JOURNAL
OF COMPUTATIONAL
Multigroup
PHYSICS
Transport
46, 237-270 (1982)
Theory
with Anisotropic
Scattering
R. D. M . GARCIA AND C. E. SIEWERT
Departments of Nuclear Engineering and Mathematics,
North Carolina State University, Raleigh, North Carolina 27650
Received October 30. 1981
The special case of a triangular transfer matrix relevant to multigroup transport theory with
Yth order anisotropic scattering is discussed. The developed theory reduces the calculation of
the reflected and transmitted angular fluxes to a sequence of one-group problems involving
only angular fluxes at the boundaries. The theory is then extended to yield similar results at
any location within a finite slab. The FN method is used to establish particularly accurate
numerical results for a test problem.
1. INTRODUCTION
In two recent works [ 1,2] concerning m u ltigroup transport theory a solution for
the case of isotropic scattering and a triangular transfer matrix was developed, and
numerical results were reported. Here we extend our previous analysis to include the
important effects of anisotropic scattering. W e thus consider, for i = 1, 2,..., A4,
(1)
where ui is the total cross section for group i and oij(l) = oijPti(r), with Pii = 1,
denote coefficients in Legendre expansions of the transfer cross sections. In addition
v,(z,~) representsthe angular flux in the ith group and
(6,,,(z)
= j’, Vj(Z,P)P,(P)4.
(2)
W e are concerned here with nonmultiplying, uii < ui, finite slabs, z E [L, R], and
thus we seek solutions to Eq. (1) subject to the boundary conditions
and
W i(R,-/-J) =Rt@),
where L&)
p > 0,
(3b)
and Ri(p) are considered specified.
237
002 l-999 l/82/050237-34$02.00/0
Copyright
0 1982 by Academic Press, Inc.
All rights of reproduction
in any form reserved.
238
GARCIA
AND
2.
SIEWERT
ANALYSIS
In [l] full-range orthogonality properties of appropriate elementary solutions and
Green’s functions were used to deduce a system of singular-integral equations and
constraints for the boundary fluxes. Here we shall develop the equivalent expressions,
generalized to include the effects of anisotropic scattering, in a more direct manner.
We first change ,D to -,L in Eq. (l), multiply the resulting equation by exp(-oiz/s)
and integrate over all z to obtain
sPBi(Pv3) - ai(P - 8) I” Wi(Z9-P> eXP(-Uiz/s) CfZ
L
=
f,il
,g
t-l)
aij(r)
p,@u)
(4)
@j,l(s/ai)~
where
(5)
B,01, s) = w,(L -cl) exp(--o&ls) - v,(R -P) exp(-aJ+)
and
We can now multiply Eq. (4) by (U - s)-’ P&), s 6? [-1, 11, and integrate over all P
to find
t-l
1” Oi @i,n(s/ui)
+
+j$,
lie
C-l
1’ uij(r)
@j,dslut)
j:
1 pan)
pI(P)
*
We let g,,,(r) denote for the ith group the polynomials introduced for one-group
theory by Chandrasekhar [3], i.e.,
h,,n~~,~(t)=
tn
+
l)gi,n+l(O
+
(8)
Vi,“-I(0
with g,,,(r) = 1 and
hi,” = 2n + l - C,Pii(n)9
(9)
where c1= u,Ju,. On multiplying Eq. (7) by j?,,(n) g,,,(s) and summing over n from 0
to Lf, we find
ui
,to
t-l
1’ B*i(O
@l,l(s/ui)
I;,,Ls)
+
$
I$:
,io
t-l)’
x Pi,,(s)- g&N = s j; 1PW, cr)Bib s) +P
uU(z)
@jJ(s/ui)
9
(10)
MULTIGROUP
TRANSPORT
THEORY
239
where
F,,,(s)= g,,,(s)+ + ci If 1G,(s,~1PAP)&
(11)
and
G,(s,~u)= f’ MO g,,,(s) Prti).
I=0
(12)
It is not diffkult to show that
(13)
where
1
A,(s)= 1 +s
I-1 w*tP>-P--s4
9
(14)
with
(15)
is the one-group dispersion function [4]. We find we can write Eq. (10) as
n,(s)x,,(s)=$~; rG,(s4401~)&
i 1
(16)
where
(17)
yij(s)
=
,I$
t-l
I= 1
1’ Pij(O
@jI.I(slu*)
Ei,l(s)
(18)
and
(19)
Here the polynomials E,,,(s) are defined by
J%,,(s)= (l/d
g,,,(s) - P,(s)13
(20)
and with E,,,(s) = 0 they can be readily computed from
(21-k
l)sEt,,(s)=sPii(O g,,,(s)+ (1-k l)Ei,,+,(s) + zEi,l-~(s)*
(21)
240
GARCIA
AND SIEWERT
We note that the functions Qji.[(s/ai) can have essential singularities at the origin, but
otherwise, they are analytic in the complex s-plane. The functions X,(s) and Y,(s)
therefore are, with the exception of the origin, also analytic in the complex s-plane.
Thus, on investigating Eq. (16) for the first group, i = 1, and assuming that c, # 0, we
see that
(22)
where, in general, ri,m, m = 0, 1, 2 ,..., 2~~- 1 are the zeros of /ii(s). The left- and
right-hand sides of Eq. (16) are analytic in the complex s-plane cut from -1 to 1
along the real axis. Thus, on letting s approach the branch cut and using the Plemelj
formulas [5], we find that Eq. (16) yields, for v E [-1, 11,
(23)
where, in general,
(24)
Thus, for v E [--I, 11, it follows that
a,c,X,,(v) = 2vB,(v, v)
(25)
L,(v) vB,(v, v) - tc, VP I ’ @,(v,P) B,(P, v) - dp = 0.
--I
P---v
(26)
and
Equations (22) and (26) can be seen to be the system of singular-integral equations
and constraints [6, 71 that define the exit fluxes for the first group wi(L, -p) and
yl,(R,,~), ,u > 0, in terms of the incident distributions L,(U) and R ,(,u). Thus,
Eqs. (22) and (26) can be solved numerically or, e.g., by the F,,, method [7-91 to
establish B,@, s). In the event that ci = 0, Eq. (25) yields B,(,u, ,u) = 0, ] ,U] E (0, I].
Considering now that B,(p, s) is known, we note that Eq. (16) yields, for i = 2,
(126)
X226>
= ; j'-1 ~G,(wu)
+ ;~21[y2*(s)
B~(P,
4
s) -P---s
-~2(s)x21(s)I
(27)
MULTIGROUP
TRANSPORT
241
THEORY
or, for vE [-1, 11,
1
L,(v) VI&@, v) - Qc* VP
=
fo21k*
Y,‘(V)
+
I -1 ,uG,tv, P) 4b
v) &
(28)
X21(v)l
and
%CJ22(V) = 2vB,(v, v) - ~*,~*,(V).
(29)
For c2 # 0, Eq. (27) yields
For c, = 0, Eq. (29) yields, for 1P ] E (0, 11,
B2@9P)
=
c&4-’
c2
~2lX,l(P),
=
(31)
0,
whereas for c2 # 0 we can solve Eqs. (28) and (30) to find B2(p, s); of course, for
either case we must first compute
W,,(s)
=
c2
Y,,(s)
(32)
+X,,(s)
which can also be written, if we use Eqs. (17) and (18), as
W,,(s)
=
5
I=0
(-W2lU)
@,,[w72)
(33)
g,,,w
If we now multiply Eq. (4) by P,(p) and integrate over all p there results, for i = 1,
%l%&/d
+ (n + 1) @1,,+&/~1)
=-(-l)“Pn+
+ n@,,,-Wl)
(34)
l)~jf,llp,t~~~lt~,s)dll.
which yields
@l,“Wl)
= (-1)”
&!l,f&)
@l.OWl)
-
C-1)”
I,,,,
(35)
where D,,,(s) = 0 and
~~l,n~l,n(s)
=
’
(n
+
+
w
l)Dl,n+lts)
+
nD,,n-,(s)
+ 11; j’ PP,Cu)B*tP,s)&
-1
(36)
242
GARCIA
AND
SIEWERT
Substituting Eq. (35) into Eq. (17), with i = j = 1, we find
An expression alternative to Eq. (37) that allows the calculation of ~D,,~(s/a,) in the
event that G,(s, s) = 0 is provided in Appendix A. We see from Eqs. (25) and (26)
that, for VE [-1, 11,
and from Eq. (16) that, for s 6 l-1, 11,
It is apparent from Eqs. (22) and (39) that a limiting procedure must be used if
X,,([,J
is required. For the case c, = 0 we note, since B,(v, v) = 0, ] v ] E (0, 11, that
Eqs. (38) and (39) reduce to the following equation for all s:
(40)
Finally, we use Eqs. (33) and (35) to conclude that
where @Pl,0(s/a2)is available from Eq. (37) and, in general, sU = u,/uI. Equations
(28) and (30) can now be solved to yield the exit distributions for the second group
w2(L, -,u) and w#?,,u), P > 0, in terms of the incident distributions L,@) and R,Q)
and the previously established B,(p, s). Note that in this way we are able to deduce
the exit fluxes for the second group directly from the incident distributions for that
group and the boundary fluxes of the first group.
We now wish to generalize the foregoing and consider the ith group. We assume
that the B,(p, s),j = 1,2,..., i - 1, have been established, and we deduce from Eq. (16)
that, for v E [-1, 11,
i-l
n,(v) vB,(v, v) - tc,vP 11, ,uG,@, lu) B,(K v) &
and
= 4 c 011F,(v)
i=l
i-1
u,cJ,,(v)
= W(c 4 - i=Ic q,&,(v)
(42)
(43)
MULTIGROUP
TRANSPORT
and, for c, # 0,
CiCi,m
I
--1
’ Pw,,m7P)
B,019
C,m)
c1 -*z
243
THEORY
i,m
= -
i-l
c
,=l
ui, K,G,rn)~
(44)
where
For the special case ci = 0, we see from Eq. (43) that, for 1.~1E (0, I],
m4 Pu)= $ ‘$ u,4,w,
j 1
ci = 0,
(46)
whereas for c, # 0, the functions W,(S) rather than just X,01), ] ,u] E (0, 11, clearly
are required before we can solve Eqs. (42) and (44) to find B,(p, s). In analogy with
Eq. (34) we now find, for j = 1,2 ,..., i - 1,
sh,,, @,,nW,I) + (n + 1) @,.“, 1(s/uj)
+
n@j,n-*(s/uj)
= -tml)” Mj,n(s)9
(47)
where
(48)
It is clear that we can write
(49)
where D,,,(s) = 0 and
(50)
On substituting Eq. (49) into Eq. (17), with i = j, we find
@,,,ds/uj>= Gy ‘(~3S) X,,(S) + 5 P,,(l) D,,,(S) p,(s)][
I=1
(51)
Further, we can deduce from Eq. (16) that for v E [- 1, 1]
+
c
k=l
(52)
U/k y/k(V)
I
244
GARCIA AND SIEWERT
and for s @ [-1, l]
j-l
+ 2
(53)
ujik[Yjk(S)-dj(s)Xj,(s)l
k=l
Again it is apparent that a limiting procedure must be used in the event that Xjj(r;,,,)
is required. Finally, if we use Eq. (49) in Eq. (45) we conclude that
wij(s>
=
@j,O(s/ai>
-
5
I=1
5
I=0
Pij(O
Pij(O
gj,JSijS)
Dj,l(sijs)
tTi,lCs>
(54)
gi,ds)9
where @j,o(~/~i) is available from Eq. (5 1). We recall that Wijo1) =X,(p) for Ci = 0,
and thus, for this case, Eq. (54) can be used in Eq. (46) to establish the desired result.
3. THE FN SOLUTION
Rather than pursue exact analysis to solve the developed singular-integral
equations and constraints for the exit distributions Wi(L, -,u) and Wi(R,p), ,u > 0, we
prefer to use the F,,, method [7-91 to construct a concise approximate solution. We
thus let d = R -L, di = aid, and write, for the ith group and ,U> 0,
Wi(Lv-P) =Ri@) exP(--di/P) + 2
ai,aPa(%
-
l)
(55a)
a=0
and
Wi(RvP)=Li@) exp(-di/P) + 5 bi,,P,(@ - 1).
n=O
(55b)
If we now use Eqs. (3) and (55) in Eq. (5) we can deduce from Eqs. (42) and (44)
that
i-l
r,aBi,a(O
+
ci
exP(--di/t)
bi,aAl,,(t)]
= C,l,(t) + C aijlfj(t)
(564
+
CiexP(-Ai/<)
%,Ai,,(Ol
=
Wb)
j=l
and
Ii?
a=0
[bi,cxBi,a(t)
ciJi(t)
+
‘2
j=l
OijJij(C)
MULTIGROUP
TRANSPORT
245
THEORY
for all <E Pi = (v~,~}U [0, 11. Here v~,~, m = 0, 1,2 ,..., xi - 1, denote the positive
discrete eigenvalues relevant to group i,
and
where
si(d9 Pu,t) = (1 - exP(-~i~/Cr)
exP(-~i~lt))l@
+ 0
and
We have also introduced
@[j(r)
=
e”“”
W,(r)
(5%
and
tgij(<) = e -uiR’t W,(-0.
Finally, the functions Ai,,
Wb)
and B,,,(r) req uired in Eqs. (56) are defined as
Ai,.(5)=j~pP,(Zlr--L)Gi(-S,a)~,
Pu+C r@[-LO),
(60)
and
<E [O, 11. (61b)
In Appendix B we report some recursion relations that establish an accurate and
convenient method for evaluating these basic functions A,,,(T) and B,,,(T).
Considering the right-hand sides of Eqs. (56), we note from Eqs. (57) that the
functions Z,(r) and Ji(<) are immediately available directly from the boundary data
for the ith group. The additional terms Zij({) and Jii(<) clearly represent downscattering contributions to the ith group. We assume now that the constants {aj,,}
246
GARCIA
and {ZJ,,,}, j=
P > 0,
AND SIEWERT
1, 2 ,..., i - 1, have been found so that the approximate results, for
Bj(Pu, S) = exp(-uj+)
I
Rj@)[exp(---dj/~) - exP(---dj/s)l
+5a=0%YUF - 1)I
(624
and
Bj(-p,
S) = eXp(-UjL/S)
I
x 5
Lj(U)[ 1 - eXp(--dj/P)
bj,apcx(2P
-
exP(-dj/s)l
- exP(--dj/s)
(62b)
l>
a=0
I
are available for j = 1,2,..., i - 1. Thus, on considering Eqs. (56) at N + 1 values of
r E Pi, say C,s, we can solve the system of linear algebraic equations
+ ci
ai,aBi,a(4.4)
exP(-di/ti,D)
bi,aAi.fx(~i,b)l
i-l
=
cizi(ti,~)
+
aijzij(ti,,)
2
j=l
and
I?
a=0
fbi,mBi,m(tiJ3)
+
ci
exP(-dilri.o) ai,aAi,n(4i,J]
i-l
=
ciJi(ti.Ll)
+
C
j=l
aijJij(ti,b)9
t63b)
for p = 0, 1,2,..., iV, to find the desired constants for the ith group {a,.,} and {bi,, }
provided we first express ZJ{) and J,(r) in terms of known quantities. We therefore
proceed to use Eqs. (59) and the various results developed in Section 2 in order to
deduce expressions that can be used in a convenient manner to compute the desired
Z!,(r) and .ZIj(c). To be specific we note that for i = 1 the right-hand sides of Eqs. (63)
are known since Z,(c) and J,(c) are given by Eqs. (57). Thus, we can solve the system
of linear algebraic equations to find {a,,,} and {b,,,}. Considering Eqs. (63) for
i > 2, we see that we must compute Z,,(c) and J,(r) for j = 1,2,..., i - 1, along with
Zi(c) and J#) as given by Eqs. (57), before we can solve the linear system to find
{ai,,} and (b,,,}. We find
(64)
MULTIGROUP
TRANSPORT
241
THEORY
where
@~*(~/a~)
= C-l)’Sj,ltsijt)@/tO(t/O*)
- (-l)‘D,t*(Sij()
M/t/(Sij‘f) =F
I
I
(65)
'f [Uj,, + (-1)' eXp(-di/r)bj,a]
a=0
Kj,/(s,jG+
T,,,
I
(68)
In Eq. (68) we have used the definitions
- (-1)’ Lj(p)[ 1 - exP(-dj/p) exP(-dj/t)l 144
Z-a,,=dh&1(
-
0
(69)
l)f’,@)&v
(70)
and
/-I
uj,l(t/ai)
=
Cm1
1’
C
k=l
ujk(O
(71)
@k+,,l(U”i)*
We note that Devaux et al. [lo] have reported a recursion relation that provides an
efficient way to compute the numbers T,,, (see Appendix B). We also point out that
U,,,(</u,) is considered known since all @:,,({/a,) required in Eq. (71) have, by
necessity, been computed in previous steps. Finally for sij< E [0, I], XG(sijt) is given
by
x~tsijOE~ I lzj(sijt)+
[2 -Aj,O(Sijr)]
$J
uj.aPat2sijt-
l)
a=0
+ lx=0
ii L”j,aGj,a(sijO- exp(-40 bj,Jj,a(sijt)l
+
sji
5
,=I
uj,‘(t/“i)
Ej,‘(sijO
19
(72)
248
GARCIA
AND
SIEWERT
where
Gj,att) = Ji PGj(L Pu)[
paw
1
1) &
- 1)-P&t-
P--l
(73)
can be computed effectively from a recursion relation (see Appendix B). For sijr &
[0, 1 ] we find
xi(sijt)
=
+A,‘1(sij43
1 yj(sijt)
-
sjidj(sijt)
5
I
+
uj,[(t/Oi)
p,(sfjt)
I=0
sji
I:1
uj,l(~/“i)
Ej,l(sijG
(74)
19
where
yj(i3 = zj(13 + i
[aj,,Aj,,t-t)
n=O
- exP(-dj/t)
bj.,Aj,,(t>]*
(75)
In a similar way we find
Jij(4
=
5
I=0
Pij(0
@JCloi)
(76)
gi,l(t),
where
(77)
with
@jy~(T/(Jr)
= G,Y’(sijG SijT) xJG(stj<) + : (-l)‘Pjj(l)
[
I=1
Here D,,(s,r),
D,T,(Sij<)P,(Sij<)]*
(78)
with DJyo(sij<)= 0, are available from
= (I + I) D~yl+
-sijthj,lDi,l(sijt)
lts(jO
+
lD,,-l(sijt)
+
(79)
M,~,(sijt),
where
/+I
MJsijg= 21+ Nj,LSijG
- C [t-l) ’bj,a+ exP(-di/t)
ai,,]T,,,
I
ui
+
a=0
(--l)‘(tl”i)
I
(80)
vj,l(t/“i>*
In Eq. (80) we have used the definitions
Nj,ltO = 1: PuPlO1){RjO1)[l - exP(--dj/p)
- (-l)‘Lj(~)[exP(-dj/~)
exP(--dj/t)]
- exP(--dj/t)]}
Q
(81)
MULTIGROUP
TRANSPORT
249
THEORY
and
j-1
k=l
where the latter, as discussed before, is available from previous steps. Finally, for
sijtE [O, 11, X,;(S,~<) is given by
~Jst,t)= $ ]Jj(sijO
+ [2-Aj,0(sijt)l
f bj.apa(2sijta=0
l)
(83)
For S,jrGC[0, l] we find
x,lT(sijO
=$A,'(sijt)
zji(sijt)
I
-
sjiAj(sijt)
5
I=0
vj,l(t/ai)
p,(Sijt)
(84)
where
Ej(O=4(0
+ aio [bj,,Aj,,(-r)-exP(-dj/r)aj,,Aj,,(r)].
(85)
Having developed the FN method to find the surface fluxes, we now demonstrate how
a slight modification of the analysis of Section 2 and the FN method can be used to
compute accurately the angular fluxes for all z E (L, R).
4. THE INTERIORANGULARFLUXES
If we change iu to -p in Eq. (l), multiply the resulting equation by exp(-o,z/s),
and integrate over z from z1 to z2, with L < zi < z2 < R, we obtain an equation
similar to Eq. (4), viz.
581/46/2-6
250
GARCIA
AND
SIEWERT
where
B:(PP s) = w,(z, 3-Pu) exP(*iZJs)
- I/liCz27-PI exP(-Qizds>
Q-37)
and
We can follow the development discussed in Section 2 to obtain from Eq. (86)
generalizations of Eqs. (42)-(44). Thus, for the ith group and for v E [-1, l] we find
i-l
J,(v)vB:(v,v)
-
tc#‘(’
,uG,(v,Co&+,
VI++=
--I
t
C
j=l
Oijw$(v)
(89)
and
i-l
a*cJ,:(v)
= 2vB:(v, v) - c
o&~(v)
w-9
j=l
and, for cI # 0,
(91)
where
w4 = 5 (- 1)’P,,(O
@ jwJ,>g,,,(s)
(92)
ms) = E ewljm @,T,W~,)~,(S).
(93)
l=O
and
l=O
For the special case ci = 0, we find, for ] ~1)E (0, 11,
(94)
We note that the expressions developed in Section 2 for the computation of @,/.,(s/ui)
can be generalized so that @,?,(~/a~)can be found in a similar manner.
We now let z, = z, z2 = R, and then use Eqs. (3b), (55b), and the approximations,
for ~1> 0,
W~-G
-~)=RtOl)ex~l*,(R --z)/ccI+ 2 Ci,a(Z)Pa(2u
- 1)
a=0
(954
MULTIGROUP
TRANSPORT
THEORY
251
and
in Eq. (87) to deduce from Eqs. (89) and (91) the first of our FN equations. Similarly
we can take z1 = L, z2 = z, and then use Eqs. (3a), (55a), and (95) in Eq. (87) to
deduce from Eqs. (89) and (91) the second of the F,,, equations. Thus, for z E (L, R)
and < E P, we find
j,
[ci,a(z)Bi,a(O- Cidi.a(Z)Ai,a(Ol
i-1
= CrZ,(G0 + c qj4,(zv 4)
j=l
and
.$I0
[di,a(z)
Bi,a(t)
-
Ai,a(4)l
CiCl,a(Z)
i-1
= c,J&, 0 + /=I
c q,J,,(z,0
Wb)
Here
1,6,Q=~~r[L~Ol)G,(-Cr)exP[-~i(z-L)/~IS,(R-Z~~,C)
+ R,(P)G,(t,,) C,@ - Z,CI,014,
(974
Il(r,C)=lo’plL,~)G,(r,~)C,(r--L,/~,r)
+ R,(P)G,(--4,~)exp[-o,(R- z)/P]Sk -L,L 814,
z1=z and z2=R,
U&, t) = eOp’
w$
L (8,
Wb)
Wu(z, 43 = e -Ofz’lW$ (-0,
Wb)
(984
and
z,=L
and z2=z.
Investigating the right-hand sides of Eqs. (96), we note from Eqs. (97) that Z,(z, <)
and J[(z, <) are available, for any z E (L, R), directly from the boundary data for the
252
GARCIA
AND SIEWERT
ith group. The terms Z&r, 0 and Jij(z, <) represent, as before, down-scattering
contributions to the ith group, and at this point the constants (a,,,} and {bi,u } have
already been established. Thus, on considering Eqs. (96) at the same N + 1 values of
<E Pi used in Eqs. (63), namely ti,B, we can solve the system of linear algebraic
equations
i-l
=
cili(zT
x
4.0)
.fl
a=0
+
C
j=l
aijzij(Z9
ti,&3)
-
Ci exp[+7i(R
-
z)/<i,o]
(994
bi,Ji,tz(ti,LJ
and
fI
a=0
[di,cz(z)
Bi,n(ti,fi)
-
CiCi,a(z)Ai,a(ti,fl)]
i-
1
= ciJi(zv cl,l3)+ C aijJ,j(Z,
j=1
’
i
u=o
<i,O)-
Ci
exp[-a,@ - Z,)/t;i,o]
(99b)
ai,nAl,a(<i,4)
for /?= 0, 1,2,..., N to find {ci,,(z)} and {&(z)) for selected values of z E (L, R)
provided we first express Zij(z, <) and Jij(z, 0 in terms of known quantities. We
observe that the functions A,,,(c) and B,,,(c) appearing in Eqs. (99) are the same as
used in Eqs. (63), and thus, the only new quantities to be evaluated are Zi(Z, <),
J*(z9t), zij(z3 09 and .Z,(z, [). Further we note that the matrix of coefficients in
Eqs. (99) is independent of z so that the solutions to these equations for many values
of z can be obtained with one matrix inversion. We now summarize the equations
that can be used to compute the desired Zu(z, r) and Jii(Z, 0. We find
'ijCz9
t) = 5 C-l 1’Plj(O @itltz, tl”l)
I=0
gi,l(O
(l(v
where
@i[(zYt/ui) = t-l)’
with
gj,l(sijt)
@LO(z9t/O,) - (-l)’ Djtl(zY si,t)
(101)
MULTIGROUP
Here DL,(z, s&,
TRANSPORT
253
THEORY
with D&(z, s&), are available from
Sij~h,,,D,?dz, SijO = (I + 1) DA,+ l(Z, Sijr) + zDL,- ,(z, S,<)
(103)
+ MjTl(z9 sijT)v
where
M,t[(z9
Sijt)
=
y
Kj,i(z,
sijt)
+
'5
a=0
I
+ t-1)’ exP[-oi(R
[cj,a(z)
-ZYtl
-
(-l)'
dj,,tz)
bj,al r=,~ - + uj,[(z9 t/ui)*
I
1
( lo4)
In Eq. (104) we have used the definitions
- Z)/P] - ew[--aj@ - Z)/Cl 1
Kj,i(z9 63 = I,’PPIOI){RjOI)[ exP[ *j(R
- (--1)‘Lj@) exP[+Jj(z - LYPI
X
[ 1 - eXP[-oj(R - Z)/Pl exp[--aj@ - Z)/tl I) Q
(105)
and
&=I
Finally, for si,r E [0, 11, X,$(z, s,,<) is given by
+ I? [Cj,a(Z) Gj,a(SijO + {dj,,(Z) - exP[*i@
a=0
-Z)/r]
bj,,} Aj,u(sijC)]
(107)
*
I
For so<@ [0, l] we find
xi(z9
sij5)
=
~ni’(sljtl
1 yj(z9
sijO
-
sjidj(sijt)
5
I=0
uj,,(z,
</ai)
p,(sijt)
(108)
I=1
254
GARCIA
AND
SIEWERT
where
yj(z9 63 = 1j(z9 t) + f
a=0
[cj,,(z) Aj,,(-T)
+ {dj,a(z) - exP[-oj@
-
zYtl bj,aIAj,a(t)l*
(109)
In a similar way we find
Jij(z7
63
=
Ii0
Put0
@j;ltz,
t/ui)
(110)
gi,L0
where
(111)
with
+
,gl
t-l)’
Pjj(O
D,Tdz,
sijt)
(112)
p,(sij6)]*
Here D,yr(z, sijc), with D,;,(z, sij<) = 0, are available from
(113)
where
Mj;,(z,
Sij()
=
21+
Nj,l(zv
ui
sijt7
I
+ C [C,,,(Z) - t-1)
a=0
+ C-l 1’$
dj,atz) - exP[*i(z
- L)ltl
uj,cx
vj,I(z9 <lo,)*
(114)
In Eq. (114) we have used the definitions
Nj,l(z90 = i,’P~dPPjtP)exd-uj@ - ZYPI
[ 1 -- exp[--u,(z- L)/P] eXP[+Jj(Z
- L)/tl I
- (--l)‘Lj@)[ev[--a/(z-L)/Pl -expbj(z -LYtlll 4
X
(115)
255
MIJLTIGROUPTRANSPORTTHEORY
and
(116)
Finally, for silt E [0, 11, X,v(z, s,~C)is given by
=$
Jj(z,s,~C)+
[2-Aj,,(siJO]
5
a=0
I
+
If
a=0
[do,&)
Gj.cx(siJO
+
dj,,(Z)P,(2SiJ<-
{Cj.a(Z>
exP[-ai(Z
-
1)
--L)/<]
aj,,)
Aj,u(s,r)]
(117)
For s,<& [0, l] we find
x,~(z.s,j5)=~ni1(sijT)
[sj(z,SijS)
I
-s~~dj(slJ~)
+
sji,$
’
1%
Vi,dz~
V,,dZ,
t/“i)Ej,,(sijO
</ai)P,(siJr)
19
(118)
where
~j(z~O=Jj(~~
t)
+
aio
[d~,a(~)~j,,(-O
+ {cj,a(z)-exP[--aj(z-LIltI uj,cxIAj,a(Ol-
(119)
5. NUMERICAL RESULTS
In order to demonstrate the computational merit of our solution we now consider a
20-group albedo problem with a lOth-order Legendre expansion.of the scattering law.
A 20-cm thick slab has an isotropically incident distribution of radiation only in the
first group and only on the surface at z = L, i.e., for ~1> 0
(120a)
and
(120b)
256
GARCIA
AND
SIEWERT
To facilitate the data handling we use a fictitious cross-section set (in units of cm-‘)
defined, for i = 1, 2 ,..., 20, by
ui = (i/10) - 0.15aj,, - 0.156i,,,
(121a)
and
U,,(Z)= (21+ l>j/[ lOO(i - j + l)]( gij)‘,
j = 1, 2,..., i
and Z=O, I,..., 10, (121b)
where
g, = 0.7 - (i + j)/200.
(12lc)
The scattering law defined by Eqs. (121b) and (12 lc) is a truncated version of the
Henyey-Greenstein phase function introduced in the field of radiative transfer [ 111.
The Henyey-Greenstein phase function is characterized by one parameter g which is
a measure of the degree of anisotropy, i.e., g + 1 implies forward scattering while
g + - 1 implies backward scattering. For a monoenergetic problem g corresponds to
the average cosine of the scattering angle. In our problem the values of g given by
Eq. (121~) correspond to moderate forward scattering and were chosen in order to
avoid negative values in the scattering law (with LP = 10).
In solving the systems of linear algebraic equations given by Eqs. (63) and (99) we
have used, for various orders of the F,,, approximation, the collocation scheme
ti.0
=
vi,B
7
p = 0, 1,2 ,...) Ki - 1,
(122a)
and
<i,fl=f + 4 cos{(2p-
2Ki + 1)Z/[2(N+
b=q,q+
l -Ki)]},
l,..., N.(122b)
The points given by Eq. (122b) are the zeros of the Chebyshev polynomial of the first
kind T N+ i-,,(2x - 1). Based on the results of our computations which closely
followed the technique discussed by Siewert [ 12) we have concluded that there is
only one pair of discrete eigenvalues relevant to each group of the considered
problem, and thus we list in Table I the positive eigenvalue for each group. We list
TABLE
I
The Positive Discrete Eigenvalue Basic to Each Group
i
1
2
3
4
5
vi.0
1.014675230187
1.013664030621
1.012702645157
1.011789497866
1.024569285561
i
6
7
8
9
10
VI.0
1.010101983620
1.009324801731
1.008590208601
1.007896906406
1.011201112487
i
11
12
13
14
15
VI.0
1.006629121797
1.006052161369
1.005511523631
1.005005991723
1.004534348940
i
16
17
18
19
20
VI.0
1.004095374943
1.003687842667
1.003310515972
1.002962148042
1.002641480584
MULTIGROUP
TRANSPORT
251
THEORY
our converged results for the exit angular fluxes in Tables II-V. Further converged
results for the angular fluxes at various positions inside the slab are shown in
Tables VI-XI. We note that to compute the angular fluxes accurately for all ,u we
used a recently proposed technique [13]. First the functions Bi,,(r) defined by
Eqs. (61) are expressed as
Bi9m(t)=2Pa(X-
l)-ci{
[~-A~,o(O]
Pa(X-
1) + Gi,a(t)J
(123)
and this relation can be used in Eqs. (56) for c =p E [0, l] to find the following
alternative expressions for the emerging fluxes:
Wi(L9
-Pu)
Ii(p) + ]2
exP(-At//J) + 2
=Ri(P)
-Ai,Ool)l
5
a=0
1
ai,aPa(2P
-
I)
+ a=0
i [ai,aG,,,(P)
- eXP(-djP)
bi,aAi,,@)]+ f j=l
I ‘? u,Z&)(124a)
and
yli(R,iu)=LicU) exP(--di/p) + 4 [J,(p) + [2 -Ai,O@)]
+ 5
[bi.,Gt,&)
5 bi,,P,(2u - 1)
a=0
- exP(--dilC1)ai,di,&)l
[ + f ‘2 aijJ&)*
a=0
j=l
(124b)
In a similar way we find for the interior fluxes
yli(z,-~)=Ri(~)exP[-ai(R
+2
]ziCz9P)
-Z)/p]
+
L2
-Ai,OOl)]
i
a=0
c,,,(z)
p,GP
-
1)
+ a=0
i [Ci.a(z)
Gi,a@>
+ {di,a(z)- exP[*i(R -z)/PI bi,aIAi,aol)l/
i-l
+
t
+
3
C
j=l
(125a)
Oijzij(z,P)
and
Ji(z9P)
+
[2
-Ai,O(P)]
I
5
a=0
di,,(z)Pa(&
-
l)
+ I? [d,,,(z) Gi,,01)+ kdz) - exP[*i(z - L)/Pl ai,aI Ai,a@)l/
a=0
i-l
+
f
C
j=l
OijJij(z,,K)m
(125b)
i= 1
5.0885(-2)
2.8195(-2)
1.8568(-2)
1.3061(-2)
9.6343(-3)
7.3536(-3)
5.7554(-3)
4.6063(-3)
3.7653(-3)
3.1192(-3)
2.6287(-3)
i= 11
4.8231(d)
4.2927(A)
3.7343(-4)
3.2384(A)
2.8225(-4)
2.4783(-d)
2.1900(a)
1.9457(-d)
1.7397(-4)
1.5668(a)
1.4137(-4)
P
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
12
4.0797(-4)
3.6660(-4)
3.2144(-I)
2.8070(-4)
2.4611(-4)
2.1722(-4)
1.9286(-4)
1.7210(A)
1.5448(-I)
1.3962(-4)
1.2641(A)
i=
1.2999(-2)
8.6681(-3)
6.2371(-3)
4.6596(-3)
3.5983(-3)
2.8516(-3)
2.3033(-3)
1.8921(-3)
1.5814(-3)
1.3378(-3)
1.1453(-3)
i=2
13
3.4993(-4)
3.1721(A)
2.8019(q)
2.4627(A)
2.1715(-4)
1.9262(-A)
1.7179(-4)
1.5394(-d)
1.3870(-4)
1.2578(-A)
1.1426(-4)
i=
5.8860(-3)
4.2785(-3)
3.2310(-3)
2.4996(-3)
1.9835(-3)
1.6080(-3)
1.3241(-3)
1.1055(-3)
9.3667(-4)
8.0287(d)
6.9424(-d)
i=3
14
3.0368(a)
2.7747(-A)
2.4677(A)
2.1823(-h)
1.9345(A)
1.7240(A)
1.5442(-4)
1.3892(A)
1.2563(A)
1.1429(-d)
1.0416(-A)
i=
The Exit Angular
3.3641(-3)
2.5789(-3)
2.0122(-3)
1.5956(-3)
1.2912(-3)
1.0640(-3)
8.8871(d)
%5096(-d)
6.4282(A)
5.5637(-4)
4.8472(-4)
i=4
The Exit Angular
II
III
15
2.6619(d)
2.4499(-d)
2.1926(-4)
1.9501(-4)
1.7374(d)
1.5552(-t)
1.3987(-4)
1.2630(-4)
1.1461(-d)
1.0459(a)
9.5611(-5)
i=
i=6
2.3537(A)
2.1806(A)
1.9631(A)
1.7554(-4)
1.5714(-4)
1.4125(A)
1.2752(-A)
1.1557(-4)
1.0521(-4)
9.6301(-5)
8.8290(-5)
2.0970(-d)
1.9547(-4)
1.7694(d)
1.5902(-4)
1.4299(-A)
1.2905(-4)
1.1692(A)
1.0633(A)
9.7107(~5)
8.9134(-5)
8.1949(-5)
i= 17
for i = 1 l-20
i= 16
i=l
1.1467(-3)
9.6248(A)
7.9986(A)
6.6715(-4)
5.6261(d)
4.8015(A)
4.1361(A)
3.5904(-I)
3.1451(-4)
2.7806(-A)
2.4661(A)
for i = l-10
1.5457(-3)
1.2701(-3)
1.0397(-3)
8.5661(-t)
7.1511(A)
6.0504(a)
5.1717(-4)
4.4586(A)
3.8823(-4)
3.4135(A)
3.0128(-4)
Fluxes vi@, -p)
TABLE
3.1535(-3)
2.4685(-3)
1.9565(-3)
1.5724(-3)
1.2862(-3)
1.0690(-3)
8.9923(A)
7.6454(-t)
6.5749(-4)
5.7091(-4)
4.9900(-4)
i=5
Fluxes y,(L, -p)
TABLE
1.8810(-4)
1.7632(A)
1.6043(-4)
1.4487(d)
1.3082(d)
1.1851(-4)
1.0775(A)
9.8307(-5)
9.0051(-5)
8.2882(-5)
7.6408(-5)
i= 18
8.8628(A)
7.5717(-4)
6.3758(-4)
5.3764(-4)
4.5755(+)
3.9355(-4)
3.4138(a)
2.9821(A)
2.6267(-4)
2.3339(-A)
2.0793(A)
i=8
19
1.6974(-4)
1.5994(A)
1.4623(-4)
1.3264(-d)
1.2026(A)
1.0934(-4)
9.9742(-5)
9.1283(-5)
8.3859(-5)
7.7384(-5)
7.1525(-5)
i=
7.0652(A)
6.1275(-4)
5.2196(A)
4.4447(d)
3.8138(A)
3.3036(A)
2.8839(-d)
2.5337(-4)
2.2431(d)
2.0022(-4)
1.7914(A)
i=9
10
1.5400-4)
1.4582(-d)
1.3393(-4)
1.2199(A)
1.1103(-4)
1.0129(A)
9.2697(-5)
8.5089(-5)
7.8385(-5)
7.2514(-5)
6.7193(-5)
i= 20
6.8284(A)
6.0047(-d)
5.1654(-4)
4.4359(a)
3.8335(-I)
3.3407(-4)
2.9317(-4)
2.5882(-4)
2.3009(-4)
2.0608(A)
1.8504(-4)
i=
2
u
0
*
z
F
!2
00
1.1146(-3)
1.5864(-3)
2.4017(-3)
4.9489(-3)
1.2386(-2)
2.6436(-2)
4.6527(-2)
7.1061(-2)
9.8380(-2)
1.2713(-l)
1.5630(-l)
i= 11
2.1534(-5)
2.6462(-5)
3.1729(-5)
3.7922(-5)
4.5360(-5)
5.4309(-5)
6.4972(-5)
7.7453(-5)
9.1751(-5)
1.0777(-4)
1.2538(-4)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
i=
P
12
1.8639(-5)
2.2831(-5)
2.7274(-5)
3.2465(-5)
3.8664(-5)
4.6087(-5)
5.4898(-5)
6.5183(-5)
7.6946(-5)
9.0113(-5)
1.0458(-d)
i=
3.4317(-4)
4.5301(a)
5.9970(d)
8.0657(-A)
1.0994(-3)
1.5012(-3)
2.0197(-3)
2.6424(-3)
3.3420(-3)
4.0868(-3)
4.8463(-3)
i=2
1.6352(-5)
1.9968(-5)
2.3770(-5)
2.8187(-5)
3.3432(-5)
3.9685(-5)
4.7079(-5)
5.5688(-5)
6.5518(-5)
7.6512(-5)
8.8584(-5)
i= 13
1.7392(d)
2.2419(A)
2.8704(A)
3.7054(A)
4.8210(-4)
6.2888(A)
8.1612(-4)
1.0452(-3)
1.3129(-3)
1.6124(-3)
1.9351(-3)
i=3
1.4499(-5)
1.7653(-5)
2.0946(-5)
2.4748(-5)
2.9242(-5)
3.4574(-5)
4.0858(-5)
4.8158(-5)
5.6478(-5)
6.5776(-5)
7.5982(-5)
i= 14
The Exit Angular
1.0772(-A)
1.3714(A)
1.7251(-4)
2.1806(A)
2.7724(A)
3.5321(-4)
4.4840(-4)
5.6385(A)
6.9894(-I)
8.5165(a)
1.0191(-3)
i=4
The Exit Angular
IV
V
5.8073(-5)
7.2858(-5)
8.9704-5)
1.1050(q)
1.3656(A)
1.6903(A)
2.0877(d)
2.5621(A)
3.1124(-4)
3.7331(-A)
4.4168(d)
i=6
1.2971(-5)
1.5748(-5)
1.8627(-5)
2.1934(-5)
2.5823(-5)
3.0419(-5)
3.5817(-5)
4.2073(-5)
4.9192(-5)
5.7142(-5)
6.5865(-5)
i= 15
16
1.1692(-5)
1.4157(-5)
1.6695(-5)
1.9596(-5)
2.2992(-5)
2.6989(-5)
3.1670(-5)
3.7080(-5)
4.3229(-5)
5.0089(-5)
5.7615(-5)
i=
i=l
1.0610(-5)
1.2812(-5)
1.5066(-5)
1.7630(-5)
2.0618(-5)
2.4122(-5)
2.8212(-5)
3.2930(-5)
3.8284(-5)
4.4252(-5)
5.0797(-5)
i= 17
4.4676(-5)
5.5763(-5)
6.8195(-5)
8.3362(-5)
1.0217(A)
1.2541(A)
1.5366(a)
1.8722(A)
2.2603(-4)
2.6973(A)
3.1785(A)
for i = l-10
Fluxes yl,(R, p) for i = 1 l-20
TABLE
1.1525(A)
1.4787(-4)
1.8566(a)
2.3364(d)
2.9545(A)
3.7446(-A)
4.7327(d)
5.9301(-4)
7.3295(-4)
8.9074(-d)
1.0630(-3)
i=5
Fluxes y,(R,p)
TABLE
9.6829(-6)
1.1662(-5)
1.3677(-5)
1.5958(-5)
1.8605(-5)
2.1698(-5)
2.5298(-5)
2.9441(-5)
3.4135(-5)
3.9365(-5)
4.5098(-5)
i= 18
3.5756(-5)
4.4432(-5)
5.4026(-5)
6.5611(-5)
7.9849(-5)
9.7307(-5)
1.1842(d)
1.4338(-A)
1.7218(-4)
2.0456(-4)
2.4019(q)
i=8
19
8.8825(d)
1.0671(-5)
1.2482(-5)
1.4523(-5)
1.6882(-5)
1.9629(-5)
2.2817(-5)
2.6477(-5)
3.0620(-5)
3.5232(-5)
4.0285(-5)
i=
2.9444(-5)
3.6442(-5)
4.4088(-5)
5.3236(-5)
6.4390(-5)
7.7979(-5)
9.4325(-5)
1.1359(A)
1.3576(-A)
1.6065(A)
1.8803(-4)
i=9
10
8.1854(-6)
9.8099(-6)
1.1446(-5)
1.3282(-5)
1.5395(-5)
1.7847(-5)
2X%86(-5)
2.3939(-5)
2.7615(-5)
3.1704(-5)
3.6183(-5)
i=20
2.9886(-5)
3.7031(-5)
4.4666(-5)
5.3710(-5)
6.4659(-5)
7.7929(-5)
9.3832(-5)
1.1253(-4)
1.3403(-4)
1.5817(-I)
1.8473(-A)
i=
z~
s!
Fi
z
2
ZI
2
;I
5
v
5
s
5
i= 12
i= 11
6.4895(-5)
7.8313(-5)
9.6221(-5)
1.2098(A)
1.5600(-4)
2.0702(A)
2.7932(-l)
3.7090(-4)
4.7031(-4)
5.6385(-4)
6.4306(-4)
P
-1
-0.8
-0.6
-0.4
a.2
0
0.2
0.4
0.6
0.8
1
5.8213(-5)
6.9873(-5)
8.5334(-5)
1.0654(A)
1.3629(-4)
1.7916(-4)
2.3930(d)
3.1505(-4)
3.9743(A)
4.7583(-4)
5.4362(-4)
4.7532(-4)
6.3596(A)
8.8218(A)
1.2884(-3)
1.9994(-3)
3.3843(-3)
6.1309(-3)
8.9005(-3)
1.0270(-2)
1.0647(-2)
1.0504(-2)
1.0471(-3)
1.4632(-3)
2.1471(-3)
3.3761(-3)
5.7980(-3)
1.1621(-2)
1.0604(-l)
3.1493(-l)
4.6327(-l)
5.6269(-l)
6.3224(-l)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
i=2
i=l
M
5.2815(-5)
6.3073(-5)
7.6589(-5)
9.4989(-5)
1.2060(-4)
1.5715(A)
2.0792(-4)
2.7154(-4)
3.4080(-4)
4.0729(-4)
4.6572(-4)
i= 13
i=4
TABLE
2.2548(-4)
2.8916(A)
3.8060(-4)
5.1900(d)
7.3619(d)
1.0971(-3)
1.6888(-3)
2.4442(-3)
3.0428(-3)
3.3742(-3)
3.4995(-3)
i=5
VII
1.3465(d)
1.6854(a)
2.1598(-4)
2.8558(-4)
3.9049(-4)
5.5598(-4)
8.1014(A)
1.1457(-3)
1.4886(-3)
1.7593(-3)
1.9369(-3)
i=6
1.1058(-4)
1.3719(-4)
1.7403(A)
2.2731(-4)
3.0636(-4)
4.2861(-4)
6.1246(A)
8.5297(d)
1.1061(-3)
1.3194(-3)
1.4720(-3)
i=l
14
4.4547(-5)
5.2700(-5)
6.3321(-5)
7.7585(-5)
9.7152(-5)
1.2459(d)
1.6205(-4)
2.0854(d)
2.5922(-A)
3.0848(A)
3.5278(A)
i= 15
4.1304(-5)
4.8650(-5)
5.8170(-5)
7.0875(-5)
8.8191(-5)
1.1228(-4)
1.4491(A)
1.8522(-4)
2.2918(-4)
2.7212(a)
3.1106(-t)
i=l6
17
3.8496(-5)
4.5153(-5)
5.3738(-5)
6.5128(-5)
8.0555(-5)
1.0185(-4)
1.3049(-4)
1.6572(A)
2.0416(-4)
2.4185(-4)
2.7628(-4)
i=
Fluxes yi(z, p) for z = L + A/4 and i = 1 l-20
4.8333(-5)
5.7443(-5)
6.9376(-5)
8.5506(-5)
1.0779(a)
1.3931(-t)
1.8270(-4)
2.3679(d)
2.9574(-4)
3.5271(A)
4.0342(-4)
i=
VI
Fluxes yl,(z,p) for z = L + A/4 and i = I-10
2.0836(A)
2.6721(A)
3.5252(-4)
4.8356(-4)
6.9193(-4)
1.0444(-3)
1.6302(-3)
2.4158(-3)
3.1081(-3)
3.5443(-3)
3.7561(-3)
The Angular
2.9442(d)
3.8406(-4)
5.1690(d)
7.2703(-4)
1.0739(-3)
1.6919(-3)
2.7893(-3)
4.1886(-3)
5.2044(-3)
5.7110(-3)
5.8661(-3)
i=3
The Angular
TABLE
3.6043(-5)
4.2105(-5)
4.9888(-5)
6.0156(-5)
7.3982(-5)
9.2930(-5)
1.1823(-4)
1.4924(-4)
1.8308(A)
2.1638(A)
2.4699(-4)
i= 18
9.3621(-5)
1.1526(-t)
1.4490(-4)
1.8724(-4)
2.4921(-4)
3.4339(-4)
4.8257(A)
6.6300(-4)
8.5588(-4)
1.0257(-3)
1.1550(-3)
i=8
19
3.3881(-5)
3.9427(-5)
4.6514(-5)
5.5817(-5)
6.8274(-5)
8.5228(-5)
1.0772(-4)
1.3518(d)
1.6515(A)
1.9474(A)
2.2208(4)
i=
8.1008(-5)
9.9037(-5)
1.2351(A)
1.5808(4)
2.0807(-4)
2.8291(A)
3.9185(-4)
5.3192(a)
6.8292(-4)
8.1998(A)
9.2922(-4)
i=9
10
3.1962(-5)
3.7055(-5)
4.3537(-5)
5.2003(-S)
6.3278(-5)
7.8524(-5)
9.8623(-5)
1.2307(A)
1.4977(A)
1.7619(A)
2.0073(-4)
i=20
8.4882(-5)
1.0339(-4)
1.2833(A)
1.6323(A)
2.1325(A)
2.8730(-4)
3.9406(G)
5.3084(A)
6.7815(A)
8.1111(-4)
9.1575(A)
i=
E
s
si
u
Q
%
0
s
M
s
i=l
4.6637(d)
6.5719(-4)
9.7087(-4)
1.5206(-3)
2.5630(-3)
4.7967(-3)
1.7611(-2)
1.0153(-l)
2.1428(-l)
3.1513(-l)
3.9787(-l)
i= 11
3.1587(-5)
3.7875(-5)
4.6145(-5)
5.7338(-5)
7.3005(-5)
9.5688(-5)
1.2911(4)
1.7643(A)
2.3717(A)
3.0662(-4)
3.7912(-4)
P
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
P
-1
-0.8
-0.6
-0.4
a.2
0
0.2
0.4
0.6
0.8
1
12
2.8323(-5)
3.3784(-5)
4.0923(-5)
5.0525(-5)
6.3863(-5)
8.3007(-5)
1.1094(4)
1.5016(-4)
2.0027(A)
2.5757(-4)
3.1754(-4)
i=
2.2300(d)
3.0006(-4)
4.1608(-t)
5.9792(A)
9.0213(A)
1.4586(-3)
2.5687(-3)
4.5595(-3)
6.8622(-3)
8.7448(-3)
1.0026(-2)
i=2
13
2.5691(-5)
3.0493(-5)
3.6736(-5)
4.5082(-5)
5.6592(-5)
7.2978(-5)
9.6668(-5)
1.2967(-4)
1.7167(A)
2.1968(A)
2.7005(A)
i=
i=4
14
IX
6.5688(-5)
8.1781(-5)
1.0381(a)
1.3507(A)
1.8139(a)
2.5304(d)
3.6685(A)
5.3863(-4)
7.6632(d)
1.0242(-3)
1.2807(-3)
i=6
5.3896(-5)
6.6465(-5)
8.3510(-5)
1.0743(-4)
1.4239(A)
1.9558(d)
2.7848(d)
4.0154(A)
5.6338(-A)
7.4781(A)
9.3504(-4)
i=l
2.1664(-5)
2.5480(-5)
3.0392(-5)
3.6885(-5)
4.5725(-5)
5.8122(-5)
7.5748(-5)
9.9953(-5)
1.3053(-4)
1.6548(a)
2.0228(-4)
i= 15
2.0087(-5)
2.3525(-5)
2.7931(-5)
3.3725(-5)
4.1568(-5)
5.2492(-5)
6.7!903(-5)
8.8931(-5)
1.1541(-4)
1.4566(d)
1.7757(-4)
i= 16
1.8722(-5)
2.1839(-5)
2.5815(-5)
3.1018(-5)
3.8023(-5)
4.7716(-5)
6.1293(-5)
7.9705(-5)
1.0281(A)
1.2921(-q
1.5710(-4)
i= 17
for z = L + A/2 and i = 1 l-20
TABLE
1.0980(-4)
1.4081(-4)
1.8419(A)
2.4721(A)
3.4365(-4)
4.9928(d)
7.5999(A)
1.1749(-3)
1.7253(-3)
2.2971(-3)
2.7919(-3)
i=5
Fluxes ~,(z,p)
2.3508(-5)
2.7771(-5)
3.3286(-5)
4.0615(-5)
5.0657(-5)
6.4843(-5)
8.5174(-5)
1.1329(-4)
1.4894(A)
1.8968(A)
2.3251(-I)
i=
VIII
Fluxes w,(z, p) for z = L + A/2 and i = l-10
1.0055(-4)
1.2860(d)
1.6813(-4)
2.2623(A)
3.1638(A)
4.6406(A)
7.1509(A)
1.1178(-3)
1.6625(-3)
2.2513(-3)
2.7862(-3)
The Angular
1.4098(d)
1.8405(-4)
2.4616(a)
3.395 l(A)
4.8879(A)
7.4368(-t)
1.2010(-3)
1.9722(-3)
2.9949(-3)
4.0077(-3)
4.8357(-3)
i=3
The Angular
TABLE
1.7530(-5)
2.0369(-5)
2.3977(-5)
2.8677(-5)
3.4969(-5)
4.3624(-5)
5.5663(-5)
7.1895(-5)
9.2203(-5)
1.1540(-4)
1.3994(-4)
i= 18
4.5600(-5)
5.5781(-5)
6.9474(-5)
8.8500-5)
1.1596(A)
1.5716(-4)
2.2033(-t)
3.1281(-4)
4.3358(d)
5.7151(-A)
7.1320(d)
i=8
1.6480(-5)
1.9079(-5)
2.2368(-5)
2.6633(-5)
3.2315(-5)
4.0085(-5)
5.0823(-5)
6.5220(-5)
8.3179(-5)
1.0369(-A)
1.2542(A)
i= 19
3.9433(-5)
4.7897(-5)
5.9192(-5)
7.4746(-5)
9.6954(-5)
1.2986(d)
1.7961(A)
2.5157(-t)
3.4496(-4)
4.5170(A)
5.6215(d)
i=9
10
1.5549(-5)
1.7937(-5)
2.0947(-5)
2.4835(-5)
2.9990(-5)
3.7000(-5)
4.6628(-5)
5.9467(-5)
7.5437(-5)
9.3668(-5)
1.1302(A)
i=20
4.1400-5)
5.0105(-5)
6.1650(-5)
7.7431(-5)
9.9770(-5)
1.3254(-4)
1.8155(A)
2.5190(-4)
3.4299(4)
4.4733(4)
5.5567(-d)
i=
f4
0
2
4
<
ii
s)
t-4
(S-)096O'L
(S-)9Sl9'S
(S-)&OP&'P
(S-)ZIZf’f
(S-)8Z9S.Z
(S-)&lZO'Z
(S-jLPE9.I
(S-)SIS&'I
(S-)LL&I'I
(9-bSIL.6
(9-h&6&.8
(S-)SO&6'L
(S-)P&SZ'9
(S-h8083
(S-3S9S9.f
(S-)6ZOS'Z
(S-)IL6I'Z
(S-)PL9L'I
(S-)ZPSP'I
(S-)88IZ'I
(S-)S9&0'1
(9-)1056'8
LI =?
(S-hLLI6.8
(S-)0900'L
(S-h6Sf.S
(S-)&SPo’P
(S-)L180'&
(S-)&OOP'Z
(S-)6616.1
(S-)LILS'I
(S-)&II&'1
(S-)SOIl'l
(9-)ZLIS‘6
91=!
(t48600'1
(S-)8106'L
(S-)ZZlW9
(S-h1S’P
(S-)LOOP'&
(S-)IL&9'Z
(Sd696O.Z
(S-)SLOL'I
(S-)8LIP'I
(S-)ZS61'1
(S-)66lOT
S1=!
(t+ZSI'I
(S-)6&86.8
(S-)6S6L'9
(S-)IZ9o’S
(S-)Z96L'&
(S-)1916*Z
(S-)&VO&'Z
(S-)0998*1
(S-)91PS'I
(S-b&61.1
(S-)98603
PI =!
(t'--)ILZ&'l
(I'-h'O&o'I
(S-)88PL'L
(S-1EIIZL.S
(S-)Zl9Z'P
(S-)88PZ'&
(S-)ZOSS-Z
(S-)O&WZ
(S-)OL89'1
(S-)Z8OP'I
(S-)ZO61*1
&I=!
(i'-)SPPS'l
(V-)&P61'1
(S-hSZ6.8
(S-)PSPSP
(S-)ZLZO'P
(S-)OIS9'&
(S-)09P8'Z
(S-)89LZ'Z
(S-)&098*1
(S-)pPPS'I
(S-)Z86Z'l
ZI =I
(b)ZOZO'I
(I-)&IOp'I
(V-)SOPOl
(S-)199S’L
(S-)1625'S
(S-k9PI'P
(S-)Z8OZ'&
(S-)S6PS'Z
(S-)POLo'Z
(S-)LOOL'l
(S-)BLZp'I
II =!
(t+8LI'Z
(tdS699.1
(b)ZI&Z‘I
(S-)S&L8’8
(S-)LIZp'9
(S-)9ZLL'P
(S-k&99'&
(S-)LO68'Z
(S-)ZZ&&*Z
(S-)&Z163
(S-)PL8S'I
d
z'k
P'O9'08'0I-
(f-)PSLL’I
(E-16SIE’l
b-1LS66.8
b-)896L'S
(V-)&60L'&
(t-)1&9P’Z
(P-)&VU'1
(P-)&6ZZ'I
(S-)OOt'6'8
(S-)9&65*9
(S-1ESS6.P
(E-h78IL’I
(E-)609Z'I
(V-)18PS’8
(V-)Z6Sp'S
(P-)OZSV&
(idL99Z.Z
(df99S'I
(P-)&&ZI'I
(S-)81&Z'8
(S-)16Z1’9
(S-h6P9’P
(E-10S91.E
(E-)U&&‘Z
(&-1Z8SS.I
(P-)SSPS'6
b-)BSPL’S
(t-)1919’&
(P-)ZPIP'Z
(P-)&899*1
(tdE9LI.I
(S-18LLP.8
(S-bILZ.9
Z=!
(E-)ILZ&‘L
(C-1CSZ9.S
(E-1ZI6L.E
(f-)PZOZ'Z
(f-)190Z'l
+)PS&O’L
(t--)LSZP’P
(V-h8P8.Z
(P-)9068*1
(V-)OSO&‘l
(S-hP9f.6
I=!
(I-b96P.Z
(I-h’I9L’I
(Z-)SOS6’6
(Z-)Z6ZP'&
(E-)O&SP'S
(E-)819Z’Z
(f-k861.1
(t-)OISS*9
(ldOLS6’E
b-kO8S’Z
b-)098L’I
?i
Z’iVO9’08’0I-
8';
9'0
P’O
Z'O
(S-)ZP8&'9
(S-)ZOLO.S
(S-)Z8&6'&
(S-)PIfO’f
(S-hS&‘Z
(S-)8L98'1
(S-h'8IS.I
(S-)ZI9Z'I
(S-)Z990'1
(9-)86&1'6
(9-hSZ6.L
81=!
(tr)SZLS’L
(tdSSE9’S
(t-)9096'&
(t-h&LYZ
(id686L.I
(V-h76VZ’I
(S-16ZVO.6
(S-jLL9L.9
(S-)&981'S
(S-)80&03
(S-)ZZLI’&
&=!
aqL
(d669p’S
(t-1EL6O.t
(t'-)&&16'Z
(b-)88663
(tdL69E‘l
(S-19SL9.6
(S-)Z9OI’L
(S-h6E'S
(S-)&681'P
(S-)ZSO&*&
(S-h6E9.Z
P=!
saxnla Jqn8uy
61=!
~o~(d‘.z)'d
oz = !
oz-11 =.i pm p/g& + 7=r
(t7)8PPI’P
(td6EZI.E
(V-)IPPZ'Z
(i-kl9S.I
(P-)6980.1
(S-kO6L.L
(S-kP6L’S
(S-)8SPp'P
(S-bP6t'f
(S-)&06L’Z
(S-16PSZ.Z
s=!
IX TI8V.L
(F)ZZSZ’&
(bdZS9t7.Z
(tdIL8L’I
(i406SZ'l
(S-)9P88'8
(S-)66PP'9
(S-)OlS83
(S-116SL.E
(S-)6586-Z
(S-)LSOVZ
(S-)6&967
9=!
8’:
9’0
P'O
Z'O
(tdZt’OZ’&
(P-)ZL&P’Z
(t48LLL.I
(I’--)9&9Z’I
(S-)S800.6
(S-)&109.9
(S-)&&00'S
(S-)LI06'&
(S-)ZIll'&
(S-)18lS.Z
(S-hO9o'Z
L=!
aql
8=!
37BV.L
JOJ(II‘Z)~~ saxnl,I qnSuy
6=!
=! pw3 p/v& + 7=2
oI=!
01-l
x
MULTIGROUP
TRANSPORT
263
THEORY
We have found that these expressions are significant improvements especially as
P-P 0 over the usual (and simpler) expressions given by Eqs. (55) and (95).
In addition we show in Table XII converged results for the group fluxes
which, for z = L and z = R, can be expressedby using Eqs. (55) and (120) as
(127a)
h(L) = 4, + a,,0
and
tiitR)
=6i,*E*(di)
+
(127b)
bi,O,
where, in general, E,(x) denotes exponential-integral functions. For z E (L, R) we use
Eqs. (95) and (120) in Eq. (126) to obtain
#i(z)=Gi,lE2[a*(z
L)l +
-
ci,O(z)
+
(127~)
di,O(z)*
Finally, we list in Table XIII our converged results for the group albedos
Ai* = 2
‘Pm,
I 0
TABLE
(128a)
-P)dc1
XII
The Group Fluxes o,(z)
i
r=L
1
2
3
4
5
6
1
8
9
10
11
12
13
14
15
16
17
18
19
20
1.0117
3.9657(-3)
2.0883(-3)
1.3228(-3)
1.3007(-3)
7.0807(a)
5.5208(A)
4.4579(d)
3.6946(-4)
3.6961(A)
2.7 104(-4)
2.3568(A)
2.0748(d)
1.8450(-4)
1.6545(-4)
1.4946(a)
l-3588(-4)
1.2423(-4)
1.1415(-d)
1.0536(-4)
z=L+A/4
r=L+A/2
3.5594(-l)
9.9306(-3)
5.0739(-3)
3.1027(-3)
3.0869(-3)
1.5694(-3)
1.1895(-3)
9.3681(d)
7.5937(-4)
7.6078(A)
5.3840(A)
4.6029(A)
3.99 10(-4)
3.5000(-4)
3.0989(-4)
2.7666(a)
2.4877(A)
2.2511(-4)
2.0486(-4)
l-8737(-4)
1.7050(-l)
6.2916(-3)
2.922 l(-3)
1.6910(-3)
1.7578(-3)
8.2100(--t)
6.1336(-4)
4.7874(-4)
3.8576(-4)
3.87 10(-4)
2.7184(-4)
2.3188(A)
2.0072(-4)
1.7580(-4)
1.5550(-4)
1.3871(-4)
1.2465(-4)
1.1273(-4)
1.0254(A)
9.3748(-5)
r=L+34/4
8.7855(-2)
3.6317(-3)
1.5888(-3)
8.98 18(-4)
9.4748(A)
4.3208(-4)
3.2196(-4)
2.5096(-4)
2.0207(-4)
2.0269(-4)
1.4228(-4)
1.2133(-4)
1.0500(-4)
9.1954(-5)
8.1325(-5)
7.2537(-5)
6.5175(-5)
5.8941(-5)
5.3609(-5)
4.9008(-5)
z=R
4.6716(-2)
1.9091(-3)
7.8 103(A)
4.2724(A)
4.5042(-4)
1.9734(-4)
1.4496(A)
1.1154(A)
8.8744(-5)
8.8221(-5)
6.1011(-5)
5.1520(-5)
4.4 159(-5)
3.8308(-5)
3.3569(-5)
2.9673(-5)
2.6427(-5)
2.3692(-5)
2.1365(-5)
1.9367(-5)
264
GARCIA
AND SIEWERT
TABLE
XIII
ATand BT
Present work
DTF69
i
Ai*
Bi*
AI
B:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
6.4394(-3)
2.4468(-3)
1.3718(-3)
9.0655(-4)
9.1002(-4)
5.1696(-4)
4.1123(-4)
3.3795(-4)
2.8449(-4)
2.8836(-4)
2.1483(-4)
1.8886(-4)
1.6797(A)
1.5080(-4)
1.3645(-4)
1.2430(A)
1.1390(-4)
1.049 1(-4)
9.7071(-5)
9.0188(-5)
7.3 lOO(-2)
2.6667(-3)
1.0693(-3)
5.7560(A)
6.0465(-4)
2.5976(-4)
1.8942(g)
1.4482(-4)
1.1456(-4)
1.1340(-4)
7.7912(-5)
6.5506(-5)
5.5914(-5)
4.83 12(-5)
4.2175(-5)
3.7144(-5)
3.2964(-5)
2.9452(-5)
2.6472(-5)
2.3920(-5)
6.4399(-3)
2.4467(-3)
1.3719(-3)
9.0668(d)
9.1011(4)
5.1708(A)
4.1134(-4)
3.3805(A)
2.8459(-4)
2.8844(-4)
2.1492(A)
1.8894(-4)
1.6805(-4)
1.5088(-4)
1.3652(-4)
1.2437(-4)
1.1397(d)
1.0497(A)
9.7135(-5)
9.0250(-5)
7.2983(-2)
2.6646(-3)
1.0678(-3)
5.7472(-4)
6.0380(-4)
2.5935(-4)
1.8912(-4)
1.4458(-4)
1.1437(-4)
1.1321(A)
7.7785(A)
6.5399(-5)
5.5822(-5)
4.8233(-5)
4.2106(-5)
3.7082(-5)
3.2909(-5)
2.9402(-5)
2.6427(-5)
2.3879(-5)
and the group transmission factors
(128b)
If we use Eqs. (55) and (120) in Eqs. (128) we find
Ai* = ai. + $a,,,
(129a)
Bjr = 26i,lEj(di) + bi,O+ fbi,,*
(129b)
and
All our numerical results are accurate, we believe, to within fl in the last digit
shown. In Table XIII we also show the results of a calculation by Renken [ 141 who
used DTF69, a discrete-ordinates code [ 151, with 40 space points and eight directions
for each half range ofp. We observe here as in the isotropic scattering case [2] what
we believe to be a slight deterioration in the DTF69 results for increasing absorption
(as the group number increases).
Regarding the convergence of our method, we have found that to establish
MULTIGROUPTRANSPORTTHEORY
265
w,(L, -,u) and ~,(R,~) accurate to what we believe to be five significant figures for
all p required in this case N = 20; for the interior angular fluxes and the integrated
quantities 4,(z), A r*, and B: we have found that N = 15 was sufficient to obtain five
figures of accuracy.
Finally, we would like to mention that we have also generated numerical results for
the 19-group problem considered in [2], but generalized to include anisotropicscattering effects of the Klein-Nishina differential scattering cross section. A set of P,
multigroup transfer cross sections was provided by Renken [ 141. Due to the truncation of the Legendre expansion of the cross section, however, the resulting
multigroup transfer cross sections turned out to be negative for some values of the
scattering angle. Thus, the familiar and challenging question of how to deal with the
solution of a strictly nonphysical problem was encountered [ 161. From a
mathematical point of view a transport equation based on cross sections that can be
negative is a perfectly valid candidate for study and can clearly yield a solution that
can be negative. One can (as we did) solve such a problem and accept, at least on a
mathematical basis, the solution-positive or otherwise. We note that in comparing
our results for the considered problem with P, scattering with those obtained by
Renken [ 141, with and without the use of the negative-flux fix-up option [ 151 in the
DTF69 code, we found excellent agreement with Renken’s results only when he did
not use the negative-flux fix-up option. A separate question is what is the relationship
of the solution obtained in this way to the physically correct solution that satisfies the
transport equation for which the Klein-Nishina cross section has not been truncated.
6. CONCLUSIONS
We conclude from our studies that the FN method is capable of producing accurate
results for the considered multigroup model. The most interesting aspect of the
method seems to be the capability of finding the reflected and transmitted angular
fluxes for a given group by using only the boundary data and established emerging
fluxes for preceding groups. This feature of the FN method is a particularly attractive
one for shielding calculations, where frequently the interior angular fluxes are not of
primary interest. For most situations the results deduced from the method of discrete
ordinates are clearly adequat-specially
when we consider the magnitude of the
uncertainties associated with the input data. For strong absorption and/or optically
thick slabs, however, increased computer time will be required by strictly numerical
methods to achieve a desired degree of accuracy-a characteristic not shared by the
FN method.
APPENDIX
A: ALTERNATIVE EXPRESSIONSFOR @j,O(s/uj)
We have mentioned in Section 2 that an alternative expression to Eq. (37) is
needed to compute @I,0(s/a,) in the event that G,(s, s) = 0 for some s. The same is
266
GARCIA
AND
SIEWERT
true, in general, for the @J~,~(s/~~)
given by Eq. (51) when Gj(s, s) = 0. It is clear that
Eq. (37) is only a special case of Eq. (51), and thus we now proceed to derive an
alternative expression to the latter. If we set n = 0 in Eq. (7) and use Eq. (49) we
obtain, for s @J[-1, 11,
where Q,(s) denote the Legendre functions of the second kind [ 171, i.e.,
(Z+
l)sQ,(s)= (I+ 1) Q,+l(s)+ @-I(S) +do,r
64’4
with
Q,,(s)= 4 log((s + 1)/b - l)),
s cz [-I, 11,
(A3)
Q&j = 4 W(l
VE [-1, 11.
(A41
or
+ v)/(l - VI>,
If we demonstrate that A,(s) and Gj(s, s) do not have common zeros for s & [-1, 1 ] it
is clear that we can divide Eq. (Al) by n,(s) to obtain the desired alternative formula
for @ji,0(s/uj)9
S @ [-I, 113in the event that Gj(s, s) = 0. First we use the summation
formulas given by Inonii [ 181 to write n,(s) and Gj(s, s) in the convenient forms
Aj(s>= (9 + ~)[QY(s)gj,u+I(S)- QY+,(s)gj,dS)I
0-W
and
Gj(svs) = (CjS)-‘(~ + I)[&+ I(S)gj,ds) -P&s) gj,y+I(S
W)
It is easy to show that the following identity holds:
(y + 1P,+,(s) Q,P(s>
- f’ds) QY, &)I = 1.
(A7)
We now multiply Eq. (A5) by Py+ ,( s ) an d use Eqs. (A6) and (A7) to obtain
ps?+ l(S) 4(s) = gj,Y+ 1(s) -
CjSQy+
I(S) Gj(s, s)*
W)
By contradiction, if we suppose that there exists some s* & [-1, 1 ] such that
/ii
= G,(s*, s*) = 0 we conclude immediately from Eq. (A8) that gj,lp+ ,(s*) = 0.
From Eq. (A6), however, we see that this would require gj,&*) = 0. Clearly, this is
not possible; otherwise from Eq. (8) we would have gj,,(s*) = 0 for all 1. We must
then conclude that there is no such s*.
MULTIGROUP
TRANSPORT
267
THEORY
In order to obtain an alternative formula for @,,,(~/a~), v E [-1, 11, we let s
approach the branch cut to find that Eq. (Al) yields, for VE [-I, 11,
v,w9 9 @,,,owJ,)
= 2vqv9 VI+ I=0
5 ~,,(w,,,(v)P,(v)- ‘&? 5 (-1)’
k=l
X
ujk(l)
•t
V
@k,l(v/uj)
l=O
6-W
Pi(V)
and
C
k=l
C
I=0
(e-1)’
ujk(l)
@k,l(v/uj)
We can use Eqs. (A9) and (AlO) to show that, for
Ujlj(V)
@j,O(V/Uj)
= 2vBj(v,
+
v ,i,
j-l
v, + v Jl,
ujj(4
Dj,l(v)
/l[
v
Q/(V)*
E [-1, 1 ] and Gj(V, V)
(A101
=
0,
Bj(Vvv>
P--v
14’
Bj(P9 v, -
Tl(v)
40
(Al
k=l
1)
‘=I
where the polynomials F,(v) can be generated with the recursion formula [ 121, for
12 0,
(21+ 1) VT,(V)= -d’,, + (I + 1) r,+,(v) + r,-,(v),
WV
where
T,(v) = 0.
(A13)
Again, if we demonstrate that J](v) and Gj(v, v) do not have common zeros for
v E [-1, 1 ] we can divide Eq. (Al 1) by Izi(V) to obtain the desired alternative formula
for @j,o(v/oj), vE L-1, 11, when G,(v, v) = 0. We let s approach the branch cut to
obtain from Eq. (A5)
409 = (9 + l)[Q,(v) g,,ip+l(v) - QY+dv) g,,ip(v)l
6414)
and
Gj(v, “I= (CjV)-‘(g + ~)[J’Y+ I(“) gj.dV) - PA”) gj,ip+ I(V)]*
(A15)
Of course Eq. (A7) is still valid for v E [-I, I],
(9 + ~W’Y+Av)Qs&) - P&v) QY+&)I = 1,
6416)
268
GARCIA AND SIEWERT
and Eq. (A8) yields
pY+
lCv>
kj:i(v>
=
1 (VI
gj,rP+
-
Cj VQ,P+
I(“>
Gj(V,
(A*7)
“1.
By contradiction, if we suppose that there exists some v* E [-1, 1] such that n,(v*) =
Gj(v*, v*) = 0 we see from Eq. (A17) that we must have gj,u+ I(v*) = 0. As before,
the possibility that gj,Jv*) = 0 has to be ruled out, and thus Gj(v*, v*) = 0 would
require Py+,(v*) = 0 in Eq. (A15). At the same time Lj(v*) = 0 would require
QF+i(v*) = 0 in Eq. (A14). But Py+,(v*)
and QY+ ,(v*) cannot be zero
simultaneously, otherwise Eq. (A16) would be violated. We must then conclude that
there is no such v*. We note that this result implies that there are no discrete eigenvalues embedded in the continuum.
APPENDIX
B: RECURSIVE RELATIONS
The functions Ai,,(r) defined by Eq. (60) can be shown to satisfy, for a > 0, the
recursive relation
di,a-l(O
+
=
2(2a
(2a
+
+
I)
lIK25+
5
I=0
l)Ai,a(O
(-l>‘Pii(f>
+
gi,!(t)
ta
+
l)Ai,u+l(G
Ta,17
(Bl)
where for forward recursion the required initial value can be computed from
Ai,O(t)
=
’
,6-i
CM1
1’ Piit0
giJ(O
FQ)
cl(O’
Here the functions C,(T) can be found from
EC,-,(O+(2/+
l>K,(O+(~+
l)To,,
l)C,+,(f3=(2~+
033)
with
Co(<) = 1 - cw*
+ u/o>.
(B4)
We recall from Sections 3 and 4 that the functions A,,,(<) are required for real 6 &
[--I, 0). We have found that the use of Eq. (Bl) in the forward direction is stable
only for c E (-LO], and thus an alternative procedure is desired. Using the Christoffel-Darboux formula [ 171 for the Legendre polynomials, we have deduced the alternative recursion relation
‘cr+lt2t+
1)Ai,m(t)+Ptz(2t+
l)Ai,,+l
CO
=
(-1>“(2/(a
+ 1)) ri,a(t),
(B5)
where
ri,O(O
=
5
I=0
C-l
1’ Piit
gi,l(t)
TO,,
9
VW
MULTIGROUP
TRANSPORT
269
THEORY
and, for 1 < a < Y + 1,
~,,,t~)=~*,,&)+
t-l)w+
l)p,w+
1) 5 t-VP,,V)
&,dWd
Pm
I=0
Since T,,, = 0 for a > I+ 1, we see that Eq. (B7) yields
ri,a(t)
=ri,Y+
a>Y+l.
l(O9
038)
We have found that backward recursion of Eq. (B5) in the manner suggested by
Miller [ 191 is stable for real < & [-1, 0). As discussed before in [2], however, such a
scheme can be time consuming for r close to the transition points -1 and 0, and for
this reason we have actually used forward recursion of Eq. (Bl) for <E
[-1.001, -1) U [0, O .OOl] without losing too many significant figures. For other < we
used backward recursion of Eq. (B5). The functions B,,,(r) defined by Eqs. (61) and
required for <E Pi can be deduced from the recursive relation
-aBi.m-,tO+
=
2t2a
Pa + 1W+
l) ci 5
l)B,.,tt)-(a
Pitt0
gi,dt)
+
T~,19
l)Bi,,+,(t)
(B9)
I=0
with
Bi,o(C;)= 2(1
-Ci)
+
ciAi,o(C)*
@lOI
Forward recursion of Eq. (B9) can be used efficiently to generate B,,,(r) for <E
and thus, the
[0, 11. For <= vi,,, it is easy to see that Bia,(~i,,,) = -CiAi,,(-vi,,),
recursion relations developed for A,,,(<) can be readily used to establish Bi,,(vi,m).
We note that the strategy adopted here yielded A,,,(<) and B,,,(r) accurate to at least
13 significant figures for a up to 40 (working in double precision with an
IBM 370/165 machine).
We now turn our attention to the constants T,,, defined by Eq. (70). We note that
the L,, are a special case (m = 0) of the more general TE,, considered by
Devaux et al. [lo], and thus we write, for a > 0 and 12 0,
l)][(@ a + 1))T,-,,, + T,,I + ((a+ 1)/P+ l))T,+~.,l
- tw + 1))T&L1.
(Bll)
T u,l+l = (z+
1)/[2(1+
In this equation a runs from a = 0 to I+ 2 (note that T,,, = 0 for /I > 12+ 1) for each
1, from I= 0 to Y - 1. To initiate our calculation we use
To.0= f
0312)
T, ,. = :.
(B13)
and
270
GARCIA AND SIEWERT
Finally, the polynomials G,,,(r) defined by Eq. (73) and required for < E [0, 1 ] can
be computed effkiently by forward recursion from
aGi,a-,(t)=
2(2a
(2a
+
l)
+
5
I=0
l)Gi.m(O + (a + l)Gi.a+,(O
1)(XPiit
gi,LO
Ta,I,
0314)
with
G,.,(t)
= 0.
(B15)
ACKNOWLEDGMENTS
The authors would like to express their gratitude to J. H. Renken of Sandia National Laboratories for
several helpful discussions related to this work and for communicating the DTF69 results listed in
Table XIII. The authors are also grateful to D. H. Roy for the interest shown in this study and to the
Babcock & Wilcox Company for partial support of this work, which was also supported in part by the
National Science Foundation. One of the authors (RDMG) wishes also to acknowledge the financial
support of the Comissb National de Energia Nuclear and the Instituto de Pesquisas Energiticas e
Nucleares, both of Brazil.
REFERENCES
1. C. E. SIEWERTAND P. BENOIST,Nucl. Sri. Eng. 78 (1981), 311.
2. R. D. M. GARCIA AND C. E. SIEWERT,Nucl. Sci. Eng. 78 (1981), 315.
3. S. CHANDRASEKHAR,“Radiative Transfer,” Oxford Univ. Press, London, 1950.
4. J. R. MIKA, Nucl. Sci. Eng. 11 (1961), 415.
5. N. I. MUSKHELISHVILI,“Singular Integral Equations,” Noordhoff, Groningen, 1953.
6. R. L. BOWDEN, F. J. MCCROSSON,AND E. A. RHODES,J. MoB. Phys. 9 (1968), 753.
7. C. E. SIEWERT,Astrophys. Space Sci. 58 (1978), 131.
8. C. E. SIEWERTAND P. BENOIST,Nucl. Sci. Eng. 69 (1979), 156.
9. P. GRANDJEANAND C. E. SIEWERT,Nucl. Sci. Eng. 69 (1979), 161.
10. C. DEVAUX, C. E. SIEWERT,AND Y. L. YUAN, Astrophys. J. 253 (1982), 773.
11. L. G. HENYEY AND J. L. GREENSTEM,Astrophys. J. 93 (1941), 70.
12. C. E. SIEWERT,J. Math. Phys. 21 (1980), 2468.
13. R. D. M. GARCIA AND C. E. BEWERT, Nucl. Sci. Eng., in press.
14. J. H. RENKEN, private communication, 1981.
15. J. H. RENKEN AND K. G. ADAMS, “An Improved Capability for Solution of Photon Transport
Problems by the Method of Discrete Ordinates,” SC-RR-69-739, Sandia National Laboratories,
Albuquerque, N.M., 1969.
16. H. BROCKMANN,Nucl. Sci. Eng. 77 (1981X 377.
17. M. ABRAMOWITZ AND I. A. STEGUN (Bds.), “Handbook of Mathematical Functions,” Nat. Bur.
Stds. Applied Math Series 55, 1964.
18. E. IN~NO, J. Math. Phys. 11 (1970), 568.
19. J. C. P. MILLER, in “British Association for the Advancement of Science, Bessel Functions, Part II.
Functions of Integer Order, Mathematical Tables,” Vol. X, p. xvii, Cambridge Univ. Press, London/
New York, 1952.