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PL9700384INSTYTUT ENERGII ATOMOWEJ INSTITUTE OF ATOMIC ENERGY

RAPORT LAE - 16/A

THE NUMERICAL ANALYSIS OF EIGENVALUE PROBLEM SOLUTIONS

IN THE MULTIGROUP NEUTRON DIFFUSION THEORY

ZBIGNIEW I.WOZNICKI

Institute of Atomic Energy, Research Group A-14 05-400 Otwock- § wierk, Poland Email: r05zw@cxl.cyf.gov.pl

The second editionExtended and substantially revised, June 1995

VOL 2 8 iis 0 c OTWOCK - SwiERK 1995

We regret that some of the pages in this report may

not be up to the proper legibility standards, even though the best

possible copy was used for scanning

Zbigniew I.WoZnicki: The Numerical Analysis of Eigenvalue Problem Solutions in the Multigroup Neutron Diffusion Theory

The main goal of this paper is to present a general iteration strategy for solving the discrete form of multidimensional neutron diffusion equations equ­ivalent mathematically to an eigenvalue problem. Usually a solution method is based on different levels of iterations. The presented matrix formalism allows us to visualize explicitly how the used matrix splitting influences the matrix structure in an eigenvalue problem to be solved as well as the interdependence between inner and outer iterations within global iterations. Particular itera­tive strategies are illustrated by numerical results obtained for several re­actor problems.

Zbigniew I.WoZnicki: Analiza numeryczna rozwlgzah problemdw wlasnych w wielo- grupowej teorii dyfuzji neutrondw

Celem pracy jest przedstawienie ogdlnej strategii iteracyjnej rozwigzywania dyskretnej postaci wielowymiarowych rdwnart dyfuzji neutrondw, rdwnowaznej ma- tematycznie problemowi wlasnemu. Zwykle metoda rozwigzywania jest oparta na rOZnych poziomach iteracji. Przedstawiony w pracy formalizrn rnacierzowy pozwala uwidoczniC wplyw uZytego splittingu macierzy problemu wlasnego oraz wspdlza- leZnoSO miqdzy iteracjami wewnqtrznymi i zewnqtrznymi w obrqbie iteracji glo­bal nej . PoszczegOlne strategic iteracyjne sq zilustrowane wynikami numeryczny- mi otrzymanymi dla kilku typdw reaktordw jgdrowych.

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necKn 3KBMBa/ieHTHOi/t 3a#aHM Ha coGcTseHHwe SHaneHMS. Ogwhho mexoa peiueHwfl ocho-

saH Ha pa3/iwhhwx ypoBHsx wTepam/ti/i. ilpeacraB/ieHUbiM b stoui paGore maTpuHHwfl &op-

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BHyrpeHHi/imn w BHeiuHnmn WTepauMsmi/i b cpefle r/ioGa/ibHoi/! urepaunn. OTAe/ibHbie uTe-

pauwoHHwe crparermi M/i/iiocTpi/ipoBaHbi c nomoujbio nwc/ieHHbix peay/ibraroB rio/iyneHHwx

nnn HecKO/ibKnx tuttob aaepnwx peaxxopoB.

Wydaje Instytut Energii Atomowej • OINTEA Naklad 65 egz Obj^to^: ark-wyd. 6,5; aik-druk. 16.

Datazloteniamaszynopisu: 22.06.95. Pr3144zdnia 1993.01.04

II

PREFACE

This report is the extended and essentially revised version of the former one and published under the same title as RAPORT IAE - 6/A (1994). New numeri­cal results are presented for the eight group thermal reactor problem (Test Problem 5) with seven thermal groups and the reliability of solutions is stud­ied for problems with symmetrical solutions. This new material is mainly in­cluded in Section 5

mNEXT PAQE(S)

left BLANK

CONTENTS

1. Introduction ....................................................... 1

2. Problem Formulation................................................. 22.1 Reduced Problem ....................................................... 42.2 Basic Problem ......................................................... 5

3. Single Splitting Iterative Strategies .............................. 63.1 Global Iterations ..................................................... 73.2 Global-Outer Iterations .............................................. 103.3 Global-Inner Iterations .............................................. 133.4 Global-Outer-Inner Iterations ........................................ 14

4. Double Splitting Iterative Strategies ..................................... 164.1 Global Iterations ..................................................... 174.2 Global-Outer Iterations .............................................. 194.3 Global-Inner Iterations .............................................. 204.4 Global-Outer-Inner Iterations ........................................ 22

5. Numerical Illustration ............................................. 235.1 Computational Aspects ................................................. 245.2 Numerical Experiments ................................................. 265.3 Discussion of Numerical Results............. 325.4 Reliability of Symmetrical Solutions ............................... 405.5 Final Remarks and Conclusion ........................................ 41

Acknowledgement ...................................................... 43

References............................................................. 44

Tables and Figures.................................................... 45

V

1. INTRODUCTION

The neutron diffusion theory is the most widely used method in the analysis of criticality of nuclear reactors. The consideration of criticality is gene­rally referred to as an eigenvalue problem for the multigroup neutron diffu­sion equations which solution provides the reactivity eigenvalue, i.e. , the effective multiplication factor and power profiles in large scale reactors subject to nonuniform loading a fuel and its depletion [1]. Hence the numeric­al solution for realistic heterogeneous reactor problems requires detailed ne­utron diffusion calculations in two- or three-dimensional geometries.

In the numerical solution of multigroup neutron diffusion equations much effort have been devoted over the last four decades to the development of ef­ficient iterative methods and implemented in numerous computer codes (see,e.g. References in [4,7]). The standard method of solution is based on the outer- -inner iteration strategy in which the discrete form of neutron diffusion equ­ations is solved groupwise by means of inner iterations stopped when either their number achieves a value assumed for each group or a given convergence criterion is fulfilled. Completing the cycle of inner iterations in all groups corresponds to one outer iteration for which the fission sources are recalcu­lated. Usually Chebyshev polynomial acceleration techniques [2,3] are used for increasing the rate of convergence of outer iterations.

The aim of this paper is to make some systematic study of iterative strate­gies for solving multidimensional neutron diffusion equations by using three levels of iterations called global, outer and inner iterations which in the solution algorithm have the following interpretation. In inner iterations the values of neutron flux are updated within groups with fixed both scattering and fission terms. On the level of outer iterations the neutron flux is com­puted with updating the downscattering term in a given outer iteration and the upscattering term is modified between successive outer iterations. After com­pleting the cycle of outer iterations, corresponding to one global iteration, the fission term is recalculated. The matrix formalism presented in the paper allows us to visualize explicitly how the used matrix splitting influences the matrix structure in an eigenvalue problem to be solved, as well as to show an interdependence between inner and outer iterations within global iterations.

With the assumption of 1 outer iteration per global one, the global-outer- -inner iteration strategy reduces to the outer-inner iteration strategy widely used. Thus the discussed global-outer-inner iteration strategy is a generali­zation of the outer-inner iteration strategy, which used for solving reactor problems with a significant upscattering allows us to reduce the number of global iterations on a factor equal approximately to the used number of outer iterations per global one.

1

In the next section the discrete form of multigroup neutron diffusion equa­tions is formulated together with describing the structure of matrices and so­lution properties. Iterative strategies based on different levels of (global, outer, inner) iterations are presented for single splittings in Section 3, and for double splittings in Section 4. The numerical illustration of different iterative strategies is given in Section 5 for several reactor problems where the efficiency of particular iterative strategies is estimated in terms of the computational work per mesh point.

2. PROBLEM FORMULATION

The time-independent multigroup neutron diffusion equations representing a system of coupled elliptic partial differential equations of the second or­der can be written, as follows

-VD (r)V<p (r)9 9

I (r)<t> (r)9 9

CE9*9 1 99

(r )4> ul , [r)<p , (r) (2.1)

g = 1,2, ..., Gwhere notation is standard [1,4]. The following group-dependent boundary con­ditions

d<p (r)D (r)— --- + a (r)0 (r) =0, ref (2.2)9 dn 9 9

are used where n is the normal outwardly directed to the boundary F.According to practice the above continuous problem is converted to discrete

representations into a mesh of fine cells. The most frequently used approach, the standard low-order finite difference approximation [1,2,3], leads to the system of linear coupled algebraic equations which can be combined into the following matrix eigenvalue problem

E (p = ixFT <p.

The order of the square matrixE = A - Sd - Su

(2.3)

(2.4)is equal to the number of energy-space mesh points, GxN {G is the number of energy groups and N is the number of spatial mesh points), where the matrices A, Sd and Su represent diffusion-removal, downscatter and upscatter terms res­

pectively. The components of the column vector <p represent approximate values of neutron flux 4> at the energy-space mesh points. The emergence spectrum of fission neutrons and a production (fission) term are represented by the column natrices X and F respectively, where superscript T denotes the transposition.

The precise structure of these matrices depends, among other things, on the ordering of the unknown fluxes in the vector <p. For the natural ordering of the unknowns in <p, row by row and plane by plane, and assumed in the remainder of the paper, the structure of particular matrices can be visualized, as fol­lows

2

[A I ■ 0 lb su su s“ 1l 1,2 1,3 . 1 , GA 0 sd 0 0 02 2,1

, Sd =

C/1 cII 0

0 . 0 0 . suG-l.GA sd sd . sd 0G G,1 G,2 G,G-l 0

V V V

X =X2

, F =F2

and <f> =?2

XG. Fg. .V

The structure and properties of NxN submatrices A in the block diagonal9matrix A are dependent on the problem dimensionality, the mesh geometry (rec­tangular, triangular, uniform, nonuniform), the method (mesh-centered, mesh- -edged, an approximation order) used for the derivation of difference equa­tions providing n-point formula which forms n nonzero diagonals in A , as well

9as the reactor material composition represented by group constants (see, for example, [7] and references given herin). All NxN nonzero submatrices in Sd, Su, X and F are diagonal matrices with nonnegative entries.

The properties of Eq.(2.3) have been exhaustively studied over the years. Under quite general conditions, Frohlich [5] has proven that Eq.(2.3) has a unique positive eigenvector <p and a corresponding single positive eigenvalue A1 (interpreted as the effective multiplication factor k ) greater than the absolute value of any other eigenvalue. For earlier developments see, for example, References given in [2,5].

The matrix E is nonsingular and monotone, i. e. , its inverse is nonnegative, where each A 1 is a positive matrix, i.e., A*1 > 0, and the eigenvalue problem

9 9(2.3) can be written in the following form

\<p = B<p (2.6)where

B = E-1XFT £ 0. (2.7)

A large class of methods for determinig the eigenvalue of largest modulus Aand the corresponding eigenvector <p^ is based on the power method [2,3,8] in which successive estimates for Aj and <p^ are generated by the process

and

<p(l + l) = ^jyjB<p(l)

\\<p(i+v\A(1+1) = X(l)-

\\<p(D\

(2.8)

(2.9)

where 1 is the iteration index and two norms either maximum !]•!! or Euclidean

3

|»|| are most commonly used. Since the largest (in modulus) eigenvalue A^ of the nonnegative matrix B is positive and simple, the power method is a conver­gent process for almost randomly chosen nonnegative starting vector <p(0), i.e.

\( 1) \ , <p( 1) —> <p as 1 —> co.The rate of convergence in the power method is governed by the subdominance ratio (according to the terminology proposed in [8])

<r(B) = U2 l/\ (2. 10)

assuming that the eigenvalues A^ of the matrix B are ordered in such a way that

A^ > |Az| £ |A3| £ ___ (2.11)

The snailer the subdominance ratio, the faster the convergence. Assuming that for sufficiently large values of 1, A(1) is approximately equal to A^, one can equivalently consider the problem

where

has eigenvalues vB , 1

‘V iand

<p( 1 + 1) = B(A^ )<p( 1)

B(Ai ) = 1 E"1XFTi

• A^/Ai and according to the above ordering

= p[B(Ai) ] = 1 > |rB z | £ | i>b 31 £ ....

<r(B) = <r[B(A ) ] = \v \.l B, 2

(2.12)

(2.13)

(2.14)

2.1 REDUCED PROBLEM

Since the rank of the matrix F is only N, the eigenvalue problem (2.6) with the order of GxN can be reduced to the variant with the order of N which ite­rates on the fission source vector [2,4]

= FV (2.15) Multiplying Eq.(2.6) by FT and using (2.7) and (2.15), one obtains the reduced

problem\ip = (hp (2. 16)

where0 = FTE_1X £ 0. (2.17)

Both matrices B = (E 1X)(FT) and 0 = (FT)(E 1X) have the same rank and their

nonzero eigenvalues are identical but this implies that both eigenvalue prob­lems given by Eqs.02.6) and (2.16) are mathematically equivalent.

The fundamental eigenvalue A^ and the corresponding eigenvector ^ which represents fission sources can be obtained by means of the power method de­scribed previously.

In actual practice Chebyshev polynomials on the flux vector or on the fission source V,1 are usually used to accelerate the convergence of the power

4

method iterations [2,3,4]. The application of Chebyshev polynomials is based on the assumption that the eigenvalues of B lor 0) are real, nonnegative and ordered in such a way that A^ > A^ - ^ & A^ & 0, and the corresponding eigenvectors form a basis for the JV-dimenslonal vector space. However, these hypotheses may be not true in some multigroup neutron diffusion problems.

2.2 BASIC PROBLEM

In the considerations given above, it was assumed that the matrix E 1 is gi­

ven explicitly. In the case of one-dimensional problems however, with the ab­sence of upscattering process, i.e., Su = 0, it is often possible to compute the matrix E 1 directly. In two and three dimensions the inversion of E has

proven impractical and therefore, a special iterative strategy must be used.Eq.(2.3) together with Eq.(2.4) can be rewritten as

Ap = Sd<p + Sa<p + JxF T<p (2. 18)

and introducing the iteration index I, one obtains

A<p(l + l) = Sd ip( 1+1) + S u<p(l) + ^jjXFr<p(l) (2.19)

or equivalently<p(l+l) = A_1[S d<p(l+l) + S u<p(l) + ^jjXFr<p( 1)]. (2.20)

The fact that Sd is a strlcly lower triangular block matrix allows us to use

the components of <p obtained in the latest iteration 1 + 1. After completing the calculations for the iteration 1+1, a new value of A(1+1) is estimated accord­ing to the formula given by Eq.(2.9). The above equation can be represented in the following form

<p(l+l) = T(A (l))<p(l) (2.21)

whereT(Am) = [I - A-1 Sd ]”1 A"1 [Su + ]. (2.22)

As can be seen the entries of T(A(1)) change their values from iteration to iteration in dependence on changes of A(1) and for sufficiently large values of i, Ml) becomes close to A^ and

T(Am) —» T(Ai) = [I - A”1 Sd ]-1 A-1 r~u[Su + ± XFT] fc 0. (2.23)

Assuming that the eigenvalues of T(A ) are ordered, as follows

VT,1 = P[TU1)] = 1 > |UT,2 1 ^ ^T.S1 ^

then for sufficiently large values of 1, the rate of convergence of iteration process (2.20) is governed by the subdominance ratio

<r[T(A)] = I v | (2.24)1 T , 2It is evident that with the absence of upscattering, i.e. , when Su = 0,

T(At) = B(At).

5

The iterative process represented by Eq.(2.20) is referred to as "fission source iteration" or alternatively as "outer iteration" however, for its im­plementation, it is necessary to have the inverses of the submatrices A form­ing the block diagonal structure of A visualized in (2.5). In one-dimensionalproblems A are three diagonal matrices which inverting can be easily done by

9the well known forward elimination - backward substitution procedure [2]. For two- and three-dimensional problems the matrices A have more than three dia-

9gonals and moreover its order is much greater than that in one-dimentionalcase, so that the inversion of A becomes impractical or simply impossible

9with respect to round-off errors. In this case an approximate inversion of A

9is usually done iteratively through a series of "inner iterations". The stra­tegies based on different levels of iterations are presented in the next sec- tions.

3. SINGLE SPLITTING ITERATIVE STRATEGIES

Inhomogeneous solutions appearing in multigroup diffusion problems are re­ferred to the iterative solution of the linear equation system

A0 = c (3.1)which can be expressed in the form

M0<t+1) = N0(t> + c, t £ 0 (3.2)where 1 denotes the successive iterates and

A = M - N (3.3)represents the single splitting of the NxN nonsingular matrix A as the classi­cal splitting of A in the theory of iterative methods. The above iterative process is convergent to the unique solution

4> = A-1c (3.4)for each #<0) if and only if M is a nonsingular matrix and the corresponding

iteration matrix9 = M-1N (3.5)

has the spectral radiusp(S) = max |y | < 1, 1 s i < N (3.6)

where are eigenvalues of '§.

*(t + U =or in terms of <p( 01

Eq.(3.2) can be written in the equivalent form

5<fr(t) + H_1c, t £ 0 (3.7)

where

From Eq. (3.3)

M"^ = (I + 9 + *?2+ ...

it follows that

A"1 = (I - M-1N)_1M_1

h;;,c, t £ 0

+ shn"1 = £ sV11 =0

= (I - &)"1M'1

(3.8)

(3.9)

(3.10)

6

hence(3.11)M'1 = (I - S)A'\

Substituting Eq.(3.11) into Eq.(3.9), one obtains

} = (I - ^t + 1)A"V (3.12)

Since by the assumption p(W) < 1, !?* approaches the null matrix when t —» » and consequently M —> A 1, and the solution of (3.8) tends to the unique solution defined by Eq.(3.4) for arbitrary 0) . In the convergence analysis

of iterative methods the (asymptotic) rate of convergenceR(S) = - lnp(i?) (3. 13)

is certainly the simplest practical measure in the rapidity of convergence for a convergent matrix 'S and especially useful for comparing the efficiency of different iterative methods [3,8].

The matrix M * can be considered in some sense as an incomplete inverse of A, approximating A-1 after t iterations, therefore M ^ is called the prein­vert ioner of t-degree of the matrix A, where M ^ = A 1 and = M 1 is

the preinvertioner of O-degree of the matrix A.Convergence behaviour is usually studied by examining the error vector de­

fined by e(t+1) = * - ,(t + i) (3.14)(0)Specifically for # =0, one obtains

hence by Eq.(3.12)

«'t*u - A-= - h;;,c

e<t,u - (3.15)Thus, the solution of Eq.(3.8) corresponds to a solution obtained with the preinvertioner ) approximating A™1 after t Iterations and for 01 =0 its

error vector with respect to the unique solution of Eq.(3.4) is determined by Eq.(3.15).

3.1 GLOBAL ITERATIONS

Assuming that the splitting of A is represented by the following matrix form

[A 1 [m 1 IX Il l i

A 0 M 0 N 02 2 2• = M - N = -

0 0 0A M Nc c c

Eq.(2.18) can be written, as follows

M#> = Ny> + Sd<p + S °<p + ixFT<p (3. 17)

and itreducing the iteration index 2, one obtains

7

(3.18)

If the splitting of A is chosen in such a way that M is a nonsingular matrix and relatively easy to invert, the above equation can be expressed as

1, z , „ . . .. z,\ . „u z , % . 1

M<p(l + l) = N<p(l) + S <p(l + l) + S <p(l) + ifjj*F (p(l).

<p(l + l) = M [S <p(l + l) + Nyfi; + S <p(l) + ^jjXF <p(l)]

or equivalently9(1+1) = V(\(l))<p(l)

whereV(A(1)) = [I - M"1Sd]"1[V + M'1(Su + ^^jjXFT)]

(3.19)

(3.20)

(3.21)

and the iteration matrix V, associated with the splitting of A defined by Eq.(3.16), has the form

V = M N =

TV 1 [H-1n 1i 1 1V 0 M-1 N 02 2 2

0 0V M-1NG G G

(3.22)

For sufficiently large values of I, A(1)

V(A (D)

A^ and

V(A^) = [I - M-1Sd]-1[V + M'1(Su + ^ XFT)]l

(3.23)

and assuming that the eigenvalues of V(A ) are ordered, as follows

Vl = p[VUl)] = 1 > '"v.gl £ 1 yv, 3 1 £ •• •

the rate of convergence in the iteration process (3.20), representing the glo­bal iteration (1) strategy, is governed by the subdominance ratio

<r[V(Ai )] = |ry 2I (3.24)

where the index 1 is referred to global iterations which are simply power method iterations and M 1 is the preinvert loner of 0-degree of the matrix A.

By an analogy to the convergence analysis of iterative methods for solving linear equation systems, one can define the (asymptotic) rate of convergence, as follows

R(V) = - ln«r[V] (3.25)

which is an useful measure for the rapidity of convergence to the dominant eigenvalue of a given matrix V in the power method [8]. Since the number of iterations required for obtaining the dominant eigenvalue is roughly inversely proportional to the rate of convergence, the arithmetical effort required for computing the dominant eigenvalue of two different matrices and by means of the power method can be evaluated by comparing the numbers of iterations obtained with a given degree of accuracy obviously, assuming that the computa­tional cost per iteration is comparable for both matrices.

8

The eigenvalue spectrum of V(A^) depends on the assumed splitting of thematrix A. When A = M - N represents a nonnegative splitting of A [9] (i. e. ,M 1 a 0, M N a 0 and NM™1 & 0) or, its special case, a regular splitting of A(i. e. , M-1 & 0 and N & 0), then V(A^) is a nonnegative matrix which ensures

that the iteration process (3.20) is convergent for all <p(0).

Usually the NxN submatrices A are defined by the following decomposition9

A = K - L - U (3.26)9 9 9 9

where K , L , U are diagonal, strictly lower triangular, and strictly upper9 9 9

triangular matrices, respectively. Moreover these three matrices are nonnega­tive. Defining the block diagonal matrix A of Eq.(2.5) in a similar way, i.e.,

A = K - L - U (3.26a)it is evident that the entries of K, L and U are those of K , L and U , for

9 9 9all g = 1.... .G, respectively. In the majority of numerical problems, itera­tive schemes are derived from splittings representing the Gauss-Seidel method [2,3,8] and defined by

M = K - L and N = U (3.27)9 9 9 9 9

or for the block matrix structureM = K - L and N=U. (3.27a)

The above equations represent a regular splitting of A and the correspondingiteration matrix £ has the form

£ = M_1N = [I - K~1L]”1 K_ 1U a 0. (3.28)

The SOR method [2,3,8] closely related with the Gauss-Seidel method, is re­presented by the following splitting matrices

M = lx[l - wK^L] and N = -[wU - (o-l)K] (3.29)o o w w

and the associated iteration matrix £ can be written asus£ = M_1N = [I - wK”1L]"1K-1 [wU - (o-l)I] (3.30)us US US

where us is the relaxation factor. It is evident that for us = 1, the above equ­ations reduce to Eqs.(3.27a) and (3.28) representing the Gauss-Seidel method. As is well know [3], p( j?y) < 1 for all 1 < us < 2, and when A is an 2-cyclic cosistently ordered matrix, there exists

2us _ =------------- (3.30a)op l + vT^pTFI

minimizing the value of p{£^). The efficient method for the estimation of p(£^) is given in [8].

The eigenvalues of V(A^) defined by Eq.(3.23) satisfy the equation

[I - M"1Sd]'1M'1(N + Su + 1 XFT)]x = i/x (3.31)

9

or equivalently(3.31a)[N + vSd + Su + 1 XFT]x = vMx

1

and substituting Eqs.(3.27a) into Eq.(3.31a), one obtains

K"1 [v(L + Sd) + U + Su + i XFT]x = i>x. (3.32)l

In the case of using Eqs.(3.29) the corresponding equations can be written, as V (X )y e [I - M"1 Sd ]~1 M~1 (N + Su + 1 XFT)]y = %)y (3.33)W 1 W W W A^

or equivalently .[N^ + T}Sd + Su + i XF ]y = TjM^y (3.33a)

land after substituting Eqs.(3.29) into Eq.(3.33a), one obtains

K'1 [t)(L + Sd) + U + Su + 1 XFT]y = 5^—y. (3.34)A^ W

From the above equation, it may be seen an implicite dependence of r> on the relaxation factor w where obviously, T) = v for w = 1. In the case of the ma­trix £ , the problem was related with finding w minimizig the value of the do-

(i)

minant eigenvalue and for an 2-cyclic consistently ordered case its value is determined by Eq.(3.30a). In the considered eigenvalue problems, associated matrices are defined in such a way that their dominant eigenvalues are equal to unity, so that in the case of the matrix V (X ), r? =1 and for the corres-CJ 1 1ponding eigenvector y^

K_1[(L + Sd) + U + Su + i XFT]y = -y . (3.34a)l

is fulfilled for all u> * 0. Thus, in this case the main problem is related with finding the value of w which minimizes the second eigenvalue equiva­lent to the subdominance <r[V (X )]. As is observed in numerical experiments, there is w minimalizing <r[V (X )] and its value is usually greater thanbest 0) 1w minimizing the spectral radius of £ in the SOR method. For 1 < u s wopt W best<r[V (X )] decreases monotonously, for w < w ^ w a strong increase of W 1 best crlt<r[V (X )] is observed and with w the disconvergence of iteration process

U 1 crltis occuring, where usually u <2. This effect will be discussed and demon-c r 1 tstrated in examples given in Section 5.

It is interesting to mention that in Test Problem 2 considered in Section 5 only this strategy is the most efficient solution method in comparison to all others.

3.2 GLOBAL-OUTER ITERATIONS

Denoting the fission source term byffi; = ^jxrVn

Eq.(3.18) can be rewritten as

(M - Sd)<p(l + l) = (N + Sa)<p(l) + f(1).

(3.35)

(3.36)

10

For fixed f (1) the cycle of outer iterations p = 1,2, ... ,P can be performed according to the following scheme

(M - Sd)<p(l+l) = (N + 1+^-1 + f(1) (3.37)or equivalently

<p(l+Z) = G<p(l+Zy-) + [I - M"1Sd]“1M'1f('i; (3.38)where G = [I - M~1 Sd ] ~1M-1 (N + Su) = [I - M'Vr1 [1/ + M_ 1 Su 3. (3.39)

It is easy to notice that

for p=l, <p(l+^) = G<p(l) + [I - M-1Sd]'1M'1f('i;,for p=2, <p(l+= G<p(l+^) + [I - M_1Sd]"1M‘1f('J/)

= GZ<p(l) + [I + G] [ I - M'1Sd]"1M'1ffi;

for p=P, <p(l+l) = GP<p(l) + [I + G + G2+ ___ + GP_1HI - M"1Sd]"1M"1f(J)

which can be written in the following general form

<p(l+l) = VUC l))<p( 1) (3.40)where

and

V(M1» = c" * mX

p

(3.41)

M‘J_n = £ GP-1(I - M'Vl'V1. (3.42)p=i

Assuming that for sufficiently large values of 1, V(A(1)) « V(), and the ei­genvalues of VU ) are ordered, as follows

Vi = ■ 1 8 8 l*’,,3l 8

the rate of convergence in the iteration process (3.40), representing the global-outer iteration (l,p) strategy, is governed by the subdominance ratio

o-tVU^] = |vy 2i (3.43)

where the index p is referred to outer iterations.As can be seen for a convergent matrix G,

GP -> 0 and M'1 x —> E'1 (3.44)(p-i)

when P —» co, hence, V(A^) —> B(Ai). Thus the matrix ^ can be considered as the pre invert loner of (.P-1) -degree of the matrix E in global-outer itera­tions. Since, with P = 1, this strategy reduces to the global iteration stra­tegy, hence, can be considered as the pre invert loner of zero-degree ofthe matrix E in the global iteration strategy.

This strategy is used in the EQUIPOISE method for which no proofs have been found to guarantee its convergence [6]. However, the necessary and sufficient condition for the convergence of global-outer iterations is this in order that

11

G should be a convergent matrix. Assuming that the nonnegative matrix G defin­ed by Eq.(3.39) has the eigenvalues x satisfying the following equation

Gx = xx (3.45)

or(N + xS + Su)x = xMx (3.46)

and using the splitting matrices of Eq.(3,27a), one obtains

K-1[x(L + Sd) + U + Su]x = xx. (3.47)

Since the diagonal dominance properties of the matrix E = A + Sd + Su imply

that the nonnegative matrix

$ = K”1[L + Sd + U + Su] (3.48)E

representing the iteration matrix in the Jacobi method is irreducible and its spectral radius is lesser than unity, then the spectral radius of the matrix G, representing in some sense the iteration matrix in the Gauss-Seidel method, is also lesser than unity and moreover p(G) < p($E) [2,3,9],

In the case of the splitting matrices of Eq.(3.29) defining the SOR method, one obtains

G y = [I - M'1Sd]'1M'1(N + Su)y = <y (3.49)w w w w

which gives us(Nu + £Sd + Su)y = £M^y

or equivalentlyK-1[£(L + Sd) + U + Su]y = ^-^y. (3.50)

The above equation shows an implicite dependence of £ on the relaxation factor w. As is well know [3], |£| < 1 for all 1 < u> < 2, and moreover exists o> mini- imalizlng the dominant eigenvalue £ , and in the case when E is an 2-cyclic consistently ordered matrix

2w = ----- ----- (3.51)

1 + Vl-r

where x = p(G) = p2 (). In a general case both matrices Sd and Su are nonne­

gative and the matrix heis no properties of 2-cyclic cosistently ordering, and there is no the precise formula for u>, however its estimation by means of the above formula is a sufficient approximation for applications. In reactor problems without upscattering for which Su = 0, $e is a block triangular ma­

trix and its eigenvalues are related only with the entries of diagonal subma­trices $ = K 1(L + U ), and each of them is 2-cyclic consistently ordered.

9 9 9 9As is usually observed in practice, the value of w minimalixing p(G ) is a

lesser than the value of w minimalizing <r[V(A )].best 1

12

3.3 GLOBAL-INNER ITERATIONS

The block structure of Eq.(3.17) allows us, with a fixed index 2, to intro­duce the following cycle of inner iterations, t = 1,2,...,Tg, for each group g

H <p (2+=A) = N <p (2+^-) + c (J)99 19 99 19 9

(3.52)

whereVJ) = ^ * s ♦ mrx, zFku) (3S3>

k = l k = g +1 k =1

is collecting scattering and fission terms for the given group g. Eq.(3.52) can be written in the equivalent form

ip (2+=A) = M_1N <p (2+^-) + M-1 c (1) = V <p (2+~) + M~1 c (1) (3.54)9 * 9 9 9 9 19 99 99 19 99

and we have

for t=2, <p (2+=A) = V <p (1) + M_1c (1)9 i 9 9 9 9 9

for t-2, <p (2+=A) = V ip (l+=r~) + M_lc (2)9 19 99 19 99

= (2) + (I + V )M-1c (2)9 9 9 9 9

for t=Tq, y (2+2) = Vlqip (2) + (I + V + I/2 +___ + t/79'1 )M-1c (2)9 99 99 9 9 9

which can be written as

where

ip (2+2) = V ip (2) + M *c (2)9 9 9 9 9

(3.55)

V s 1ZT9 = (M_1N )Tq9 9 9 9

(3.56)

M"1 S M'1 = £99 g , (Tg-l) % g g (3.57)

and

is the preinvert loner of (Tg-2)-degree of the matrix A .9

Defining

V =

(9 1 ft-1 i [c ( 2 21i ly o M'1 0 c (222 2 2

• . r1 = • and c( 1) = •

0 0cq(22

the group equations (3.55) can be condensed into one equation1

orV>(2+22 = Vip(l) + M c( 1)

ip(l+l) = Vip(l) + M'1[S<V(2+22 + S uip(l) + ~jjXFJip(l)]

(3.58)

(3.59)

(3.60)

13

and hence

where<p(l + l) = VU(l))<p(l)

V(\(l» = [I - M~1 Sd ]"1 [y + M~1 (Su + 1 -xft ) ].\(1)

(3.61)

(3.62)

Assuming that for sufficiently large values of 1, WiX(l)'.

ordered, as follows

= p[ V( A )] = 1 > |D

V(A^), and the ei­

genvalues of V(A^) are ordered, as follows

V,1 1 ' V,2 V,3the rate of convergence in the iteration process (3.61), representing the global-inner iteration (l,t) strategy, is governed by the subdominance ratio

<r(V(Ai )] = |i>y>2 (3.63)

where the index t is referred to inner iterations, and Tg may have different values for each g.

As can be seen for each convergent matrix V

V —> 0 andwhen Tg

M"1 —> A'1 (3.64)hence

T(AJ (3.65)V(Ai)

where T(A^) is defined by Eq.(2.23). Thus when values of Tg are increasing for all g = 1,2,...,G, the eigenvalue spectrum of V(X(1)) tends to the eigenvalue spectrum of KAO)).

When Tg = 1 for each g = 1,2, ... ,G, this strategy reduces to the global iteration strategy, M 1 = M 1 can be considered as the preinvertioner of zero-

-degree of the matrix A in the global iteration strategy.This global-inner iteration strategy is equivalent to the strategy, known

in the literature under the name of outer-inner iteration strategy (where in­dex 1 is referred to outer iterations and t to inner iterations), implemented in the majority of computer codes; see, e.g, References given in [4,7,20].

Usually the SOR method, defined by splitting matrices of (3.29), is used for accelerating the convergence of inner iterations and since A are 2-cycllc

9consistently ordered matrices the optimum relaxation factor tv , minimalizingoptthe spectral radius of the associated iteration matrix V = £ can be determin­tved by means of the formula (3.30a). However, as can be observed in practice the use of tv = tv in the iteration process does not lead to the minimaliza-„ opttion of tr[V(A ) ] which occurs with using tv = tv > tv in all groups.1 best opt

3.4 GLOBAL-OUTER-INNER ITERATIONS

The matrix V(A(J)) of Eq.(3.62) related with a cycle of inner iterations performed in all groups, g = 1,2,...,G, can be expressed, as follows

_TV(A(1)) = G + [I - M'1Sd]'1M"1^jjXF‘ (3.66)where

14

(3.67)G = [I - M"1Sd]'1[V + M”1 Su]With the above form of V(ACli) the global iteration scheme can be written as

<p(l + l) = G<p(l) + [I - M'1Sd]"1H"15^XFTym.

Now introducing the scheme for outer iterations p = and using the re­lation (3.35), one obtains

<p(l+^) = G<p(l+^~) + [I - M'1Sd]‘1M'1f('j; == GP<p(l) + [I + G + G2 + . . . + GP-1][I - M'1Sd]"1M_1f('i;

and for p = P

where<p(l+l) = V(M l))<p( 1)

vu(D) = gp + M(p_n xrnXFT

and8‘J_n = l GP-1[I - M"1Sd]'1M'1.

p=i

Assuming that for sufficiently large values of i, V{A(1))genvalues of V(A^) are ordered, as follows

(3.68)

(3.69)

(3.70)

V(Ai), and the ei-

V, * pl$Ui)1 = 1 " ' l%,3l s ••••

the rate of convergence in the iteration process (3.68), representing the ge­neral form of global-outer-inner iteration (l,p,t) strategy, is governed by the subdominance ratio

crtVU^] = |vy 2I (3.71)

and the matrix is the pre invert loner of (P-2)-degree of the matrix E, with the inner iteration process included into the matrix M .

Thus in inner iterations the values of <p are updated within groups with fixed both scattering and fission terms. On the level of outer iterations the values of <p are computed with updating the downscattering term in a given ou­ter iteration and the upscattering term is modified between successive outer iterations. After completing the cycle of outer iterations, corresponding to one global iteration (equivalent to a power method iteration), the fission term (also called the fission source), 0 = FT<p, is recalculated.

It is evident that the strategies described in Sections 3.1, 3.2 and 3.3 are special cases of the global-outer-inner iteration strategy obtained with assuming P = 1 and/or Tq = 1 for all g = 1,2, ,..,G. Of course, when P —> co

GP —> 0, -> E'1 and V(AJ —> B( ). (3.72)

As can be seen the structure of the matrix V(A^), expressed explicitly by Eq.(3.69), is strictly related with the choice of the splitting matrices M and

N, and the values of P and Tq determining the degree of relevant preinver- tloners respectively, and each of them influences the subdominance ratio <r[V()]. However, their implicite interdependence makes serious difficulties

15

in theoretical investigations of behaviour of the subdominance ratio in depen­dence on the assumed splitting of A = M - N and values of parameters P and Tq. Hence, it is clear why an empirical approach, for evaluating iteration parame­ters in different strategies, is used in actual practice. Iteration strategy parameters are usually assumed as dependent on the type of reactor.

However, as can be concluded from experience [11,12,13] and the analysis of numerical results presented in Section 5, the choice of splitting matrices M and N = M - A has a dominant influence on the efficiency of the solution of neutron diffusion theory problems.

4. DOUBLE SPLITTING ITERATIVE STRATEGIES

In the linear equation systemA<p = c (4.1)

the matrix A can be expressed in the formA = P - R + S (4.2)

called the double splitting of A [10] and if P is a nonsingular matrix, this splitting leads to the following scheme spanned on three successive iterates

P<p(t + i) R<f>(t) S0(t'U + c, t > 0 (4.3)or equivalently

= p 1R0(t) - p-1S0(t_1) + P~ 1 c, t > 0. (4.4)

In the convergence analysis of this scheme, the same approach can be used as in the case of iterative methods based on a single splitting of A = M - N for which M is assumed as a nonsingular matrix. Eq. (4.4) can be written in the following equivalent form of double order [10]

rJ t +1) <P

Denoting by4>

A t + 1 )

(t)

r^r, -P sl„n r,<t><P rp-V

[ I , 0 ,(t-1)L* L0■

r,(t + l )9 .(t)

r,( t )<P Tp‘V

vu ., $

A t-1 )L*, V =

0

(4.5)

(4.6)

and

one obtains

W =rP'1R, -P *S

I,(4.7)

*(t + 1) = W(U + v, t > 0. (4.8)

Thus the necessary and sufficient condition, which ensures that the itera­tive method given by Eq. (4.8) tends to the unique solution vector <j> = A 1 c for all vectors 0> and 1) , is that the spectral radius of the iteration matrix

16

W, p(MO, was less than unity. It is Interesting to notice that if S is a non- singular matrix, the matrix W is also nonsingular and

Defining

then

-l 0, I

-s'1?, s'1?

P, o

f—-

cnF

M = and IN =0, P P, o

A = MP-R, Srp- p

(4.9)

(4.10)

(4.11)

represents single splitting of the matrix A and corresponding to the double splitting of A given by Eq.(4.2), where

W = W'V (4. 12)

By an analogy to considerations given in Section 3, it is evident that

%t) = d + w * w2+ • • • + wblM'1 = £ IT1 W"1 (4. 13)0

or equivalentlyfijj = (I " ^t+1)*_1 (4.13a)

is the preinvert loner of t-degree of the matrix A and with p(W) < 1 and t —> co

W'1, -4 A'1. (4.14)

It is evident that when S = 0, the double splitting of A reduces to the single splitting of A.

The construction of iterative algorithms based on the double splitting of A is detailed discussed in [9,10,11,12].

4.1 GLOBAL ITERATIONS

Assuming that A = P - R + S represents a double splitting of A in a 9 9 9 9 9

given group g, one can write

P <p (1+1) = R <p (1) - S <p (1-1) + c (1), 9 9 9 9 9 9 9

P 9(1) = P <p (1), t > 0

fp , 09'<pga+iy -s '

9 Rp (V 19 4-re an

9o, PL 9 v a)L 9 J

k- 0 <P (1-1)L 9 J 0where c^(1), collecting the scattering and fission terms for a given group g, is defined by Eq.(3.53). Eq.(4.15) can be expressed as

M * (1+1) = IN * (1) + w , t > 0 (4. 16)9 9 9 9 9

17

$ (1+1) = W • (1) + W_1w , t > 0

where9

V (1+1)'

9 9 9 9

V (i) [c (1)

$ (1+1) = 9 , 9(1)= 9 , w = 99 <P ( 1)9

9 <p (1-1 )9

9 0

P , O' R , -S P”1R , -P_1S "(M — 9 , IN = 9 9 and Vf = M 1 IN = 9 9 9 9g 0, P 9 P , o 9 9 9 . I, 0L 9 J L 9 J

The above group equations can be condensed into one equation

9(1 + 1) = \U\(1))9(1)

(4.17)

(4.18)

(4.19)

(4.20)

similar to Eq.(3.20), where

WU(l)) = [I - M'1sd]'1[Mr + W_1(SU + x7I7xfT)]

is a matrix of order 2xGxN and

9^(1+!)' 9^(1)' 'Mi

9(1+1) =9 (1+1)2

, 9(1) =

9 (1)2, M =

M 02

0

$ (1+1)G 9g(1)

1

(4.21)

(4.22a)

{W1 ' 0

w =^2 0

. Sd =S2,1 0 0

0 ' rG s=.2 ■0

s=,=-, 0

(4.22b)

Su =

0 s s s1,2 1,3 . 1 , C

G-l , G

0

and F = (4.22c)

where $ (1 + 1), $ (1), M and Vf are defined in (4.18) and (4.19), and the sub-9 d 9u 9 9matrices in S , Su, X and F have the form

k, l

rsd , o] [su , ol "X Fk , 1 , s“ = k, 1 > % ~ 9 and F = 9

'o o k, l 0, 0 9 0 9 0(4.22d)

where the NxN submatrices Sd , Su , X and F are those of matrices Sd, Su,k, 1 k, 1 g g

X and F, respectively, and defined by (2.5).

18

When the matrix W is convergent, then for sufficiently large values of I, A(1) ~ A^ and

W(A( 1)) —> WCA ) = [I - W"1Sd]'1[»r + M_1(SU + | XFT)]1 l

and assuming that the eigenvalues of W(A^) are ordered, as follows

(4.23)

V, ‘ p[WU.)l = 1 ' 5 'Va1 =

the rate of convergence in the iteration process (4.20), representing the glo­bal iteration (1) strategy for double splittings, is governed by the subdomi­nance ratio

f[W(A^)] = |t>H 2I (4.24)- I

where the index 1 is referred to global iterations. The matrix (M is the pre­invert loner of zero-order of the matrix A represented by the form

fA 1 [M 1 [IN1 1 1A 0 M 0 IN 02 2 2

= M - IN = -0 0 0

A M INc G G

where the matrices M and IN are defined by (4.19).g gIt is evident that all iterative solutions based on using the double split­

ting of A are related not with the problem of (2.3) but with the following equivalent matrix eigenvalue problem with doubling order

E$ = (4.26)where

E = A - Sd - Su (4.27)

and all matrices and $ are defined by (4.22) and (4.25).In Test Problem 2 considered in Section 5 this iteration strategy is ex­

tremely efficient in comparison to all others.

4.2 GLOBAL-OUTER ITERATIONS

The equation (4.20) can be rewritten, as follows

(M - Sd)i(l + 1) = (IN + Su)*(l) + f(1) (4.28)

where nowi(l) = ^-j-yKFVi; (4.29)

denotes the fission term.Similarly as on Section 3.2, the cycle of outer iterations p = 1,2,...,P

can be performed with fixed f(1), according to the following scheme

(w - sd)4>n+-; = (in + su)$('i+-—) + id) (4.30)p p

or equivalently

19

(4.31)

where

$(i+~) = mi+—) + [i - w_1sd]"1w'1f('i;P P

G = [I - M~1 Sd ]~1 W-1 (IN + Su) = [I - IM"1Sd]"1 [If + W_1SU] (4.32)

Hence, for p = P,

9(1 + 1) = GP9(l) + [I + G + G2 + ___ + GP_1ni - IN"1Sd]"1IM"1f('i;

which can be written in the general form

9(1+1) = \n\(l))9(l) (4.33)where .

wu(l)) = gp + M, , TrrrKF (4.34)( P-1) Ml)

and p1 = Vf1 [I - K‘1Sd]'1l'1. (4.35)

(P-i) up=i

Assuming that for sufficiently large values of 1, Xl{\( 1)) « W(A^), and the ei­genvalues of W(Aj) are ordered, as follows

Vl = P[W(X!)] = 1 ' '"w,2' £ ,3' £the rate of convergence in the iteration process (4.33), representing the global-outer iteration (l,p) strategy for double splittings, Is governed by the subdominance ratio

<r[V(Ai )] = |i>w 2I (4.36)

where the index p is referred to outer iterations.For a convergent matrix G,

GP —> 0 and W_1 —> E"1 (4.37)( p-i )

when P —> oo, and hence, W(A^) —> B(A^), where now

B(A ) = 1 E 12?ET. (4.38)1 Al

The matrix M * ^ is the preinvertioner of (P-1)-degree of the matrix E de­

fined by Eq. (4.29) in global-outer iterations. Since, with P = 1, this strate­gy reduces to the global iteration strategy, hence, M can be considered as the preinvertioner of zero-degree of the matrix E in the global iteration strategy based on double splittings. The necessary and sufficient condition for the convergence of global-outer iterations is this that G should be a con­vergent matrix.

4.3 GLOBAL-INNER ITERATIONS

Referring back to group equation Eq.(4.16), for a fixed index 1 one can in­troduce the cycle of inner iterations, t = 1,2, ...,Tg, for each group g, as follows

M 9 U+=i) = IN 9 (l+~) + w (1) (4.39)99 19 99 19 9

20

or* (1+=A) = N-1IN * (i+^A) + N-1w il) = W 9 U+^A) + W_1w (!) (4.40)

g I g ggg 1 g g g gg J 9 gg

Hence for t = Tg

$ (1+2) = rTg* (2) + (I + If + W2 + ___ + JTTfl_1)M"1w (2) (4.41)g gg gg ggg

which can be written as

where

and

is the preinvertioner Defining

* (1+1) = W 9 (1) + M_1W (2) g g g g g

(4.42)

W = If*9 = (W-1IN ) Tg g g g g

(4.43)

W'1 ■ ft1 = £*g g.(Tg-i) g g

(4.44)

of (Tg-2)-degree of the matrix A .g

[ifi

fw'1i

v^(1)'

if =if2 0

, S'1 =K' 0

and v(1) =vjl)2

0 0

WG (!)

the group equations (4.42) can be condensed into one equation

9(1+1) = W9(l) + W-1w(1) or

9(1 + 1) = W9(l) + vrl[Sd9(l + l) + S u9(l) +

and hence9(1 + 1) = V(.X(1))9(1)

(4.46)

(4.47)

(4.48)

whereW(A(1)) = [I - M'1Sd]'1[K + W_1(SU + —^XFT)]. (4.49)

Assuming that for sufficiently large values of 1, W(X(l)) ~ W(A^), and the ei­genvalues of W(A^) are ordered, as follows

vw,i = p[W(\)] = 1 > = '"w,3the rate of convergence in the iteration process (4.48), representing the global-inner iteration (i,t) strategy for double splittings, is governed by the subdominance ratio

o-tVU^] = |Dh 2I (4.50)

where the index t is referred to inner iterations, and T may have differentg

values for each g.As can be seen for each convergent matrix W and with Tq —> co for all

1 * g * G,

21

(4.51)Vf —> 0 and M-1 -1

andWCA ) -» TU ) (4.52)

where TU ) is the matrix of the basic problem defined previously by equations (2.21) to (2.23) and now reformulated, as follows

*(1 + 1) = TU (!))*(1)where

TU( 1)) = [I - A 1Sd]'t A_1(SU + ~ jKFT j

but for sufficiently large values of 1, \(1) —>A^, and

(4.53)

(4.54)

TU(D) —> T(A ) = [I - A"1 Sd ]'1 A"1 [ Su + ~ XFT] ^ 0. (4.55)1 i

When Tg = 1 for each g = 1,2, ... ,G, this strategy reduces to the global iteration strategy, hence, IM ^ = W 1 can be considered as the preinvertioner of zero-degree of the matrix A in the global iteration strategy

4.4 GLOBAL-OUTER-INNER ITERATIONS

The considerations analogous to those given in Section 3.4 lead with p = P to the following equations

where

and

where

*(1+1) = Wi\(l))*(l)

w"1 = E Gp"1 (I - m” 1 sd ] ~1 w~1(P-1) uP=1

G = [I - W-1Sd]_1 [W + M_1SU]

(4.56)

(4.57)

(4.58)

(4.59)

and W 1 is the preinvertioner of A in the cycle of inner iterations.

Assuming that for sufficiently large values of 1, W( Afi)) ~ WU ), and the eigenvalues of W(A^) are ordered, as follows

= 1 > luWi2 W, 3

the rate of convergence in the iteration process (4.56), representing neral form of global-outer-inner iteration (l,p,t) strategy for double tings, is governed by the subdominance ratio

the ge- split-

<r[ S( A ) ] = |w |. (4.60)

The matrix W * ^ is the preinvertioner of (P-1)-degree of the matrix E in the

global-outei—inner iterarlon strategy (the inner iteration process is included into the matrix M 1 given in Eq.(4. 45)) and with P —> *

GP 0, ^ —» E~1 and W(Aj ) —> B(A^). (4.61)

22

5. NUMERICAL ILLUSTRATION

In this section the convergence behaviour of iterative strategies, discussed in two previous sections, is illustrated in a large number of numerical exper­iments on various types of reactor problems.

As was already mentioned, particular levels of iterations in the global- -outer-inner iteration strategy play a special role in the iterative solution of a given problem. In inner iterations the values of <p are updated within energy groups with fixed both scattering and fission terms. On the level of outer iterations the values of <p are computed with updating the downscattering term in a given outer iteration and the upscattering term is modified between successive outer iterations. After completing the cycle of outer iterations, corresponding to one global iteration and equivalent to a power method itera­tion, the fission term \f)(l+l) = FT<p(l) is recalculated for the next global

iteration.The rapidity of convergence to the dominant eigenvalue of a given matrix V

in the power method is dependent on the separation of the largest eigenvalue from the rest of the eigenvalue spectrum and can be examined in terms of (as­ymptotic) rate of convergence [8] defined in Eq.(3.25), that is

R(V) = - ln<r[V]

Since the number of iterations required for computing the dominant eigenvalue, with a given degree of accuracy, is roughly inversely proportional to the rate of convergence, the efficiency of different iteration strategies can be evalu­ated by comparing observed numbers of iterations obviously,with the assumption of a similar arithmetical effort per iteration.

However the structure of matrices appearing in the global-outer-inner ite­ration strategies, represented by Eq.(3.68) in the case of single splittings of A (or by Eq. (4.56) when double splittings of A are used), is strictly re­lated with the chosen splitting matrices as well as with the assumed numbers of outer and inner iterations (P and Tg, respectively) determining the degree of relevant preinvertioners. Each of them influences not only the subdominance ratio <r[V(Ai) ] (or <r[))) but also contributes, in a different way, to the arithmetical effort per global iteration.

Thus the efficiency of different strategies should be rather estimated by a comparison of the computational work (expressed by the total number of flops) required for obtaining the solution with the same convergence criteria. There­fore, the numbers of global iterations obtained for particular iterative stra­tegies will be converted to the total number of flops per mesh point, defined in the next subsection.

23

5.1 COMPUTATIONAL ASPECTS

All computations were carried out by means of the following codes

HEXAGA [11], HEXSOR and HEXSLOR

solving the two-dimensional multi-group neutron diffusion equation (2.1) by means of difference approximation based on 7-point mesh-edged difference for­mula in uniform triangular mesh imposed on a 120-degree parallelogram region. In the case of the natural ordering of mesh points, row by row, coefficient submatrices A of the matrix A defined in Eq.(2.5) have only seven nonzero

9diagonals forming a tridiagonal block structure suitable for the implementa­tion of 1-line algorithms. For solving inhomogeneous problems with seven dia­gonal matrices within inner iterations, the AGA prefactorization method [10, 11], the Gauss-Seidel method and the 1-line Gauss-Seidel method are used as splitting algorithms in HEXAGA, HEXSOR and HEXSLOR codes, respectively. The convergence of global iterations in all codes is accelerated only by using successive overrelaxation processes in inner iterations. In the case of HEXAGA the double successive overrelaxation process [10,11], representing a double splitting iteration strategies discussed in Section 4, is implemented as the most efficient acceleration procedure, where the same value of relaxation fac­tor w is used in both forward and backward sweeps. In both HEXSOR and HEXSLOR the well known SOR procedure (equivalent to the single successive overrelaxa­tion process [10]), representing a single splitting iterative strategies dis­cussed in Section 3, is used.

As is demonstrated in [10], accelerating the convergence of the Gauss-Seidel method by means of the single successive overrelaxation process is more effi­cient than by using the double successive overrelaxation process, that is, in­versely as in the case of HEXAGA algorithm.

Computations for each problem were continued until the maximum pointwise relative change of fluxes between global iterations, e^, and the relativechange of k , e , were less than prescribed numbers, where k is defined, eff k ef fas follows

k (1+1) = A(1 + 1) = S 0(1+Udveff v

and the integration of fission sources tp(l + l) = FT<p(l) is approximated by the

trapezoidal method. In the case of the global-outei—inner iteration strategy represented by Eq.(3.68) and Eq.(4.56)

c = max <P

and

(p (1 + 1, p=P, t=l) - <p (1, p=P, t=T)n n

<p (i+i, p=p, t=i)for all n = 1,2,... A/xG (5.1)

24

(5.2)CkKec(1+1) ~ Kee(1)

where l,p,t are iteration indices in global, outer and inner iterations, res­pectively. The values of P and T (where T = T for all 1 s g £ G) are assumedgto be constant in the whole iteration process for a given problem. The index 1 plays a role of an iteration index in the power method used for the calcula­tion of the dominant eigenvalue of a given matrix which structure is dependent on the assumed values of P, T and the used (single or double) splitting of A.

It is evident that this strategy with P = 1 and T = 1 reduces to the global iteration strategy represented by Eq. (3.20) and Eq. (4.20); with P > 1 and T = 1 to the global-outer iteration strategy represented by Eq.(3.40) and Eq. (4.33); with P = 1 and T > 1 to the global-inner iteration strategy repre­sented by Eq.(3.61) and Eq.(4.48) and equivalent to the well known outer-inner iteration strategy used commonly in solving neutron diffusion problems.

In this paper the majority of results was obtained with the following stopping criteria

e s 10 5 and e s 10 6 (5.3)ip k

and iteration strategies were analyzed for the following numbers of inner ite­rations per outer one

T = 1,2,3,5,8,12 and 20.

The comparison of the efficiency of different iterative strategies, used for solving reactor problems, is very often done by comparing the total number of inner iterations required for obtaining the solution with given convergence criteria. In problems with a simple scattering model, the total number of in­ner iterations is roughly proportional to the number of arithmetic operations. However for problems in which neutrons are scattered throughout a large number of energy groups this proportionality is not satisfied and the total number of inner iterations may be not a satisfactory measure of arithmetic effort. Therefore in this paper, the efficiency of particular iterative strategies is estimated by a comparison of the computational work expressed by the total number of flops required for obtaining the solution with the same stopping criteria. A flop is defined as the amount of work associated with arithmetic effort for doing the arithmetic statement [ 15]

au ' au 1 fc.) * ,5'4)

which can be considered as a convenient unit of computational work in computer calculations.

The total number of flops per mesh point for a converged solution can be obtained from the following formula

25

NFL = NGLOB x NCOEFF (5.5)

where NGLOB is the number of global iterations and NCOEFF is a number of flops per global iteration per mesh point which can be expressed, as follows

NCOEFF = N + 2 + NOUT x (N + N +NxNINNx NMFL) (5.6)FIS SCAT XFIS C

whereNOUT - number of outer iterations per global one (?),NINN - number of inner iterations per outer one (T = T for all g = 1.2....G),

9NMFL - number of flops required by a method used for solving 7-point formula

including one flop for computing the flux relative error. NMFL = 14 for HEXAGA (with the double overrelaxation process) and NMFL = 10 for both HEXSOR and HEXSLOR.

N , N and N are numbers of flops related with computing fissionFIS SCAT XFISsources, scattering terms and a fraction of fission neutrons, respectively;N^ = G is the number of energy groups and the number 2 is related with thecalculation of k and the normalization of fission sources. Thus these four

ef fparameters are dependent on the type of reactor problem and their values for test problems considered in this paper are given in Table 1.

5.2 NUMERICAL EXPERIMENTS

Computations were carried out for the series of models derived from six test problems (two fast and four thermal reactor examples) taken mainly from the literature. The first four test problems were calculated in the single precision on a PC computer and two last on the Convex C3210 computer with the single precision for HEXAGA and the double precision for HEXSLOR.

In all analized strategies the same number of inner iterations and the same value of relaxation factor w were assumed in each energy group for the whole iteration process. Thus each numerical experiment is considered as a separate problem defined by the assumed number of inner iterations per outer one and the value of relaxation factor w. In computations for each such problem the null starting vector <p( 01 is used, in the case of a double splitting of A im­plemented in HEXAGA, 11 =0 is assumed. However with the calculation of the starting fission source vector 11 = FTy(°’ (according to Eq.(2.15)) all com­ponents of the vector 0> were put to unity. Thus, the starting distribution

of fission sources was assumed to be piecewise flat in fissionable subregions.The observed numbers of global iterations and the corresponding computa­

tional work (expressed by the number of flops per mesh point) in a dependence on the number of inner iterations, as a parameter, are plotted versus the relaxation factor w for each iterative strategy used in solving a given test problem. The values of the optimum relaxation factors u> , minimizing theoptspectral radii of block-diagonal iteration matrices V defined by Eq.(3.22) (or

26

W defined by Eq.(4.22b) in the case of HEXAGA) and satisfying the following inequality [10]

, . HEXAGA . HEXSLOR HEXSOR1 < w < w < w <2opt opt opt

are marked in figures by vertical dashed lines at the abscisa axis, consecu­tively for HEXAGA, HEXSLOR and HEXSOR.

The value of w is determined by means of the formula given by Eq.(3.30a)opt

for HEXSOR and HEXSLOR. However it should be mentioned that the iteration ma­trices V in HEXSOR are not 2-cyclic consistently ordered matrices [3] but as is demonstrated in [10] computig w by means of Eq.(3.51) provides a suffi-o ptciently good estimate of w for needs of practice. In the case of HEXSLORoptiteration matrices are 2-cyclic consistently ordered and the estimation ofw can be done by means of the sigma-SOR algorithm [8]. The value of w opt optminimizing the spectral radius of the iteration matrix Vf, appearing in the double overrelaxation process used in HEXAGA, have been determined by means of the technique described in [10].

Test Problem 1 (TP1)

This well known benchmark problem is derived from the four group fast reac­tor SNR300 [17,p.824,Fig.2] with 30-degree symmetry sector and the full range of downscattering. Computations were carried out in a 120-degree rhombic sec­tor depicted in Fig.1 for three models of mesh division:TPla - 18x18=324 m.pts (mesh points) with the mesh step equal to 6.4665 cm, TPlb - 35x35=1225 m.pts with the mesh step equal to 3.23325 cm,TPlc - 69x69=4761 m.pts with the mesh step equal to 1.616625 cm.

Macroscopic group constants (correspondingly to the notation used in Eq.(2.1)) are given in Table 2. Numbers given inside hexagons in Fig.1 corresponds to the material composition numbers specified in Table 2.

The global-inner iteration strategy (1 outer iteration per global one), equivalent to the well known outer-inner iteration strategy, is mainly used as a solution method. The behaviour of the number of global iterations versus relaxation factor w (where the same value of w is used in all energy groups), obtained with using the stopping criteria (5.3) and for different numbers ofinner iterations, is depicted in Figs. 2,3 and 4. Numbers at curves correspondto the numbers of inner iterations T to be fixed in all energy groups.

It is evident that for T = 1 (1 inner iteration used in each group) this strategy reduces to the global iteration strategy represented by Eq. (3.20) for both HEXSOR and HEXSLOR, and by Eq. (4.20) for HEXAGA but with w > 1 (because for w = 1 the double splitting used in HEXAGA reduces to the single one and represented by Eq.(3.20), [10]), and in this case

27

max for all n = 1, NxG (5.la)e<P

<p (1 + 1) - <p (1) n n

<p (i+i)

As can be seen in Figs. 2, 3 and 4 that when the number of mesh points is increasing, greater numbers of inner iterations 7 are required for reducing the number of global iterations. However with some values of T a further de­crease of the number of global iterations is not observed.

As was mentioned in Section 3.1, the existence of

u> = u < 2, (5.6)c r 1 t

for which a disconvergence occurs, is observed in these figures and its value is dependent on T and the used splitting. As the number of mesh points is in­creasing the value of w decreses for HEXAGA and increases for both HEXSORcrltand HEXSLOR. It is well known that iterative solutions of inhomogeneous linear problems solved in the inner iterations of both HEXSOR and HEXSLOR are conver­gent for all 0 < (j) < 2, whereas the eigenvalue problem implemented in these codes can be not solved when values of w being in the range u> < w < 2 arecrltused in inner iterations.

The corresponing total computational work represented by the number of flopsper mesh point (determined by means of Eq.(5.5) and the data from Table 1), asa function of the relaxation factor w, is depicted in Figs.2a, 3a and 4a.

As can be seen in the case of HEXAGA the results obtained with T = 1 aresuperior in a comparison to all others.

The comparison of HEXAGA results for Test Problem lb (1225 m.pts.) obtainedwith 1 and 2 outers/global is demonstrated in Figs.5 and 5a.

The values of k obtained for this reactor problem in dependence on the ef f

number of mesh points are given below.

Problem No. of m.pts. k«ff

TPla 324 1.12168TPlb 1225 1.12441TPlc 4761 1.12521

Test Problem 2 (TP2)

This problem, analized extensively by the author in the seventies, is the four group fast reactor model SNR2 with 30-degree symmetry sector and the full range of downscattering. Computations were carried out in a 120-degree rhombic sector depicted in Fig.6 for two mesh divisions:

28

TP2a - 35x35=1225 m. pts with the mesh step equal to 9.5552 cm,TP2b - 69x69=4761 m.pts with the mesh step equal to 4.7776 cm.Macroscopic group constants are given in Table 3.

The behaviour of the number of global iterations and the corresponding num­ber of flops per mesh point versus relaxation factor w, obtained for the glo­bal-inner iteration strategy (1 outer/global) and the stopping criteria (5.3), are depicted in Figs.7 and 7a for TP2a and in Figs.8 and 8a for TP2b. The va­lues of k for this problem are equal to 0.999447 for TP2a and 1.00098 for TP2b.

It is interesting to notice that for this reactor example only the global iteration strategy, that is with 7=1, is most efficient in a comparison to others, especially for HEXAGA.

Test Problem 3 (TPS)

The four group thermal reactor problem HTGR [16,p.228-237] with 60-degree rotational symmetry sector is shown in Fig.9. Neutrons are downscattered only to the next energy group and macroscopic group constants are given in Table 4.

Since the option of 60-degree rotational symmetry boundary condition is not implemented in used codes, computations were carried out for the full reactor region consisting 1687 m. pts with the mesh step equal to 20.9 cm and zero bou­ndary condition on the outer boundary of reactor.

The results of the global-inner iteration strategy obtained with the stop­ping criteria (5.3) are demonstrated in Figs. 10 and 10a and with the stopping criteria

e s 10 4 and e s 10 5 (5.7)<p k

are depicted in Figs.11 and 11a.It is interesting to notice that the change of convergence criteria implies

some changes in curve profiles and w which value equal to about 1.38 withcrltthe stopping criteria (5.3) increases to about 1.46 with the stopping criteria (5.7) for both HEXS0R and HEXL0R. The best results are again obtained for HEXAGA with 7=1.

The value of k obtained for this problem is equal to 1.11671. ef f

Test Problem 4 (TP4)

This example is the modified version of the MOATA four group thermal reac­tor problem [18] with three thermal groups and two regions. The original ver­sion of the M0ATA problem, oriented in rectangular geometry [18], have been used by the authors of work [19] in their analysis and developments of itera­tive strategies (see References in [19]) for solving problems with upscatter- ing and illustrated numerically Just on this reactor model. Since in the lit­

29

erature there is an avaricious number of examples of reactor problems with up- scattering for triangular (hexagonal) geometry, the author adapted this reac­tor problem to needs of this work by using original group constants, neglect­ing a thin wall around the core region and deforming the reactor region to the 120-degree parallelogram shown in Fig. 12; where reactor dimensions were chosen in such a way in order to obtain k « 1, as was in the original version of the MOATA problem. Macroscopic group constants are given in Table 5. Computa­tions were carried out for three models of uniform triangular mesh division:

TP4a - 19x12=223 m.pts with the mesh step equal to 4 cm,TP4b - 37x23=851 m.pts with the mesh step equal to 2 cm,TP4c - 73x45=3285 m.pts with the mesh step equal to 1 cm.

The behaviour of the number of global iterations and the corresponding num­ber of flops per mesh point versus relaxation factor w, obtained for the glo­bal-inner iteration strategy (1 outer/global) and the stopping criteria (5.3), are depicted in Figs.13,14,15,13a,14a and 15a for the assumed mesh divisions, respectively. The inspection of these figures and those for Test Problem 1 indicates a similarity in the behaviour of numbers of global iterations when the number of mesh points and the number of inner iterations per outer one are increasing.

The HEXAGA results obtained with 1 and 2 outers/global and shown in Figs.16 and 16a allow us to conclude that using 2 outers/global leads to some reduc­tion of the number of global iterations in a comparison to the strategy based on 1 outer/global for the model of upscattering used in this reactor problem, which was not observed for the TPlb problem (Figs. 5 and 5a) with r.he absence of upscattering.

In order to evaluate how the upscattering model used in this problem influ­ences the rate of convergence of global iterations, the problem has been firstly examined without upscattering for the mesh division corresponding to TP4c model (3285 m.pts). The comparison of results obtained in computations performed for T = 1,2 and 3 with original upscattering and without upscatter­ing are shown in Fig.17 for HEXAGA and in Fig.18 for HEXSLOR. As can be seen in these figures, differences for HEXSLOR are small and In the case of HEXAGA are almost the same, which allows us to conclude that the MOATA upscatter is insignificant and rather can not be considered as a representative model for the convergence analysis of solutions for reactor problems with upscattering.

It seems to be interesting to consider, how increasing values of upscatter­ing cross sections may effect the behaviour of convergence. In order to esti­mate such an effect, computations were performed by means of HEXAGA and HEX­SLOR for the TP4c problem (with 3285 mesh points) which upscattering term has been changed by using in both regions the values of z" ^ and z" ten times greater than their original values, and in the core region z" was increased

30

by a factor 106. The above changes of values of upscattering cross sections

imply some modification of £r according to its definition, that is9

Sr = 1“ + V Zsc (5.8)

9 9 q¥9> 9-49where Ea is the absorption cross section. For this artifically introduced sig-

9nifleant upscattering model the values of Zr and E° , are given also in9 9—>9

Table 5a. Of course, a similar modification of £r have been done in the case9

considered without upscattering.The comparison of results obtained with 1, 3 and 12 inner iterations per

outer one for the original and this significant upscattering are shown in Fig. 19 for HEXAGA and in Fig. 20 for HEXSLOR. As can be seen, increasing the number of inner iterations implies the increase of the number of global itera­tions when the strategy with 1 outer/global is used, however in the case of 1 inner per outer iteration the numbers of global iterations are comparable for both models of upscattering. Other results obtained only with this significant upscattering are depicted in Figs.21 and 21a by an analogy to Figs.15 and 15a demonstrating results obtained with the original upscattering. Further compa­rison results for HEXAGA can be find in Figs.22,22a,23 and 23a.

The values of k obtained for this reactor problem in dependence on the number of mesh points are given below together with effects cased by changes of a scattering model in the TP4c problem.

Problem No. of m.pts.

keff

originalupscatter

withoutupscatter

significantupscatter

TP4a 228 1.01277TP4b 851 1.01265TP4c 3285 1.01238 1.01767 0.78912

The behaviour of convergence with a realistic model of significant upscat­tering is analized in the next problem.

Test Problem 5 (TPS)

This problem corresponds to the Test Problem 4 which layout is depicted in Fig. 12 and differs only with the model of group structure, that is, the set of four group data is replaced by the set of eight group data with seven thermal groups and given in Table 6. The mesh division corresponding to TP4c (3285 mesh points) has been assumed as the computational model.

The results obtained by HEXAGA and HEXSLOR for strategies with 1, 2, 3, 5

31

and 8 outer iterations per global one are shown in Figs.24,24a to Figs. 28,28a, respectively.

As can be seen in these figures that for a given number of inner itera­tions, increasing the number of outer iterations per global one implies the decrease of the number of global iterations. The case with 1 inner iteration per outer one, corresponding to the global-outer iteration strategy (which with 1 outer iteration per global one reduces to the global iteration strate­gy) is demonstrated in Figs.29 and 29a for HEXAGA, and in Figs.30 and 30a for HEXSLOR. The behaviour of the number of flops per mesh point with 1 inner ite­ration per outer one is almost independent on used numbers of outer Iterations per global one, as is demonstrated in Figs.29a and 30a.

The change of the model of macroscopic group constants from 4 to 8 groups implies the change of the value of k from 1.01238 to 1.03928.

Test Problem 6 (TPS)

This test problem taken from [20] is a three group thermal reactor example with the downscattering only to the next group and a small mesh refinement described by 240x240 = 57600 m.pts., where the mesh step is equal to 0.125 cm. The layout of reactor is depicted in Fig.31 and macroscopic group constants are given in Table 7.

This problem have been used by the author of [20] in studying the efficiency of the line cyclic Chebyshev iterative method using hyper- or parallel-line inversion with the highly vectorizable point Chebyshev iterative method on the CDC Cyber 205 vector computer, where computation results presented in [20] were obtained with the stopping criterion

e s 5xl0"3 (5.9) <P

The computations of this problem were here carried out with the above stop­ping criterion on the Convex C3210 coputer by means HEXAGA running in the single precision arithmetic and HEXSLOR running in the double precision arith­metic because for the single precision arithmetic, oscillations of relative flux errors are occuring in HEXSLOR computations already with c ~ 0.01 andthe solution can not be obtained for lower values of e . The results of compu-

<f>tat ions obtained for the iterative strategy using 1 outer/global and the stop­ping criterion (5.9) are demonstrated in Figs. 32,32a,33,33a and 33b. The com­parison of results obtained by HEXAGA for the strategy using 1 and 2 outer iterations per global one is shown in Figs.34 and 34a for a few numbers of in­ner iterations per outer one.

32

5.3 DISCUSSION OF NUMERICAL RESULTS

The examination of this large number of numerical experiments allows us to evaluate the efficiency of particular iterative strategies as well as to con­clude whether an optimized iterative strategy can be find.

In iterative strategies used here systematically for solving various re­actor problems, the convergence of global iterations is accelerated only by means of successive overrelaxation processes, where the same value of relaxa­tion factor w is used in each energy group. It is observed in all figures that the rate of convergence is minimized with w which value is greater thanbestthat of w minimizing the spectral radius of the iteration matrix V (or W inoptthe case of double splittings used in HEXAGA).

In all eigenvalue problems formulated in previous sections, associated ma­trices are defined in such a way that when A(I) —> its dominant eigenvalues v are equal to unity and therefore the second eigenvalues v play a role of the subdominance ratio <r determining the rate of convergence.

The example of a behaviour of some eigenvalues versus the relaxation factor w, representative for the HEXSLOR algorithm, is demonstrated in Fig.35. In the case of HEXAGA the behaviour of eigenvalues of the matrices, representing ite­rative strategies based on double splittings of A (and discussed in Section 4), as a function of w is qualitatively similar to that shown in Fig.35, how­ever with values of w , w and w occuring in the interval betweenopt best crlt1 and 1.2.

The dashed curves "Rr" and "Rc" illustrate the behaviour of the spectral radius of the iteration matrix V, equivalent in this case to the iteration matrix of Eq.(3.30) in the 1-line SOR method, where p(V) achieves its mini­mum at the point "l” with w^ [8]. The dashed straight line "VI" corresponds to the dominant eigenvalue vy of the matrix V(A ) representing the matrices defined by Eqs. 3.21,3.41,3.62 and 3.69. This dominant eigenvalue vy is real and equal to unity, and its value is independent on w. The dashed curves "V2", "V3" and "Vs" represent the second, v , third, v , and s-th, v , eigen- values of V(A^), respectively, where vy g and i>y ^ are usually real and posi­tive eigenvalues in some range of w’s, whereas ^ is a complex eigenvalue.

The behaviour of eigenvalues of V(A^) versus w has a such nature that some eigenvalues, among them vy g and vy 3,are a strictly decreasing function of w, whereas others, among them vy , are a strictly increasing function of w. In the cutting of the curves "V2" and "Vs" at the point "2" occuring with wbestthere is the largest separation of subdominant eigenvalues from the dominant eigenvalue i>y ^ and the subdominance ratio <r( V() ] achieves its minimum value maximizing the rate of convergence for global iterations. Since = 1 the curve of <r[ V(At) ], marked by the continuous line in Fig. 35, coincides with thecurve "V2" until to the point "2" and for the values of w’s greater than wbest

33

it coincides with the curve "Vs" corresponding to the modulus of the complexeigenvalue , where at the point "3" occuring with ^ , <r[ V( A^ ) j = 1. Forw £ u the complex eigenvalue v becomes dominant and the power methodc r 11 v, sdoes not converge [8].

Matrices V(A^) appearing in particular iterative strategies, as a result of used splittings of A, are characterized by a different behaviour of eigenva­lues l>v i versus u for 2 s i < n and the point "2" corresponding to the mini­mum of <r[ V( A ) ] = i>v n \ may have different positions on the vertical line lo­cated near by w . Dependently on the assumed numbers of inner iterationsbestper outer one and/or outer iterations per global one, the point "2" may move to positions above or below the line "82" correspomding to the second eigen­value v ^ of the matrix B(A ) and representing <r[B(A )].

The structure of the matrix B(A^) defined by Eq. (2. 13) is related with the explicite form of E 1 = (A - Sd - Su) 1 and when there is the upscattering its

second eigenvalue v is usually lesser than the second eigenvalue v (re- presented in Fig. 35 by the dased straight line denoted by "T2") of the matrix T(A ), associated with the basic eigenvalue problem defined in Section 2.2, and related with the explicite form of A 1, in the case of its implementation.

Obviously, with the absence of upscattering (S = 0), T(A ) B{ A ).The use of a splitting of the matrix A, allowing us to avoid the computation

of its inverse, leads to the class of iterative strategies discussed in pre­vious sections. In the case of single splittings of A, iterative strategies are represented by the matrices defined in Eqs.3.21,3.41,3.62 and 3.69, which second eigenvalues ^, with w = 1, are greater than the second eigenvalue v of the matrix T(A ) associated with the basic eigenvalue problem. Fur-T , 2 1thermore the values of ^ are dependent on the used splitting of A and for the programs considered here the following inequality

HEXACA „ HEXSLOR „ HEXSOR V < V < VV , 2 V, 2 V , 2

is fulfilled with w = 1. The corresponding values of vV,2

(5.10)

at the point "2",determining the minimum value of crtVlA^)], satisfy as well a similar inequality where

, . HEXACA ^ HEXSLOR ^ HEXSOR ^ _1 < w < w < w <2.best best best

(5.ii:

As was pointed out in Section 3 increasing the number of inner iterationsper outer one, T, and/or the number of outer iterations per global one, P,implies that relevant preinvertioners approximate much better the matrix A 1and/or E 1 , and the eigenvalue spectrum of V(A^) approaches the eigenvalue

spectrum of T(A^ ) and/or B(Aj), so that the point "2" is located at the line"T2" or "82". But this means that the subdominance ratio <r[V(A )] achieves thefixed values of <r[ T(A ) ] = | v | or <r[B(A )] = \v I, where usually a few1 T, 2 1 B . 2first subdominant eigenvalues of B(Aj) and T(A^) are real and positive.

34

As was demonstrated in numerical experiments the increase of inner itera­tions T or outer iterations P reduces the number of global iterations. However with sufficiently large values of T and/or P a further decrease of the number of global iterations is not observed. For instance, an increase of inner ite­rations per outer one above 8 in TPla (324 m.pts) and TPlb (1225 m.pts) does not decrease the number of global iterations which achieves its asymptotic va­lue equal to about 25 global iterations and corresponding to <r[T()] in this case. Thus, 8 inner iterations used in each group and providing the asymptotic number of global iterations can be considered as equivalent to the infinitynumber of inner iterations. For TPS the values T =5 and T =8 can be consi-a adered as equivalent to the infinite number of inner iterations for HEXAGA and HEXSOR (or HEXSLOR), respectinely, in a large range of w's. It allows us to conclude that preinvert loners of 5- and 8-degree approximate the matrix A 1

quite well in these problems.As can be seen, the decrease of mesh size in the TP1 and TP4 problems im­

plies increasing the values of T providing the asymptotic number of global iterations equal to about 25 and 30 global iterations for TP1 and TP4, respec­tively. However in the case of HEXAGA it occurs that only a few inner itera­tions per outer one (or global one if 1 outer/global is assumed) allows us to reduce the number of global iterations below its asymptotic value.

For the TP2a problem with 1225 m.pts. the asymptotic number of global ite­rations with the convergence criteria (5.3) is equal to about 650 (Figs 7 and 7a) and all iterative strategies with T > 1 provide the number of global ite­rations not lesser than 650 in a broad range of w's. However, the most inter­esting result is observed for the case with T = 1 for which the number of glo­bal iterations is significantly reduced below its asymptotic value, especially for HEXAGA providing the solution after 72 global iterations with w « 1.19. This reduction of the number of global iterations occurs also with T = 1 for both HEXSOR and HEXSLOR, however not such significantly as in the case of HEX­AGA, and the solution is obtained with about 480 and 560 global iterations for HEXSLOR and HEXSOR respectively. In this case the convergence behaviour of the HEXAGA solution obtained with 1 inner/global is similar to the convergence be­haviour appearing with the solution of linear equation systems illustrated by the curves "Rr" and "Rc" shown in Fig. 35. This means that the vertical line at the point "2" which abscissa determines w is tangent to the curve "V2" re-bestpresenting in this case the behaviour of the eigenvalue of the matrix U(Ai) defined by Eq.(4.23).

As can be seen in Figs. 8 and 8a showing results for TP2b (4761 m.pts. ) ob­tained by HEXAGA and only HEXSLOR, decreasing the mesh size leads to a change of curve profiles. The number of global iterations for HEXAGA decreases below 650 with 2 and 3 inners/global whereas for HEXSLOR increses to about 650 al­

35

ready with 1 inner/global. However, the best results are again obtained by HEXAGA with T = 2.

Thus, iterative strategies based on the splitting of A (mainly on the dou­ble splitting of A) may, with small numbers of inner iterations T, provide so­lutions with a greater rate of convergence than that occuring in the asympto­tic range represented by constant value of <r[T( A^ ) ] (or cr[B(A )]) characteriz­ing the strategy defined by Eq. 2.21 (or Eq.2.12) whose implementation requires the explicite form of matrices A 1 (or E 1). This effect can be observed in

other problems as well.It should be mentioned that in the neutron diffusion theory codes of common

use which iterative strategies are usually based on Chebyshev accelerationtechniques, inner iteration solutions are obtained by minimizing the spectralradius of each submatrix V in the block diagonal iteration matrix V definedQin Eq.3.22 and being a component in the structure of the matrix V(A^) associ­ated with a given eigenvalue problem. The minimalizat.ion of each p(V ) leads

9to using, in the iteration process, values of w different in each energyoptgroup but the spectral radius p{V) is determined by the maximum value of p(V )

9for all 2 s g s G. This approach provides matrices V(A^ ) with real dominant eigenvalues, which is more suitable for an efficient use of Chebyshev accele­ration techniques. It was observed in numerical experiments that when the op­timum values of the relaxation factor u> were used in each group g, the number of global iterations was about two times greater than the number of global iterations obtained with using in all groups the same value of w and coi—optresponding to the maximum value of p(V ) for all 2 £ g < G, as is assumed in

9the iterative strategies analized in this paper. However, as is demonstrated in all numerical results, minimizing p(V) does not imply the minimalization of cr[V(A )] which is just achieved with using in all groups w = w > w and1 best optin this case nearly all eigenvalues of V are complex with its modulus, equal to w - 1 for HEXSLOR, greater than the value of p(V) obtained with wbest optand equal to w - 1 for HEXSLOR.opt

As can be seen in the TP1, TP2 and TP4 problems the values of o , wopt bestand w increase as a mesh size is decreasing and quite often the values of u> and w are dependent on assumed stopping criteria. This effect isbest c r 1 tclearly demonstrated in Figs.10 and 11 for the TP3 problem in which the change of stopping criteria causes changes of curve profiles and the value of wcrlt

The results for the TP1, TP2 and TP3 problems obtained by HEXAGA, HEXSOR and HEXSLOR in single precision arithmetic on PC computer with using u andbest1 outer iteration per global one are summarized in Tables 8 and 8a. The re­sults of the VALE program [21], available in the literature only for TP1 and TP3, are also given in Table 8 for comparison reasons. The iterative strategy implemented in VALE with 1 outer/global is based on using 1-line SOR in inner

36

iterations (similarly as in HEXSLOR) where either single or double error mode flux extrapolation is done occasionally on outer iterations with or without Chebyshev acceleration. The VALE computations were performed in the 30-degree symmetry sector for TP1 (the 120-degree sector was used in HEXAGA, HEXSOR and HEXSLOR) and in the 60-degree sector with the rotational symmetry boundary condition for TP3 (the full reactor region was used in HEXAGA, HEXSOR and HEX­SLOR). In calculation of the number of flops per mesh point for VALE, the va­lues of NCOEFF from Table 1 were assumed the same as those for HEXSLOR.

The comparison of results from HEXAGA computations obtained for the TPlb problem (1225 m.pts.) with using 1 and 2 outers/global is shown in Figs.5 and 5a. The use of 1,2,4 and 6 inner iterations per outer one in the strategy using 2 outers/global gives 2,4,8 and 12 inner iterations per global one and these last four numbers were assumed as numbers of inner iterations per outer one in the strategy with 1 outer/global, which allows us to compare results obtained with the same inner iteration effort per global iteration for both strategies. As can be seen in these figures, for this problem with the absence of upscattering both strategies are equivalent from the point of view of inner iteration effort per global iteration. Furthermore, increasing the number ite­rations per outer one reduces the number of global iterations to the same as­ymptotic value equal to about 25 global iterations for this problem.

The model of upscattering used in the TP4 problem does not imply any dif­ficulties in the convergence of global iterations which behaviour shown in Figs.13,13a,14,14a, 15 and 15a for the strategy using 1 outer/global is similar to the convergence behaviour in the TP1 problem and demonstrated in Figs.2,2a, 3,3a,4 and 4a. The comparison of results obtained in computations with this original upscattering and without upscattering, shown in Fig.17 for HEXAGA and in Fig.18 for HEXSLOR, indicates a similar behaviour of global iterations, which allows us to conclude that the MOATA upscatter is rather insignificant. However, as is demonstrated in Fig.16, using greater numbers of inner itera­tions per outer one in the strategy using 2 outers/global leads to some de­crease of the number of global iterations.

As can be seen in the comparison of results demonstrated in Figs. 19 auid 20 or in Figs.15 and 21 as well as in Figs.22,22a,23 and 23a, introducing a slg- nificat upscattering in the TP4c problem causes a significant increase of num­ber of global iterations and its aisymptotic value. In this case, the use of strategies with more than 1 outer/global allows us to reduce the number of global iterations.

The results obtained for the TP4 problem with different models of upscat­tering are summarized in Tables 9 and 10.

The realistic eight group model of significant upscattering with seven thermal groups is detailed anal!zed in the TPS problem. The results obtained

37

by HEXAGA and HEXSLOR for strategies with 1,2,3,5 and 8 outer iterations per global one are depicted in Figs.24,24a to Figs.28,28a, respectively.

It is observed in these figures that increasing the number of outer itera­tions per global one implies the decrease of the number of global iterations whose asymptotic values amount 96,52,36,23 and 16 for 1,2,3,5 and 8 outer ite­rations per global one, respectively. Minimum numbers of global iterations for different numbers of inner iterations per outer one are achieved in HEXAGAwith the same value of w and the number of global iterations is greaterbestfrom its asymptotic value only with 2 inners/outer. In the case of HEXSLOR, an unusual behaviour of the number of global iterations is observed with 2 in- ners per outer where the minimum number of global iterations is achieved with u> < to . Only 3 and 5 inners/outer allow to reduce the number of global ite-o p trations below its asymptotic value. In terms of computational work, using 1 inner/outer in HEXAGA for all numbers of outer iterations per global one provides solution with the number of flops per mesh point three times lesser than for 2 and 3 inners per outer giving comparable results, and five times lesser than for 5 inners per outer. In the case of HEXSLOR the minimum number of flops per mesh point is achieved with 1 and 3 inners/outer and is about one and half times lesser than that with 5 inners/outer. The results obtained for this problem with values of w's close to w are given in Table 11.be s t

The case with 1 inner iteration per outer one, corresponding to the global- -outer iteration strategy (which with 1 outer iteration per global one reduces to the global iteration strategy) is demonstrated in Figs.29 and 29a for HEX­AGA, and in Figs.30 and 30a for HEXSLOR. The behaviour of the number of flops per mesh point with 1 inner iteration per outer one is almost independent on the used number of outer iterations per global one, as is demonstrated in Figs.29a and 30a.

For this problem HEXAGA provides the solution with twice lesser number of flops per mesh point in comparison to the solution obtained by HEXSLOR.

Computations for the TP6 problem described by nearly sixty thousand mesh points were carried out on the Convex C3210 computer only by means of HEXAGA and HEXSLOR. When HEXSLOR was running with single precision arithmetic equiva­lent to 8 decimals, oscillations of relative flux errors were occuring alreadywith c ~ 0.01 and it was impossible to obtain the solution with e £ 0.001.

<P <pBut it is interesting to notice that for HEXAGA running with single precisionarithmetic, convergence oscillations are not observed and the solution can beobtained with e s 10 5.

<PNumbers of global iterations obtained for the 1 outer/global strategy with

u’s close to w for different values of e and numbers of inner iterations best <pper outer(global) one are given in Table 12 for HEXAGA with single precision arithmetic and in Table 13 for HEXSLOR with double precision arithmetic equi­

38

valent to 16 decimals. The convergence behaviour of k and other results in a dependence on are shown in Tables 14 and 15 where the same HEXAGA results but obtained with using double precision arithmetic are presented in Table 14a for comparison reasons. As can be observed, there are only small differ­ences in values of k obtained by HEXAGA with using single and double preci­sion arithmetic.

The results of computations obtained for this problem with 1 outer/global and 0.005 are plotted in Figs.32,32a,33,33a and 33b.

As car be seen in these figures the increase of inner iterations per outer one reduces the number of global iterations which asymtotic value, equal to 8 iterations with s 0.005, is achieved in HEXAGA computations with 30 inner iterations per outer one (Table 12). In the case of HEXL0R computations it is necessary to use much more inner iterations per outer one because with 50 in­ner iterations per outer one the solution is obtained after 20 global itera­tions (Table 13).

It is interesting to notice that in the case of HEXAGA computations (Fig.32 or Fig.33), 2 and 3 inners/outer reduce the number of global iterations to the same level and an unusual behaviour of number of global iterations is observed in HEXSL0R computations with 1 inner/outer, which is a quite different from the behaviour observed in all other problems. This strategy with 1 inner/outer in HEXSLOR provides the solution with the computational work about two times lesser in a comparison to other strategies but two times greater than in the case of the HEXAGA solution.

In the paper [20] the efficiency of iterative strategies implemented in the DXY program is estimated by comparing, among other parameters, the total num­ber of inner iterations, where different numbers of inner iterations are used in particular groups. For this three-group test problem the best results were obtained in [20,p.18] for the strategy based on the red-black line cyclic Chebyshev iteration method providing the solution, with s 0.005, after the total number of inner iterations equal to 315+162+99 = 576. In the solutions obtained by HEXAGA and HEXSLOR the corresponding total numbers of inner itera­tions are equal to 123+123+123 = 369 and 352+352+352 = 1056, respectively (Tables 14 and 15).

The comparison of results from HEXAGA computations obtained with using 1 and 2 outers/global is shown in Figs.34 and 34a for different numbers of inner iterations per outer one. The use of 1,2,3 and 5 inner iterations per outer one in the strategy using 2 outers/global gives 2,4,6 and 10 inner iterations per global one and these last four numbers were assumed as numbers of inner iterations per outer one in the strategy with 1 outer/global in order to com­pare results with a similar inner iteration effort per global iteration.

39

As can be seen in Fig. 34 both strategies give different numbers of global iterations with small numbers of inner iterations per outer one but the curves for the strategy using 2 outers/global with 3 and 5 inners/global coincide with the curves for strategy using 1 outer/global with 6 and 10 inners/global, respectively. In terms of computational work depicted in Fig.34a it is evident that both strategies with 1 and 2 inners/outer are equivalent and provide the best results in comparing to remaining ones for this problem with the absence of upscattering.

In any case for the problems anal ized in this paper the strategy used in HEXAGA with 1 inner iteration per global one is still superior or competitive to strategies using greater numbers of inner iterations per global one.

5.4 RELIABILITY OF SYMMETRICAL SOLUTIONS

In order to obtain some information on the reliability of solutions pro­vided by both programs HEXAGA and HEXSLOR, it seems to be interesting to check the symmetry of flux distribution in symmetrical solutions.

Since in the TP1 and TP2 problems there is the 30-degree symmetry sector of reactor configuration, it is possible to compare the values of fluxes at points located symmetrically with respect to the shorter diagonal of the 120- -degree rhombic area of solution used in both HEXAGA and HEXSLOR, which cor­responds exactly to the test of 60-degree symmetry. This task is done by de­termining the maximum relative error of symmetry

c = 2x sym

V,, i

- <pi m, n<p. , j

+ <p1 m, n(5.12)

for all mesh points and groups, where <p and <p are values of fluxes atl , J m, nmesh points (i,j) and (m,n) located symmetrically with respect to the shorter diagonal of the 120-degree rhombic layout.

Computations were performed by means of HEXAGA and HEXSLOR running on the Convex C3210 computer with double precision arithmetic and the values of esymfor solutions obtained for the TP1 and TP2 problems with different mesh divi­sions, values of w, numbers of inner iterations per outer one and s 10 5

satisfied in two successive global iterations are summarized in Table 16.Additional results for the TPlc and TP2a problems obtained with c 5 10 2 and

-4 vi 10 , satisfied also in two successive global Iterations, are presented

in Table 16a.As can be seen in these tables, in all considered cases HEXAGA provides

solutions with the maximun relative error of 60-degree symmetry e lessersymten orders in comparison to solutions obtained by means of HEXSLOR. In solu­tions obtained with s 10 , the fluxes at points located symmetrically with

40

respect to the shorter diagonal of the layout have the same values with 15 significant digits for HEXAGA and 5 significant digits for HEXSLOR. In the case of HEXSLOR the values of e are comparable in the magnitude to the va-symlues of used for stopping the iteration process.

These HEXAGA results, with exellent properties of solution symmetry, allow us to suppose that HEXAGA may provide much more reliable solutions than those obtained by means of HEXSLOR.

5.5 FINAL REMARKS AND CONCLUSION

In the numerical results obtained for considered problems, it can be ob­served that there is some range for numbers of inner iterations T, T_ s T s T with Til, for which the solution can be obtained with a comparable iteration effort (represented by the total number of inner iterations) and roughly pro­portional to the computational work (expressed by the number of flops per mesh point) for problems with simple models of scattering. In the case of Test Problem 6, as can be seen in Tables 14 and 15 or in Fig. 32a (repeated in Figs. 33a and 33b with rescalling the axis w), this range is 3 s T ^ 20 for HEXAGA and 2 s T s 30 for HEXSLOR but in this problem the minimum number of flops is obtained with T = 1 for both programs, where HEXAGA provides the solution with two times lesser number of flops per mesh point.

Thus,how to choice the number of inner iterations per outer one inside this range of values of T or a criterion for stopping inner iterations, providing solutions with a comparable computational work, seems to be an open question. The upper bound of this range T corresponds to values of T dependent on an used mesh division, for which the number of global iterations achieves its as­ymptotic value determined by the subdominance ratio <r[T(Ai) ] (or <r[B(A )] when upscattering exists), so that using values of T > T leads to a superfluous increase of computational work.

It was observed in all considered examples that the increase of the number of inner iterations per outer one, T, is accompanied by increasing values of w **or which no convergence occurs. This effect is clearly demonstrated in Fig.32 showing the results obtained by HEXAGA and HEXSLOR for the TP6 problem.

In HEXAGA computations (see also Fig. 33 with rescalling the axis w), with T = 1, u> » 1.198 and with T = 30 changes its value to w « 1.202, forcr 11 crltHEXSLOR with T = 1, w ~ 1.97 and with T = 50 changes its value to w ~crlt crlt1.99. It is interesting to notice that in the case of HEXAGA the value ofu>bOSt ** 1 19? is rather insensitive to changes of T, whereas for HEXSLOR, itis observed moving the value of w to greater values as T increases. As canbestbe seen in Fig.32, for HEXSLOR with T = 1, the value of w coincides nearlybestwith ** 1-952 and is slowly increasing as T increases where achieves thevalue equal to about 1.975 for 20 s T s 50.

41

is aBy inspection of Fig.35, it is evident that the determination of ^best simple problem from the theoretical point of view. If for the curves denoted by "V2" and "Vs" in Fig. 32 the functional dependence on w is known, then the abscissa at the point "2" for which both curves are cutting gives the solution for to . However this functional dependence on to is never unknown in advancebestand finding even its approximate form can be a very much time consuming taskor impossible at all. One of possible aproaches to determining to may belooking for a dependence of to on to where for a priori estimate of tobest opt optthere are efficient techniques [8,10].

From the results obtained by HEXSLOR for the problems considered here, it is rather difficult to find a formula approximating sufficiently well the va­lue of to in all cases. As was already mentioned u changes its value inbest besta dependence on T in the TP6 problem as well as in other cases, it may be im­possible to distinguish w from to as, for example, can be observed inbest optthree variants of the TP1 problem or in the TPS problem when the strategy with 2 inners per outer is used.

However in the case of the HEXAGA algorithm some regular deviation of tobestfrom to can be observed in considered problems and moreover y is ratheropt optinsensitive to the used number of inner iterations per outer one. The follow­ing formula

1.2 - toto = to + p{W )------- ^ (5. 13)best opt 1 2

provides a quite good approximation to the true value of y^ for a large class of reactor problems where pOf^) is the spectral radius of the iteration matrix W with y = 1 and denoted by Vf , and y is the optimum relaxation fac-1 opttor minimizing the spectral radius of W in the double over-relaxation process used in HEXAGA [10,11,12]. The estimate of y in HEXAGA can be obtained byoptmeans of the efficient technique described in [10].

Finally, it should be emphasized that the main object of this section was to give some numerical illustration of discussed strategies by using splitting methods, obviously not all possible existing. In solving reactor- problems with upscattering by means of the global-outer-inner iteration strategy, the number of global iterations is decreasing as the number of outer iterations per glo­bal one P increases. For problems with a significant upscastter this reduction of global iterations is more dependent on the used value of P and with small values of P, the following relations for subdominance ratios hold

<r[7(Ai )] « {<r[V(Ai )]>P or «r [&U )] = (<r[ W(A ) ] >P .

However, in the case of problems with the absence of upscattering, it can be observed in numerical experiments that the global-outer-inner iteration stra­tegy and its variants (global-outer and global-inner iteration strategies) provide comparable numbers of global iterations.

42

The analysis of large number of numerical results obtained for considered problems, representing various types of fast and thermal reactors, allows us to conclude that solutions with the smallest computational work can be obtain­ed by using the iterative strategy with only 1 inner iteration per global one in HEXAGA which algorithm is based on one of simplest models of the AGA pre- factorization methods with the double overrelaxation process as an accelera­tion technique [9,10,11]. But this leads to the conclusion that the choice of splitting matrices M and N = M - A has a dominant influence on the efficiency of the method used for the solution of neutron diffusion theory problems and the observed convergence behaviour of HEXAGA solutions indicates evidently a significant potential of splittings derived from the AGA prefactorization methods.

In comparison to HEXSLOR, HEXAGA provides solutions with 2 to 5 times les­ser number of flops for considered test problems. In the case of the VALE [21] and DXY [20] programs with Chebyshev polynomial acceleration which results are available only for the TP1, TP3 and TP6 problems, results obtained by HEXAGA are about two times superior from the viewpoint of computational work. The DXYresults for the TPS problem are available only with c £ 5x10 3 and it would

f_sbe interesting to compare results obtained with £ 10 for this problem.

It seems that a comparison of solutions for the TP2 problem, obtained by HEXAGA, DXY, VALE as well as computer programs based on strategies developed recently, could be a valuable estimation of validity of these programs.

In summary it is worth to mention on a possibility of obtaining HEXAGA so­lutions with single precision arithmetic for large reactor problems for which, as was demonstrated in the TP6 problem, the HEXSLOR solution can be obtained only with using double precision arithmetic. The problem for the a priori de­termination of ubest in HEXAGA is sufficiently well solved for applications. Furthermore, the examination of solution symmetry in the TP1 and TP2 problems, for whose HEXAGA provides solutions with the maximum relative error of 60- -degree symmetry ten orders lesser in comparing to HEXSLOR solutions, allows us to suppose that HEXAGA solutions, on the one hand characterizing with an increased rate of convergence, on the other hand may be much more re­liable than solutions obtained by means of the SLOR splitting implemented in HEXSLOR.

ACKNOWLEDGEMENT

The author would like to thank Drs. S.Potempski, H.D.Phuong and H.Wojcie- chowicz for their useful discussions, comments and expert programming assist­ance. Thanks are also due to Professors J. M. Barry and J. P. Pollard for the accessibility of the eight group macroscopic constants for Test Problem 5.

43

REFERENCES

1. J. J. Duderstadt and L.S.Hamilton, Nuclear reactor analysis, Wiley, New York, 1976.

2. E.L.Wachspress, Iterative solution of elliptic systems and applications to the neutron diffusion equation of reactor physics, Prentice-Hall, Englewood Cliffs,N.J,1966.

3. L. A. Hageman and D.Young, Applied iterative methods, Academic Press, New York,1981.

4. D. R. Ferguson and K.L.Derstine, Optimized iteration strategies and data ma­nagement considerations for fast reactor finite difference diffusion theory codes, Nucl. Sci.Eng.,64,593-604, 1977.

5. R. Frohlich, Positivity theorems for the discrete form of the multigroup diffusion equations, Nucl.Sci.Eng.,34,57-66,1968.

6. M. L Tobias and T.B.Fowler, The EQUIPOISE method - A simple procedure for group-diffusion calculations in two and three dimensions, Nucl. Sci. Eng.,12, 513-518,1962.

7. C. H. Adams, Current trends in methods for neutron diffusion calculations, Nucl.Sci.Eng.,64,552-562,1977.

8. Z. I.Wo2nicki, The sigma-SOR algorithm and the optimal strategy for the uti­lization of the SOR iterative method, Mathematics of Computations, 206,619- -644,1994.

9. Z. I. Wo2nlcki , Nonnegative splitting theory, Japan J. Indust.. Appl. Math. , 11,289-342,1994.

10. Z. I.WoZnicki, Estimation of the optimum relaxation factors m partial fac­torization iterative methods, SIAM J.Matrix Anal. Appl., 14, 59-73,1993.

11. Z. I.WoZnicki, HEXAGA-II-120-,-60,-30 two-dimensional multigroup neutron diffusion programmes for a uniform triangular mesh with arbitrary group scattering, Rep. KfK 2789,Kernforschungscentrum Karlsruhe, Karlsruhe,Germany 1979.

12. Z. I.WoZnicki, HEXAGA-III-120-,-30 three-dimensional multigroup neutron diffusion programmes for a uniform triangular mesh with arbitrary group scattering, Rep. KfK 3572,Kernforschungscentrum Karlsruhe, Karlsruhe,Germany 1983.

13. Z. I.Wo2nicki, Two- and three-dimensional benchmark calculations for trian­gular geometry by means of HEXAGA programmes, in Proc. Internal. Meeting on Advances in Nuclear Engineering Computational Methods, Knoxville, IN, April 1985, American Nuclear Society.

14. Z.I.Wo2nicki, On numerical analysis of conjugate gradient method, Japan J. Indust.Appl.Math.,12,487-519, 1993.

15. G.H.Golub and C.F.Van Loan, Matrix computations, Johns Hopkins Press,Balti­more, 1983.

16. Benchmark Problem Book, Rep.ANL-7416,Suppl. 2,228-237, 1977.17. Benchmark Problem Book, Rep.ANL-7416,Suppl. 3,1985.18. J. P.Pollard, Rep.AAEC/E269 (Australian Atomic Energy Commission),1974.19. J.M.Barry and J. P.Pol lard, Performance of a nonstationary implicit scheme

for solving large systems of linear equations, Nucl.Sci.Eng.,92,27-33,1986.20. I. K.Abu-Shumays, Vectorization of transport and Diffusion Computations on

the CDC Cyber 205, Nucl.Sci.Eng.,92,4-19, 1986.21. Rep.ORNL-5792,1981.

44

Table 1. Parameters In Eq. (5.6)

Test N N N NProblem c FIS XFIS SCAT

1 4 4 4 62 4 4 4 63 4 4 4 34 4 4 4 85 8 8 8 496 3 3 3 2

45

Table 2. Macroscopic group constants for Test Problem 1 [ 17, pp. 799-800]

Material compos!tion

Groupnumber

g

D9 sr9

vZf9 rd9 —>9 ♦ 1 Id9—*9 + 2

£dq—fcg *3

1 1 2.87679E+00 2.82040E-02 1.18780E-02 7.68000E-01 2.35970E-02 4.07910E-06 4.44930E-082 1.57085E+00 5.27470E-03 5.32520E-03 2.32000E-01 1.61530E-03 4.23090E-08 -3 7.22490E-01 1.76120E-02 1.04710E-02 0. 4.68380E-03 - -4 9.64200E-01 2.S5460E-02 2.66110E-02 0. " - -

2 1 2.87654E+00 2.87820E-02 1.49430E-02 7.68000E-01 2.32620E-02 4.64510E-06 4.99680E-062 1.57136E+00 6.04910E-03 7.68870E-03 2.32000E-01 1.57180E-03 4.07240E-08 -3 7.12710E-01 1.95100E-02 1.48090E-02 0. 4.34140E-03 - -4 9.42980E-01 3.37140E-02 3.81590E-02 0. - -

3 1 2.28561E+00 3.59590E-02 7.74270E-03 7.68000E-01 3.20710E-02 3.88800E-06 4.50390E-082 1.17193E+00 5.88550E-03 1.08250E-04 2.32000E-01 2.77760E-03 9.00180E-08 -3 6.32480E-01 1.60410E-02 2.97420E-04 0. 5.89710E-03 - -4 8.18360E-01 1.33490E-02 8.46870E-04 0. - - -

4 1 2.50307E+00 2.48140E-02 0. - 2.29460E-02 1.03200E-06 1.04890E-082 1.31468E+C0 1.64120E-02 0. - 3.76870E-03 7.03610E-12 -3 5.74280E-01 7 21220E-02 0. - 8.68150E-03 - -4 6 15370E-01 1.6868CE-01 0. - - - “

5 1 4.61S42E+00 1.3159QE-02 0. - 1.29420E-02 S. 87900E-07 6.99030E-03

2 2.90183E+00 1.45590E-03 C. - 1.28710E-03 4.36330E-12 -3 1.02118E+00 4.60010E-03 0. - 3.45330E-03 - -4 1.72963E+00 7.8E600E-04 0. - - - -

Table 3. Macroscopic group constants for Test Problem 2

Materialcomposition

Groupnumberg

D9 if9

vlf9

zdg—>g + l

zdg—*g+2

Zdg—>g + 3

1 1 2.20000E+00 2.24000E-02 1.11000E-02 9.05000E-01 1.63000E-02 1.81000E-04 1.70000E-072 1.34000E+00 1.05000E-02 5.95000E-03 9.00000E-02 5.90000E-03 3.30000E-08 -3 9.73000E-01 1.20000E-02 7.79000E-03 5.00000E-03 2.98000E-03 - -4 7.99000E-01 2.73000E-02 2.33000E-02 0. - - -

2 1 1.98000E+00 2.50000E-02 4.11000E-03 9.05000E-01 2.08000E-02 1.83000E-04 8.70000E-082 1.18000E+00 1.12000E-02 5.07000E-04 9.OOOOOE-02 7.90000E-03 2.70000E-09 -3 8.74000E-01 9.85000E-03 7.01000E-04 5.00000E-03 2.19000E-03 - -4 8.00000E-01 1.49000E-02 2.65000E-03 0. - - -

3 1 1.01000E+00 5.80000E-02 0. - 5.19000E-02 8.57000E-05 2.00000E-082 6.08000E-01 6.15000E-02 0. - 3.77000E-02 2.27000E-11 -3 4.53000E-01 8.76000E-02 0. - 1.2S000E-02 - -4 3.30000E-01 3.20000E-01 0. - - -

4 1 4.25000E+00 9.24000E-03 0. - 6.61000E-03 4.33000E-05 9.50000E-092 3.24000E+00 6.62000E-Q3 0. - 4.61000E-03 9.70000E-12 -3 2.00000E+00 7.31000E-03 0. - 5.80000E-03 - -4 1.70000E+00 2.03000E-03 0. - - - -

5 1 3.03000E+01 1.85000E-02 0. - 7.81000E-04 1.91000E-0S 6.OOOOOE-092 1.84000E+01 1. 12000E-02 0. - 4.38000E-04 2.10000E-12 -3 1.31000E+01 7.97000E-03 0. - 2.10000E-04 - -4 6.74000E+00 4.22000E-03 0. - - - -

Table 4. Macroscopic group constants for Test Problem 3 [ 16, pp. 234-235]

Materialcomposition

Groupnumber

g

Dg 9vl?

9 *9 £d9 —>9 ♦ 1

1 1 2.65000E+00 1.1120SE-02 5.58000E-05 9.67500E-01 1.09000E-022 1.37000E+00 7.97900E-03 3.61000E-04 3.25000E-02 3.28000E-033 1.34000E+00 2.99280E-02 9.88000E-04 0. 2.04000E-024 1.31000E+00 1.09000E-02 4.76000E-03 0. -

2 1 2.65000E+00 1.09586E-02 5.58000E-05 9.67500E 01 1.09000E-022 1.37000E+00 4.19800E-03 3.61000E-04 3.25000E-02 3.28000E-033 1.34000E+00 2.12360E-02 9.83000E-04 0. 2.04000E-024 1.31000E+00 2.84000E-03 4.76000E-03 0. -

3 1 2. 3S000E+00 1.23977E-02 9.83000E-05 9.67500E-01 1.23000E-022 X. 21000E+00 5.29200E-03 6.36000E-04 3.2SOOOE-02 3.67000E-033 1. 19000E+00 2.43590E-02 1.74000E-03 0. 2.28000E-024 1.16000E+00 5.35000E-03 8.39000E-03 0. -

4 1 2.35000E+00 1.23984E-02 9.31000E-05 9.67500E-01 1.23000E-022 1.21000E+00 5.33600E-03 5.73000E-04 3.25000E-02 3. 67000E-033 1.19000E+00 2.42240E-02 1.57000E-03 0. 2.28000E-024 1.16000E+00 4.99000E-03 7.56000E-03 0. -

5 1 2.35000E+00 1.23986E-02 8.97000E-05 9.67500E-01 1.23000E-022 1.21000E+00 5.35000E-03 5.34000E-04 3.25000E-02 3.67000E-033 1.19000E+00 2.41410E-02 1.46000E-03 0. 2.28000E-024 1.16000E+00 4.77000E-03 7.05000E-03 0. -

6 1 2.35000E+00 1.23991E-02 8.65000E-05 9.67500E-01 1.23000E-022 1.21000E+00 5.37400E-03 4.95000E-04 3.25000E-02 3.67000E-033 1.19000E+00 2.40S80E-02 1.36000E-03 0. 2.28000E-024 1.16000E+00 4.55000E-03 6.54000E- 03 0. -

7 1 1.64000E+00 1.77139E-02 0 - 1.77000E-022 8.50000E-01 5.33218E-03 0. - 5.33000E-033 8.32000E-01 3.31197E-02 0. - 3.31000E-024 8.21000E-01 1.06000E-04 0. - _

48

Table S. Macroscopic group constants for Test Problem 4 [18,p.12]

composition numb«r6

D9

zr9

vXf9

rd9—>9* 1

id9—>9*2

original upscatterl-mgZu Euq*l —>g 9*2—>9

1 1 2.15776E+00 9.41612E-02 3.73274E-04 7.53564E-01 9,37282E-02 0. - -

2 9.04249E-01 6.02354E-02 3.30167E-03 2.46436E-01 4.88109E-02 B.7S701E-03 5.09412E-03 1.66342E-073 3.69404E-01 6.81026E-01 3.96448E-02 0. 6.48822E-01 - 4.77318E-02 -4 1.65875E-01 1.12777E-01 9.77554E-02 0. - -

2 1 2.22483E+00 2.24931E-02 0. - 2.24898E-02 0. - -

2 9.67802E-01 4.23208E-03 0. - 4.22309E-03 4.23284E-08 1.58544E-03 0.3 8.90918E-01 9.82699E-02 0. - 9.65788E-02 - 8.07923E-03 -4 9.16907E-01 8.32790E-03 0. - -

Table Sa. Macroscopic group constants for Test Problem 4 with significant upscatterlng

Materialcomposition

Groupnumberg

Er9

significantz"9*1—*9

upscatterlmgzu9 + 2—*9

1 1 9. 41612E-02 - -2 6.02354E-02 5.09412E-02 1.66342E-013 7.26073E-01 4.77318E-01 -4 7.08705E-01 - -

2 1 2.24931E-024.23208E-03 1.12539E-01 8.10410E-02

234

1.58544E-02 0.8.07923E-02

Table 6. Macroscopic group constants for Test Problem 5

Mater. Groupnumber D if f Xcomp.

g9 9 9 9

i 1 1.46338E+00 4.76988E-02 9.98480E-04 1.OOOOOE+OO2 6.64659E-01 2.28751E-01 9.84187E-03 0.3 4.21452E-01 7.57320E-01 3.74418E-02 0.4 2.86852E-01 9.99381E-01 4.28166E-02 0.5 2.07771E-01 6.37374E-01 6.61408E-02 0.6 1.44426E-01 1.29060E+00 9.92365E-02 0.7 1.10424E-01 1.88S58E+00 1.33412E-01 0.8 8.29912E-02 1.74149E+00 2.03107E-01 0.

i 1 1.24224E+00 4.61472E-03 0. -2 8.55065E-01 1.26264E-02 0. -3 7.76724E-01 6.18582E-02 0. -4 7.88051E-01 6.49914E-02 0. -5 7.91263E-01 1.87279E-02 0. -6 8.05448E-01 2.38339E-02 0. -7 7.72072E-01 2.00360E-02 0. -8 1.08991E+00 1.66264E-02 0. -

50

Table 6. (continued}

Mater. Groupnumber Zd Zd zd zd Zd zd

comp.g

9—*9+ I g—*g+2 g—>g + 3 g—>g*< g—*g*s 9—>9*6

i 1 4.65811E-02 2.30480E-04 5.22233E-0S 7.62686E-05 0. 0.2 1.37274E-01 3.28173E-02 3.65639E-02 7.56773E-03 3.39985E-03 2.75796E-033 3.30509E-01 3.04949E-01 5.30115E-02 2.12912E-02 1.40595E-02 -4 7.15333E-01 1.05292E-01 4.00932E-02 2.53505E-02 - -5 3.688S2E-01 8.74426E-02 5.02595E-02 - - -6 4.81418E-01 1.19740E-01 - - - -7 6.56025E-01 - - - - -

2 1 4.S9988E-03 0. 0. 0. 0. 0.2 1.21501E-02 3.24111E-04 1. 14455E-04 6.90137E-06 1.97710E-06 1.08630E-063 4.16156E-02 1.59270E-02 1.36140E-03 4.52799E-04 2.38788E-04 -4 5.05304E-02 3.20156E-03 1.10054E-03 6.09639E-04 - -5 9.99063E-03 2.19810E-03 1.08201E-03 - - -6 5.57058E-03 2.18569E-03 - - - -7 3.5708SE-03 - - -

Mater. Groupnumber z“ z" z” z“ Zu z“comp.g

g*l —)g 9*1—>9 9 + 3—>9 9 + 4—>9 9+S—>g 9 +8 >9

i 2 8.11520E-03 4.00278E-05 1.33927E-05 9.08439E-06 9.20614E-06 1.20187E-053 8.35905E-02 7.00571E-03 2.19485E-03 1.80514E-03 2.07293E-03 -4 7.94302E-02 1.72763E-02 1.22979E-02 1.25768E-02 - -5 6.04366E-01 2.51265E-01 2.23986E-01 - -6 8.75925E-01 3.39181E-01 - - - -7 1.02710E+00 - - - - -

2 2 2.17343E-03 2.07989E-07 7.63545E-09 6.93883E-10 2.47682E-10 1.02013E-103 9.43152E-03 3.10424E-04 4.2S664E-05 2.69027E-05 2.21881E-05 -4 4.97703E-03 4.50238E-04 2.75652E-04 2.31993E-04 - -5 1.53456E-02 5.94654E-03 4.45838E-03 - - -6 9.89648E-03 5.94789E-03 - - - -7 5.45336E-03 - - - - -

51

Table 7. Macroscopic group constants for Test Problem 6 [2,Tables 8.7 & 8.10]

Materialcomposition

Groupnumberg

D9 9

vt*9

%9 Ed9—»9 + l

1 1 1.40000E+00 4.OOOOOE-02 0. 1.00000E+00 3.00000E-022 8.00000E-01 8.OOOOOE-02 5.OOOOOE-02 0. 4.00000E-023 4.00000E-01 2.50000E-01 5.OOOOOE-O1 0. -

2 1 1.50000E+00 8.OOOOOE-02 0. - 7.00000E-022 5.00000E-01 1.500Q0E-01 0. - 1.50000E-013 1.00000E-01 2.00000E-02 0. - -

3 1 3.00000E+00 1.OOO0OE-O2 0. - 0.

2 2.00000E+00 2.OOOOOE-O1 0. - 0.

3 1.00000E-01 3.00000E+00 0. - -

Table 8. Results for Test Problems 1, 2 and 3 obtained with 1 outer/globalstrategy for c. = 10"S

9and ek = 10"6.

Problem No. of inner

HEXAGA VALE* )

and No.of ItersNo. of

globalsTotal No. of No. of

globalsTotal No. of

mesh points perouter No. of

innersflops

per m. ptNo. of inners

flops per m. pt

i 18 72 1296TPla 2 19 152 2432324 3 24 288 4418

4 19 304 3344

1 33 132 2376TPlb 2 20 160 25601225 3 16 192 2944

4 21 336 3696

1 56 224 40322 32 256 4096

TPlc 3 21 252 38644761 3,7,4.5 34 714 -7500

5 20 400 59208 21 672 9744

TP2a 1 72 288 5184

1225 2 662 5296 847363 657 7884 120888

TP2b 1 107 435 7704

4761 2 193 1544 247043 550 6600 101200

1 40 160 2760TP3 2 55 440 68751687 3 55 660 9955

5,5,4,4 25 450 4825

e s 10"49

1 27 108 19442 26 208 3328ck i 10 53 32 384 5888

*) The results obtained by VALE [21) for TPla.b.c are quoted from [21, p. 34] (because those given In [17, p.833] were obtained with = 5x 10~5 and

= 5x10 6) and for TP3 are quoted from [17, p.269).

53

Table 8a. Results for Test Problems 1. 2 and 3 obtained with 1 outer/globalstrategy for = 10~s and = 10 6.

Problemand No. ofmesh points

No. of Inner iters per

outer

HEXSOR HEXSL0R

No. of globals

Total No. of inners

No. of flops

per m.ptNo. of

globalsTotal No. of Inners

No. of flops

per m.pt

1 65 260 3640 58 232 3248TPla 2 33 264 3168 40 320 3840324 3 22 264 2992

8 22 704 7392

1 132 528 7392 92 368 5152TPlb 2 68 544 6528 58 464 55681225 3 46 552 6256 48 576 6528

8 21 672 7056 30 960 10080

1 247 988 13832 195 780 109202 122 976 11712 110 880 10560

TPlc 3 80 960 10880 72 864 97924761 5 54 1080 11664 58 1160 12528

8 33 1056 11088 38 1216 1276812 21 1008 10416 33 1584 16368

1 538 2152 30128 476 1912 26768TP2a 2 655 5240 62880 690 5520 662401225 3 720 8640 97920 700 8400 95200

1 684 2736 38304TP2b 2 677 5416 649924 /bl 3 710 8520 96560

1 398 1592 21094 312 1248 16536TP3 2 311 2488 28923 205 1640 190651687 3 179 2148 23807 85 1020 11305

5 66 1320 14056 62 1240 13206

1 206 824 10918 165 660 8745c < 10"1

<P 2 157 1256 14601 122 976 11346C 5 10'5 3 86 1032 11438 54 648 7182k 5 42 840 8946 39 780 8307

54

Table 9. Results for Test Problem 4 obtained with 1 outer/global strategy for e*10 5 and c = 10-e

Problemand No. ofmesh points

No. of Inner Iters per

outer

HEXAGA HEXSOR HEXSL0R

No. of globals

Total No. of lnners

No. of flops

per m. ptNo. of globals

Total No. of lnners

No. of flops

per m. ptNo. of

globalsTotal No. of lnners

No. of flops

per m. pt

TP4a 1 21 84 1554 47 188 2726 41 164 2378

228 2 23 184 2990 42 336 4116 38 304 37243 23 276 4278 30 360 4140 28 336 3864

1 38 152 2812 88 352 5104 78 312 4524TP4b 2 24 192 3120 59 472 5782 53 424 5194851 3 24 288 4464 42 504 5796 37 444 5106

5 23 460 6854 35 700 7630 32 640 6976

1 60 240 4440 188 752 10904 155 620 8990

TP4c 2 34 272 4420 124 992 12152 83 664 8134

3285 3 24 288 4464 87 1044 12006 63 756 86945 23 460 6854 59 1180 12862 48 960 104648 19 608 8854 39 1248 13182 35 1120 11830

TP4c 1 63 252 4662 202 808 11716 155 620 89903285 2 78 624 10140 205 1640 20090 180 1440 17640with

significant 3 57 684 10602 112 1344 15456 90 1080 12420upscatter 5 89 1780 26522 101 2020 22018 89 1780 19402

Table 10. Results obtained for Test Problem 4 (3285 m. pts. ) with thesignificant upscatter for = 10 5 and = 10 6.

No. ofouter Iters.per global

No. of Inner Iters per

outer

HEXAGA HEXSL0R

No. of globals

Total No. of Inners

No. of flops per m.pt

No. of globals

Total No. of Inners

No. of flops

per a.pt

1 63 252 4662 155 620 89902 78 624 10140 180 1440 17640

1 3 57 684 10602 94 1128 129725 89 1780 26522 89 1780 19402

1 32 256 4544 78 624 85802 43 688 10922 91 1456 17290

2 3 30 720 10980 44 1056 118805 47 1880 27730 44 1760 18920

1 22 264 4620 55 660 89102 30 720 11340 60 1440 16920

3 3 21 756 11466 30 1080 120605 34 1360 29988 30 1800 19260

1 15 300 5190 35 700 93102 19 760 11894 37 1480 17242

5 3 14 840 12684 18 1080 119885 22 2200 32252 18 1800 19188

56

obtained for e * 10 5 and e = 10 8.V *

Table 11. Results for Test Problem 5 (3285 m.pts.) with seven thermal groups

No. ofouter Iters.per global

No. of Inner Iters per

outer

HEXAGA HEXSLOR

No. of globals

Total No. of lnners

No. of flops

per aptNo. of

globalsTotal No. of lnners

No. of flops per m.pt

1 65 520 11635 141 1228 207272 106 1696 30846 274 4384 621983 77 1848 31031 72 1728 22104

1 5 82 3280 51414 69 2760 322238 82 5248 78966 102 6528 7211412 82 7872 115702 95 9120 9756520 96 15360 221472 96 15360 160032

1 33 528 11484 77 1232 218682 55 1760 31460 133 4256 59052

2 3 40 1920 31840 37 1776 224965 43 3440 53492 36 2880 332648 44 5632 84304 53 6784 7441212 52 9984 146224 51 9792 104244

1 22 528 11374 53 1272 223132 38 1824 32414 90 4320 59490

3 3 27 1944 32103 25 1800 225235 30 3600 55830 25 3000 345258 36 6912 103284 37 7104 77737

1 14 560 11270 31 1260 199952 24 1920 32760 55 4400 57475

5 3 18 2160 34650 16 1920 231205 19 3800 57855 16 3200 359208 23 7360 108675 23 7360 79235

1 11 704 14982 22 1408 243322 19 2432 42902 36 4608 62856

8 3 15 2880 47310 14 2688 334045 16 5120 79136 14 4480 51324

57

Table 12. The number of global Iterations (1 outer/global) for Test Problem 6obtained by HEXAGA with different values of and w"s close to wbest

No.of inner iters, per w

e*5outer iter. 0.01 0.005 0.001 0.0001 0.00001

1.1960 99 133 182 275 3811 1.1965 100 123 180 273 367

1.1970 142 156 214 317 430

1.1960 62 73 104 148 1982 1.1965 61 71 101 140 183

1.1970 71 78 112 163 no conver.

1.1965 66 74 93 120 1573 1.1970 61 71 85 111 161

1.1975 51 70 85 126 161

1.1965 38 43 54 70 905 1. 1970 35 41 49 65 88

1.1975 31 21 46 64 108

1.1965 26 29 36 47 748 1. 1970 25 28 33 42 58

1. 1975 19 27 29 40 68

1.1965 14 17 21 28 3612 1.1970 13 16 20 26 32

1.1975 12 14 19 26 54

1.1975 10 11 14 19 3020 1. 1980 9 11 14 18 23

1. 1985 11 12 15 19 98

1.1970 7 8 10 13 1930 1.1975 7 8 10 13 15

1.1980 7 9 10 14 22

1.1970 7 8 10 14 33

501.1975 7 8 10 13 181.1980 7 8 10 13 411.1985 7 8 9 12 19

58

Table 13. The number of global Iterations (1 outer/global) for Test Problem 6 obtained by HEXSLOR with different values of and u's close to «beBt

No.of Inner Iters, per ti

VI

outer Iter. 0.01 0.005 0.001 0.0001 0.00001

1.9625 251 336 414 564 7251 1.9650 256 352 401 536 655

1.9675 357 363 483 544 660

1.9650 324 341 387 472 5472 1.9675 314 324 342 372 518

1.9700 297 311 386 413 511

1.9650 275 290 323 368 4163 1.9675 250 268 297 326 353

1.9700 241 268 290 342 372

1.9650 169 178 198 226 2531.9675 145 159 181 202 218

5 1.9700 144 156 176 191 2271.9725 159 165 176 210 241

1.9650 98 104 117 134 152

81.9675 90 95 107 120 1311.9700 82 85 104 113 1351.9725 93 97 104 125 145

1.9650 67 71 80 92 104

121.9675 62 66 74 82 901.9700 62 65 71 83 921.9725 64 66 76 85 97

1.9700 41 43 47 52 5820 1.9725 38 40 43 51 55

1.9750 40 41 47 51 591.9725 27 28 31 35 40

30 1.9750 26 27 31 34 391.9775 27 29 32 37 42

1.9750 20 21 23 27 30

501.9775 19 20 23 26 291.9800 19 20 22 25 281.9825 21 21 23 27 29

59

Table 14 HEXAGA results obtained in single precision for Test Problem 6(57600 m.pts.) for 1 outer/global strategy with the best relax­ation factor w and for different values of e .

B 9

*9 WBNo. of lnners per

outer

No. of globals

TotalNo. of lnners

No. of flops per

m. pt.kef f

1.1965 1 123 369 6396 0.797374 6.95x10'*1.1965 2 71 426 6674 0.804614 5.05xl0"41.1970 3 71 639 9230 0.810810 7.80xl0'S1.1970 5 41 615 9020 0.810739 2.62x10'*

5xl0"31.1970 8 28 672 9688 0.810984 1.20xlOS1. 1970 12 16 576 8224 0.810702 1.05x10'31. 1980 20 11 660 9350 0.810208 1.llxlO'31.1975 30 8 720 10160 0.810937 1.27x10'3

1. 1965 1 180 540 9360 0.807413 1.20x10'*1. 1965 2 101 606 9494 0.809427 1.45x10'*1. 1970 3 85 765 11050 0.810959 1.14xlO'S1. 1970 5 49 735 10780 0.810945 7.17x10'7

10~31. 1970 8 33 792 11418 0.811018 5.35xlO"S1. 1970 12 20 720 10280 0.810924 2.34xlO'S1. 1980 20 14 840 11900 0.810919 2.42x10'*1.1975 30 10 900 12700 0.811182 2.35x10®

1.1965 1 273 819 14196 0.810896 6.79xl0~61.1965 2 140 840 13160 0.810961 1.53x10'®1.1970 3 111 999 14430 0.811152 1.07x10"®1. 1970 5 65 975 14300 0.811155 1.19x10"®

10"41. 1970 8 42 1008 14532 0.811152 2.38x10"®1. 1970 12 26 936 13364 0.811157 2.86x10'®1. 1980 20 18 1080 15300 0.811133 2.46x10~S1. 1975 30 14 1260 17780 0.811160 9.54x10"®

1. 1965 1 367 1101 19084 0.811144 8.94xl0"71. 1965 2 183 1098 17202 0.811145 1.07x10'®1. 1970 3 161 1449 20930 0.811165 6.56x10'71.1970 5 88 1320 19360 0.811160 7. 15x10'7

ID"5 1.1970 8 58 1392 20068 0.811164 2.74x10"®1. 1970 12 32 1152 16448 0.811169 3.22x10'®1.1980 20 23 1380 19550 0.811162 2.21x10®1.1975 30 15 1350 19050 0.811162 1.06x10"®1. 1975 50 18 2700 37980 0.811162 1.67x10"®

60

Table 14a. HEXAGA results obtained In double precision for Test Problem 6(57600 a.pts.) for 1 outer/global strategy with the best relax­ation factor w and for different values of c ..b 9

c* WBNo. of inners per

outer

No, of globals

TotalNo. of inners

No. of flops per

m. pt.k.rr Gk

1.1965 1 123 369 6396 0.797224 6.95x10'*1.1965 2 71 426 6674 0.804162 5.05x10'*1.1970 3 90 810 11700 0.810837 2.15x10'®1. 1970 5 41 615 9020 0.810582 2.61x10*

SxlO"31.1970 8 28 672 9688 0.810829 1.34x10'®1. 1970 12 16 576 8224 0.810546 1.05x10"31. 1980 20 11 660 9350 0.810052 1.llxlO"31. 1975 30 8 720 10160 0.810781 1 27x10"3

1. 1965 1 180 540 9360 0.807263 1.20x10"*1.1965 2 101 606 9494 0.809272 1.45x10"*1. 1970 3 107 963 13910 0.810992 2.58x10"®

mo 1.1970 5 49 735 10780 0.810789 6.30xl0"7-R1.1970 8 33 792 11418 0.810860 5.24x10

1. 1970 12 20 720 10280 0.810768 2.19x10'®1. 1980 20 14 840 11900 0.810762 2.42x10'*1. 1975 30 10 900 12700 0.811026 2.36x10"®

1. 1965 1 273 819 14196 0.810740 7.96x10'®1.1965 2 140 840 13160 0.810805 1.55x10"®1.1970 3 121 1089 15730 0.810997 8.14x10'71. 1970 5 65 975 14300 0 810997 1.80x10"®o1.1970 8 42 1008 14532 0.810995 5.92xl0-71.1970 12 26 936 13364 0.810999 5.46x10"®1. 1980 20 19 1140 16150 0.810990 1.70x10"®1.1975 30 13 1170 16510 0.811001 5.58x10"®

1. 1965 1 364 1092 18928 0.810986 6.13xl0"71. 1965 2 181 1086 17014 0.810987 1.26x10'®1.1970 3 144 1296 18720 0.811067 2.70xl0"71.1970 5 81 1215 17820 0.811006 3.48xl0*7

10*S 1.1970 8 52 1248 18928 0.811006 1.60xl0"71.1970 12 32 1152 16448 0.811007 1.85x10"®1. 1980 20 23 1380 19550 0.811005 1.90x10"®1.1975 30 15 1350 19050 0.811007 1.31xl0"71.1975 50 17 2550 35870 0.811005 1.75x10'®

61

Table 15. HEXSLOR results obtained In double precision for Test Problem S (57600 m.pts. ) for 1 outer/global strategy with the best relax­ation factor w and for different values of c,.

G* WBNo. of lnners per outer

No. of globals

TotalNo. of lnners

No. of flops per

m.pt.k.r, Gk

1.9650 1 352 1056 14080 0.814759 6.30x10"®1.9675 2 324 1944 22680 0.810975 1.90x10"®1.9675 3 268 2412 26800 0.811008 3.55x10"®1.9700 5 156 2340 24960 0.811009 2.32xl0"7

5xl0"31.9700 8 85 2040 21250 0.811009 1.37x10'®1.9700 12 65 2340 24050 0.811008 4. 56x10"71.9725 20 40 2400 24400 0.811007 4.64xl0~71.9750 30 27 2430 24570 0.811007 3.68xl0"7

1.9650 1 401 1203 16040 0.812520 4.46x10"®1.9675 2 342 2052 23940 0.810995 9. 76x10"71.9675 3 297 2673 29700 0.811007 3.18x10"®1.9700 5 176 2640 28160 0.811007 5.32x10"®

10"31.9700 8 104 2496 26000 0.811007 1.98x10*71.9700 12 71 2556 26270 0.811007 1.45xl0"71.9725 20 43 2580 26230 0.811007 5. 57x10"®1.9750 30 31 2790 28210 0.811007 8.32x10"®

1.9650 1 536 1608 21440 0.810765 5.42x10"71.9675 2 372 2232 26040 0.811007 1.91xl0"71.9675 3 326 2934 32600 0.811007 4.82x10"®1.9700 5 191 2865 30560 0.811007 1.76x10"®

1G"41.9700 8 113 2712 28250 0.811007 2.63x10'®1.9700 12 83 2988 30710 0.811007 8.68x10"®1.9725 20 51 3060 31110 0.811007 8.75x10"®1.9750 30 34 3060 30940 0.811007 8.07x10"®

1.9650 1 655 1965 26200 0.810939 1.37x10"®1.9675 2 518 3108 36260 0.811007 7.50x10"*11.9675 3 353 3177 35300 0.811007 6.10x10"*°1.9700 5 227 3405 36320 0.811007 4.35x10"*°

io"5 1.9700 8 135 3240 33750 0.811007 1.86x10"®1.9700 12 92 3312 34040 0.811007 1.40x10"®1.9725 20 55 3300 33550 0.811007 2.05x10"'°1.9750 30 39 3510 35490 0.811007 1.27x10"®1.9800 50 28 4200 42280 0.811007 2.14x10"*°

62

Table 16. Maximum relative error of 60-degree symmetry for the solutionobtained for 1 outer/global strategy with s 10 8.

TestProblem

andNo. of HEXAGA HEXSL0Rinners No. of No. of

No. of perouter w global c w global e

mesh pts. Iters. Iters. -y-

i 1. 13 19 3.99xl0"16 1.65 59 7.OOxlO*08TPla

3 1. 15 25 3.54x10'16 1.6 34 5.77xl0~°8324

8 1.15 25 2.84xl0~16 1.6 27 6.llxlO*06

1 1.158 34 4.05xl0~18 1.76 105 3.13xl0*°8TPlb

3 1.175 18 4.55xl0~16 1.8 48 7.41xl0~°71225

8 1. 16 24 3.30x10*16 1.8 30 9.24xl0*°8

1 1.18 59 1.21xlO~1S 1.87 191 5.21xl0~°S

TPlc 2 1.18 32 1.07xl0~18 1.87 104 3.41xl0*°8

4781 5 1.185 17 8.40xl0~18 1.88 51 5.88xl0~°6

12 1. 18 22 1.86x10*18 1.888 33 5.80xl0~°6

1 1. 185 75 7.09x10*16 1.75 413 2.31xl0*°8TP2a

2 1. 1 666 3.48x10*16 1.65 702 6.35xl0*°81225

5 1.0 663 1.76x10"16 1.5 725 1.91xl0*°6

1 1. 185 108 1.04x10*18 1.85 685 5.llxlO"08

TP2b 2 1. 197 249 9.49x10*18 1.84 678 1.33xl0*°6

4781 5 1.1 691 7.06x10*16 1.72 730 3.89xlO*06

8 1.2 709 9.22xl0"16 1.6 718 3.13xlO*06

63

Table 16a. Maximum relative error of 60-degree symmetry e for the solutionSyB-2 -4obtained for 1 outer/global strategy with c s 10 and c s io

f <P

TestProblem

andNo. of mesh pts.

No. of HEXAGA HEXSL0Rinnersper

outer VNo. ofglobaliters.

ceye wNo. ofglobaliters.

£eye

1 1. 18 28 9.00x10*'" 1.87 87 3.80xl0*°2TPlc

2 1.18 15 1.03x10*15 1.87 47 2.56xl0*°24781 8.90xl0-18 4.04x10*°"5 1. 185 8 8 25c s 10"2

<P 12 1. 18 8 6.94xl0~16 1.888 14 3.98x10*°"

1 1. 18 46 9.70x10*16 159 1.91xl0*°4TPlc 6.88x10*16 4.86x10*°"2 1. 18 26 1.87 904781

5 1.185 14 9.66x10*'" 8 43 4.53x10*°"c 3 10"4

<P 12 1. 18 18 9.89x10*'" 1.888 27 4.59x10*°"

TP2a 1 1. 185 35 1.11x10*'" 1.75 142 2.56x10*°"

1225 2 1.1 36 3.14x10*'° 1.65 108 1.48x10*°"

c a io"2V

5 1.0 16 1.70x10"'° 1.5 56 1.07x10*°"

TP2a 1 1.185 62 8.71x10"*° 1.75 286 2.22x10*°*

1225 2 1. 1 454 2. 17x10*'® 1.65 504 5.75x10*°"

e s io"4V

5 1.0 438 1.75x10*'° 1.5 501 1.57x10" 06

64

symmetry boundary condition

symmetryboundarycondition

vacuumboundarycondition

vacuum boundary condition

hexagon side length h = 6.4665 cm

Fig. 1 The layout of Test Problem 1 with 30-degree symmetry sector [17,Fig.2 in p.824]

65

Num

ber of

globa

l itera

tions

350 --

1 outer/global

---------- HEXAGA---------- HEXSOR---------- HEXSLOR

250 -

200 -

150 -

100 -

50 —

Fig. 2 No.of global iterations vs. relaxation factor w forTest Problem 1 with 324 m. pts.

66

Num

ber of

flops

per me

sh po

int (xl

OO

O)

35

1 outer/global

--------- HEXAGA--------- HEXSOR--------- HEXSLOR

25 -

20

15

s r

1.0 1.2 1.4 1.6 1.8 2.0CO

Fig.2a No.of flops per mesh point vs. relaxation factor w

for Test Problem 1 with 324 m.pts.

67

Num

ber of

globa

l itera

tions

350

1 outer/global

--------- HEXAGA--------- HEXSOR--------- HEXSLOR

250 -

200

150 -

100 -

1.2i r ~T I r

1.4l—l p

1.6

! IT i r f i—r

1.8CO

T

Fig.3 No.of global iterations vs. relaxation factor w forTest Problem 1 with 1225 m.pts.

68

Num

ber of

flops

per me

sh po

int (xl

OO

O)

35

1 outer/global

--------- HEXAGA........ - HEXSOR--------- HEXSLOR

25 -

Fig.3a No.of flops per mesh point vs. relaxation factor wfor Test Problem 1 with 1225 m. pts.

69

Num

ber of

globa

l itera

tions

350

1 outer/globali

HEXAGAHEXSOR

, | | , | | | | , | | | | |—|- i—r~T—f t—i—r—r1.0 1.2 1.4 1.6 1.8

CJ

Fig. 4 No.of global iterations vs. relaxation factor w forTest Problem 1 with 4781 m.pts.

70

Num

ber of

flops

per m

esh p

oint

(xlOO

O)

35

30 -

25 -

20 -

15 -

10 -

5 -

0 — 1.0

1 outer/global

--------- HEXAGA--------- HEXSOR--------- HEXSLOR

i i r1 [~i i r1.2

\

i | i i r1.4

12',«<\\

II i: i:ui:n ,; i: i;u

\208\\

Fig.4a No.of flops per mesh point vs. relaxation factor u

for Test Problem 1 with 4781 m.pts.

71

Num

ber of

globa

l itera

tions

175

150 -

HEXAGA

1 outer/global2 outer/global

125 -

100

CJ

Fig.5 No. of global iterations vs. relaxation factor w for Test Problem 1 with 1215 m.pts. Comparison of HEXAGA results with 1 and 2 outer/global.

125

72

Num

ber of

flops

per m

esh p

oint

(xlO

OO

)

35HEXAGA

1 outer/global2 outer/global

CO

Fig.5a No.of flops per mesh point vs. relaxation factor w for Test Problem 1 with 1225 m. pts. Comparison of HEXAGA results with 1 and 2 outer/global.

T--1.25

73

symmetry boundary condition

symmetryboundarycondition

vaccuaboundarycondition

vacuum boundary condition

hexagon side length h = 9.5552 cm

Fig.6 The layout of Test Problem 2 with 30-degree symmetry sector

74

Num

ber of

globa

l itera

tions

1400

1 outer/global

--------- HEXAGA--------- HEXSOR--------- HEXSLOR1200

1000

Fig.7 No.of global iterations vs. relaxation factor w forTest Problem 2 with 1225 m.pts.

75

Num

ber of

flops

per me

sh po

int (xl

OO

O)

350

1 outer/global

HEXAGAHEXSORHEXSLOR

200 -

150 -

100

Fig.7a No.of flops per mesh point vs. relaxation factor wfor Test Problem 2 with 1225 m.pts.

76

Num

ber of

globa

l itera

tions

1400

1 outer/global

—— HEXAGA -------- HEXSLOR

1200

1000

Fig.8 No.of global iterations vs. relaxation factor w forTest Problem 2 with 4781 m.pts.

77

Num

ber of

flops

per me

sh poi

nt (xlO

OO

) 1 outer/global

HEXAGAHEXSLOR

Fig.8a No.of flops per mesh point vs. relaxation factor wfor Test Problem 2 with 4781 m.pts.

78

&0-degrea rotational ayeweetry boundary condition

hexagon side length h = 20.9 cm

Fig.9 The layout of Test Problem 3 with 60-degree rotational symmetry sector [16,p.2291

79

Num

ber of

globa

l itera

tions

700

600 H

1 outer/global

--------- HEXAGA--------- HEXSOR--------- HEXSLOR

500 t '

0 -j—r | f--y-

1.0 1.1i | r i r i |—t— '• 'i ' : ' j i—r—1.2 1.3 1.4

CO

Fig.10 No.of global Iterations vs. relaxation factor w forTest Problem 3 with 1687 m.pts.

80

Num

ber of

flops

per me

sh po

int (xl

OO

O)

70

60

1 outer/global

HEXAGAHEXSORHEXSLOR

• i

r~T—r1.5

CO

Fig.10a No.of flops per mesh point vs. relaxation factor wfor Test Problem 3 with 1687 m.pts.

81

Num

ber of

globa

l itera

tions

700

600 -

1 outer/global

--------- HEXAGA--------- HEXSOR---------- HEXSLOR

500 H

Fig.11 No.of global iterations vs. relaxation factor w for Test Problem 3 with 1687 m. pts. , e = 10 4, e =10

<f k

82

Num

ber of

flops

per me

sh po

int (x

lOO

O)

70

60

50

40

30

1 outer/global

HEXAGAHEXSOR

— — — HEXSLOR

/ < z t

/ \/ \

'

- z x

i | i i i i | r 1.3 1.4

i r1.5

CJ

Fig.11a No.of flops per mesh point vs. relaxation factor w forTest Problem 3 with 1687 m.pts. e

<P10-4 ek 10-s

83

20c■ 12cm

2 0cm

24c

Fig.12 The layout of Test Problem 4

40c m

[18,p.11]

84

Num

ber of

globa

l itera

tions

350

300 -

1 outer/global

--------- HEXAGA............ HEXSOR--------- HEXSLOR

250 -

200

150

100

50 -

\ 1

x 1

• • •

/' ' V/;

/ /

I I I I

i I i i [~~i i i i | i r r1.0 1.2 1.4

”1—i—i—i—i—i—i—i i—r1.6 1.8 2.0

Cd

Fig.13 No.of global iterations vs. relaxation factor w forTest Problem 4 with 228 m. pts.

85

Num

ber of

flops

per me

sh po

int (xl

OO

O)

35

1 outer/global

--------- HEXAGA--------- HEXSOR--------- HEXSLOR

25

ii

// /

- -, , I , r

18 2.0CO

Fig.13a No.of flops per mesh point vs. relaxation factor wfor Test Problem 4 with 228 m.pts.

86

Num

ber of

globa

l itera

tions

350

300 -

1 outer/global

---------- HEXAGA--------- HEXSOR---------- HEXSLOR

250 -

Fig.14 No. of global iterations vs. relaxation factor w forTest Problem 4 with 851 m.pts.

87

Num

ber of

flops

per me

sh po

int (xl

OO

O) 1 outer/global

HEXAGAHEXSORHEXSLOR

Fig.14a No.of flops per mesh point vs. relaxation factor wfor Test Problem 4 with 851 m.pts.

88

Num

ber of

globa

l itera

tions

1 outer/global

HEXAGA- HEXSOR- HEXSLOR300 -

250 -

100 —

Fig. IS No. of global iterations vs. relaxation factor u> forTest Problem 4 with 3285 m.pts.

89

Num

ber of

flops

per me

sh po

int (xl

OO

O)

351 outer/global

HEXAGAHEXSOR

1.0 1.2 1.4

j,t3;

i!

~r—r [ r' t i ; | —7-*-]—1—r1.6 18 2.0

0J

Fig.15a No.of flops per mesh point vs. relaxation factor wfor Test Problem 4 with 3285 m.pts.

90

Num

ber of

globa

l itera

tions

175

150 -

125 -

100 —

75 -

50 -

25 -

1.00

HEXAGA

1 outer/global2 outer/global

1.05 1.10 1.15 120T r

00

Fig.16 No.of global iterations vs. relaxation factor w forTest Problem 4 with 3285 m.pts. Comparison of HEXAGAresults with 1 and 2 outer/global.

i—1.25

91

Num

ber of

flops

per me

sh po

int (xl

OO

O)

35HEXAGA

0 —| |—,—,—|—|—i—i i i | r f1.00 1.05 1.10

t—|—r120 1.

GO

Fig.16a No.of flops per mesh point vs. relaxation factor wfor Test Problem 4 with 3285 m.pts. Comparison ofHEXAGA results with 1 and 2 outer/global.

92

Num

ber of

globa

l itera

tions

350

300 -

1 outer/global

HEXAGA--------- original upscatter--------- without upscatter

250 -

200

150

100

50

i i i i r1.00 1.05

T-j-r1.10

TT-T“1.15

i—i—r~~j~1 20

CO

Fig.17 No.of global iterations vs. relaxation factor w forTest Problem 4 with 3285 m.pts. Comparison of HEXAGAresults with original and without upscattering.

7-----

1.25

93

Num

ber of

globa

l itera

tions

350

300

250

200

150 -

100

50

1 outer/global

HEXSLOR--------- original upscatter--------- without upscatter

01.70

i r~i r—i—r—|—i—i—i r—]—r-r“T—i—r~~>—E'~r1 75 1.80 1.85 1.90

CJ

Fig.18 No.of global iterations vs. relaxation factor w forTest Problem 4 with 3285 m.pts. Comparison of HEXSLORresults with original and without upscattering.

1.95

94

Num

ber of

globa

l itera

tions

350

CO

Fig. 19 No.of global iterations vs. relaxation factor w forTest Problem 4 with 3285 m. pts. Comparison of HEXAGAresults with original and significant upscattering.

1.25

95

Num

ber of

globa

l itera

tions

350

1 outer/global

HEXSLOR

170 1.75 1.80 1.85 1 90 1.90J

Fig.20 No.of global iterations vs. relaxation factor w forTest Problem 4 with 3285 m.pts. Comparison of HEXSLORresults with original and significant upscattering.

96

Num

ber of

globa

l itera

tions

350

1 outer/global

---------- HEXAGA--------- HEXSOR---------- HEXSLOR300 -

\ i'i

CJ

Fig.21 No.of global Iterations vs. relaxation factor w forTest Problem 4 with 3285 m. pts. and significantupscattering (1 outer/global).

97

Num

ber of

flops

per me

sh po

int (xl

OO

O)

35

30 -

25 -

20 -

2

15

10

5

1 outer/global

--------- HEXAGA--------- HEXSOR--------- HEXSLOR

1

'U

' ill§

: :vI I

I I ! I

i r i—r—r1.0 1.2

i—] i T i i ]__r—r ~i |“1.4 1.6 1.8

T I 12.0

CO

Fig.21a No.of flops per mesh point vs. relaxation factor wfor Test Problem 4 with 3285 m.pts. and significantupscattering (1 outer/global).

98

Num

ber of

globa

l itera

tions

350

1 outer/global

HEXAGA

300 -

250 -

200 -

Fig.22 No.of global Iterations vs. relaxation factor w forTest Problem 4 with 3285 m.pts. and original upscat-tering (1 outer/global), for HEXAGA.

l1.25

99

Num

ber of

flops

per me

sh po

int (xl

OO

O)

35

1 outer/global

HEXAGA

Fig.22a No.of flops per mesh point vs. relaxation factor u>

for Test Problem 4 with 3285 m.pts. and originalupscattering (1 outer/global), for HEXAGA.

100

Num

ber of

globa

l itera

tions

350

1 outer/global

HEXAGA

300 -

250 -

Fig.23 No.of global Test Problem upscattering

Iterations vs. relaxation factor u for 4 with 3285 m.pts. and significant (1 outer/global), for HEXAGA.

11.25

101

Num

ber of

flops

per me

sh po

int (xl

OO

O)

Fig.23a No.of flops per mesh point vs. relaxation factor wfor Test Problem 4 with 3285 m. pts. and significantupscattering (1 outer/global), for HEXAGA.

i

125

102

Num

ber of

globa

l itera

tions

1 outer/global

--------- HEXAGA-------- HEXSLOR

300 -

250 -

- ' i

100

50 -

CO

Fig.24 No.of global iterations vs. relaxation factor w forTest Problem 5 with 3285 m.pts. and seven thermalgroups (1 outer/global).

103

Num

ber of

flops

per me

sh po

int (xl

OO

O)

70 -

10 -~i

0 -4-

1.0 1.2

1 outer/global

---------- HEXAGA-------- HEXSLOR

' 5i ■ i i ■i i

'i i ''

\ '1 'I ''i'V

X

i\I nI

i

__| , . T [-—p-,--- 1- 1 1 - —|--- r 1 r—r—14 16 18 2.0

CO

Fig.24a No.of flops per mesh point vs. relaxation factor wfor Test Problem 5 with 3285 m.pts. and seven thermalgroups (1 outer/global).

104

Num

ber of

globa

l itera

tions

350

300 -

2 outer/global

--------- HEXAGA-------- HEXSLOR

1.0 1.2i—[—i—r i ] i r -r—i—|—i—i—i—i1.4 1.6 1.8 2.0

CJ

Fig.25 No.of global iterations vs. relaxation factor w forTest Problem 5 with 3285 m.pts. and seven thermalgroups (2 outer/global).

105

Num

ber of

flops

per me

sh po

int (xl

OO

O) 2 outer/global

Fig. 25a No.of flops per mesh point vs. relaxation factor wfor Test Problem 5 with 3285 m.pts. and seven thermalgroups (2 outer/global).

106

Num

ber of

globa

l itera

tions

350

3 outer/global

----- HEXAGA---- HEXSLOR

Fig.26 No.of global iterations vs. relaxation factor w forTest Problem 5 with 3285 m.pts. and seven thermalgroups (3 outer/global).

107

Num

ber of

flops

per me

sh po

int (xl

OO

O)

Fig.26a No.of flops per mesh point vs. relaxation factor wfor Test Problem 5 with 3285 m.pts. and seven thermalgroups (3 outer/global).

108

Num

ber of

globa

l itera

tions

350

300 -

5 outer/global

---------- HEXAGA-------- HEXSLOR

250 -

150 -

t—|—i

Fig.27 No.of global Iterations vs. relaxation factor u> forTest Problem 5 with 3285 m.pts. and seven thermalgroups (5 outer/global).

109

Num

ber of

flops

per me

sh po

int (x

1000

)

70

i60

J 5 outer/global

....... .... HEXAGA5 -------- HEXSLOR

50 -

3

40

30 -

20

10 -

2

I ' i

~i r i i r1.0 17

7 r1.6 18 2.0

CJ

Fig.27a No.of flops per mesh point vs. relaxation factor ufor Test Problem 5 with 3285 m.pts. and seven thermalgroups (5 outer/global).

110

Num

ber of

globa

l itera

tions

350

300 -

8 outer/global

--------- HEXAGA-------- HEXSLOR

250 -

200 —

150 -

100 -

CO

Fig.28 No.of global iterations vs. relaxation factor w forTest Problem 5 with 3285 m.pts. and seven thermalgroups (8 outer/global).

Ill

Num

ber of

flops

per me

sh po

int (xl

OO

O)

70

_] 8 outer/global

--------- HEXAGA-------- HEXSLOR

1 0 1.2 1.4 IS 1 8 2 0LV

Fig.28a No.of flops per mesh point vs. relaxation factor wfor Test Problem 5 with 3285 m. pts. and seven thermal groups (8 outer/global).

112

Num

ber of

globa

l itera

tions

350

300 -

HEXAGA

1 outer/global2 outer/global3 outer/global 5 outer/global 8 outer/global

250 H

1.22

Fig.29 No.of global iterations vs. relaxation factor w forTest Problem 5 with 3285 m.pts. and seven thermalgroups. Comparison of HEXAGA results obtained with1 inner/outer for 1,2,3,5 and 8 outer/global.

113

Num

ber of

flops

per me

sh poi

nt (xl

000)

70 —r

HEXAGA

---------------- 1 outer/global----------------- 2 outer/global----------------- 3 outer/global— — — 5 outer/global— — 8 outer/global

Fig.29a No.of flops per mesh point vs. relaxation factor wfor Test Problem 5 with 3285 m.pts. and seven thermalgroups. Comparison of HEXAGA results obtained with1 inner/outer for 1,2,3,5 and 8 outer/global.

114

Num

ber of

globa

l iter

atio

ns

350HEXSLOR

300

250

200

150

100

50

0

------------ 1 outer/global------------ 2 outer/global----------- 3 outer/global------ ---- 5 outer/global— — 8 outer/global

4

/ / / '/ / /

/ / //

//

tt-t'i | rn i | t r i i | i i i nTTr! | i i r-ryTrn i | rr 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95

CJ

Fig.30 No.of global iterations vs. relaxation factor w forTest Problem 5 with 3285 m.pts. and seven thermalgroups. Comparison of HEXSLOR results obtained with1 inner/outer for 1,2,3,5 and 8 outer/global.

2.00

115

Num

ber of

flops

per me

sh po

int (xl

OO

O)

70 -

60 -

-

HEXSLOR

1 outer/global2 outer/global3 outer/global 5 outer/global 8 outer/global

\ /

10!

I

0 —1 70

i r175 1.80

i—i—rh—r i r ~t~ |-i—i185 190

T 1 p--,----

195 2.00Cv

Fig.30a No.of flops per mesh point vs. relaxation factor wfor Test Problem 5 with 3285 m.pts. and seven thermalgroups. Comparison of HEXSLOR results obtained with1 inner/outer for 1,2,3,5 and 8 outer/global.

116

4> - o

0 8 16 72 80MESH NUMBER

mesh size h = 0.125 cm

Fig.31 The layout of Test Problem 6 [20,p. 18]

117

Num

ber of

globa

l itera

tions

400

350

1 outer/global

---------- HEXAGA---------- HEXSLOR

300 -

250 -

200 -

1.10 1.15 1.20

2'

G . 1'

x r v':;v\;

!i 'x:

\ i ' ;\ i ' !

' . 12

20

30

li\

\ I ' iX \ t

\i \ Jp \ <

' v i

50 \ i

l—p ■ “ i "~r i i |" i i _~r rL r i r185 LOO 1.95

CO

Fig.32 No.of global iterations vs. relaxation factor w forTest Problem 6 with 57600 m.pts.

T2.00

118

Num

ber of

flops

per me

sh po

int (xl

OO

O)

35

30 -

1 outer/global

----- — HEXAGA---------- HEXSLOR

0 u—i r~r—|—i—i—r1.10 1.15

t r1.20

,5030', -

i ' i <i1 i 11 i 11 i

\ \

\\2s

1

1!

Y:iiJ20(

1.90 1.95CO

Fig.32a No.of flops per mesh point vs. relaxation factor wfor Test Problem 6 with 57600 m.pts.

l2.00

119

Num

ber of

globa

l itera

tions 350

300

250

200 -

150 -

100

50

1 outer/global

HEXAGA

0 —i i i—i | r1.12 1.14

-I— [ I P" T' T'T'T -r—r-T—J-l r 1 T1.16 1.18 l 20

GO

Fig.33 No.of HEXAGA global iterations vs. relaxation factor wfor Test Problem 6 with 57600 m.pts.

120

Num

ber of

flops

per me

sh po

int fxl

OO

O)

35

1 outer/global

HEXAGA

Fig.33a No.of HEXAGA flops per mesh point vs. relaxation factor wfor Test Problem 6 with 57600 m.pts.

121

Num

ber of

flops

per me

sh po

int (xl

OO

O) 1 outer/global

HEXSLOR

20 -J

15 -

10 -

T~~ r -

Fig. 33b No.of HEXSLOR flops per mesh point vs. relaxation factor wfor Test Problem 6 with 57600 m.pts.

122

Num

ber of

globa

l itera

tions

350

HEXAGA

300

1 outer/global2 outer/global

250 -

CO

Fig.34 No.of global iterations vs. relaxation factor w forTest Problem 6 with 57600 m.pts. Comparison of HEXAGA results with 1 and 2 outer/global.

123

Num

ber of

flops

per me

sh po

int (xl

OO

O)

35

30 -

25 4

20 -

15 -

10 -

5 -

HEXAGA

1 outer/global2 outer/global

0 ---[-t-m^TTT-rrn-r'T'~t~t-7-7—\—r r : -j-rT-r-r-1.15 1.16 1.17 1.18 1 19 1.20 1.21

CO

Fig.34a No.of flops per mesh point vs. relaxation factor w for Test Problem 6 with 57600 m.pts. Comparison of HEXAGA results with 1 and 2 outer/global.

124

Eig

enva

lue m

odul

us

1.2)

//

//

/

CJ

Fig.35 A typical behaviour of eigenvalues vs. relaxation factor w.(B2 h ',2 ' ' T2S 1UT,21' II>z,III

>

X- V2 = <5 hi % < w Sf III

< M

Rr.Rc » pCJE )).

125