Annals of Nuclear Eneroy, Vol. 7, pp. 297 to 306 Pergamon Press Ltd 1980. Printed in Great Britain

AN ERROR ESTIMATE FOR THE MULTIGROUP M E T H O D W I T H A P P L I C A T I O N T O S H I E L D I N G

C A L C U L A T I O N S

P A U L NELSON

Institute for Numerical Transport Theory, Department of Mathematics, Texas Tech University, Lubbock, TX 79409, U.S.A.

I. I N T R O D U C T I O N

The multigroup method is perhaps the single most widely used technique for approximately determining the energy distribution of particles in a system modelled by linear transport. There is a vast amount of practical experience in using the method, but there is a real paucity of work involving theoretical studies of the adequacy of the multigroup approximation. In fact the article of Bellini-Morante and Busoni (1972) represents the only such work of which the author is aware; however these researchers consider the initial- value problem for time-dependent transport, and con- sequently their results are not relevant to the steady- state situations of interest in the present work. In addition the cited work was done in the context of an ~l -norm, which is not appropriate for the shielding problems which motivate the current work (cf. Section 2).

The question of convergence and error estimates for the multigroup method is far from academic. For example, in deep penetration problems involving media with deep minima in cross sections, the multi- group method is known (e.g. Becker, 1975) to have potential for significant inaccuracies. Recognition of such difficulties has stimulated development of the multi-band method of Cullen (1974) [see also Niko- laev (1976) for citation of several Russian works con- cerning a related method], In any event, a theoretical study of convergence for the multigroup method would hopefully provide some insight into the ulti- mate capabilities and limitations of the method. This article is presented as an initial effort in this direction.

Probably the simplest transport situation in which the multigroup method is of practical interest is that of slab geometry. However, slab geometry frequently poses annoying pathologies associated with particles travelling nearly parallel to the slab faces. In order to avoid such pathologies, and concentrate on the energy dependence, we choose to consider a variable- energy version of the rod model which has often been employed in conjunction with invariant imbedding

(Bellman and Wing, 1975). In Section 2 we describe this model in detail, and give an existence-uniqueness theorem in the appropriate function space as sug- gested by our concern with shielding problems. The multigroup approximation is formulated in detail in Section 3, and the appropriate convergence criterion is given in Section 4. Section 5 contains a brief dis- cussion of considerations concerning the reference spectrum used in defining the multigroup approxi- mation.

Our convergence theorem for the multigroup method is based on a discrete version of the cele- brated Lax Equivalence Theorem. This result is for- mulated in Section 6, with its proof relegated to the Appendix; in this section we also state and outline a proof of the convergence theorem for the multigroup method. Sections 7 and 8 are concerned with estab- lishing, respectively, the stability and consistency results necessary to apply the Lax Equivalence Theorem. The detailed proof of the convergence theorem is given in Section 9, along with further dis- cussion of its hypotheses. In Section 10 we show that the associated error estimate suggests the maximum possible error in a given multigroup approximation is proportional to the maximum fluctuation of the total cross section over all energy groups, and present an example intended to test the reality of this conclusion. Finally, Section 11 contains a few conclusions and suggestions for further related studies.

2. THE C O N T I N U O U S - E N E R G Y ROD M O D E L

We write the governing equation in the form

~x + ~(x, v)~

-- fvk(x,v',v) (x,v')dv' + S(x,v),

subject to boundary conditions q~(0, v) = fo (v), v > 0,

(b(a,v)=L(v), v <0,

(1)

(2)

297

298 PAUL NELSON

vated this work, functional of the x = a ,

where a > 0 andf0,f~ are known. Here x e [ 0 , a] and v ~ V where the velocity space V is some interval in ( - oo , oc) with left and right endpoints denoted, re- spectively, by Vm~° and Vmax.

In reactor work one eventually would be interested in computing a quantity of the form

~fv4,(x,v)g(x,v)dvdx.

However, in the shielding applications which moti- one is more often interested in a

outcoming flux at the 'outside' wall

v+ 4,(a, v)g(v) dr, (3)

where g is given, and V+ = {v E V: v > 0} . This sug- gests use of the norm

HfH = max ( If(x,v) ldv, (4) O<_x<_a Jv

in the sense that equation (3) is a continuous linear functional relative to this norm (for bounded g).

Let 4, be a solution of the boundary-value problem (1), (2) which is continuously differentiable in x. Then 4, satisfies the integral equation

4, = T4, + w. (5)

Here the integral operator T is defined by

Tf(x, v ) = s g n ( v ) F e x p ( - s g n t v ) f ; a ( s , v ) d s )

x fv k(x', v', v)f(x', v') dr' dx', (6)

where

= ~ ( v ) = ~'0, v > 0 , a, V < 0,

and the known function w is given by

= sgn(v) f~' exp(--sgn(v) W(X~ V)

fx ~ a(s, v) ds)S(x', v) dx' + f~(v) ×

exp(-sgn(v) f f a(s, v) ds). (7) ×

We denote by C1 the set of real-valued functions f on [0, a] x V such that equation (4) is finite and the mapping x---~f(x,.) is continuous from [0, a] to L,0x(V). We assume the cross section cr is bounded and Lebesgue measurable on [0, a] × E and satisfies the

condition that the mapping x --~ a(x, v) is continuous, uniformly in (x, v)~ [0, a] x V. From these assump-

tions it follows that w ~ C1 and

Ilwll -< IS(x, v)[ dvdx + v)[ dv ,

provided that each of the integrals on the right is finite. Thus it is reasonable to consider equation (5) as an operator equation in C~.

We assume that k (=differential cross section) is bounded and Lebesgue measurable on [0, a] x V z, and that the inequality

{~2fvk(X,V' ,v)dv:(x ,v ' )~[O,a]xV} sup , )

= c < 1. ~8)

holds. Therefore there exists a constant c' such that

vk(X,V',v)dv' <~ c' < vc

for all (x, v) E [0, a] × V. (9)

we further assume for given • > 0 there exists 6 > 0 such that Ix - Xo[ < 6 implies the inequality

k(x, v', v) - k(xo, v', v) < • (lo)

holds for all v, v' e V. Condition (8) is the requirement that the underlying

medium be submuhiplying. The other conditions are basically of a technical mathematical nature, and are satisfied by any reasonable data, except for the re- quirement of continuity in x. The latter can be weak- ened to require only piecewise continuity in x, but we shall not include this generality in order to avoid the attendant notational complexities.

Lemma 1 For arbitrary g E C1 and positive integer n, let

W.(g) = o<_x<af~'"f[fv'" fv H[V(x - X1)''"

× H [ v . - l ( x , - x - x.)] exp(-sgn(vO

× f i ' a(S, Vl)dS)...exp(-son(vn_1)

x f / i - ' ~(s, v,_,) d~). k(x,, v , v)...

× k(x., Vn, v.- t ) [g(x. , v . ) fdv . . . .

× dv I dv d x n . . , dXl,

An error estimate for the multigroup method 299

where H (y) = 0 or 1 according, respectively, as y < 0 or y >_ O. Then W,(g) _< c"a,,~allgll.

Proof

For n = 1 the desired result follows by carrying out the integration in the order dv dvldX~, with appro- priate use of equation (8) and other more-or-less obvious estimates. (In this case the exponential factors are nonexistent, and therefore are to be replaced by unity as is customary for empty products.) For n > 1 we use equation (8) to estimate

C( + fv ) { S: I: £ • f H[vI(x, - x2)] . . . 3v

follows from the result of Lemma 1 that II1"011-< C"~maxallgll- As a consequence T is a bounded linear operator on C1 into itself, with spec- tral radius not greater than c < 1. The fact that equa- tion (5) has a unique solution t~ ~ Cx now follows from standard theorems of functional analysis.

Now i f f c C~, then the boundedness of k and the conditions involving equations (9) and (10) imply that

fvktX, v)f(x, v')dr' t ,

is continuous in x, for arbitrary fixed v ~ E It follows that TO can be differentiated in x by Leibnitz' rule, and that the x-derivative is continuous in x. It then follows by elementary algebra that 4, satisfies equa- tions (1) and (2). This completes the proof of Theorem 1.

H [v, _ 1 (Xn - 1 - - Xn)]

x e x p ( - s g n ( v l ) f ~ ' a(s, v t )ds) . . .

•.. e x p ( - s g n ( v . - 0 f ; ° ' 0"(S, Vn- 1) ds )

X O'(X1, V l ) . k ( x 2 , v2, u 1 ) . . , k ( x n , Vn, V n_ 1)

× Ig(x,,, v.) ldv, . . , dvn- 2 dx . . . . dxl} dr' 1 . %

If we now evaluate the integral with respect to xt , easy estimates yield the inequality W.(g) <_ el'V._ l(g). This yields an inductive proof of the desired conclu- sion.

Theorem 1

Under the above assumptions, if w ~ C1, then there exists a unique continuously differentiable (in x) solu- tion c~ ~ C1 of the boundary-value problem (1), (2).

Proof

First we show that T maps Ct into itself, then we show that the integral equation (5) has a unique solu- tion 4~ in Ct, and finally we demonstrate that this q~ is actually a continuously differentiable solution of (1), (2). I f fE C~, then it follows from equation (8) and the existence of an upper bound for a that I[TfLI is finite, with the norm defined by equation (4). Furthermore continuity of x--~Tf(x, .) from [0, a] to 5°1(V) fol- lows from the boundedness of a, and the condition involving equation (9). Thus T maps C1 into itself.

For arbitrary g cC1, it is readily seen that LI'I"g [[ -< W,(g), where 14I,(0) is as in Lemma 1. It

3. T H E M U L T 1 G R O U P A P P R O X I M A T I O N

Let th be a solution of the boundary-value problem (1), (2) for the continuous-energy rod model. Let J ~ {1)mi n = V 0 ~ V 1 <( V 2 ~ . . . ~ V G - I ~ V G ~--- Vmax}

be a partition of V. We generically denote by 19 = (v 9_t,vg), 1 _< g _< G, an interval in J . If we integrate equation (1) over each interval 19 ~ J , the result can be written

0-7 + ~g(x)4;. = ~gAx),~Ax) + &(x), j = l

a = ~,2 . . . . . G, i / l )

where

= [ . ~(x,t,)dv, g = 1,2 . . . . . G, (12) o

ag(x) = £ , a(x, v)~b(x, v) dv/~9(x ),

g = 1,2 . . . . . G, (13)

~gJ(x) = f,o f , , k(x,v', v)d)(x, v')dr' dv/c~s(x),

g , j = 1,2 . . . . . G, (14)

and

Sg(x) = I. S(x,v)dv, g = 1,2 . . . . . G. (15) o

The multigroup equations are defined by equations (11)-(15) except with the exact solution $ replaced by some a priori known reference spectrum q5 o = 4o (x, v). We indicate the corresponding quantities in a nota-

A.N.E. 7/4-5 H

300 PAUL NELSON

tion similar to the above, except with ' ~ ' replaced by ' - ' . Thus the multigroup equations are

&5 ° G -~ + e,(x)~, = y. k,j(x)~j(x) + s,(x).

j = l

g = 1,2 . . . . . G, (16)

where

2, f ~o(x) a(x, v)4'o(X, v)dv/ = 4'o(X, v) dr,

9 = 1,2 . . . . . G, (17)

~.j(x)=£°f , kCvv',v)4'o(x,,,')d~,'

x d v / ~ 4'o(X,v)dv, g , j = l , 2 , . ,G (18) ,; l j

and S+(x) is still given by equation (15).

Clearly the ~i satisfy the boundary conditions

~+(a) = (_ f~(v) dv, v o~< 0 a / u

~0(0) = (. fo(v) dv, v+_, >_0. (19) , a /

These equations do not specify a boundary condition for any 9 such that O~(vo_t,vo), and indeed no boundary value is a priori known for any such ~o. In order to avoid this difficulty we henceforth assume that all partitions J to be considered are such that v = 0 is a boundary point of one of the subintervals.

Equations (6) along with the boundary conditions

t " ~°(a) = t . f~(v) dv, v o<_0,

g

~°(0) = f. fo(v) dv, v . - i _> 0, (20)

define a two-point boundary-value problem in G dependent variables. The existence question for this boundary-value problem will be settled as a side result of other matters to be studied in the ensuing pages.

4. CONVERGENCE CRITERIA

Consider a sequence {J.} of partitions of V such as described in the preceding section. Each such parti- tion consists of G. subintervals of V, indexed as l.,g, g = 1, 2 . . . . . G.. For each value of n we define a pro- jection operator P. : C1 ---* C([0, a] )~" ( : G.-tuples of

continuous real-valued functions on the interval [0, a]) by

P . f (x) =

( f , f ( x , v ) d v . . . . . fj, ~ f {x , v )dv ) . (21) n i . . .

We shall always consider C([0, a]) G- endowed with the norm

Gn

[r(fl,f2 . . . . . fG.)]] = max Y" IL(x)l. (22) 0 < ~ c . < a g_~=l

A sequence If . l such that each f. ~ C([0, a]) ~- is said to converge discretely to f ~ C1, if

lim [rf. - p.f][ = 0. n ~ z o

Let qS. = (~.,1,~.,z . . . . . qS.,~.) be the solution of the two-point boundary-value problem (6) and (20) corre- sponding to the partition J . . Our basic objective in the remainder of this paper is to find conditions under which ~. exists, at least for sufficiently large n, and I~. ~ converges discretely to the solution 4' of the continuous energy problem (1) and (2).

The basic motivation for interest in the discrete convergence described in the preceding paragraph is as follows. It is natural to use the approximation

fv ° " h(v)4'(x,v)dv ~ ~ h(vo)~.,~(x) (23) i f - 1

to compute the quantities on the left in terms of the nth multigroup approximation, where v 0 e I..+. Sup- pose [4'.I converges discretely to 4' in the sense defined above, and let

~o. ,g(h)=max{[h(u)-h(v)[ : u, vel . ,gl (24)

be the fluctuation of h over I., o. If h and the sequence {~¢.} of partitions are such that

max{]o.,0(h)l: g = 1,2 . . . . . G.}---~0asn----~w_,

then the quantities on the right-hand side of equation (23) converge to the left-hand side as n ~ ~ . In par- ticular this holds if Vis compact, h is continuous, and the partition norms

llu~.,,ll = max{Iv.,,+1 - v.,+[: g = 0, 1 . . . . . G.-1} (25)

converge to zero as n ~ * .

s. REVERENCE SPECTRA

The reference spectrum 4'o = 4'o(X, v) in equations (17) and (18) is selected by various forms of artistry.

An error estimate for the multigroup method 301

Probably the most common is as the solution of some integral equation of the form

[B 2 + ~(v)]q~o(V) = fv k(v', v)4)0(v')dv' + S(v),

where ~r, k and S have been assumed constant (or piecewise constant) in x. The selection of the buckling constant B 2 is also largely a matter of art. A slight modification of this approach would lead to taking q5 o as the solution of

[B 2 + or(v)] 4~o(V) = fv ks(v', v)q~o(V') dr '

+ -k~. kr(v, v')Oo(V') dr',

where ks and kv are, respectively, the differential cross sections for scattering and fission, The idea is to find the largest eigenvalue kcft for which this equation has a nontrivial solution, and to take the associated eigen- function as the reference spectrum.

The preceding are only some of the ideas which have been used to generate reference spectra. See Bondarenko (1964, esp. section 1.3), Clark and Han- sen (1964, esp. p. 201) and Bell and Glasstone (1970, esp. section 4.5) for other possibilities. All of these methods have the property that the reference spec- trum is constant over a fixed material region. This can be a large source of error, because the velocity depen- dence of the actual flux can vary very significantly with location. The multiband method can be viewed as one method of building such spatial dependence into the multigroup cross sections (Cullen, 1975).

Because of the wide variation in methods for determining a reference spectrum, it is desirable that a study of convergence make minimal assumptions on this quantity. We shall strive toward this objective.

6. STATEMENT OF THE

CONVEaCENCE TnEOnEM

Let [,¢,} be a sequence of partitions of ,V, such as introduced in Section 4. For each n let the quantities f~, and F, be defined, respectively, by

Ft, = max{la(x,v) - c~(X, Vo)l: 0 < x < a,v

a n d v o i n 1,, o, g = I , . . . , G , } , (26)

F . = m a x { f G [ k ( x , v ' , v ) - k ( x , vo, v)ldv:

O<_x<a,v '

and v0 in I.. 0, 1 < g < G.}. (27)

The notation V+ = { v e V : v > 0}, V_ = {veV:v < 0} will also be needed in the following, We can now state our basic convergence theorem, whose proof is the major purpose of the remainder of this paper.

Theorem 2

Suppose S ~ S I ( [ O , a ] x V), focSf l (V+) and f~ ~ Y~I(V_). Then the multigroup problem defined by equations (16)-(18) and (20) (with appropriate ad- ditional subscripts 'n') has a unique continuously differ- entiable solution ~n E C([O, a]) Gn. I f furthermore the solution 0 of the continuous-energy problem (1), (2) is such that the mappin 9

x ---,~ (x, .)

is continuous from [0,a] to L,m(V), and {J,}, a and k are such that {f~nl and [F.] converge to zero as n---. ~ , then {q~, I converges discretely to 4).

The detailed proof of Theorem 2 is given in Section 9, after we develop the needed machinery in this and the following two sections. A basic role in the proof is played by a discrete version of the Lax Equivalence Theorem (Lax and Richtmyer, 1956). We now develop the terminology necessary to state and prove this result.

Consider an underlying exact equation of the form

Lf = g, (28)

where L is a (not necessarily bounded) linear operator from some complete normed linear space ~, onto a second such space M, the element g of ~ is known, and f in ~, is the unknown which is hopefully deter- mined by this equation. Let [a.'~ and [~ . ) be sequences of complete normed linear spaces, and sup- pose that for each n there exist projection operations P, and R, mapping ,~ and ~ into a , and ~, , respect- ively. The sequence f,((~,, P,)] is termed an approxi- mation to ~,, and similarly for I (M,, R,)I. A sequence ~f, 1, such that each f , is in (,., converges discretely to an e l e m e n t f o f , if ]If. - P,f]l---*0 as n---, ~ . Similar terminology applies to sequences Ig. ] such that each g, is in .~,.

Consider a sequence {L. '~ of linear operators such that each L, has domain in u, and range equal to ~, . Such a sequence will be termed a discrete approxi- mating sequence of linear operators, or discrete ASLO. Such a discrete ASLO is discretely stable if the family {L, -1', is eventually uniformly bounded (i.e. if there exist constants C, N > 0 such that n _> N implies L.- 1 exists as a bounded linear operator satisfying IlL. - II -< C). A discrete ASLO [L.I such that L.- ~ exists for all sufficiently large n, is said to be discretely

302 PAUL NELSON

convergent at such g in ~ if {L~-IR, g} converges dis- cretely to some element of (,, and is strongly discretely convergent at such g if IL,-tg, I converges discretely to some element of ~ for any sequence [g, *~ converg- ing discretely to g.

We are fundamentally interested in approximating our problem (28) by a sequence of problems of the form

L.f. = g,, (29)

where [L, I is a discrete ASLO, the elements g, in .~, are known and the approximating equations (29) hopefully determine a sequence [f,] of solutions which approximate the solution of the exact equation (28). Such a sequence (29) is termed a discrete linear approximation. It is discretely consistent with equation (28), for a given f in the domain of L, if P , f is in the domain of L. for all sufficiently large n, and the sequences IL, P J ] , [g,] each converge discretely to Lf.

We are now able to state the promised discrete version of the Equivalence Theorem, The proof of this result is given in the Appendix.

Theorem 3

I f the sequence {P. } of projection operators for the domain space is uniformly bounded, then a discrete ap- proximating sequence of linear operators which is strongly discretely convergent at some element of ~ is necessarily stable. Conversely, if {L, 1 is a discretely stable discrete approximating sequence of linear opera- tors, and g in b is such that the exact equation (28) has a solution f in a with [LnP, f} converging discretely to Lf, then {L. I is strongly discretely convergent at g, and furthermore ILg lg. I converges discretely to f for any sequence {gn } which converges discretely to g.

The proof of our fundamental Theorem 2 proceeds as follows. In the next section we show how the continuous-energy problem (1), (2) and the multi- group approximation (16}-(18) and (20) can be fitted into the above abstract framework as equations of the form (28) and (29), respectively. Furthermore, the defi- nitions are such that our space C~ plays the role of (¢, and the notion of discrete convergence to an element of a agrees with that introduced in Section 4. We then proceed to establish discrete stability of the ASLO corresponding to the multi-group approximations, following which discrete consistency is proved in Section 8. Theorem 2 then follows from the converse (second) part of Theorem 3, as described in detail in Section 9.

7. STABILITY

We first show how to consider the problem (1), (2) in the framework described for equation (28) in the preceding section. Let L be defined by

L f = (f+(0,-),f-(a, .),~- + a f - Kf), (30)

where f+ (f_) is f restricted to [0, a] × V+ (respect- ively [0, a] × V_ ), and

Kf(x , v) = fv k(x, v', v)f (x, v')dv'.

The domain of L is taken as those f i n C,( = ~z) which are continuously differentiable in x and satisfy the condition

Of ~-x 6 5aI([0, a] x V).

It is clear that the domain of L is dense in C1, and it follows from Theorem 1 and the discussion preced- ing it that the range of L is equal to ~q¢91(V+) x ~p l (V_) x ~(~l([0, a] x g). We take the latter as our concrete instance of the space / in the preceding section.

The spaces C([0, a] )G" play the role of the ,z, in the abstract setting of the preceding section, and the pro- jection operators P, are as defined in Section 4. The operators [L, l for the multigroup method are given by

L , f = ~[fm+ ,(0) . . . . . f~.(0)], [fl(a) . . . . . f,.(a)], (

I ~f, ~2xx + 0, A - ( K . f ) , . . . . .

?fa"-+?x 6 a ° f a , - ( K . f ) a , , ] } . (31)

Here f = ( f l . . . . . fG.)eC([O,a]) G', m = m , is such that Vr,+l = 0, 80 is defined by equation (17), and

G (K,f),(x) = ~ koj(x)f~(x), g = 1 . . . . . G,,

j - I

where/~o~ is defined by equation (18). (In the definition of both 80 and k0~ appropriate additional subscripts 'n' must be inserted.) The appropriate range spaces for L. are

/,,, = R p" × R m" X I;(~l([0, a]) 6-,

where p, = G , - m.. The projection operators R,

An error estimate for the multigroup method 303

mapping ~ I (V+) x ~ I ( V _ ) x 5zl([0, a] x V) into t h e s e / , should be taken as

tire0 T R.(f0,f . , f ) = (v) dv , L L" ' l . .o d o = M.+ 1

[f, f("v)d']]2,} With the above definitions the equations (16) and

(20) which (hopefully) determine the multigroup ap- proximation 49. can be written as

L. ~b. = R.(fo,f. , S), (32)

whereas the continuous-energy problem (1), (2) has the representation

L49 = (fo,f~, S). (33)

In order to use Theorem 3 we wish to show that the multigroup approximation

L.~ . = (h +,h- ,H)

has a solution 49. in C([0, a]) °" for every (h +, h- , H) in ~ . , and furthermore that there exists a constant M (independent of n) such that

l'~.H <- M{ ff fv'H(x,v)'dvdx + fv+ lh+(v)ldv

+ fv_ 'h - (v ) ldv} ' n = 1,2 . . . . . (34)

Lemma 2

Under the assumptions detailed above, for every n = 1, 2 , . . . and (h +, h - , H) in E., the multigroup approximations

G = ~ ~gj(X)~n,j(X) + ng(x), g = 1,2 . . . . . G .

j= l

have a unique continuously differentiable solution ~. in C([0, a]) 6~ satisfying the boundary conditions

49.,g(a) = h~-, g = 1 . . . . . m. h + 49..,(0) = g-re., g = M , + 1 . . . . . G n.

Furthermore, this solution satisfies the inequality (34).

Proof

The proof of existence and uniqueness for the multi-

group approximations follows the same outline as the proof of Theorem 1. That is, first we show that if 49. ~ C([0, a]) G" is a continuously differentiable solu- tion, then it satisfies a system of integral equations of the form

~b. = T.49. + w., (35)

where w. c C( [0, a] )~", and

IIw, II < II(h +, h - , H)II~.. (36)

One then shows that T. maps C([0, a])G" into itself, and satisfies the inequalities

II(T.)~LI _< cktrmaxa, k = 1,2 . . . . . (37)

The key to proving equation (37) is use (as in Lemma 1) of the inequality

1 G. (~j(X) g=lE k0J(x) ~ e < 1, j 1,2 . . . . . G,.

In turn this inequality follows easily from equations (8), (17) and (18), One then shows directly that ~b. is continuously differentiable. Finally, the inequality (34) follows from equations (35), (36) and (37), with M = 1 + 6m,xaC/(1 - c). We leave the problem of fill- ing in the details to the interested reader.

8. CONSISTENCY

Lemma 3

I f the solution 49 of the continuous-energy problem (1), (2) is such that 49 e CI, the mapping

x --~ ~x ~ (x,-) is continuous from [0, a] to 5el(V), (38)

and {OCn}, k, a are such that f~ . - - ,0 and F . -~ 0 as n - * ~ (where ~n and F. are defined, respectively, by equations (26) and (27)), then the muhigroup approxi- mations (16) and (20) are discretely consistent with the continuous-energy problem (1), (2)for this O.

Proof

The proof of discrete consistency requires that we show, in the above notation, that [L.P.491 and { R. (fo ,f. , S)} each converge discretely to L49 = (f0, f . , S). But {R.(f0, f . , S)} converges dis- cretely to (fo, f . , S), virtually by definition of discrete convergence. That [L.P.49] converges discretely to L49 means I[ L . P . 4 9 - a.L49]]-,0, g simple calcula- tion shows the latter is equivalent to requiring con-

3 0 4 PAUL NELSON

vergence to zero of the truncation errors

+ #°(x) f t ¢(x,v) d v - £ a(x,v)c~(x,v)dv n , e n , g

°" f, - ~ kgj(x) dp(x, v) dv j = l ~,~

+ fs,., ; k(x, v', v)¢(x, v') dr' dv dx. (39)

But if ¢ satisfies equation (38), then we have

Ox ¢(x, v) dv n , o

= t i m 1 a~-0 ~ .,. (x', v) dx' dv

lf] T M = (x, v) dv + lira Axx

n.~ Ax~O

c75 (x, v)l dv dx'

Similarly we find

~0 f,. ¢(x, v) dv - f,°,o

and

a(x, v)dp(x, v) dv

< f~. fj L¢(x, v)l dr, n , 9

koj(x) ok(x, v') dr' g = 1 . , o

- f/..o fv k(x,v',v)ck(x,v')dv' dv

r.£ 14 (x, v)l dr.

It follows that the truncation error satisfies

IIL.P.q$ - R.L¢I[

gfv _< (~. + F.) [¢(x, v)[dv

This completes the proof of I ,emma 3.

dx. (40)

9. P R O O F O F T H E C O N V E R G E N C E T H E O R E M

Proof of Theorem 2 Consider the multigroup approximations and the

continuous-energy problem formulated as equations (32) and (33), respectively. It follows from Lemma 2 that the {L, i are a discretely stable ASLO. That IL .P . 4$1 converges discretely to {L4b'j follows from Lemma 3, and as observed in the proof of Lemma 3 it is trivial that [R.(fo,f . , S)', converges discretely to (fo,f.,S). Discrete convergence of {~b.l to ¢ then comes immediately from the converse portion of Theorem 3.

It is of interest to explore further the hypotheses of our convergence theorem, particularly the require- ments that f~. and F. converge to zero and the assumption (38).

If a is continuous, V is compact, and Ib¢oll---* 0, where I[oCn] I is defined by equation (25), then it follows that f~.--*0. If we make the additional assumption that the mapping

(x, v') ~ k(x, v',. ) (41)

is continuous from [0, a] x V to ~°~(V), then the same result holds for F, . In the case V = ( - ~ , oo), suppose that

l imv, ,a , , = o % l i m v . ,0 = - m , n ~oo n~c,

and

lim m a x { v , , o - v , . o _ l : g = 2 . . . . . G , - i } = O. n~z~

If

lim a(x, v), lim a(x, v)

each exist uniformly in x, and a is continuous, then f~.---+ 0. The same is true of F. provided we assume that each of the limits

lim k(x, v', .)

exists in ~ ° l ( v ) , uniformly in x ~ [0, a], in addition to equation (41) being continuous from [0, a] x V to ~l(v).

With regard to (38) we simply note that if S ~ C~, and, again (41) is continuous from [0, a] x V to £~'l(V), then it follows from (I), (8) and the bounded- ness of a that ¢ satisfies (38). Note that continuity of k implies continuity of (41) if Vis compact.

An error estimate for the multigroup method 305

10. T H E E R R O R E S T I M A T E

From equat ion (A.2) in the Appendix, along with the estimates in Sections 7 and 8, we see that the discrete error associated with the nth mult igroup ap- proximat ion satisfies

] lP .4~ - ~ .1]

I1 -/- O 'maxaC/(1 - - c ) ] a ( ~ ' ~ n -~- r.)114~11, ( 4 2 )

at least provided the exact solution q~ satisfies equa- tion (38). F r o m this error estimate it clearly follows that the mult igroup approximat ion converges dis- cretely to the exact flux, provided the exact flux satis- fies equat ion (38) and f2. + F, converges to zero. Per-

gence of the mult igroup method. There natural ly arises the question of whether it is also a necessary condition. We have been unable to answer this defi- nitely, a l though it seems likely that the answer is yes. (However, it also appears likely that an example for which the mult igroup method fails to converge will be fairly complex.) On the other hand we shall now present an example which shows that the uniformity of convergence emphasized in the preceding para- graph may fail in the event that f2. + F, fails to con- verge to zero, even if the norm of the mult igroup part i t ion does converge to zero.

For the example we take a slab of thickness a = 10, a strictly absorbing medium (k = 0), Vm,~ = 1, and is

a(x,v) = a(v) = ~ 2 , 1 - (2m - 1 ) - 1 _< V < 1 -- (2m) -1, m = 1,2 . . . . .

[ 0,1 ( 2 m ) - 1 _< v < 1 - (2m + 1 ) - 1 , m = 1 , 2 . . . . .

haps just as importantly, one concludes that the rela- tive discrete error (i.e. II P. 4~ - ~ . ll/]l 4, H) converges to zero as ~ . + F., uniformly over the class of all problems with fixed a, t~ and k satisfying our assump- tions and such that 4~ satisfies equat ion (38). Thus the problem of determining a priori a mult igroup struc- ture (i.e. a part i t ion J ) which is guaranteed to meet a specified relative discrete error tolerance for all prob- lems in some reasonably large class has been trans- formed to the presumably simpler problem of determining a mult igroup structure for which the as- sociated value of fl + F is less than a known multiple of the error tolerance, In addit ion this result also strongly suggests the strategy of choosing the multi- group structure so that the quantit ies

o.,o(tY)+~o.,Olfvk(' , ' ,v)dv]

The value of 1)mi n and of a for negative velocities is irrelevant to our considerations, so long as Vmm is finite. We choose the reference spectrum as qSo --- 1. Let {G.} be a sequence of real numbers such that G. --~ o0 as n ~ or, and suppose J , = {I)mi n = < Un, 0 < V,.X < ... < V..a. = 1 -- (2n) -1 < v..a = 1} defines a sequence of mult igroup part i t ions such that 11,¢.[1 ---* 0 as n ~ c~. Note that ~ . --- 2, F. - 0, so that ~2. + F. does not converge to zero. For arbitrary but fixed n, consider the boundary-value problem (1), (2) with fo = f l =- 0 and

= ~ ' 0 , 0 < v < 1 - (2n) -1 fo(v) [ 2 n , 1 - (2n)-1 < v < 1;

we denote the solution of this problem by q~,, to indi- cate explicitly the dependence on n. At x = a this exact solution is given by

I 0 , 0 < v < 1 - ( 2 n ) - 1 ,

4~(10, v) = J 2 h e -2° ,1 - ( 2 n ) -1 < 1 - ( 2 r n - 1) -1 _< v < 1 - (2m) -1, m = n + 1 . . . . .

~,2n, 1 - (2n)-1 < 1 - (2m)-1 < v < 1 - (2m + 1)-1, m = n . . . . .

[see equat ion (24)] are equal for each group Therefore we estimate

g = 1 . . . . . G. ;i.e. the fluctuations of cr and ( 2n ~ 1_ 1

qS(10, v) dv >_ ,.--. 2m - 1 2m dv +

fv k( , v) dv °~ ~ c

= 2 n E _l ,.=. 2m(2m - 1)

should be equi-distributed over the groups. n ~ l l We have shown that convergence to zero of > - > dx

f~. + F. is a sufficient condit ion for (discrete) conver- - 2 m =. ~ - 2 = 2"

3 0 6 PAUL NELSON

But we also find

1) , , 6.,G,,= 2n 2 m - 1 2m > 2(n + 1) -> m = n + l - - 4'

and thus the mult igroup approximat ion gives

~b,,o,(10) = exp ( -10 f f . . o ° ) _< e -2"5.

Therefore the relative discrete error satisfies

l iP.@.- ~.11 > 0.5 -- e -2"5 > 0.4, 114,.11

which shows the uniform convergence property fails.

Belleni-Morante A. and Busoni G. (1972), J. Math. Phys. 13, 1146.

Bellman R. E. and Wing G. M. (1975) An Introduction to lnvariant Imbedding. John Wiley, New York.

Bondarenko I. I. (1964) Group Constants for Nuclear Reac- tor Calculations. Consultants Bureau.

Clark M. E. and Hansen K. F. (1964) Numerical Methods of Reactor Analysis. Academic Press, New York.

Cullen D. E. (1974) Nucl. Sci. Engng 55, 387. Cullen D. E. (1975) Nucl. Sci. Engn 9 58, 261. Lax P. D. and Richtyer R. (1956) Communs pure appl.

Math. 9, 267. Nikolaev M. N. (1976) Nucl. Sei. Engn O 61,286.

A P P E N D I X

!1. C O N C L U D I N G R E M A R K S

The most significant of our results undoubtedly is the indication that the error in a given mult igroup calculation depends on the group structure through the fluctuations (26) and (27) in the total and differen- tial cross sections, respectively. Of course, our results establish this only as an upper bound for the error, and only for the simple rod model. Fur ther theoreti- cal studies for more realistic geometries, and also computa t ional experimentation, are needed to estab- lish the true significance of these fluctuations relative to the accuracy of the mult igroup method. It is some- what surprising that the min imum value of the cross section does not appear in our error estimate; in fact our convergence theorem and error estimate do not even require that the total cross section be bounded away from zero. However, the deep minima usually associated with large inaccuracies in deep-penetrat ion problems tend to lie in the resonance region, and therefore the resulting errors could equally well be at tr ibuted to wide fluctuations in the cross section within a given group.

Our results also suggest that the bands in a multi- band approximat ion should be selected to be approxi- mately of equal extent. This suggestion warrants further study within a context devoted specifically to the mul t i -band approximation.

R E F E R E N C E S

Becket M. (1975) Nucl. Sci. Engng 57, 75. Bell G. I. and Glasstone S. (1970) Nuclear Reactor Theory.

Van Nostrand Reinhold.

Proof of the discrete Lax Equivalence Theorem (Theorem 3)

In order to establish the first statement of the theorem, suppose that [L,] is a discrete ASLO which is strongly discretely convergent at some g e ~, and, additionally, sup- pose, for purposes of a proof by contradiction, that ~,L. I is unstable. It follows that there is an increasing sequence [mp l of integers such that for each integer p there exists hpe.~,,,~, with Ilhp[l=l and IlL,.~hpll>P. Let the sequence {g.} be defined by

=}'R.g.~ + p-l/2hp if n =mp forsome p,

g" [, R.g, otherwise.

Then both Ig. ] and {R,g~ converge discretely to g. By the strong discrete convergence of IL.I, the sequences L,- l g. I and I L,- 1 R. g*, both converge discretely. If f0 ~ ~,

is the difference of these discrete limits, then

p~oo Jp Lmp hp - P,,,f0ll = 0. (A.1)

But this is not possible, as - 12 - 1/2 - 1

lip L,.,, hp - Pm,,,foll >- lip L,.,, h, ll - I[Pm,,fo[[

> pi:2 _ IIP,.,,Jo][,

and the right-hand side of this expression is unbounded in p by virtue of the fact that [P, I is uniformly bounded.

For the converse portion, let f, = L. -~ g, for n suffi- ciently large that L. -~ exists as a bounded linear operator, where [g. } is an arbitrary sequence converging discretely to g 6 .~. Then we have

f, - P , f = L,-I{(g. - R.g) + ( R . L f - L.P,d)]. (A.2)

The desired result now follows from discrete stability and discrete consistency. This completes the proof of Theorem 3.