ths Eaaiu diagonal but also off the main diagonal.
27
If LB^U W&S a simple diagonal matrix, i*ee,
U—^S , than Tg would be the null matrix and
the iteration matrix £4 would be also the null
matrix for which C> CE )=G e In such a case the EVA
two-sweep iterative method becomes a direct, mcstJioU,
whereas х.Ъе Gauss-Seidel iterative sethod never doesso» This method is then represented by the equation
and th© two-sweep equations (44) have the followingform ,
Such a case occurs in the one=«dira£msional diffusion
equation /three-point difference equations/s Шеге Ш©matrix L has only one positive aubdiagonal, thematrix U has only one positive auperdiagonal, so thatLDg U~ Pg and Tg;=O . The equations C*4a) represente. practical form of the Gaussian elimination methodused for solving systems of linear equations withtridiagonal matrices« This numerical procedure, knownas the method of sweeps, was introduced into realtorcomputations at KAPL by R«He Stark j j j e la the SovietUnion i t was developed under the паше of "dispersion"or "driving-through" method [s]• Wilkinson [9] has ahowathat this procedure is numerically atabl© with respectto rounding errors when the matrix A is diagonallydominant end tritiiagonel „ Ss teas ions of №i@ method to
2S
systems of linear equations with block tridiagonal
natrices have been developed by Cuthill and Varga [iOJ.
Thus we see that the EWA two«-sweop iterative
ncthod presented in the form of the equations (44)
is an extension of the aethod of sweeps- used up to
noF only for the direct solution of systems of linear
equations with tridiagonal matrices - to the case
of multi-diagonal /or, generally, full/ matric©se
In further considerations VJ© shall assume that
T E is not the null matrix and In the case of the
tvro- anil three—dimensional diffusion equations the
corresponding matrices T^ have respectively two
and six non-negative diagonals symmetrically located
above and below the main diagonal» In order to be
more precise, wo shall restriat to the applications of
this two-sweep taethod only for n s n matrices A=k-lrU
such that. n^S. I a S*?. matrix A is a particular
case of the tridia«onal matrix/ and the product of
the matrices L and U has some non~zero off-diagonal
entriesa
We have the following theorem
Theorem 10
Let, the Jacobi matrix 3S~ & (L*U) be & non-negativei r reduc ib le ПКП matrix with O>2. and zero diagonale n t r i e s such t h a t 0<^CE>)44 and l e t $Ą be the mm-negative Ei'A matr ix of (43) such t h a t С&{ОThen e i t h e r
29
or
To prove the theorem we observe that the aatris
is uon-oegative, and there exists a non-negative
veetor X such that
"4 D; 1и)or equivalently
= A E X , (Aba)
where A E—C(A)>0. Using (э4> , (зт) and (зэ) wehave from (45 a)
We are concerned only with the case ^.£^0 » so that
the Ш А two-sweep iterative method does not reduce to
a direct onee Thus (4б) givea us
Since the matrix 2) is non-no gat iv© and irreducible
end the matrix Tg is non-negative, the matris
is also non-negative and irreducible for every number
\P^0 , and its spectral radius ^(СтЕ0^)}>С СВДis a strictly increasing function of ij* and tends to
infinity with тЗ>-?»со /see Theorem 3/e Prom (46a) w©
30
see thai
If P CK) 1 3 1 1 i » e oonclude from the monotone increasing
behaviour of (jugCtfy) that " X T " ® os* A £ = Ł
th© case trtxea C K ^ С Ю <C , for the sasae reasons
U™' "V ^ U e пенсе O ^ A E ^ ' J which completes the
proof of the theorem*Tho ease ^CB^)>^ cannot bo included in our
theorem» However, it is of no interest for practicalapplications,, For Q C$)7> -\ 9 the matrix A in theequation (i) does not satisfy the assumption С16a)and in this case of the EWA two-eweop iterative method,the diegoael matris
may have some negative entries or, what is eveaa zero entry , so that the matrix D ^ is singularsIt is illustrated in examples given Ъ©1от°»
No%?, we shall show how a diagonal matrix Pg canb© obtained, so that the condition (зо) is satisfyedo
Frora the equations (34) , C^") s^^ (39/ we get thefollogins relation
Dg{U) * dioq (UK-
lisich gives us P^^O and for tbe l<-th diagonalentry of Pg the formula
3i
We see that the recursive formula (48a^ enables
us to calculate all the entries of the diagonal matrix
Pg aad thea those of B E » that ia ^ Ц ^ ^ Ц - р Ц .
Knowing %he matrix D e ^ Q ean calculate the off—«diagonal entries of IDpU which are simply the
eatries of Tg 0 Tney ean Ъв aspreaaed as follows
4 - 0 for
an
It may be worth emphasising that the simplicity of
successive calculations of the entries Рц>о1ц
ti.j is due to strict triaagularity of the matrices L
and U ,
By mesas of the formula (48a) we determine entries
of Pg and consequently the entries of Dg in a few
esaaples of the matrix A l С ( Ю is calculated
from Lemma I*
A,
Example^!
Vi-ł-l-i 444
44-11
L.51 209
32
Example 2
34 -I 4
4 3 4 4i 4 Ъ 4
4 4 4 5!
Example _ 3
2 4 4 4
4 2 4 44 4 2 44 4 4 2
4 2 4 -i-14^-i4 4 4 2
Р
1V* 4520
2«7
"30
J_
p ч
33
Да can Ъ« seeas Jacob! matrices ia Examples 3and 4 have spectral radii CCS)>1 and hence, aaeaphasized before» some entries of the matrix Dg arenegative or гсго; the la t te r implies that the matrixBg, is singular /a<&6 Exsaple 4/.
B. f ae ASA JEwo-Sweep Iterative, Method
fbo splitt ing (32) of a nonsingular nxn matrixД uaod in the ©ТА method can be generalized to
wbere the RXO matrices ^ L and U are definedin (i6> and
i^Q for alt
i j - 0 foraU i > i .
(50)
We assume, similarly as in the Ш А method, that
the matrices k=dma, (К i'O and =<iiQ<J СрцО satisfythe following condition
for all <4l4n. so that
34
is also a non-negative diagonal matrix in which j^
for all 14i 4 r t and Вд ^ 0. Thus, xm can writet&e matrix A as follows
А—ВА-(1*нУ(1нф+ рА+ нч-qWe have th© following ident i ty
Now, w© express the non-negative matrix (1+Н)В
as a sum of non-aegative matrices, i.e.,
where ^^^(СЫ-Н^ВдСи^)) i s the main diagonal of
(1Ж)Вд ( i n t o and the matris Тд+Н4+Ą4 has zero
entr ie s on the smin й i atonal and i t s моп-zero entr ies
are those of (1*Н)11д (LJ+Ą).
If w© choose the matrices [\ , H amd Q, in sueha w sr that
^ - R 4 ^ н-н4 and q «
tsien w© get
Ą ] [ S Ц -тА i—
35
where
Siaee (Ь"И) and (iHQ) are strictly lower and upper
triangular noa-negativ© matrioes, respectively, and
Вд ^ 0 j then on the basis of Theorem 5 we conclude
that the matrix Мд is non-singulav and- Н д ^ О «Thus,
the equation (б?) represents a regular splitting of Д
and th© iterative method associated with this splitting
can be written as
•т ]fes)
- [i-ńJowD] К [HWOD»] Tis the iteration matrix for this method,,
Similarly as in th© Ж/А method, th© iterative method
given by (S9j oam easily Ъе realized in practice by
the two-sweep procedure for assy initial vector
36
(64)
Since (1*Ь1;1)д ami .ид vU's'Q.i are lower and upperstrictly triangular matrices, respectively t then successive components of jb^ can be calculated recur-
sively for increasing Indices in the £_ОГУДГ<1and successivo components of i 4 + t ) сал be eal
7culated recursively for decreasing indices in the
We shall call this iterative aetiiod the AGA jjiterative get hod, and the matrix cft defined
in (6O) - the AGA matrix associated -with the matrix Aof the equation Ci) e
The AGA method represented by the equations (6i)is a general foro of the two-sweep iterative n©thode
By the choice of H s O and £ ( ^ 0 the AGA iterative
method reduces to the EV/A onee As mentioned above„the
EWA iterative method reduces to a direct one only for
a tridiagonal matrix A « The AGA iterative method can
be reduced to a direct one for any matrix n bj a. parti»
cular choice of locations of positive entries in matri-
ces H and Q[ , so that Тд гг 0 e The correspoad»ing equations have the following form:
]
and
above direct method for the solution of
equations fey triangularization, аз the particular case
of the AGA two-sweep method, is simply equivalent to
the Gaussian eliasination raethod» It will be illustrated
on an example given below» However, the application
of this direct method to the numerical solution of matrix
equations obtained Ггош the finite difference approxima-
tion to elliptic partial differential equations in two
and three dimensions is impractical because of the large
computing effort and large storage capacity necessary
for calculation on digital computers» Another important
disadvantage is the possible appearance of instabilities
due to rounding errors in recursive calculation of
entries of Вд i H ^ Q, and Тд о Thus, the direct
method has rather a theoretical significance for us.
In further considerations we shall assume that
H,,(c|_ and 'Тд are not null matrices and as for the EWA
iterative method we have the following theorem,,
Theorem t Д1
Let the Jacob! matrix Ъ^ К (L+U") be a non»-negativs irreducible n»n matrix with П > & and
zero diagonal entries such that 0<^(J!V)4 andlet JX( b® th© non-negative MkA satris of (во) suchthat ^ (Л4) > 0 . Then either
38
CK<? СД^К 4 and
or
4= SThe proof of this theorem is similar to the proof
of Theorea iOa Since the matris A j is mra-negative,
there exists a non«-negative vector у such that
i—* -J г -i т~*
or
Вд {[I-CL+WD; 4]where Ад= ^ №4) >0.
Using
(L+H)Dj((lHQ)—PA+TA+H+Q. and
we obtain from Сб2а)
К*
But from the assumption that Ад "^0 9 we caa write
Since the matrix jj is non-negative aad irreducible,
and the matris Тд is non-negative, th© matris:
39
4 [u U+1> T A] (64)
is also non-negative and irreducible for any number
с(егА
•\j* O , and its spectral radius P С^дОЗО) ® (
is a strictly increasing function of "$* /Theorem 3/e
Ргош (бЗ&} » w e s e e that ^фгдС Ą'")) = 1 » И 0(B)=?|tve conclude from the monotone increasing behaviour of
*A^*) that —^r = 0 or А д — 4 о For the
same reasons, if 0 < ^ ( B ) < 4 then - ц ~ > 0 that
is ОК.Ад<£ i which completes the proof of the theorem»
The general AGA two—sweep iterative method given
by ths equations (6t) represents a class of methods
which differ among themselves by assumed locations of
non-negative entries in strictly lower and upper trian-
gular matrices H and Q .
Let W y be the set of indices (i^j/ such that
Н ц % О s and Vl^ s similar set for the matrix Q .
A particular version of the AGA two-sweep method depends
then on the choice of W H and W Q O Let US assume that4 1 W
the sets Ш ц and W Q are giveno We shall show that
this allows us to calculate the entries of all the matrices
appearing in the corresponding version of the AGA iaethode
At the same time it will be shown below that Ы ц may
not contain the indices of the first column and W||
those of the first row.Prom the equations (52) , (55) and (56) we have
40
where
and off-diagonal entries of (L*H)Dl (IHQ^ sr© thoseof TĄ*H*C| • Since L+H and U+^ are strictlylower and upper triangular satrices, then we obtainthe following fonaulae
1-4
Pu d Sfi>
=У"!,s~ Ps,s
. . . . П
(65a)
t=«0 for
d«
H65b)
О for
d
t;
.
, we see that the set of indices
is complementary to the aet W^
of the ©Htries
on the whole
set of istdie©s ^S<j ••.-! П »Ргош th© above formulae we aee that independently
of the in i t ia l choice of \^н and Wą the ©ntri©a ofH in the f irst column and the entries of Q_ in the
first row are always equal to zero» At the ваш© timeit is seen that the above equations allow us to calcu-late the entries of the matrices Рд /and Од331 R— ?Ą / S
TĄindices
H and Q, for successive rows with increasing*
I or for successive columns with increasing
indices ] » It is worth noticing that the entries of
the given L - th row or ] -th column have*to be
calculated simultaneously for all the above matricese
As the illustration of the AGA method, we now
consider a general 4*4 matris A e
We assume the following locations of positive
entries is H and
H
0 0 0 0
0 0 0 00 0 0 0
Я
0 0 00 0 00 0 0
42
so that
Ą A
о о о о
0
For the above choice of locations of positive entriesin the Beatrices H and Q_ we now determine theentries of Рд /and В д /, Тд ^ H and Q. by meansof the formulae C65aBb,o,d; for Ssaraplos i sod 2from the previous section
Bxanmle i2
i.li
4
±'5
43
'АЛ'49 Ю
"АЛ 62,?
А,2~5
'~55dvT 36
55
In both examples, the last entries of xJ are
smaller thsn the corresponding entries of Jig for
the S'/A iterative method. This decrease of the entries
of XJ follows from the appearance of non-negative
aatrices H and Q. in the AGA method.
We novr consider the case when the AGA iterative
method reduces to a direct method for which Тд is
the null matrix that is> in the general case, W H contains
all indices of the strictly lower triangular matrix H
and W ą contains all indices of the strictly upper
triangular matrix Q. o
For the general 4x*i matrix A considered above
we save
o o o o0 0 0 0
о h w о о
0 0 00 °0 0 00 0 0
0
О
44
and
P + H
0 0 0 0
0(67a)
Por the above locations of positive entries in th©
we determine the entries ofmatrices H and
/and D A /, H and Ą from the formula©
for the same ^samples 1 and 2 e The aero entry
Example 2 follows from the fact that t^e matrix
is singular ( 0
±n
Esaąpla i
P
3,3 3
5
Л.3^5
34
Ł5
45
d О
matrix A"proee&nr© allows us to find the inverse"*4 tmmodiately by a» Inversion of triangular
-A -i
Having determined the matrices Вд ^ H andalready above in Example * t?e can now calculate thematrix A"| as follows
ą0
0
0
-4
f0
0
л4030
-4
- |s35"
0s4
""5h
"3
00
4,j
"Z
00
0
'1
46
ч0и0
0
45
45
0
0
i40ŁО
sj _
5a5
i44Ą
3
"z
0Ąв
k54 •2.
0
0
iĄ
2.
0
0
0
\
г
4
4
a4
4
44
г-i
л
Ą
42
Before leaving this section we should like to empha-
size that in this paper we are concerned with only
iterative versions of the two—sweep methods; that is, we
restrict the application of these methods to the nxn
Eatrioes A ^ K —L — U (n>2D such that the products of
L and U give the matrices with non-zero off-
«diagonal entries and consequently Tg or Тд are not
null matrices a The above restriction allows always to
determine all the entries in the matrices D(= and T^
or DA,TA^H and Q for §CB)44 .When LU isonly a diagonal matrix, we obtain always the direct
method/TE5=O or Т А^О,Нэ0 and As=0 /andthe diagonal matrix В Б ^ В Д is nonsingular if and onlyif cCBKl.
Finally, it should be noticed that Beauwens [i6]
proved that the matrices Bg and Вд have indeed po-
sitive diagonal entries whon A is strictly or irredu-
cibly diagonally dominanto His results are an important
complement of this papers
47
c * GoE^taglaon of the Two-Sweep I t e r a t i v e Methods
wi.th_th.eu Jacoibi,iiiand ,the i Gaaaa-Seidel Methods
In. this section w@ point out that on the basisof the regular splitting theory, it is possible toеощраге the spectral radii of iteratioa matrices forall iterative methods described abovee
Por em- needs we shall recall certain propertiesof matrices is the fona of Lecuaas e
For any nxn matrix В we have
where ,
It oaa b© ©asily proved,, Let В have the reel oroomplex eigenvalue Л sad the oorrespoadiog ©ig@nveotor
О
From the above aquatics it follows
ia generals ,
48
But from the definition of the spectral radius we
have that
S i I lm
which completes the proof•
Lemma, ,4
Lat E and F be two ПНП matrioes» Then,
In order to prove this Lemma, we assume that EF
has an eigenvalue A=£0 aixd a corresponding eigen»
vector кфО , so that
Since A=^=0 and ^4 f cO 1tt follows that
Hence, A is also ш eigenvalue of FE „ If on©
or tooth E шй F are singular, then 0 is an
eigeuvalu® of EF sad FE because
49
Tlius, we see that the matr ices EF and FE have thssame eigenvalues and as the conclusion of t h i s f a c t ,we have
which oompletea the proof .
Lemma _5
Let E = tóij) and г^Суц) Ъе two non-negativematrices such that for any 44 i»j ^ П either Sior e i j ^ f i j = O. If @ CF)>0 then
It is easily seen that for such I and J forwhich £{;">0 we have
Let us denote
where we consider only those entries of F for which
ij > О » then
/in particular, it can occur that E=stoCF /, and from
Theorem 4 we have
50
Sinceo£>4 and (F^)>Q fey hypothesisB we obtain
that
proves the Lemma.»
LeaiaajS
Let B ^ L + U be a non-»negafciv© irreducible
matrix such that L and U are strictly lower and
upper triangular matrices, respectively; and moreover,
the matrix L has at least one positive entry in
each column expect the last one and the matrix U has
at least one positive entry in each row except the last
one a Then the matrix
сх-ит'а-iySo.It is easy to verify that the matrix
has еы">0 for i=s=n and any 4 4 J 4 n ~ ^ ®**d *&© matrix
has -Ь- у 0 for any Ш4П-4 and issrn.Henoe, i t followsthat the matrix
has the positive last rov? and th® matrix
5i
has the positive last еоЗяшп, but this implies that
which completes the proof of Lemma 6 e
We now tiara our attention to the iterative methods
based ©a the regular splitting of the matrix A of
the equation CO » that is,.
We represent A in the form
where M ^ 0 and H . 0 , and corresponding to this
regular splitting we have an iterative method
In the iterative methods described in this paper
we are not always able to compare the matrices N
/except in the Jacobi and the Gauss=Seidel methods/, but
we can always compare the matrices N e One might
expect that the "closer" H is to A , the faster
the method will converge. We shall discuss the regular
splitting theory from the viewpoint of tbe influence
of M 4 on ę CM"4N).We begin with the following theorem.»
Theorem^jg
Let A^=H^H4=M2,-1^2.be two regular splittings of
~*
52
' ^ НA~*>/Cb if Н ' ^ Н а > 0 then
Proof: We know from Theorem 6 that (M N) <-i is
monotone with
to prove that
respect to 9 (A" N) and i t suffices
The assumption
can be written as follows
or
Since the nonsingular matrices I+A"
are non-negative9 thenand
o.which is equivalent to
Счч)
Since the non-negative matrices HĄ and п% maybe singular, we must consider the following cases
> . A " 4 A 4 N Ł Сна)
and
A"V
As all the matrices in the above expressions are non
-negative, we have from Theorem 4 that
and
It is obvious that both eases (75b) and (?6Ъ) are
included in the above inequalities»
Prota Lemma 4 it follows that
and from Lemma 3 we can conclude that
which implies„ Ъу Theorem 6
Thus, we see that the proof of Theorem 12 is completedo
Now we consider the case of A " ^ 0 f that is,when
the real ПХ П matris: A defined as in &б) is
irreducibly diagonally dominants We hav© a still stronger
theorem which is very important in further applications о
54
Theorem 13
Let A — Hf*N^385H^-of A , where A"*>Q
be two regular splittings
• If H[4 > H^ > О then
ATo prove this theorem we notice tliat A / 0 implies
that for any saatris N > 0 ^ A N has ©t least one
positive column, and Q СА~*Ю У О « Hence, by Theorems
б and 12, we can conclude that
~
We know trota Theorem 12 t.
4 Hi4 V- 0 implies the 1ь
Moreover, the asstsmption A ^"0
ae inequality
for any matrix азай i fimplies that2,^ N4 0 then
Such is the oase with the Jacobi end the Gauss-Seidel
methods, where ^ 0 ^ M Q . ^ 0 e The inequality
Ng,^ N4 > О implies H"Ą
Ą Ъ М ' ^ 0 , but as will be
shown on an exasple, the converse statement need not
be valide The matrices M{ and Ng, may have different
locations of positive entries in spite of the fact that
M 4 ^ M^ ^ Q . However, we shell show that
A (M2.-N.T) A is a positive matrix when M ^ M ^ O .
55
The matrix A" (N^-M^) A can be expressed as
follows i
=«A"4Na~MOA'4-
ЧА+ГО (M?-ttf XA+lW A~4=-
or
Since by Iiypothesis M4 -Щ_ ^ O , w© obtain
> O
But the inequality (79*) implies that in the inequality
А"4 М г A4 N a A"4H4 A"4 H a > 0 (so)
all positive entries of А Ща А"' Н& are greaterthan the corresponding entries of A"4 Ug, A~* HA andin the inequality
A~4 A~4 > A"4 A~4 >Q Mall positive entries of A Ml&A" N4 are greater thanthe corresponding entries of A"4 W4 A~^ HA , so we
56
can conclude on the basis of Lemma 5 that
(82.)
and i4 1 Сс С A4 N a A'1 Mj) > ? СL*
Hence, by using Lemmas 3 and 4 similarly as in the proof
of Theorem 12 we obtain
Na)>?(A"4N^ > 0and by Theorem 6
which completes the proof of Theorem 13»
Now, assuming the Jacobi matrix as the starting
point, we compare the spectral radii of iteration matrice[
for all considered methods solving the equation (i) p
the matrix
defined as in (i6) is irreducibly diagonally dominant,
that is, A > O „ Ilowever, v/e restrict ourselves to
the matrices A , obtained by applying the finite
difference approximation to multi-dimensional diffusion
equations, for which the associated Jacobi matrices Ъ
setisfy the requirements of Lemma 6O
57
i , The Jacobi
CK^CBKL C&5)2„ The Gąuss-geidel point iterative method
ifVf к!^ м ^ ^ о . (as)
Sine© ^O^L^—G^froia Theorem 5 we have that
.... Ь I 0 ,so H^>H~3>0« Since Mrj^Mfr>0 then by Theorem 9 wehave
3* The ЕГ./Л t\vo~s\veep i t e r a t i v e method
tĄ= ME N E > 0 .
Since J j g^O is a diagonal matrix and L+U is a non«-negative irreducible matrix, then by Lemma 6 we canconclude that Н~^>0 <> Moreover, D ^ ^ R ' ^ O implies
56
—łthat all positive entries in the last row of D^ L
are greater than the corresponding entries of K"4
because kii>d,ii 0 for at least i = rt , sothat
with strict inequality for all entries in th© last
rot?s Но псе , we obtain
о-Е
4>а-Л^кг4^о,that is, M g > Mą ^ O e Thus, we hav® by Theorem 13
that
The11AGA_tT7o sweeT) iteratiye^ ime it.hod
The presence of noa^negative matrioes И and
ia the defittition of Ид implies that В ^
59
t?liere с!д^ ^Eti fos" a* l e a s t i-*5*1^ /for instance,see Examples i and 2 in two last sectioas/* Hence,wecan ooneitłde that
-DA CUH)]"4D;4>[i-ĄL]~Ą! 0with strict inequality for all entries in the lastrowa Since
thon wo conclude that Нд > M p > 0 a a d consequentlyЪу Theorem 13 we have
Thus, we have proved the following tbeoremj which istlie extonsiOR of the Stein-Rosenberg theorem /Theorem S/to the two—sweep methods for the irreducible oase,when0^ ( 3 ) ^ > 4 and the matrix JD satisfies the
requirements of Lemma 6 e
Theorem 14
Let the Jacobi matrix w & ^ K Cl+U) Ъе a noa«=nega»tive irreducible ПХП matrix with П > Е and aerodiagonal entries such that L has at least one posi-tive entry in each column below the main diagmal,U has at least one positive ©ntry in each row abovethe main diagonal and LU has some positive off«°diago-nal entries and such that СК^Сй)^ i . Let <h\ bethe Gausd-Seidel matrix of (в б) , £,Ą be the KU к
60
matrix of (б?) and £4 b© the AGA snatris of
•Qsea, one and only one of the following relations is
valids
CBX4.
fhe second point of this theorem has been proved In
Theorems 8, 10 and 11« The ease ^CS)^i^ which is
of no Interest for practical applications, cannot be
included in tho two-sweep methods. As shown in the
examples, who» ^Сз§)>1 , the diagonal matrices3}c cmd 1)д с ш have some negative entries or theycan even be singular0
To give a mimerical illustration of this theorem,w© consider the А%Ц matrix A from Sxsmple i forwhich A *>0 has been calculated in Section В «The spectral radius Q (H* Ю ^or each method is cal-culated £тош the formula of Theorem 6
к ti) or its bounds are evaluated on thebasie of Lajgma i* The positive entries of Tg and Тдare calculated from the formula© (4Sb) anti (65d) Brespectivelyо
61We
ą -i -4 4-4 ą -\ -i
-4 - 4 4 - 4
- 4 - 1 - 1 4
l a > ^® Jaoobi isoint i t f l w t i w ^ t h ^
0 4 4 4
4 0 4 44 4 0 44 4 4 0
S4
A
Л
4г4
4
i4
2
4
4
4
2
5
44
4
4
В4
4
3
4
4
Ц
4
3
2* The Gauas-Seldel
G Ą Ą Ą
0 0 4 40 0 0 40 0 0 0
i»• ч -- а «г
i i iiteratiye i method
5
о г 5 40 4 3 4
0 4 г 4о А г з
6 2
The EWA two»-sweep iterative method
"о о
Ц
о о0 О ~ц Tj"
о 363
0 Ц 60 °
л 30 4& %и 60 60 &0
0 30 6460 60 60
О & ŚśL 2L60 60 60
П -S JLI чбU 60 60 60
4e The AGA two-sweep^iterative method
0 0 0 0
0 0 | О
о о || о
О ii 50 35и 60 60 60п 45 65 35u 60 60 60О 50 ! 0
60 60 60Q A5 85 35
60 60 60
The locatirnr of positive entries of Тд correspond
to the choice of locations of positive entries in the
matrices H and Ą as in (бб** and (б7) •
63
As aentionod before, we see from this example that
the assunption M Ą^ Mg ^ O does not always imply
ia spite of the fact that the last inequality
implies always М д ^ М ^ ^ О /see the matrices Мдand HE/.
It should be remarked that in the two-»s\reep
aothod Ш е matrices H obtained as the product of
nonsingular, nou*-neg&tiv© upper and lower triangular
matrices are positive and, moreover» w© have
This is the reason that we observe a significant
decrease of all positive entries of A N when
passing from the Gauss-Seirtel to the two-sweep methods,
whereas for the Jacob! and Gauss-Seidel methods the
last colucras in the matrices К Ng and Д" HQ
are the same /see for instance the example given above/e
This effect explains, in a sense, the stronger reduc-
tion of the spectral radius by the two~»s\7eep methods than
by the Gauss-Seidel method in relation to the Jacob!
method j as \?il! be shovm in later chapters „ this reduction
is very often still stronger v.'licn §(£>} is closer to unity»
Thus, the application of th© two-sweep iterative
methods to the solution of practical problems in which
« (B*) very often only slightly differs from nnity
offers considerable advantages in the form oi .aster
convergence o
To complete this section we shall give another
theorem extending Theorem 14 for cases when A ^ 0 ^
that is, when the Jacobi matrix Ъ can b© reducible©
Let tlio Jacobi aatrix B = K \.i-"+U) be а
gative nxn matrix with П>2» and zero diagonal
entries such that LU has some non-negative off»
-diagonal entries and ^(E')<-'i. Further,let
0I4 be the Gnuss-Seidel uiatrix of isd) , S^ be the
b't'A matrix of i&i) , and $^ be the AGA matrix of
СбО e Then all the abovo matrices are convergent ana
The proof of this theorem follows at once. Since
i'or A" 0 we have
then by using Theorem 12 vre obtain
Tvhich completes the proof of the theorem
IV. AFPLICA2I0N OP THE OVERREL XA3?IO>3 PROCESSES
In this chapter we shall describe briefly the
overrolaxation method, as related to the Osuss-Seidel
point iterative method /th© detailed analysis of
this me Mv i earn be found is Chapter 4 of Reference i/t
ашА wo shall show how similar processes can be applied
to the two-sweep iterative methods. The application
of suoh processes in both the Gauss-Seidel and the two»
-sweep methods for a certain ohoioe of the relaxation
factor reduces the spectral radius of iteration matrix
oes which results in the acceleration of convergence,
in many cases quite considerable.
A* Success iте Qve rre1asat i on PointmIterative „ „Method
/SOR Method/
Starting directly with the Gauss*»3©id.el method of
(2'i) we ean write
and as R.— COL is a nonsingulsr matrix tor any ohoio©
of the p^"jMnet©r CO, this ean take the fona
The parameter Ш is called the
w© shall oall this iteratixr© motliocl ths
66
о ve r re 1axation /under re taxation/ jgolnt 1Летг_а1ь1уе mg_thodQft for brevity, the SOH method, where the use ofcorresponds to overrelfixation /underrelaxation/Bandwe shall call the matrix
' ^ R ^ U C O I ] (90
successive relaxation jąolnt matrix»When GO=I this ,iterative method reduces exactly to the Gauss-Seidelpoint iterative method, and the iteration matrix (9i) ,is then equal to <Ц given by (26^ a
Now, we ar© concerned with the answer to the question Iwhether there exists a value of CO which would decrease jjthe speotral radius ^G^co) in comparison with <?0N)»
Varga [4] , using Theorem 9 . as the result of theregular splitting theory, pointed out that for the ' ,range CK<2d4-l a value of 03 which minimalizes thespectral radius Q(&OC>) is C O = L
Defining вз
i t is evident tlsat this splitting of A Is regular forall 0<CO4A o if 0<CO<4? then N^V* \ 4 ^ 0and as A ^0 we obtain directly from Theorem S that
i i * for allni
n
67
Thus, undsrrelasation (P sCO-Ci j i s of no interestin practical applications. If ^C -cc»4) is different i ableas a function of CO in the vicinity of C0s=4 and
theis we can conclude that th© use of <д> greater thanunity would decrease the spectral radius < ó*-co) in
comparison with ^ С«.Л» Indeed, Young [l3] gave thefirst rigorous and comprehensive treatment of thisproblem for a large class of matrices» Varga [4] hassliown that many of Young's results can Ъе derived bytha application of theorems due to Perron and Frobenlus.Young *s achievement was to find the optimum value of a)for matrix equations obtained from finite differenceapproximations to ® large class of e l l ipt ic partialdifferential equationse He obtained that the optimumrelaxation factor 03 which Biinimalizes the spectralradius
nun
Denoting ^о.
for 0<co<.c5
for CD = 55
for o5<
we have
is real and S?Q.<L4.
is real and "VQ«ss-
is complex and
Thuaf the matrix Х ш is convergent for all 0<&><2>;Ц
For G3=O^ Л'со»^1 and Cclc^saOj^^i. It Is interesting to
consider the behaviour of Q CCEQ^) as a function of CO
68
shown in Pig* 2. The nature of the behaviour of
shows that the overestimation of со by a small amount
causes a smaller increase of уа-оэ) thai does the
of o> by a comparable amounte
B e Successive Overrelaacation Two-Swee^ Iterative Methods
The application of the overrelax&tion processes to
the two-sweep iterative methods о an be raade either for
one sweep or for both sweeps simultaneouslye We shall
consider both cases below»
1 „ Overrel.aauation in pni9i p ^
a/ The Ш А Single Suocessive 0verrela3c:ation Two-Sweep
Iterative Method /the, EWA Siacie SOR_Method/
Using the overrelasation process to the baclmard
substitution sweep, we obtain directly frosa the two*»
«sweep equations
and siaoe I-ooCgU ' i s a nensingular matrix for any ichoice of the relaxation factor CO „ Шеи xi® have |i
tor any initial vector
Wo shall ©all Ш в laetfflod Ш © WA eln^le .guqcessive
relaecation /^^S^^l^SlASE/ two^swee^ iteyatjlve method
69
or Cor brevity the M A single 30R method and we shall
call the matrix
'' [ ^ [l SS£]U]'' [ ^ [l- \SS£]\ -(co-l)l] (9?)the EWA .single successive УвДааММй matrix e The value
(JO—1 gives us the ША two<-s-preep ttremsfeive methodgiven by the equations (44) and the i terat ion matrix(97) reduces then to $ц given by (.43) ,
b/ The AGA Singla Successive Overrelgsatioa Two -SweepIterative Method /the AGA Single SOS Method/
Starting directly from the two-*swsep equations
for the AGA iterative method we obtain, similarly as
for the ESfA single ЯОГ: ч,..', the corresponding equations
and•
9 8 )
for any initial vector ф .
We shall call this iterative method the tfA^guopesgivg oyerre ,laxat i on /uafle rr elaacation/iterative motbod or for brevity the №kk. sin^L® ^ E mgtfeodвой we shall call tbe ra&trts
70
the Д5А single successive relaxation matrix* Assuming
again U)=s='i , we see that this mathod reduces esaotly
to the ША twe«sweep iterative method expressed by the
equations (6:0 end the iteration matris (ДОО) is then
equal fco Д 4 givea by Сбо") а
o / Uisj3>ua.sioja...o^,AbQ_Choioep o f ,pT>tiiquniiiiiiiiCO'iiiii[iValues
The question now ea*iaes whether there exists any
value of the relaxation factor CD which minim&iizes
the speoiral radii of the iteration matrices ^
Aco * To give an answer to the above question, we
first consider the imderrelaxation range,
Expressing the matrix Sco a s
and
we see that not?, in spite of the faot that A >0 and
> 0 for all C0>0 , the splitting A^H^ Jaot regular for all CO)*O p except for 63==^ , beoausr
^EOO is aot a non«-negative matrix., However, we shall
show that the re^lar splitting theory ©en be applied
in this o&se as a result of the faot that A Wg to is
a non-negativ© matrix for all 0<<:а>4 i - Using the
regular splitting of A givea by (40) in the WŁ two-
-sweep iterative method, that is
Since all inverse matrices and T^ in the above
equation are шш-negatlve* then A^Neco *S also а ucm»
-negative matrix for all 0^.00^ !• Moreover, as can
be зеев from (lOl) » at least all diagonal entries of
A" N decrease as <x> increases in the ramge of
О'чСхЗ^ i. Нехш©, we can conclude that
is a strictly decreasing function of CD for ©11
But, by Theorem 6, we know that
is monotone with respect t@ CA^NgjJ) for ©II
1в ша analogous way w© osa obtain sisilp?
for the AGA single SOR męthodj that isB
min ę (/С4 Мдаэ)— fi
72
and
foroli
where А — Н д ^ ^ а the splitting of Л in the MiA singleSOB method and
соФ 0.
Further conclusions can be obtained Ъу using
Theorem 13* As eaa be seeas th© matrices M in all
the above splittings of A satisfy the following
inequality
> MQL ^O for- 0<O)ii.
Although the corresponding matrices N in these splitting'
are not always non-negative, nevertheless we have
always A " 4 M ^ O for all O<oo4 -1 aod Ъеппе we
шау use the result of Theorem 13, that is, If A > 0>then
We now give th© results obtained above in th© form
of the following.theorem»
Theoremmi6
Let А^счСХ-Зэ) , where iD™3'^ ^L+U/ ±3 а а.^„_
-negative /Jacobi/, irreducible and convergent n«H Lr
~4. -4
matrix» smd К L ss& К U are s t r i c t l y lower as«tupper teiangalas* matrices, respectively* fheia, th©relasat ies matrices Ł^^ Иш as$ Лс*? giv©m by(Sfl шг&С&ОО} , respectively, are e©iav©Brg©&t for a l l0<0i> 4 i . aad
Moreover, i f (К&Э|4ОЗ»2,4 4 then
Tiie iaequal i t ie s 6.03) follow immediately e i therfrom the consideration of the matrices A №004 a a d
А"4 ^оа а O^^W^^A^Noa^^CJ) and the us© of Theoreia 9or from ih& consideration of tiio matrices N(^1 aaei
C M ^ H ^ O ) the use of Theoresa 13* ' results of tb i s ш©огеш ш*© of l i t t l e pract ica l
interest шш. have a theore t ica l siipiifiess&©© ог».1ув
Similarly; аь in the oase of the usual SOB method , оэа©siigbt ©spsot that i f <|С€щ) and ^О^ш) asfe dif ferea-t ieble ш a faaotioa o i © i s tls© vicisai.ty of oo=4^
есад <о ала' & ^ а и <оШеи Ш® as© ©f cii^S §Е®Ш%®Т than i m i t y
to QCi|) ®s«i ^Cftł) »
The author must confess that he has failed to
find the exact formula for optimum value of CO in the-)
general casee However, be has observed ежрэПшев tally
tb© asistaa©© ©f the optiauia value of ai gr®«t@r than
tmlty ws& h® has fo«nd that for both tlie Ш & aad tile
AGA single SOfi methods the following inequality is
true
CO ie Ш @ optimal value of CO and Ce3 m Q X is the
value of cO .for wliieb the spectral radius of the
iteration matrix is oqual to uuity* Aa &, rule the values
QJ aad О Э т ш 1 sr© different for tooth methods о
In practical applications of these methods to the
solution of the two-dimensional diffusion equations,
the author has observed esperimentally the following
of the spectral гайii depending on the choice
Ш which ©aa be compared with tlie accurate analysis
hf foang ija th© oase of the SOS raethod /cf * Section
A of iMs Chapter/e
For the EWA single SOE meihod, there is at least
r@al eigeavalua Vg such that JS?ĘUS
4<co<5> c\
for
75
In the case of the ASA single SOR method when
ths non-negative matrix Тд has positive entries
Ibeated asymme tr ieally with respeet to the msiia disgq
nal, there is at, least one eigenvalue ^д such thata n d itp w s 3 oljseyved that]\?<J =»
for 'kc0<O> VAis real sad
for uO=CO "9^ is Г@Ё&1 aadŚ5-1
for ^<СО<^,пах ^Ąia complex and
Tor CC»«aCOmo.s \>д is ooraples and IVAI SS
lor Ш>С^пщ.х ^ д 1 3 complex and
MoreoverB- i t was observed experimentally that Ц (j§<&)behaves irregularly in comparison to ^С^ш") that is,in certain oases CCSĆS)^. С**ой) ®пй ^ a other caa@s
СооГ) У9 С^шJ ч whereas for cCftgj) we alweys hair©)<S C^co^ a n d ^СА©)4?С«1«>- M»^© deta i l s wi l l
be given in the analysis of a model problem sadnumerical ©samples in nest chapters•
Before the end of th is cJsapter we shall s t i l ldescribe the us© of the overrelasation process simulta-neously in both sweeps of the two*»sw©ep iterativeIt was applied by the author to reactor eelewlatiosmbringing oonsiderabl© advantages» ia ©©rtaisa
76
2 о 0verrelasat io?LAIL. J^JilL ^Tg
a/ The EWĄ Double Succegg_lve_pyexTelKsationI t e r a t i v e Method _/the ША_РоиЬ1е SOR Method/
Applying the overrelasatioa process simultaneouslyt© the elimination and the backward sleeps, w© obtainfro© the two-sweep equations (44)
for ащг initial vectors ф®^ endTills saethod provides advantages ia the range of overrela-xati©a„ that is, for Stp > i aau -S?. > i whereand -2- 1 ©re relaxation faotors in the forward andbackward sweeps s We shall call this method the ВИГА doublesuccessive oveyrelasation two-aweeg iterative mstuodor for br®Tity the Ш -double SOE ^thod 0
Since I-StpLB^E and I - ^ B g U are aoasingularffiatriees for any olio ice of the r^lasmtiou factorsand £1ф t We can write the following
tSa® vector m E and toe i t e r a t i o n matrices J i e
Е йереий ова Sip тй £1ф t aad Я.ффО e Evidentlyfor •SŁj&as!, th i s method reduces t© the MA simgl© SOfi
TT
Eiethod, and vHg= G ^ and ол^=О.
From the author's experience acquired in the
application of this method in reactor calculations, it
follows that tills method with the proper choice of
relasatloas factors ^fi aad ^ф converges not only
faster than the EISA single SOR method, but very often
faster thaa the AGA single SOS method. The author
observed that the best results are obtained whan
(407)
where u> is the optimum relasatioe factor in Ш©
S U single SOS method. This will be illustrated by
numerical examples given in Chapter ¥Ie
b/ Thg_AGA Double Sucoessive Overrelaxation Two-Sweep
Iterative. Method /the AGĄ Double SOR^Method/
Similarly, ss in the ША doable SOU raet&od, we can
use the overrelasatlon pr©e©ss to both sw©@p® off the
AGA method given by the t^o=sweep squatioas (ei) 9 that
is
for any initial
78
The abovs equations can be condensed to the following
equation
^ ^ ^ Л Ó09)
where the vector ГПц and Ш © iteration matrices Л д
and ł«^ depend on ilА and J ^ , anfi $1ффО- We
shall oall this method the Ш double successive
or for brevity
the ДОЛ double SO|| method„ With ilA^-4 this method
reduces to the AGA single SOS method, and
The aut ho r mad© a few trials, using this method
in praotieal applications; however, he'did not observ©
advantages in coBipai'isoa to the AGA single SOR method^
¥e MODEL РЕОВЬШ! ANALYSIS
In order to get some information and to illustrate
the eoaiplieated natiar© of the behaviour of Q C&QS) and
^ С«^ЙЭ) as functions of CO in- cojspari'soa with the
batmvioar of Ц C«Łco) ( we BOW give the numerical analysis
of a model ргоЫеш which is representative for the
partial differ©ia©e equstions of the elliptic typ©,»
W© shall consider the numerical solution of the
two—dimensiossal diffusion equation^ io©eB тю seek the
approximations to the function $ бцу) defined in the
bounded unit rectangle lisich satisfies the following
elliptic partial differential equatioa
79
where К is the inside of the rectangle < X * O ^ 0<1)<Л
shown in Pig* 3, wits the boundary conditions
aad P is tb.e boundary of H , The given parameters
XJ s 5. ш а ^ *^e source S ar© assumed to b®
constant in the whole region Я+Г.
Usuallys xm irapose a uniform /or nonuaifora/
rectangular mesh of sides Д Х ^ and ^Yn o a *>bis
rectangle where 1 4 П 4 ^ к and l ^ m ^. Mv^W_*v •
and Ну being the smsabers of vertical aad borizonta.l
line B®y&®nts9 respectively e Instead of attesptiiag to
find the function ф (*?Ц) satisfying C&&o) for all
OK X < X and 0<1Ц<^г , and tae boimdsry condition
Clii") , we seek the approsimstions to this function ф'Оод)
at the mesh points obtained bj the intersections of the
vertical and horlaontal lines imposed он the whole
rectangle "R+Г 9 Although ther© ar© а пишЪег of
different ways /for instance,, see Chapter 6 of E®f©renc©
of finding suofe approsimatlotos of ф СХ>У1 B
a five—poiat diff©renc© ©ąwatioa d@riir©d. bj
the function ф С^Ц) ia the Taylor series %m bm®
variables*, With the assumption that ф (ЛъЩ) Шш a bounded
derivativs of the tftird ordsr Ш ® t®sna bsgimiing with
tae third ©rd©r derivative o@a be ne.glsc,t© o 1toisew@
obtain the wsuai oentral diff©r©aee quo&iosats to approxi-
mate the partial derivstives rtioh for the iasi&e
so
of a uniform s*©etaagular шеаЬ have the following form
,&
То simplify the discussion of numerical results, wa
chose sueh a mesh that its lines coincide witii
of the r«»©tas3igl©, that isg Д Х ^ Х
y Thias, ire s©©fe approximations to the fussotiomaф (Ж\Ц) fos1 ш©зЬ points eoisseidiEg with the ooraersof the rectasigl© аюй labeled by the numbers 192,3 and4 in FigoS s ©a the other handt the boundary conditions( l i i ) iwrp •=*© jF4 axis of syaunetr ' on the. outer boundariess
so that w© ©btaisi the following oaatraS. differene©ai«ioti©Hts to ©pprosifflate the part ia l derivatives 'at thecorner raasSa points;
of liaear
Si
i
f xf/в C4=$(A*f/Dfor
In matrix notation, tbe linear equations (xii)
can be T^itten in the foria
where
-г кэif
-Zj0
к-z
о-ц-гк
от
к
с/
.Счоv.'ith tlie definition (iб) we ksve
"к0
О
О
Q
кО
о
О
О
ко
оО
О
к
-г
О
А
£0
оО
0
f
оО
0А
6О
О
0
О
О
О
О
1
0
О
О
f O
Of0\00
Ths main purpose of the numerical analysis Tor
this model problem is the investigation of th® behaviour
of spectral radii of iteration matrices, as functions
82
ox CO , for different iterative methods which may
be used to obtain the nuiaerical solution of the
equation (i!5) ; т?е are not interested in the vector
solution ф of (ilś) « As irill be seen, the okoice
of such a model exaspl© allots us to investigate the
values of spectral radii also close to unitye To
шаке this roialysis of the model problem more interesting,
we shall investigate the spectral radii of iteration
matrices as functions of % with different values
of Дх/Ду „ For this purpose, we fix the values of
В and ДЦ , assuming B = i and AJ^=^ ?then
It can be easily verified that the matrix A is
irreducibly diagonally dominant for ^ ^ ® £ m d К У®*
We shall now consider particular iterative methods,
Beginning with the Jaoobi point iterative method in
which Ъ ~ К [L -UJ iS the non-°negative irreducible
saatrls, we have the foolowing characterietio polynomial
wher© the roots jUL of this polynomial ar® the eigenvalues'
of Ъ э For the model problem, we simply obtaira
геи)
S3
Thus, we sea that the matrix Ъ Ьаз tv,o pairs of
nonzero eig"nvrlues equal in absolute values but of
different signs Sor f4s sad
In the Gauss-Soidol point itorativo method, wo
write
where ^=[I-K Li К U and the roots Я^. of thecharacteristic polynomial (i22J are the eigenvalues
of X A * №© obtain £гош tho solution of this polynomial
that
'inns, d.{ has only nou~nog£itive eigenvalues and
.":.• above result obtained for the model problem is
truo for the whole class of the Jacobi matrices, that
is, for consistently ordered 2-oyolio matrices /see
Reference 4/o In this case for tke SOR metfood, w© know
from the previous chapter that ^С°"со) ©^*aiHS the
;ii niюнга value when йЭ=оЗ and
It should Ъе noticed , as can be easily verified, that
for bounded values of i (0<^ < r s*0 . when
we have
г
9 w® oojusidor tlie beliaviour of the spectralof i terat ion Bj&trioes its the two^sweep methods
for the saodol problem u
The ssatriees P E ^ B E ^ R - P E «aod TE for the EWAtwo—sweep i terat ive methods have the folljwiag form
G O O D
О Р г ,0 0
о о0 0 0
w о о о0 dg,2O ОО О d « 0
0 0 0 0
о о t a 5oО ł^O О
0 0 00
©btaisi from the formulae
•А к
.it.1.2.
85
| t can be easily seen that in the EWA iteration matrix
tli© first aad last columns sr© zero because these
eolttoms are aero is T E ; tlms9 in our ease the matrix
8ц t has at tlie most two nos^zero eigenvalues.
Evidently, we obtain from
that
whore
and
I t i s interesting that fov bounded values of 5Ll!£)<CX<e£>)
•p—§> iO we have ф-©- О and ©oasequesitly
Thus9 we obtained the result different, froze tbat i s theGaass-Seid©! method /see (i26| / . This р!а©аош©5аоа еааbe easily explainede Namely, w© ©toserv© that witia theincrease of heterogenity of the Etesh /the values of -fcoDsiderably differing froia ujsity/t increases the dominance
86
of three from five uon-sero diagonals in tbe matrix A
given by (116) - then the BYA two»sw©ep iterative method
tends to the direct method„ In the Gauss-Seidel point
iterative method, on tbe contrary, for strongly non-
uniform saeshes very slow convergence is observed e
These effects will be also shown in practical applica-
tions presented in Chapter VI *
Now, w® discuss the AGA two—swoop iterative methods,
Assuming the following locations of positive entries
in the matrices H and ^ , t?e obtain
0
00
0
"о о0 0
0 0
0 0
0
0
0
0
0
0
i
1i
óo0
0
3
VI4
3
0
0
0
0
0
0
0
0
0
0
0
0 i
о с
d0
00
Э
D
0 0
0
dzl
00
0
dс
)
"o0
00
00
0
d.
0
0
0
0
0
0
0
0
0
0
0
inhere т iav© from the fonaulae C65a«di)
S7
к
d4,i
Since in Тд only the second coluran is non^negative
/others аго жего/, tl., niatris
has the same non=°neg©tive column whose the diagonal
entxy is equal to the spectral radius @(^V). Thus,
have
and
Я о
4
056)
as сап Ъс easily verified, for bounded values
Thus,siHilarly as for the E.VA two-sweep iterative
nethod, vre see that the AGA tv/o-s veep iterative metliod
ooconos close to the direct method for greater values
of f-The spectral radii of the iteration matrices in
various methods are shotn as functions of 51 in
Pijs 4 and 5 for the model problem v/ith tv:o values
f=4 (Дх/Дц ™ i) and |=а9(Л^/Д4=5)« It can be seenf ( Д / Д ц i ) | / 4 )that both Q CA^j and ^ C&4) decrease more rapxdly
than ^C^-l) a s functions of Z- and this effect is
more pronounced fdr т«=9 than for £=H » Moreover,
тс observe froH tl;cse fisures that
Figures 6, 7, 8 and 9 present the behaviour of
^ 4 S ^^00) and ^ Cwb©) as functions ofooCO
for j-*s{ and f = 9 with fixed values lEj^CXCMand Z.*= O.I e it is seen from the above figures
that the behaviour of Cfegs) d i f ; f e r s iro™ that of f CX53)that is, for certain values of f and 2 3 ^ 3 ^ $ &and for others ^ ( ^ 5 ) ^ CicSb) *3 a n d t h e behaviour
и * ^ 4QCO-* ^ " U 4 ° ии"- oJ ^ 3 П О 1 Ч > s e n s i t i v e to thoo v e r e s t i n a i i o n of uSg_ then to the anclcros t ir iat ion
of tog by s n a i l amounts, contrary to die case of
llovrever, r/e observe s i u i l a r behaviour
01 vvoCfo) a M © СеЬоэ) » a n i i v : e have air/ays
oieover, i t is vovtlnvhilo to iiotic© that
шЕ<Ссо& a n d с5д<со^ оГ'.ге author observed t h a t tho above ineqv.r. l it ies r o r c
.-.iitisficd in o i l couaidcred cases of U3in;2 the two—
-j\.44o;> i t e r a t i v e nethods .
-'i^nres 10, 11 and 12 ш-о tho il"\i!Ouvation of
§ c $ c ^ crs functions of S. for throe values of •£ equal to
1, D and 2?, respectivelye I t is clearly scon that for
.£.=.1 v;e have cC^S)^§C*4£j i n t l i c "hole raivje of zlvhorens xor f = 9 v;ith Z.>O.OŁE a n d for £=^£5 '-i
000i5 C^^ tt^X>0.00i5 %ro sec thatAdditionally, v:e see that for certain values oi"
4
с GRO < a^) and ^ сел <^ CŁGB)
that is, the tv,'o-sra>ep iterative methods .rithout the
90
. tjrrelaxation process (ćO = i) are more effective than
the usual SCR method. It should be noticed that the
с /res c C ^ O s^d C Cjo3 ) for -р=>-25 are almostidentical to the curves for у-= zTi , v.hercas the
spectral radii of the matrices in two-sweep methods
continue to decrease as j increases.
Moreover, as we see from these figures, the appli~
cation of the overrelaxation process to the Gauss-ósidol
point method and to the AGA. two-sweep method leads to
a stronger decrease of the spectral radius than in the
case of the EtfA two-swoep method»
The effects described above will be observed in
numerical examples given in the nest Chapter,
VI. APPLICATION OF THE TWO-SWEEP MKTHODS TO THE SOLU-
TION OF THE NEUTRON DIFFUSION EQUATIONS AND NUMERI-
CAL RESULTS
Having given above the theoretical foundation of
the two-sweep methods we shall describe them now from
the viewpoint of practical applications to the solu-
tion of the neutron diffusion equations„ In applying the
diffusion equations to reactor calculations, we will
restrict ourselves here to cylindrical or plane two-
-dimensional problems as fairly typical« Tłiree-fUmensio-
nal ргоЫеша can be арргоасвес in a similar way*
i'h& continuous problem has been posed in Chapter I
and now, we proceed to its discrete form* This leads
to the system of linear aquations whose matrices have
properties interesting to us, As was mentioned in
91
Chapter V, to do this wo firfait iiuposa a mesh of hori-
zontal and vortical linos on a bounded rectangular
.;o;,v'>.i R \.itli bounanxy Г suoh that all internal
interfaces and external boundaries coincide with mesh
i.lacj /soo l«4ii. i3/. The internal interfaces delimit
subregions for which the diffusion coeffi-
Ъ and cross sections Ł.. aro assumed to bo
constant. The intersections of the horizontal and ver-
tical nosh Iinc4 define the yet \V of mosh points on
"R+ P and we seek the solution ф (г) only on this
. .• t , Tf a point f*n belongs to tho set W and is tho
uit 01 intersection of the Щ -th row and th« П -th
( 1\:' i of tho mesii linos, then we denote by ip n an
.ipproxit-iation for ф ( r n ) • Replacing the partial do-
i л natives iu the equation (i) given in Chapter I by
Г:-.,-; difference quotients (ii2^ we obtain the finite-»
--difference equations /for derivation of the finite-
- ;li ffcrenoo equations, see References 4 and 6/e Briefly?
lor ф п at a given inner point involving only its four
neighbours the following formula is valids
\ here with the above normalization of this equation
/the coefficient with ^ n + > 4 . is set equal to unity/,
L n ^ lin and \Nn arc i4inctions of the diffusion
coefficients D's and of the mesh size; к depends
on the mesh size, Ds and the removal cross sections
S ; and Cr» contains the fission and scattering
source tevus and is the function o£ D and of the
mesh -izo. Similar equations may bo derived for the
mesh points belonging to the external boundaries for
which the condition (ii) is satisfiede Because each
inner point is coupled to four other adjacent points
at the most, the difference equation &39) is called
a fiyefpoint fqrmulao
The system of the equatiobs (i39) for the whole
set W can be written in the form of the raatrix equa-
tion (i) , i.e«,t
Eeforing specifically to the mesh configuration given
in Figure t3, the MN x Ш matrix A has positive dia-
gonal ontries and non-positive off—diagonal entries ,
and tho components of the column vectors ф and С
are indexed as follows
|SS
Щ
m'n 'm
Ы
1HMH
Сm
93
According to our definition of the "--"trix A as the
i:atrix зим, that is,
A-K-L-U• mi
wo see that Kr» are Ш е entries of the diagonal ma-
trix K^ l^ and* ILn are the entries of two dia-gonals, respectively, in the strictly lower triangular
matrix L , and the strictly upper triangular matrix
U has unit entries on ono diagonal and W n entries
on the other onea But as X У О . then
кп>1п + 1^*^п +1
so that A is a strictly diagonally dominant matrix»
The author has written a few codes solving the
equation (i39^ by the two-sweep iterative methods»
First oode "EWA-II" is based on the EWA two-sweep
iterative method [ii] e Next four codes "AGA-II«=An ,
"AGA-II-B", «A&A-II=Cn and "AGA-II-Dn aa-@ particular
versions of the AGA two-aweep iterative Ketho«ie All
the codes are written in GISR ALGOL III for the GIBE
computer and they require omly on® raap&etię tap© uiaito
UHfortunately, because of comparatively small storag©
capacity of the GIER computes the codes hsv© Ъъь~ ć;-
veloped oiily for two ener^" groups without tht» u
tering process aad with the. fflasimum p^naissible
of mesh points equal to 1365« Ш® initial valises of
the flukes sr© zero ±n Ь о Ш groups Mid Ш©'initial
sources are assumed to be plane la th® core regionso
The integration of sources is porformed hj ^
Now, we shall give the results obtained i'rom these
codes for two examples. For the first example taken
from Appendix В of iteierence 4 the matrices « e i ^ Ą
and Тд for each code are given for illustration, and
the numbers of arithmetic operations as well as atoms©
requirements ere discussed. For the second example takeu
Ггош Reference 14 also the effect of nonuniform moshos
on the rapidity of convergence is studied„
We consider the numerical solutions of tho follow-
ing two-dimensional elliptic partial iifferential
equations, typical of those encountered in reactor
engineering;
where X is the square 0 <?<.{/<.2*i shown in Fig» 14
with boundary conditions
where Г is the boundary of R . The given functions
D.j1L and S are piecewise constant, w»*łi the
values given in Table Ia
Thus, ws are considering the numerical solution
of a problem with iEternal interfaces, which are repre-
sented by dotted lines in Figure 14. la this particular
problem we encounter three physically different materials
9 5
Although evny piecev/ise constant source SvXY) could
-quniiy v/ell have been used here, wo have specifically
.•l-.odea $(*;/)« О to simplify the discussion of nu-
merical results to lollow* It is clear that a minimum
of 1*3 nsssh points is necessary to descriLa tiles**
different, regions by иеапэ of a nonuniform швз'ь
Numbering the mesh points as shovm in Figure 15
;\ul uyins the approximation based on a five-point for~
-.jiila, v;o can derive th© raatris equation
Y.-iiich is a discrete approximation to G.40) and
•..aero the ISX 46 matrix A has the following loca-
tions of non-sero entries denoted b> crosses«
X Xx x x
x x xX X
XX.X
X
X
XX
X
X
XXX
X
X
XXX
X
X
XX
X
X 1XX
XX Xxxx
x x xX X
XX
XX
XX
XX
X Xxxx
x x xX X
The above block partitioning of the matris A results
from considering all the mesfe points of each successive
horizontal mesia line as a bloc
The first purpose of this example is to give the
illustration of positive entries in non^xaegaiiv© ma-
trices Tg or H , ^ and T^ j wbich represent a
versioa of the tovo-s^eep iterative method used in a
given code.Now , w© shall show these matrices in particular
X
_
XX
XX
як
X
жX
X
яX
X
Xж.
кs
к
ж.z
жл
•
i
9?
жs
XX
•X
—
i
sX
ЛX
X
XX
X.i
1
£X
ZЖ
98
XX
ж.X к
XX
Xж.
як
ж
XlX
X
я
•
ж.
99
XX
XX
X
ж
X
XЖ
XX X
X ж
X XX X
• XX X
X X
ioo
кX X
Ж X
X
я X
S X
s
•
s
ж
ж
жж.s
X
1 0 1
5 e "AGA-II-D"
ss.
Ж XxX
X
s:X
X •X X
•x жs
X&
X
ж. к
£X X
X X
X
sжх\
XК 2
X
s
X
ж
ж
The second purpose of this example is to give
numerical results concerning numbers of iterations
for various iterative methods used in particular co-
des» Since the unique" vector solution of (±42*) obvio-
usly has zero components, the error in any vector
iterate ф ** arising from aa iterative method to
solv© (i4g) is just the vector itself ! To standar-
dise the iterative methods considered, ф ^ss always
chosen to have components all equal t© 10"„ Having
selected an iterative method;, w© iterate until am reach
such that each component
o The results of
different iterative methods applied to the Matrix pro»
ble© ($.42) mc® given in two tables» In Ч&Ы® II the
results are obtained without applying the ovexrelasatlon
process and in Table III — with applying toe overrela»»
sation process, where for the Ш&-11 double SOE we use
the first positive integer j
Ш•t is less t&im unity in
103
Prom the oomparison of results given in Table II,
it is clearly seem that the two—sweep Iterative methods
have cons Шагай! 0 advantages over the point Gaus^Seidel
aeihod» As was observed in the* analysis of the model
problem, w© have
& «In this example we have for the EWA-II
method ^ ( Ю ^ ^ C ^ O and lor the AG£»XI«A method
cC&iJss! <§^{Хд) e in terms o£ the asymptotic rat© of
convergence "R^CB) used as the simplest practical
measure of rapidity of convergence of a convergent
matrix В end defined as follows /see Reference 4/
v/e can say that the ЕИОД-IX iterative method is 9
times and the AGA-II-A iterative method is 13O5 times
asymptotically faster than the point Gatass<=Seidel itera«°
tive methoda In the case of the AGA»XI«D saet,hod \ч® have
к a 4i,8j thus» this method is almost 42 times asymp»
totioally fsster than the point G©usa=»Seid©l iterative
method, a
The application of the overrolassstioa process to
the two-sweep iterative methods provides further iiapro»
vement of the rapidity of convergence, whi©H is observed
ia Table IIIO It is worth noticing that for this example
the application of the overrelasation process in both
the ШД-II and AGA-II«=B methods accelerates the rapidity
of convergence less effectively than' ia the case of the
remaining two-sweep methods e
104
However, tho author observed that using the EWA-II
double SOB method provides considerable advantages In
ESE^ esses even in comparison to tte AGA-II-A single
SOB method5 for instance, see Example 2*
Now, w© shall discuss briefly the above two—sweep
iterative methods from the viewpoint of th© total
arithmetic effort and coefficient store requirements in
comparison to the point SOR method» With the five-point
approximation, in the point SOR method we must store
simultaneously th© entries of four diagonals of К (L+U)
and also the components of two vectorss that is, ф and •
К" С * For a rectangular mesh R^ consisting of M
гсл'в, each with M mesb points and assuming that the
order of four diagonals of K~ (Ы-lT) is almost the
same as tho order of vectors equal to MM we see that
this method requires the reservation of в 14 1 locations
in Ш в fast memory of a digital computer for storing all
the coefficients of the five-point difference equations«
In terms of arithmetic oalculatio&R, the above method
requires at most six multiplications and sis additions
per mash point per iteration with the assujaptioa that
all the coefficients are calculated at the beginning of
toe iteration process and stored in the computer for
the plioi® iteration process 0
la t-tie case of the two«s^@ep iterative methods the
storage requir©H3©ats as well as the aaumber of aritfometFie
operations per mesbi point per iteration ar© increased,,
Howeverв the ргоЪ!@ш of inorea&esi storage requirements
in .the two=»sw©ep iterative jnstbods cam b© ©asily solved
by separate execution of ©eob sweep in the fast memory o
la other words, assusniog that all the eoeffi©I©Hts are
105
calculated at the beginning of the iteration process
ал<1 stored in the auxiliary memory of a computer, only
these coefficients вееезззж? *osr execution of a given
sweep are seat to the fast memory of a computer in the
whole iteration pr©eesee For instance, in the case
of the E57A-1I two—sweep iterative method for the forward
elimination s\ieep /calculation of the veotcr jfb / we
must reserve only 6MM locations in the fast memory
and for the backward substitution sweep /calculation
of the vector ф / - only 4 H N locations,, ?hus, we
soe that this method from the viewpoint of storage
requirements is equivalent to the point SOR method.
Storage requirements and the numbers of arithmetic
operations for the iterative methods considered in this
paper are given in Table IV„ The results of Table IV
are obtained on the basis of recursive formulae for
tho components of vectors ft) and (p derived directly
for each two«»sweep iterative method from the five»point
formula (i39; 9 The positive entries of И which coin-
cide with the positive entries of L are assumed as
common coefficients in the recursive formulae0
To illustrate the application of the above iteration
methods to reactor calculations, two-group calculations
were performed for a U -iig0 thermal reactor |l4J »
The reactor configuration studied is a rectangular core
surrounded by a rectilinear water reflector as stoowi
ia Figo 16 covering the quarter of the reactor,, Half of
the oore dimension is 15 си and the thlclmess of the
106
water reflector is 20 cm on all sides. Two—group cross
sections are given in Table V o The fluxes at the outer
boundary of the reflector were put equal to aero*
Iteration calculate -л in the two-sweep iterative methods
is initiated with the initial flux assumed to be zero
in the whole reactor and with flat initial sources in
th© oore region; it is terminated when the relative
change of the criticality factor becomes less than
5xtO*"6« In the case of two-sweep iterative methods th©
above convergence criterion is equivalent to the rela<°
tive change of the flus in each mesh point less than
10 % The results are compared here with the solution
of the £ive«™poisit difference aquations by TWENfY С1ШГО
[is] with the ваше convergence criterion. The iteration
method used in WENTY ШШГО is based on the SOE method,
The results obtained by TWENTY G&fi№> with the initial
flux assumed to be flat in the whole reactor ar© taken
frora Reference 14 „
W© shall consider numerical results obtained fros
th© above iteration methods for a few oases differing
by the choice of the number of ®esto points as well as
the shsp© of the m©she Тие results are shotm in Tables
¥X and ¥11 в Th© mesh intervals htsve been asssamecl ©qual
for sny regiosa ©long в giv©m asis in ©11 ; ©es©se Th©
notation (b^")sCs*8)=i256, for instasc@8 means that
eight m&śh int&Tv&ls are telsesE both im Ш е oore ©rad the
reflector eloag both ass:®ss чйаегваз the notation (iCS+le) x
s (4*4) «я 256 Ш©ШЙ© that sisteesa mesb iiatervals ar© t®Jtea
al©ag the a^ie у «m$ f©esr alonsg the asis X Ш th©
©ore and the r©flootor0 In spit© of the fact that the
<j>f ja©sSa pointo for wlaiola th©" fluxes are sousght
iOl
is equal to 256 in both eases, we have for the core
and the reflector region No* 2: ^ =к(ДХ/ду)*тнс1
in ihe first case and ^акСд^Ду^гга^б in the second
case* The number of source iterations is shown in the
fourth column of Table VI for each cod© in the case
of throe meshes diffaiing by the number of шези points
but having ^=«4 in the core and the reflector region
No 2, As can be seen, we obtain the smallest number of
source iterations for the EWA—II double SOR /where
О.ф = Qa=sSl/ f whereas in the former example be-
results were obtained for the AGA-11 single SOR met.*
The affeot of the increase of -p on the rapidity
of convergence in different methods is illustrated in
Table VII. The cases studied have the same number of
mesh points equal to 256 but different values of
f « ( A ^ v / ) a , In the cases (e+8)x&+8) , (£6+1б) х (4+4)
and (32*32)x(2+2) the corresponding values of у are
equal to i, 16 and 256, respectively, in the core and
th© reflector region Noe 20 As was shewn in the analysis
of the model problem, we observe in this example that
the increase of j- implies the increase of the number
of iterations in the point SOR method, whereas in. the
tvro—sweep iterative methods, used here without th© over-
relaxation process , we have the inverse effect, ttoafc
is, the decrease of the number of iterations © The reason
for it is that in matrices A , as the values of f
increase, three from five пота-zero diagonals are шоте
dominating and the two»»SY7®ep iterative methods ax© closer
to the direct method*, It should be mentioned that in
the point SOB method9 initial sources and fluxes are
th© same as in the two«°swa©p iterative methods ш&
108
values of CO for the point SOR method given in Table Vlif
were calculated from Young*s formula Сэз) using the
values ^(Х*) calculated with five significant digits..
VIIо CONCLUSIONS
Prom the analysis of the model problem and
cal results presented in the forraer chapter we see
that the application of the tx?o»sweep iterative methods
to the numerical solution of neutron diffusion equa-
tions allows us to obtain considerable time savings on
digital computers in comparison to the point SOR method.
The author performed many critical calculations of thertaal
reactor configurations by means of the two-sweep
iterative methods /applying mainly the EWA-IX double
SOR and the AGA-IT-A single SOR methods/ and always
obtained a rapid convergence* It follows frons his
observations that in th© range of the spectral radii
, 4? close to unity the AGA-II-A single SOR provides
th.© best results, but for somewhat smaller values of
^C<Ll) quits good results are obtained by the fflA«II
double SOEa The toro«swe©p iterative saetbods give the
best results for nonuniform meshes and this makes them
©specially attractive, ©s compared to Ш © usual SOR
ia numerical solutions of reactor problems with
geometrical configurations, heterogeneous
properties sad weak energy group couplingo
Tfee possibility of independent realisation of both
sweeps ia the fast memory of a coapuier allows use in
the oas© of both Щ & » Ц double SOU and ЖА«П-А sisigl®
109
SOR8 to uso the same storage capacity аз in the point
SOS method /see Tab!© IV/e 1'he increased number of
arithmetic operations in the iwo«sweep methods ia
comparison to the point SOR method is undoubtedly a
disadvantage from the viewpoint of numerical effort»
However, in the application of the two-swoop iterative
methods suoh as EVFA-II double SOR or AGA-II-A single
Só4 to solving multi~group diffusion equations by tho
source Iteration method thi. 'sadvantage is almost
uiinotioea.blee For a multi-group model the calculation
of the souroe term at each niosh point in a given group,
contains fission, down» and up-scattering processes
and it can require in all methods a greater number of
arithmetic operations than those given in Table IVeSQ
that the relative increase in the operation шипЪег for
the two»»sweep methods becomes less important«, In this
caae the number of source iterations can be used as
the direct measure of rapidity of convergence when
comparing these methods to the point SOR method» It
should be mentioned, as the author observed,, the us©
of zero fluxes for all energy groups is the best initial
guess for the two-sweep methods» Moreover, the use
of zero initial guess in the two-sweep mathods is always
possible, whereas such an initial gueąs for th© point
SOS method in some oases would give rise to erroneous
results о
It is interesting to notice that, as 4h© author
observed, very often the application of the two-sweep
iterative methods without the overrelaxation process
gives a lesser number of source iterations than the
point SOR methods with the optimal value of CO for
iio
the assumed convergence criterion in spite of the fact
that the spectral radii in the point SOS method are
smaller than tta© spectral radii in the t "Q—sweep
methods without, overretaxation*,
The theoretical problems encountered in the
determination of the optiaum relaxation factor CO
in the ease of the two-sweep single or double SOB
methods hav© not bean solved up to now» However, the
approximate values of the optiravia relaxation factor
may be determined experimentally« Studies on the deter-
mination of the optimum relaxation factor are being
continueds
Ono might suppose that other accelerating pro-
cedures, for instance, based on the Chebyshev polyno»
mials, can provide further improvement of the rapidity
of convergence in the i-wo—sweep iterative methods о
Finally, it should be mentioned that the appli-
cation of the two-sweep methods to the hexagonal meshes
seems especially proms ing <,
ACKNOWLEDGMENTS
I ага indebted to Professor Janusz Mika for his
-•mtimmd interest, encouragement and scientific
dswoe ia th© course of preparation of this work»
oulu like also to acknowledge numerous valuable
..uggeetions r©«©iv©d from other members of Professor
* group© I аш sspeoially grateful to Miss Krystyna
for the help In preparing the final versionof the Ejanuscript о
I l l
are also due to Dr Rei; Yoehlieh andhis group froa Kernforschimgsrentr^-im Karlsruhe fortheir kind is terest and belpful comments during шуvisit at Earlsrafae and to Dr Robert Baau'wens fromUniversito Libre de Bruxelles for proving some importantresults еоваогвей tfith the application of the Ef/Aasid the MA t>TO*«swe<ap methods,* 9
ттш i
112
Region
1
2
3
В (x,y>
1.0
2,0
3.0
I C*,Y)
0*02
0*03
0,05
SCx,y)
0
0
0
TABLE I I
Method
poiat Jacobi
point Gsusa-Sei'iel
AGA-II-AДЗД-Х1-В
AGA-II«D
Number of
iterations
10020
5010
564
371
189
305
120
Spectral
radius
C999Q8
0,99816
0o98374
0*97544
0.95217
0e97018
0 «,92530
ЧШШ 111
Method
point SOBE№XX eingl© SOEWi\fll double SOKAGiip»IX»,& s i n g l e SOSдедр»хх«в single SOBAG/i,-II-C slngl© SOSJUBA^II-B slngl© SOB
Number ofiterations
i 3 S404380
26
iOO
oa21
a> or Si
1.617?i.171.091«4S61.231*3561.S96
Spectralradius
0.91770.977
0,7290e9110e7050e572
TABLE IV
Metbod
point SUE
EWAP-IX singi© sonEW&-IX double SOE
A6AF»II«-A single SOE
AGA»II»B single SOU
AfiA«XI«»Q singi© SOE
AGA-XI-D single SOS
Kmnber of veotors of orderШ storied In fast memory
in forwardelimination
sweep
6
6
6
6
8
8
10
in baokrvardsubstitution
sweep
4/
5
6
6
7
Number of arltii®ctiooperations per Besfopoint per it-oration
multiplioa*»tlons
67
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