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Approximation of integralquantities in nuclear reactorphysicsKrzysztof Andrzejewski a & Janusz R. Mika ba Institute of Atomic Energy, Nuclear Safety AnalysisDpt. , 05-400, Otwock/Swierk, Polandb University of Natal, Mathematics Dpt. , Durban,4001, South AfricaPublished online: 01 Dec 2006.

To cite this article: Krzysztof Andrzejewski & Janusz R. Mika (1991) Approximationof integral quantities in nuclear reactor physics, Transport Theory and StatisticalPhysics, 20:2-3, 221-236, DOI: 10.1080/00411459108203903

To link to this article: http://dx.doi.org/10.1080/00411459108203903

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TRANSPORT THEORY AND STATISTICAL PHYSICS, 2 0 ( 2 & 3 ) , 221-236 (1991)

APPROXIMATION OF INTEGRAL QUANTITIES

IN NUCLEAR REACTOR PHYSICS

Krzysztof Andrzejewski

Institute of Atomic Energy

Nuclear Safety Analysis Dpt.

05-400 Otwock / Swierk

Poland

Janusz R. Mika

University of Natal

Mathematics Dpt.

Durban 4001

South Africa

ABSTRACT

A method is proposed for calculating appoximate values of those integral quantities used in nuclear reactor calculations which require a transport - theoretical formulation. A priori error estimates are given in the form of lower and upper bounds. The method is compared with the multiple collision approach. The numerical calculations of the absorption blackness in a slab shows that the present method leads to the reduction of the computing time needed to achieve a prescribed accuracy. We present the method in its one - velocity version, mainly for simplicity. It can be extended to multi - group, or even continuous energy dependent case at the cost of the considerable complication in the procedure algebra.

221

Copyright @ 1991 by Marcel Dekkcr, Inc

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222 ANDRZEJEWSKI AND MIKA

1 . INTRODUCTION

A central problem of applied mathematics is to supply solutions to functional equations. Symbolically, qiven an equation

( 1 . 1 ) Ax = f

where the function f and the operator A are specified in an appropriately chosen function space, the task is to find the function x which is an exact solution or, which is much more often the case, an approximation to x. In many applications, however, we look not for the function x but for an integral containing x i n a linear fashion. In such cases it is convenient to consider (1.1) in a Hilbert space and represent that integral as an inner product of x with a given function g. say. Thus the quantity of interest is the linear functional with respect to x 1 1 . 2 ) n = i x , g >

where < . , . > denotes the scalar product in a qiven Hilbert space. It was recoqnized quite early that the variational method is

exceptionally well suited to supply approximate values of the fuctionals of the form ( 1 . 2 ) . The field of nuclear reactor physics is no exception because many quantities of interest in nuclear reactor design and operation are represented as linear functionals. A comprehensive account of variational methods in nuclear reactor physics is given by Stacey 1 1 1 .

In the seventies Barnsley and Robinson [ 2 1 and Cole and Pack [31 introduced an approach in which approximate values of linear functionals are sought toqether with lower and upper bounds representing a priori error bounds. That approach was subsequently modified by Cole, Mika and Pack [ 4 - 6 1 who have pointed out that the crucial point in approxlmating linear functionals containinq solutions to linear equatlons of the form ( 1 . 1 ) is a proper approximation to the inverse operator. A comprehensive analysis of the new procedure was qiven by Mika [ 7 1 and by Bond and Mika [ 8 1 .

In our paper the implementation of this procedure is closely related to the classical Chebyshev approximation that ensures minimal value of the norm:

n

for qiven n.

The feasibility of the newly developed approach is demonstrated on a classical problem of absorption blackness, taken from nuclear reactor calculations. It turns out that it is an improvement of the method already proposed by Stacey [ 1 1 and by numerous authors referred to in his book. At the same time it is also strongly related to the multiple collision method. We hope that the proposed approach wlll be useful in calculatinq transport - theoretical parameters used i n nodal and diffnsion calculations.

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INTEGRAL QUANTITIES I N NUCLEAR REACTOR PHYSICS 223

2 . APPROXIMATION TO LINEAR FIJNCTIONALS

We will consider a linear operator equation

( 2 . 1 1 A x = f

in a Hilbert space H with the inner product < . . . > . We assume that A is bounded and positive definite from which it follows that A is invertible. If the operator under consideration is bounded and invertible but not self-adjoint then by operating on both sides of

(2.1) with the adjoint operator A* we obtain the new equation with

the operator A A which is positive definite. Such a procedure is often used in linear algebra since methods devised for invertinq symmetric matrices are much more efficient than those applicable for general matrices unless the matrix is ill-conditioned. Further discussion of this topic can be found at the end of paragraph 5 .

*

Let the quantity to be calculated be the inner product

( 2 . 2 ) n = < x , @

where g E H and x is an unique solution of nonhomoqeneous eq. ( 2 . 1 1 . Following the recipe suqqested by Pomraning [ 9 1 and Lewins [ l o 1 and used by Stacey 1 1 1 we can write the functional in the following equivalent form

( 2 . 3 ) 0 = < @ , $ > + < f - A * , $ > + < x - m , $ - A Y >

If i9 is an approximation to the solution of the equation ( 2 . 1 ) and Y is the approximate solution of the equation

( 2 . 4 ) AY = g

then the first two terms may be treated as an approximation to n since the last term is a small quantity of the second order.

Cole, Mika and Pack [ 4 - 6 1 proposed to 90 a step further and to write (2.2) in another equivalent form

( 2 . 5 ) 0 = I + E

( 2 . 6 ) I = <m,g> + < f - A * , * > + < V ( f - A * ) , r A W

( 2 . 7 ) E = < ( A - ' - V l ( f - A m ) , g - A Y >

The operator V is expected to approximate the inverse A-' of A . The functional I is a better approximation to n since the error

term E is of the third order of smallness. Additionally, by usinq the Schwarz inequality we obtain

(2.8) I E l 5 F : = I I A - ' - V I I . I l f - A Q I I . I l ~ - A Y l l

that allows to write the inequalities

( 2 . 9 1 I - F S 0 i I + F which qive a priori lower and upper bounds for the approximate functional 1 .

From ( 2 . 8 ) it is seen that quality of approximation depends on how well the functions 5 and P and the operator V are chosen. In this paper we will take simply 5 = \v = 0 and show that required accuracy of 1 may be achieved by the choice of an appropriate operator V .

where

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ANDRZEJEWSKI AND MIKA 224

The question of optimizing m and '# from a given class of simple functions will be discussed in section 6.

3. APPROXIMATE INVERSE

Since A is assumed to be a positive definite bounded linear operator i n a Hilbert space it can be represented by the spectral integral

( 3 . 1 ) A = J A d E ( A ) ; O < m < M < m m

where E 0 .1 is the family of projection operators related to the

spectrum of the operator A but the detailed structure of the spectrum is not important here.

The inverse A-' of .A can be written as

M

M

(3.2)

As an operator V approsimatinq A-' we will take a polynomi31 P ( A ! of degree n ? 0. Then

n

This shows that for self-adjoint operators which admit the spectral representation (3.1) the problem of aDproximatinq the inverse by a polynomial P ( A ! in t h e uniform n o r m is equivalent to

the problem of the uniform approximation of the function - by a

polynomial P ( A ! on the interval [ m , M 1 .

n 1 h

n The latter is a classical 1:hebyshev approximation problem which

has a unique solution for an arbitrary continuous function of h that may be obtained to any qiven accuracy by 'one of the exlstinq numeri.ca1 alqorithms. In our case there is a closed form solution found oriainally by Chebyshov. This solution can be expanded i n ternis of Chebyshev polynomials as it was shown i n [ 7 , 9 1 , so that polynorn~e.1 o f best approximation ,P is qiven i n the form

1 1 1 p ( h ) = - I- + -1 0 2 m M (3.4)

where

( 3 . 5 )

The optimal error of approximation as defined by (3.3) with

p repiaced by Pn is qiven by n

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INTEGRAL QUANTITIES IN NUCLEAR REACTOR PHYSICS 225

( 3 . 6 )

We employ the followinq strategy. Given an operator A we find (or estimate) the values m and M and fix n such that

( 3 . 7 )

bas a prescribed value. Then we find by iteration the polynomial P ( A ) from (3.4) and use it to calculate the approximate functional

n

( 3 . 8 ) I = CP t A ) f . g > n n

which represents wlth the error not greater then F from (3.7) n

In this section for the estimation of the norm IIA-‘- “1 an approach has been used based on spectral representation of self-adjoint operators. Howewer, other techniques can be used for that purpose based directly on perturbation theory for bounded operators 112,131.

4 . NEUTRON TRANSPORT THEORY APPLICATION

To demonstrate the feasibility of the proposed approach we consider a classical problem of the absorption blackness. This and similar quantities are needed in nodal or diffusion calculations but have t o be evaluated by transport theoretical methods.

We consider a uniform infinite slab of thickness 2a measured i n mean free paths with the scattering ratio c < 1 . The slab is irradiated from the left by a cosine current of unit intensity. The absorption blackness T is defined as the total number of neutrons absorbed in the slab. Hence it is expressed a s the integral of the neutron flux multiplied by 1 - c . Thus it can be treated as a linear functional.

Assuming the isotropic scattering we can describe the above phyfical situation by the integral equation

( 4 . 1 )

where

( 4 . 2 )

According to the definition of the blackness we assume the isotropic anqular distribution of neutrons entering the slab at t = -a.

The volume source f of neutrons in (4.1) is then calculated by integration and gives

(4.3) f(t) = E ia+t)

The source is normalized so that the total current of neutrons etnter ing the s!ak per cnit surface is 1.

We will consider t.he equation 1 4 . i ) in the Hilbert space of 5quzre siimmable functions over the intervsl [-a,a! with the inner prodnct

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226 A N D R Z E J E W S K I A N D M I K A

a

( 4 . 4 ) <x,y> = Jx(tly(t)dt; x,y E H

where the bar denotes the complex conjuqation. -a

Introducinq operators a

( 4 . 5 ) A = I - K ;

we can write ( 4 . 1 ) in the operational form

(4.6) Ax = f

(Kxi(t) = +sE ( It-s I)x(s)ds; -a

The kernel of the inteqral operator K is symmetric and has a square-summable singularity at t = s . Thus K is a bounded self- adjoint operator. To estimate the norm [IK)) of K we calculate the integral

(4.7) fE,( It-sIIds = iE*lt-s)ds + fE,(s-tids -a -a t

t dE2 a dE2

1: = J -.&t-s)ds - J -(s-t)ds = E (t-s) ds -a

= 2 - E (a+t) - E (a-t)

where we have used the recurrence relation

n = 1 , 2 , . . . . . . .

From ( 4 . 3 ) it follows that

( 4 . 8 ) 1i"l 5 7 : = C(L-E (a))

The operator K is semi-definite positive, that is, (4.9) <Kx,x> ? 0; X E H

To prove the above inequality we consider the Hilbert space of functions square summable over the interval (-a,,m) with the inner product

a, ( 4 . 1 0 ) <x,y>(*, = s x(t)y(t)dt; X,y E H

a, -a,

and the operator K defined by a,

a!

( K x)(t) = s E l ( It-sl)x(s)ds a!

-03

The operator K is bounded and 11K11 5 c, which follows from ( 4 . 8 1 m

We define the Fourier transform operator L in H by a,

l a , s x ( s le-Lstds (4.11) (Lx)(t) = -co

which is a unitary operator such that L-,= L' We observe that

1 2 (Lz)(t) = arctq t ? 0 ; z(t) = E (Itl)

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INTEGRAL QUANTITIES IN NUCLEAR REACTOR PHYSICS 221

rising the convolution theorem and the above inequality we have for any x E Hm

(4.12) < K x , x > = ( L - ' L ( K x),x> = < L ( K X ) , L ~ > = C < L ~ * L ~ , L X > 0 3 m m m 03 m 2 a,

1 m

= & arctq t ILx(t) I z dt 2 0 -m

The inequality ( 4 . 9 1 follows from (4.12) since H is isomorphic to a subspace of H containinq functions vanishinq outside the

interval [-a,al.

( 4 . 5 1 , the inequalities

m

Using (4.8) and 14.9) we can write for A , as defined by

which show that m = 1-y and M = 1 . These values substituted in (3.5) give

(4.14) 1 - F 1 + f l - y 6 =

The blackness is given as the inner product < x , g > where g(t)

(4.15) l l f l l : = 0 =

then from ( 3 . 7 1 , (4.3) and

(4.16) F = - n 2(1-y )

-

1"' [ $(Ez(a+tl 12dt

4.14) we obtain

6" = J"'* yw( 1-c)

2 1-Y

which represent.s the error if r is approximated by a

The present approach may be compared with the multiple collision method in which the quantity of interest is calculated by taking into account neutrons that have underqone a finite number of collisions. This means that the solution to the integral equation

x - K x = f is approximated by a truncated Neumann series

+ K"f in) - x - f + Kf + . . . . . or, in other words, the inverse

A-1 = ( I - K ) -I

is approximated by the operator

W ( A ) = I + K + . . . . . . . + Kn n

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226 ANDRZEJEWSKI AND MIKA

S u b s t i t o t i n q h' (A) f o r P (A) i n (4.17) we o b t a i n a f u n c t l o n a l n n

w e have

( 4 . 1 3 )

Comparinq ( 4 . 1 6 ) and (4.19) w e see t h a t

which 1s less t h e n I / : f o r a l l 0 < y < 1 . W e w i l l d e m o n s t r a t e t h e a c t u a l numer ica l r e s u l t s i n t h e next s e c t i o n .

5 . NiJMERICAL RESIJLTS

To c a l c u l a t e t h e v a l u e s of I g iven by ( 4 . 1 7 1 and g iven by ( 4 . 1 8 1 we need t o c a l c u l a t e t h e q u a n t i t i e s S d e f i n e d r e c u r s i v e l y

a s f o l l o w s

1 5 . 1 ) S ( t ) = f ( t ) = E ( a t t ) ;

I1

0 a

S ( t l = ( K S )!t) = f E ( I t - s l ) S ( s ) d s ; n = 1 , 2 , . . . . . n d n 2 . 1 1,

-a and t h e n R g iven by

n

With t h i s i i o t a t ion Q can be w r i t t e n as ri

+ Rn ( 5 . 3 ) Q = R + R + ' " . '

and I as

( 5 . 4 1 = a R + 2. R + . . . . . + a R

n o 1

I n o o 1 % n n

d h e r e t h e c o e f f i c i e n t s a a r e c a l c u l a t e d from ( 3 . 4 ) . I n t h e p r e s e n t c a s e w e have

n

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INTEGRAL QUANTITIES I N NUCLEAR REACTOR PHYSICS 229

which for the first three values of n gives

1-y /2 I = - R,; 0 1-y

1 2 6 4 6 I = - i -

4 6 ( 1 - p ) R t + - 4 6 Y

I =

Similarly we calculate higher order approximations. The calculations of the integrals in ( 5 . 1 ) and (5.2) were

performed with the use of integration subroutine QNC3 and spline interpolation subroutines SPLINT and SPLIFT of Sandia Laboratories. The singularity of the function E was treated as suggested by

Stamm'ler and Abbate [111. The error of integration was estimated

to be less than 5.10-6. The results are collected in Tables 1-3 which for various

values of c and a give the approximate functionals I n and Qn

together with the error estimates. better

approximation than Q , particularly for large values of a and c.

The error estimates F and G are quite conservative since they are

obtained with the use of Schwarz inequality. The actual results are much more accurate as can be seen from the tables which contain the values of G' up to n = 7 and of IF, up to n = 4.

It is seen from the tables that I n gives always a much

n

n

For small values of c, I gives already an exact value (within

the error of integration). For the multiple collision method we need to take Q to achieve the same accuracy as can be seen from

Table 1 which contains the results concerned with c = 0.1 For higher values of c the accuracy drops down and to achieve

the accuracy of the order of lo-' we would have to take of n > 4.

On the other hand, the cross-sections used in nuclear reactor design are not exact and the accuracy of calculation of 3-4 percent is generally accepted. The inspection of the tables shows that from such a point of view I can be always accepted and, for c not too

large, even I gives reasonable results.

The most significant results are obtained for c = 0.9 in Table 3. Although such value of c would not likely be of practical interest, it shows very clearly the advantage of the proposed approach over the multiple collision method. For instance for c = 0.9 and a = 2 the latter converges extremely slowly and for low values of n gives absurd results. On the other hand, since I = 0.223952, F = 0.013619

we have

2

values

0.210333 5 r s 0.237571

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230 ANDRZEJEWSKI AND MIKA

Table 1 Converqence of the blackness calculation f o r c = 0.1

c = 0.1 a = - 5

n Qn Gn In

0 , 3 5 1 2 7 7 , 0 2 8 8 1 8 , 3 6 3 9 5 8 1 , 3 7 2 3 2 2 . 0 0 1 9 4 @ , 3 7 3 6 2 1 2 , 3 7 3 6 1 4 . 0 0 0 1 3 I , 3 7 3 6 9 9 3 , 3 7 3 6 9 3 . 0 0 0 0 0 9 , 3 7 3 6 9 9 4 , 3 7 3 6 9 8 . 0 0 0 0 0 1 . 3 7 3 6 9 9 5 . 3 7 3 6 9 9 . 0 0 0 0 0 0

c = 0.1 a = 1 . 0

n Qn Gn In

0 , 4 2 2 8 8 0 . 0 5 3 5 0 9 , 4 4 2 5 6 0 1 , 4 5 3 7 1 2 . 0 0 4 5 5 6 , 4 5 6 1 4 4 2 . 4 5 6 0 6 1 . 0 0 0 3 8 8 . 4 5 6 2 5 9 3 . 4 5 6 2 4 3 . 0 0 0 0 3 3 , 4 5 6 2 5 9 4 , 4 5 6 2 5 7 . 0 0 0 0 0 3 , 4 5 6 2 5 9 5 . 4 5 6 2 5 9 . o o o o o o

c = 0.1 a = 2.0

n Qn Gn I n

0 , 4 4 7 5 1 5 , 0 8 6 7 0 1 , 4 7 1 3 4 4 1 . 4 5 2 7 4 3 . 0 0 8 3 4 5 , 4 8 5 8 9 2 2 , 4 8 5 7 0 4 . 0 0 0 8 0 3 , 4 5 5 9 8 5 3 , 4 8 5 9 6 1 . 0 0 0 0 7 7 . 4 8 5 9 8 5 4 . 4 8 5 9 8 3 . 0 0 0 0 0 7 , 4 8 5 9 8 5 5 , 4 8 5 9 8 5 . 0 0 0 0 0 1 7 , 4 8 5 9 8 5 . o o o o o o

Fn

, 0 1 4 4 0 9 .00@251 . 0 0 0 0 0 4 I O @ O @ O @ . o o o o o o I 000000

- - - - - - - - -

Fn

, 0 2 6 7 5 4 , 0 0 0 5 9 5 . 0 0 0 0 1 3 . o o o o o o . 0 0 0 0 0 0 . O O O O O @

- - - - - - - - -

Fn

. 0 4 3 3 5 0

. O O 1097

. 0 0 0 0 2 8

. 0 0 0 0 0 1

. o o o o o o

. o o o o o o

.000000

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INTEGRAL QUANTITIES IN NUCLEAR REACTOR PHYSICS 231

Table 2 Converqence of t h e blackness c a l c u l a t i o n for c = 0.5

c = 0.5 a = . 5

n Qn Gn

. 195154 ,253612 .271546 .277082 ,278794 ,279324 ,279488 ,279539

. - - - - - - - - - - -

. 112556 ,037895 .0 12758 ,004295 ,001446 ,000481 ,000164 .000055

c = 0 . 5 a = 1 . 0

n Qn Gn

,234933 ,320577 ,353209 ,365844 ,370767 ,372691 ,373443 .37 373%

,236795 I 100816 ,042923 ,018274 ,007780 ,003313 .001410 .000600

In Fn

244681 .056278 278219 ,005755 279678 ,000589 279601 .000060 279561 ,000006

.000001

.000000

.000000

In Fn

322024 . 118397 372074 .016314 374260 .002248 37408 1 .000310 373957 .000043

,000006 .000001 .000000

c = 0 . 5 a = 2 . 0

n Qn G n In Fn

0 .2486 19 1 ,346475 2 ,381598 3 .405421 -! ,413214 5 ,416167 6 ,418329 7 ,419030

_ _ _ _ _ _ _ _ _ _ _ _ _ _ - - _ _

,419563 ,201908 ,097165 ,046759 .022502 ,010829 ,005211 ,002508

,363935 ,209782 ,418499 ,034114 ,419159 ,005548 ,419146 .000902 ,419655 ,000147

.000024

.000004

.000001

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232 A N U R Z E J E W S K I AND MIKA

T ? b l e 3 C o n v e r q e n c e of the b l a c k n e s s calculstion f3r c = 0.9

n Qn

0 , 0 3 9 0 3 1 I . @ 6 0 @ 7 6 2 , 0 7 1 6 9 7 3 , 0 7 8 1 5 4

5 , 0 8 3 7 5 1 6 ,084861

4 . 0 8 1 7 4 Si

7 , 0 8 5 4 8 8

c = 0 . 9 a = 1 . 0

n Qn

0 , 0 4 6 9 8 7 1 , 0 7 7 8 1 8 2 , 0 9 8 9 6 4 3 . 1 1 3 7 0 1 4 . 1 2 4 0 3 8 5 . 1 3 1 3 0 7 6 . 1 3 6 4 2 5 7 . 1 4 0 0 3 1

c = 0 . 9 a = 2 . 0

n Qn

0 , 0 4 9 7 2 4

2 , 1 1 1 5 9 9 3 . 1 3 2 3 8 8

5 . 1 6 2 0 7 7 6 . 1 7 2 7 0 3 7 . 1 8 1 2 8 7

1 , 0 8 4 9 5 2

4 , 1 4 8 8 7 6

G n

.068221 , 0 4 1 3 4 4 . 0 2 5 0 5 5 , 0 1 5 1 8 4 .@092@2 . 0 0 5 5 7 6 0 0 3 3 7 9

. 0 0 2 @ 4 8

I n Fn

, 0 6 9 1 3 4 9 , 0 3 4 1 1 1 , 0 8 5 5 8 1 , 0 0 7 8 0 3 .086ae: ,001795 ,086575 .0004@8 .@em6349 . @ 0 0 0 9 3

. 0 0 0 0 2 1

.000005

. 00@00l

tn

2 0 9 5 1 5 1 6 0 5 6 3 1 2 3 0 4 8 0 9 4 2 9 8 0 7 2 2 6 6 055381 0 4 2 4 4 2 0 3 2 5 2 5

I n Fn

. 1 2 4 0 4 4 . 1 0 4 7 5 8 , 1 5 2 1 0 8 , 0 3 6 4 8 5 , 1 5 2 9 7 3 . 0 1 2 7 0 7

1 5 1 4 0 7 , 0 0 4 4 2 6 . 1 4 9 6 8 8 , 0 0 1 5 4 1

. 0OOC.37 , 0 0 0 1 8 7 . 0 0 0 @ 6 5

tn

5 8 5 7 0 3 5 0 7 3 4 7 4 3 9 4 7 4 3 8 0 6 8 0 3 2 9 7 5 3 2 8 5 6 3 8 2 4 7 4 2 5 2 1 4 3 2 4

In fn

2 1 0 7 0 2 , 2 9 2 8 5 2 2 3 8 2 9 5 . ! 3 5 9 9 6 2 2 8 9 7 7 , 0 6 3 1 5 4 2 2 7 0 4 1 , 0 2 9 3 2 8 2 2 3 9 5 2 , 0 1 3 6 1 9

, 0 0 6 3 2 5 . @ 0 2 9 3 7 , 0 0 1 3 6 4

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INTEGRAL QUANTITIES IN NUCLEAR REACTOR PHYSICS 233

which shows that I = 0.270702 gives I- with an error not greater

than 0 , 0 2 6 8 6 9 . It is amazinq to observe that such an excellent result was obtaine? by takinq into account only the virgin neutrons and totally neglecting scattering.

The authors feel that the present method might be of practical value in nuclear reactor calculations provided it is extended to other than plane geometries (particularly to the cylindrical geometry) and made to include the enerqy dependence.

If the integral equation is replaced by, for instance, the system of multiqroup integral equations, the resulting operator is not self-adjoint due to the slowinq down. In a case like that we have to adopt the following procedure, already mentioned in the beginninq of paragraph 2: If the original equation with a non-selfadjoint operator is

0

A x = f ( 1 . 1 ) *

we operate on both sides with the adjoint operator A to obtain the new equation

where

is a self-adjoint operator and A = A* . Such a procedure is often used in practical calculations but is not advisable if A is ill - conditioned, which in our case would mean that y , as defined by ( 4 . 8 ) , is close to one. Thus, for practical applications we have in mind, the operator A will be not ill-conditioned and the proposed

B x = A

B = A A *

procedure can be used in energy dependent cases. It should be also noted, that the present

proposed method is not applicable for the eigenva the operator A in eq. ( 1 . 1 )

6 . OPTIMIZATION OF 5 AND 'JJ

I n the last section we have shown that on

version of the ue calculation of

can achieve a reasonable accuracy using low order approximations to A-' which require a limited numerical effort for integration. From (2.7) it is seen, however, that the error is reduced when instead of @ = Q = 0 one takes some appropriate approximation.

We choose a simplest possible case @ = const and Q = const. To optimize the error we choose and rY such that

l i f -@Aul l = min

( 6 . 1 ) 11&rYAull = min

where u(t) f 1 for -a 5 t 5 a. From the definition of the norm we have

(6.2)

whence

Ilf-mAuII" = [IAu(IZ@" - 2 < f , Au>m + I I f I I "

( 6 . 3 )

In a similar fashion we obtain < 6, Au>

( 6 . 4 ) ' J J =

IIAUII' opt

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A N U R Z E J E W S K I A N D M I K A 234

The improved zero order approximation to 0 as defined by (2.6) is yiven by

(6.5) lo = I CtC,g) + 11' < f - $ P.U,-> + 11 i: K-$optAu,g-'!, A'u!> I J opt opt opt opt

1 - y / 2 where @t) = (l-c)u(t) for -a 5 t 5 a and yo= - The improved error estimate is obtained from (2.8) in the form

To implement the modified formulae ( 6 . 5 , and (6.6) we need the additional integrals. From ( 4 . 5 ) and (4.7 I we 0btai.n

(Au)!t) = ( I - c ) + - E fast: + E la--t) : = h!t) " [ 2 2 1 from wkich we nave

a <Au.u: = h(t) dt = Za!l-ci + - - cE 12al d

-a Fu 1- the r

< j , A u \ = Sa/it)hlt)dt = ( 1 - c ) fE (a+t)dt t & c p , < a ~ - c ; L I dt + ? 2

--u --u -a

where w was defined i n ( 4 . 1 5 ) we see frow tae above formula- +

the 3 n l y additional ~rtegra: which has ti be evalcatec n u m e IS!

U

Ez(a+t)E (2-t Id.. -a

Thus we see that the new values I ?,re obtained in 3 r e ; 0

inexaensive manne:. However, they give a larqe improverne>t of the resuits, as car, be seen fror;. Table 4 .

For reasonable values s f c and a I 1 ; systemst:cal!y b e t t e r

than ! bct possibly, worse thar ? . Whei! c and/or a %?comes l a r q e

we observe the loss of a c c ~ i r a c y i r . i when is due ta the fect- that

i is obtained as a swn of a number of negative and positive ?:erms.

This is particularly well seen for ~ 0 . 9 and a=2.0. Ir that case

2.

0

0

2 = 0 . 2 1 0 7 0 2 and I = I + 1 0 . 1 1 1 0 8 - 1 0 . 1 1 2 4 5 2 = 0 . 2 1 0 7 0 2 - 3 . 0 0 1 3 7

=0.209332. Thus I is superior to I . 0 0

0 0

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INTEGRAL QUANTITIES IN NUCLEAR REACTOR PHYSICS 235

Table 4 Comparison of l o and In

n I n F n In Fn

c = 0.1 0 3 6 3 9 5 8 , 0 1 4 4 0 9 , 3 7 3 6 9 2 . 0 0 0 0 4 7 a = 0 . 5 1 . 3 7 . 3 6 2 1 ,00025 1

. 3 7 3 6 9 9 . 0 0 @ 0 0 4 - - - - - - - - - - - - - - - - .. .- - - - - - .- - - - - - - - .- - - - _. - - - - - - - - - - - - - - -

c = 0 . 1 0 - 4 4 2 5 6 0 . I326754 . 4 5 6 2 3 4 ,000 1 8 0 a = 1 . 0 1 4 5 6 1 4 4 . GOO595

2 , 4 5 6 2 5 9 . 0 0 0 0 1 3

c = 0 . 1 0 , 4 7 1 3 4 4 , 0 4 3 3 5 0 , 4 8 5 3 6 5 0 0 0 4 1 4 a = 2 0 1 . 4 8 5 8 9 2 . 0 0 1 0 9 7

2 , 4 8 5 9 8 5 . 0 0 0 0 2 8

c = 0 . 5 0 , 2 4 4 6 8 1 ,056278 , 2 7 9 4 0 1 . 0 0 1 2 @ 6 a = 0 . 5 1 . 2 7 8 2 1 9 . 0 0 5 7 5 5

2 2 7 9 6 7 8 . GO0589

c = 0 . 5 0 , 3 2 2 0 2 4 . 1 1 8 3 9 7 , 3 7 2 9 7 6 , 0 0 5 7 6 4 a = 1 . D I , 3 7 2 0 7 4 , 0 1 6 3 1 4

2 , 3 7 4 2 6 0 , 0 0 2 2 4 8

c = 0 . 5 G , 3 6 3 9 3 5 , 2 0 9 7 8 2 , 4 1 7 3 0 2 . 0 1 5 7 1 8 a = 2 . 0 1 , 4 1 8 4 9 9 , 0 3 4 1 1 4

2 , 4 1 9 7 5 9 . 0 0 5 5 4 8

c = 0 . 9 0 , 0 6 9 0 4 9 , 0 3 4 1 1 1 , 0 8 6 1 7 8 , 0 0 1 9 6 6 a = 0 . 5 1 . 0 8 5 5 8 1 , 0 0 7 8 0 3

2 ,086882 , 0 0 1 7 8 5

c = 0 . 9 0 . 1 2 4 0 4 4 , 1 0 4 7 5 8 , 1 4 7 3 1 2 . 0 1 6 6 6 8 a = 1 . 0 1 . 1 5 2 1 0 8 , 0 3 6 4 8 5

2 , 1 5 2 9 7 3 . 012107 ____-_________-_--_-____________________------------

c = 0 . 9 0 . 2 1 0 7 0 2 , 2 9 2 8 5 2 , 2 0 9 2 6 8 , 0 8 6 1 4 0 a = 2 . 0 1 , 2 3 8 2 9 5 . 1 3 5 9 9 6

2 , 2 2 8 9 7 7 , 0 6 3 1 5 4

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236 ANDRZEJEWSKI AND M I K A

If the modified first-fliqht method qivinq I is not

sufficiently accurate we may go to I with n = 1,2, . . . . and/or take more and more sophisticated approximations to @ and @. It would be difficult to qive qeneral recommendations and the strategy should depend ona particular problem considered. However, for large values of c and a one has to be careful1 with approximate functionals containing large negative and positive terms cancelling each other.

7.REFERENCES

0

n

1. W.M. Stacey: "Varational methods in nuclear reactor physics",

2. M.F. Barnsley and P.D. Robinson: Bivariational bounds, Proc.

3. R.J. Cole and D.C. Pack: Some complementary principles for linear

Academic Press, New York 1974

Roy. Soc. London A 338 (19741, 527-533

integral equations of Fredholm type, Proc. Roy. S O C . London A 347 (1975), 239-252

4. R.J. Cole, J.R. Mika and D.C. Pack: Complementary bounds for inner products associated with non-linear equations, Proc. Roy. Soc. Edinburgh, 96A (19841, 135-142

5. J.R. Mika, D.C. Pack and R.J. Cole: Optimal bounds for bilinear forms associated with linear equations, Math. Meth. Appl. Sci. 7 (19851, 518-531

6. J.R. Mika and D.C. Pack: Approximation to inverses of normal operators, proc. Roy. SOC. Edinburqh 103A (19861, 335-345.

7 . J.R.Mika: Spectral method of approximating normal operators in Hilbert spaces, South African Mathematical Congress, Stellenbosch, 2-4 November 1987.

8. J.R. Mika and R.A.B. Bond: Method of approximate inverse for solvinq equations in Hilbert spaces (Submitted for publication)

9. G . C . Pomraning: A derivation of variational principles for inhomogeneous equations, Nucl. Sci. Eng 29 (1967). 225.

10.J. Lewins, Time-Dependent variational theory, Nucl. Sci. Eng. 31 (1968), 160.

11.R.J.J.Stamm'ler and M.J.Abbate: Methods of Steady-State Reactor Physics in Nuclear Design, Academic Press, London 1983

HPcPivPd: M d r c h 5 , 1 9 9 0 Rr,visrd: J a n u a r y 3 . 1991 Acceptpd: J a n u a r y 3 0 , 1991

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