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Multigroup linear transport in the exponential half-space—I
J. Quant. Spectrosc. Radiat. Transfer Vol. 37, No. 1, pp. 85-96, 1987 Printed in Great Britain. All rights reserved
0022-4073/87 $3.00+ 0.00 Copyright © 1987 Pergamon Journals Ltd
MULTIGROUP LINEAR TRANSPORT IN THE EXPONENTIAL HALF-SPACE--I R. DANIEL* and G. C. POMRANING School of Engineering and Applied Science, University of California, Los Angelvs, CA 90024, U.S.A. (Received 18 July 1984) Abstraet--A continuum set of singular eigenfunctions is constructed for the multigroup linear transport equation in the exponential atmosphere. This eigenfunction set is then shown to be complete for solving the subcritical half-space exit distribution problem.
1. I N T R O D U C T I O N
The exponential atmosphere t has received major attention in the continuing: development of transport theory methods for the solution of linear transport theory problems in media where the single scatter albedo is a continuous function of the optical depth variable. Particular emphasis was placed on the development of the method of singular eigenfunetion expansions. The following form of the one speed linear transport equation with isotropic scattering was considered:
#(a/Ox)~/(x, I~) + O(x, #) = (co/2)exp(-x/s) I ~ #(x, ~) dl~,
(1)
3- l where co and s are both positive constants and characterize the exponential media in question. A continuum set of singular eigenfunetions was constructed 3 and shown to be complete4 for solving the half-space exit distribution problem consisting of the transport equation, equation (1) subject to the boundary conditions ~(0, #) = f ~ ) , lim
O
(2)
~(x,#)=O, - 1 ~
(3)
X~oO
Through an expansion of the solution in these eigenfunetions and the use of an orthogonality relation, a singular integral equation was derived 3 for the exit distribution ~b(0, - p ) , 0
#(O/c~x)d/(x, I~) + ~" d/(x, p) = {[exp(-x/s)]/2}C. I t ¢/(x, g) alp, ,/- t
(4)
where #(x, g) is a column vector with its gth element #s(x,/~) representing the angular flux in group g; Ir is a diagonal G x G matrix with the gth diagonal entry Yz corresponding to the total *Present address: 10535 Wilshire Boulevard, No. 401, Los Angeles, CA 90024, U.S.A. 85
R. DANIEL and G. C. POblRANING
86
collision cross section in group g; C is a general G x G matrix with its entries C o corresponding to the scattering cross section from group j to i. We shall assume throughout that the groups are numbered in the following order: E~ ~< E~_ ~. . . . . ~< E~. We shall also assume that the optical depth transformation has been carried in such a way that E c = 1. In Section 2 we shall construct eigenfunctions for equation (4) which vanish as x --, ~ . In Section 3, we shall consider the half-space exit distribution problem and prove the completeness of the eigenfunction set of Section 1 for its solution. Finally in Section 4, we make some concluding remarks and recommendations for further investigation. In the companion article, ~2 we consider the problem of obtaining the exiting group fluxes from the half space.
2. C O N S T R U C T I O N
OF E I G E N F U N C T I O N S
We consider the ansatz for elementary solutions: 0r(x,/~) = ¢ ~ ( u ) e x p ( - x / v ) + X , ~ ) e x p ( - x/og),
(5)
where o9 - 1 = v- ] + s - ~. Inserting into equation (4) and equating the vector coefficients of the exponential linearly independent spatial modes, we obtain the vector equations O- - / ~ v -l 0 • Ikv~) = 0,
(6)
(~- --/m)-l 0) • Xv(/~) = ( C • ~ 1 ¢~(/t) d/~)/2,
O=(C.
(7) (8)
Before we present the solution of equations (6)-(8), we first define the interval ~ to be that segment of the real line contained between the points zero and ~-~. Clearly ?x c (0, 1). We also define the interval ~rg to be the segment contained between Y.g-_J~ and Y.~-~. Then for v E ~s, for some g between 1 and G, equation (6) has the distributional vector solution 0,(U) = a(v). 6 ~ , v),
(9)
where #(/~, v) is a vector distribution with elements 6j(la, v) = 6(U - v ~ j ) ,
j = 1, 2 . . . . .
G,
(10)
and where a(v) is a G x G diagonal matrix valued function with its entries given by au(v ) = as(v)ru,
= 0,
j = g, g + 1. . . . .
G
otherwise
(11)
his here is the familiar Kronecker delta, and as(v ) is an arbitrary test function. Inserting for ~,(/~) from equation (9) into equation (7), we obtain X,(#) = 2 - ' 0 - -
UOg-' 0)-'" C" a(v). f " d/~ 6(U, v). j- I
(12)
Since, in general, we may assume that det C # 0, equation (8) implies then the condition ~
X , ( u ) d u = O.
(13)
I
Inserting for X ~ )
from equation (12) we obtain the requirement 0--
2-,
du'(Z
-
• C • a(v)
•
d/~ ~(~, v).
(14)
For a given v e %, equation (14) represents G homogeneous equations for the ( G - g ) unknowns G. In general, it may be assumed that these reduced rows of C are linearly independent. This then implies that the only
a j ( v ) , j = g , g + 1. . . . . G with the coefficients being Cu, i = 1. . . . . G , j = g . . . . .
Multii~'oup linear transport--I
87
solution for the unknowns is the trivial one leading to trivial eigvnfunctions. Avoiding this construction, we rewrite Xv00 as Xv(p) = (co/2) (co~" - / ~ l ) -=. C. a(v). I ~ d/~ &(p, v) - &(p, co). ~(v), j- I
(15)
where 6(p, co) is a diagonal matrix with its entries given by
~ u 0 ~ , c o ) = ~ -co~:j)~,~, j = 1,...,G,
(16)
and Z(v) is a vector valued function to be determined from the requirement in equation (13). If coEl <1
for j = l . . . . ,G
and all v e U y g ,
then given a(v), equation (13) is satisfied by a unique choice of Z(v). Inserting for X~(/~) from equation (15) into equation (13), we obtain the solution for ~(v) as
J,(v)=(co/2)f~,
• c
• a(v)
A necessary and sufficient condition to guarantee that co Y./< 1 for j = 1. . . . . G and all
•
d/u~(/~, v).
(17)
v ~ U 7g
[i.e. all v E (0, 1)] is that
[s/(s + I)]Y~, < 1. (18) In summary, we have constructed a degenerate continuum set of eigenfunctions given by equations (5), (9), (15) and (17), with the spectral parameter v covering the interval (0, 1), provided the inequality in equation (18) is satisfied. The degeneracy of this set is due to the arbitrariness in the choice of the matrix elements in a(v). We can express this degeneracy in a more convenient way by rewriting q,,(/~) for v ~ Ys as G
0,~) = ~ aj(v)OJO, v),
(19)
J-s
where the elements of 0JO, v) are all zeroes except for the jth one which is given by ~b~(~, v) = 6 (/.= - v Y-j).
(20)
Each of the vectors #JO, v) generates an eigenvector solution of the transport equation for v ~ yj given by ~,(x, ~) = 0'(~, v)exp(-x/v) + XJ(~, v)exp(-x/co) (21) with XJ(#, v) = (co/2)(co~ -/~ ~)-'' C. e~- 6Q~, co). U(v), (22) where e / is a unit vector with only the jth element different from zero and equal to one, and is uniquely determined from f~ 0X~(~,, v)d~ =
ZJ(v) (23)
aS
/,i
~U(v) = (co/2) | (cox - #0)-' d/~- C. ~ (24) J- I provided equation (18) is satisfied. The general solution of the multigroup transport equation can now be written in the two equivalent forms #,(x,/~) ffi
fo'
dv
~v(x,/~)
(25)
or
¢/(x,#)--j.~f~jdvaj(v)#~(x,/,t).
(26)
88
R. DANIEL and G. C. POMRANING
The condition in equation (18) is certainly restrictive with regard to the exponential decay parameter s. A different ansatz is needed to construct eigenfunctions if it is violated. This may be achieved by adding more spatial modes, such as the ansatz N
dpv(X,p)= Z X , , ( v , p ) e x p [ - x ( v
l+ns
')]
(27)
n=0
where N is to be determined depending on s. We do not have, however, any explicit regarding it. In the next section, we define the half-space exit distribution problem, discuss its solvability, and prove that for a unique choice of ~(v) the superposition in equation equivalently equation (26) solves this problem. This property of the eigenfunction set is referred to as half-range completeness.
results unique (25) or usually
3. COMPLETENESS OF THE EIGENFUNCTIONS SET FOR SOLVING THE HALF-SPACE DISTRIBUTION PROBLEM The half-space exit distribution problem consists of solving equation (4) subject to the boundary conditions ~ ( 0 , p) = f(p),
limO(x,p)=0,
x~oo
0 < p .< J.
(28)
-l~
(29)
We seek a solution ~k(x, p) which is differentiable in x and integrable in p. The boundary value problem can be converted into a set of integral equations for the scalar group fluxes qbg(X), g = 1. . . . , G defined by q~g(X) = 2~
dp Og(X, p).
(30)
1
In matrix notation, this set of integral equations can be written as
dp(x) = 2 -j
f0
dx' e x p ( - x ' / s ) M
(Ix - x'l)" C. ¢(x') + 2~
;
dp I~(x, p) • f(p),
(31)
where ~ ( I x - x'l) and I~(x, p) are diagonal matrix valued functions with entries MggQX -- x'[) =
E I ( ~g
IX - - X" I ) ,
(32)
Rgg(X, IA) --~ exp ( - Z e x /p ),
(33)
and E, (0) is the familiar first order exponential integral function of argument 0. A unique solution for ¢(x) is known to exist 's with qbg(x)eL2[O, oo]~ C[0, oo], l~
(2)fg(p)eL'[0,1],
g = 1. . . . . G.
The corresponding unique solution '3 of the boundary value problem for ~,(x, p) is then given in terms of ¢(x) by @(x,p)=(4~#)-'
dx'exp(-x'/s)N(x-x',p).C.~(x')+R(x,p).f(p),
@(x,p)= --(4rip) l f ; d x ' e x p ( - x ' / s ) i ( x
-x',p).C.¢(x'),
O
(34) (35)
Clearly, the superposition in equation (25) is a solution of equation (4) which is differentiable in x and integrable in p. Moreover it satisfies the boundary condition in equation (21). Thus, if for a unique choice of a(v), this superposition satisfies the boundary condition at x = 0, given by
Multigroup linear t r a n s p o r t ~ I
89
equation (28), and if for this same choice of a(v), the corresponding superposition for the scalar flux vector O(x) given by ~(x) = 2n
dft
;0'
dv~,(x, p)
(36)
is the unique solution of equation (31) with its elements (~e(x)~ L2[0, oo] f)C[0, oo], then the superposition in equations (25), or equivalently (26), represents the solution in equation (34) and (35), and the eigenfunctions set is, in this sense, half-range complete. We shall prove this half-range completeness property by first taking the half-space Fourier transform of equation (31) (essentially, it is a Laplace transform). Second, through analysis in the transform plane, we shall show that provided the inequality in equation (18) is satisfied then the desired solution for ~(x) can be written from the inversion integral as q)(x) =
2~zJ=,~f,j dvb,(v)exp(-x/v)e/,
(37)
where bj(v) are uniquely determined functions of v on ?j, forj = 1. . . . , G. Moreover, we show that these functions bj(v) are the unique solutions of the set of singular integral equations that arise for the expansion coefficients aj(v) when we require that the superposition in equation (26), or equivalently (25), satisfies the boundary condition in equation (28). Since the corresponding superposition for the scalar flux factor from equation (26) [or equation (25)] has precisely the same form as equation (37) with bj(v) replaced by aj(v), it then becomes clear that the half-range completeness property is guaranteed for a unique choice of the expansion coefficient given by
aj(v) bj(v). =
(38)
Our proof will be a generalization of a proof already carried out for the one speed (one group) isotropic scattering case. 4 We now define extensions of ~,(x, #) and ~(x) to all x by
fV,(x, #),
x >1 o, x < o,
(39)
~b+(x) - 27t I' q,+ (x, ~) d~. d- I We also define their half-space Fourier transforms as
(40)
~+ (x, p) - \ 0 ,
~;+ (~' P) = I ~ (Ix exp(i~x) g,+ (x, p),
(41)
~+ (~) = f~oo (ix exp(i~x) 4+ (x).
(42)
Multiplying equation (4) by exp(i~x), integrating over x from 0 to oo and using the definitions in equations (39)-(42) we obtain (~ -- i/t~ U)~; + (~, p) = # ~, (0, g) + (47t)-' C . ~ + (~ + ifl),
(43)
where p =
s-1
(44)
Multiplying by 2~z(~ - i#~ 0)-I, integrating over p from - 1 to 1 and using equations (40) and (42), we obtain the half-space Fourier transform of equation (31), ~+ (() -- G(() + 2 -I F(~). C. ~+ (~ + ifl),
(45)
where we have defined the column vector G(q) by G(~) = 2~t
I'
-I
~t(~: - i~# 0)-I. O (0, #) d#
(46)
90
R. DANIEL and G. C. P o ~ N ( }
and the diagonal matrix 0=(~) by ~:(~) =
f,
d#(Ii - i~# a)-'.
(47)
--1
Since q~÷~(x) s L210, 0o] N C[0, 0o], then in the upper half plane, its half-space Fourier transform ~+g(~) is analytic, vanishes as [~1 ~ 0o and is therefore uniformly bounded. To analytically continue ~+ (~) to the lower half plane we shall make use of equation (45). To this end, we need to understand the properties of G(~) and g:(~). We shall assume that ~f~(~), g = 1. . . . . G, is a H61der function on [0, 1]. It then follows that G~(~) is analytic in the entire lower half plane cut from ( - i oo, - i Z s ] and has in general a logarithmic singularity at ~ = - i Z g . (The only exception would be if ~fz(~) vanishes as p ~ 1.) Next, the function F~g(~) is analytic in the lower half plane off the cut ( - i oe, - i Z ~ ] and has a logarithmic singularity at ~ = - i Z v With the domain of analyticity of G~(~) and Fgg(~) identified in the lower half plane (Ira ~ < 0) for all g, we can, through equation (45), determine the domain of analyticity in the lower half plane of the analytic continuation of each component of ~÷ (~). We have G
r~+~(~) = Gg(~) + 2-' Fu(~) ~ C~r~ ÷j(~ + ifl),
g = 1. . . . . G.
(48)
j=l
From the analyticity of ~÷j(~) in the upper half plane for all j and the analyticity of G~(~) and Fg~(~) for ~ ¢ ( - i o o , - i ) , it is possible to construct, through a recursive argument using the formula in equation (48), the analytic continuation of ~÷g(~) to the lower half plane cut from (-ioo, -il. Moreover, it is clear from equation (48) that q~÷g(~) is discontinuous across the cut ( - J o e , -iZ~). It remains to identify whether the segment [2-iZg, - i ] belongs to its domain of analyticity. Clearly, from equation (48), if Z I - 1 < fl then 4~÷g(~) is analytic for ~ ~ [-iY.~, - i ] since ~ + ifl belongs to the domain of analyticity of q~+j(~) for allj. This condition on X~ and fl is precisely what we had imposed earlier in the construction of eigenfunctions in equation (18). To summarize our results up to this point: q~+,(~) is analytic in the entire lower half plane cut from (--io0, -- iX~]. Moreover, from equation (48), $+~(~) has a logarithmic singularity at ~ = - i Z ~ , and by recursion at all the points ~ = - i X ~ - infl, n = 1. . . . , oo, and ~m~.= - i Z ~ - imfl, m = l , . . . , oo, for all g' # g . Our aim is to deform the path of the inversion integral of O+~(~) along the real axis, into an integral around the cut ( - i oo, -iX~]. To justify this deformation we need to establish the following two facts: (1) That the inversion integral along the connecting contour, that is the semi-circle with radius R and center at the origin, vanishes as R ~ oo. By Jordan's Lemma, ~4 a sufficient condition for this is that I~+~(~)1 be bounded for R large enough by a function of I~1 which vanishes as Ill ~ ~ . (2) Denoting by q~+~s(~) the limiting values of q~+~(~) as ~ approaches the cut ( - i ~ , - iZ~] from the right and the left, respectively, we need to show that the inversion integral around the cut, i.e. /" - iI:~ (2hi) -~ J-,o~ d~ [$ ++,(~) - q~~.,(~)]exp(- i~x)
(49)
exists for all x >t 0. We shall now devote two subsections, 3.1 and 3.2, for proving facts (1) and (2) respectively. 3.1. A uniform bound on I$+,(¢)1 We shall establish fact (1) by proving that I$+,(~)1 < As,/l~ I,
(50)
As, being a large enough positive constant and ~ being anywhere in the lower half-plane not belonging to some ~ neighborhoods of the points ~g, n = 0, 1. . . . . ~ , and ~,~g,, m = 1, . . . , ~ , all g' :# g. To prove equation (50), we shall use equation (48) and some properties of Gs (¢) and Fs~ (~). For the case g = G, it has been shown ~ that IG~(£)I and Ig~(~) I are uniformly bounded in the lower
Multigroup linear transport~I
91
half plane cut from the ~ neighborhood of the point - i by B,/I~I, with B, a large enough positive constant. Through a slight modification of their proof, it can be shown that for any g and any such that I~ + i ~ l > ~, IG~(~)I < Bs,/l~l,
(51)
IF,~(~)I < B,,/I~I,
(52)
where B~, are large enough positive constants. Inserting these estimates into equation (48), we have, for all ~ outside the e neighborhood of - i ~ , the following inequality G
Iq~+~(~)l ~< (B~,/I~I) + (2-~B~/I~I) ~ C~l~+s(~ + i[j)l,
g - 1. . . . . G.
(53)
To proceed with the proof, we subdivide the lower half plane, cut from the E-neighborhoods of the points ~,g = -iY.g - in[J, g = 1 , . . . , G, n = 0, 1. . . . , ~ , into an infinite number of strips Sin., of width [j with Sin,, = {~: - ( m + 1)[j ~< Im~ ~< -re[j}, m = 0, 1. . . . . oo. Since for all g, I~+g(~)l is uniformly bounded in the upper half plane by some large enough positive constant M, we have from equation (53), for ~ ~ So.,, G
I~+s(~)l <~ (B~c/l~l) + 2-~M(Bs,/I~I) ~, C~,
g -- 1. . . . . G.
(54)
j=l G
If we define Dg,0 = Be, + 2 -~ M B ~ ~ C~, then, for ~ E So,, we have from equation (54). j=l
I~+~(¢)1
~Ds,o/l~l,
g = 1. . . . .
G.
(55)
Similarly, using this estimate and equation (53), we can easily deduce that for ~ ¢ Su the following estimate holds:
I~+~(~)1 ~
g = I,...,G,
(56)
where Ds, t is some large enough positive constant. Continuing in this way, we can show that, for any m and ~ e S,,,
I~+,(¢)1 ~
=
1.... ,G.
(57)
where, again Dg,m is a large enough positive constant. If we can show that the sequence {Dg,m}~.t can be taken to be bounded, then clearly for ~ UmS,,~ we would have the result
1~+8(~)1 ~
(58)
Dr, m .
ra
Since q~+~(¢) is bounded in the ~ neighborhoods of -i~.s., g' # g, we conclude then that there exists a large enough positive constant Az, such that I~+~(~)1 ~< As,/l~l,
(59)
for all ¢ not in the E neighborhoods of the points l , t - - - - i E g - i n [ j , n = 0 , 1 , . . . , ~ , and ~,,,g. = - i Y . e -ira[j, m = 1. . . . . ~ , g' ~ g. This is precisely the desired result in equation (50). To prove that the sequence {Dg,m }~. ~can be taken as bounded, we return to equation (57) which implies that I~+~(~)1 is bounded in every strip S,~, m >I 1. This then allows us to define Us,,. - sup I$+s(~)l, ~s.~
g = 1. . . . . G;
m >t 1
(60)
which, from equation (53) satisfies G
U,o,, <~(B,,/rn[j) + 2-'fBs,/m[j) ~ CnUs,,,,_ ,, j-I
m >1 1, g = 1. . . . . G.
(61)
92
R. D ~ L and G. C. POMRANING
Next, recalling Co = s u p g,)
and defining U,,. = sup U~,r., B, = sup Bs,, g
g
we obtain from equation (61) the inequality
Uo. <~(B,/m~) + CoG& U<,._,/2m~,
m i> 1.
(62)
A recursive inequality of this form, was shown 4 to imply lira U~m= 0.
(63)
From equation (62), this implies that (-1o, vanishes at least as fast as m - '. The same result necessarily holds regarding ]q~+s(~)[ for ~ E S,,~. Consequently, since for ~ s Sin,, ]~l-~= O(m-z) as m ~ ~ uniformly in Sin,, we conclude from equation (57) that the sequence Dg,,, is indeed bounded. Thus, proving the estimate in equation (59) or (50). An alternate way to derive this same result is to realize that, from equation (63), we have the uniform boundedness by some constant of Iq~+g(~)[ for
eEu&, for all g. Using this information in equation (53), we readily deduce the result in equation (59). We have therefore shown that the inversion integral vanishes around the infinite semi-circle. The next subsection is devoted to proving that the inversion integral around the cut ( - i o o , -iY.e] exists. 3.2. Convergence of the inversion integral around the cut We consider the inversion integral of q~+g(¢) around the cut ( - i v , -iY~g] in equation (49). We first recall that q~+g(~) has logarithmic singularities at the points ~g. and ~s',.. Moreover, Gg(~) and F~(~) are H61der continuous on the open interval ( - i ~ , -iZs). Using equation (48), it can be easily shown through a recursive argument, similar to previous ones, that this implies that q~_g(¢) is Hflder continuous on every closed interval of ( - i v , - i Y . s ) not containing the logarithmic singularity points ep and ¢s',." Therefore, the discontinuity Ag(~) = q~+g(~) - q~+s(~), has only logarithmic singularities on the cut ( - i v , - i Y s ] at the points ~s. and ~s',., and is otherwise H61der continuous. It follows that As(~ ) is integrable and square integrable on [ - Jr, - iY.s] for any finite r > x:s" The existence of the inversion integral in equation (49) can be easily shown to be guaranteed if IAs(~) I is integrable on [ - i oo, -iX:g]. To prove this, we first obtain bounds on Iq~+s(~)l in the neighborhoods of its logarithmic singularities. Applying equation (45) recursively n times, we easily obtain the relation
H(¢ +im'fl)
#J+(~)=G(~)+
.G(~ +im[3)+
H(~ +im'[3) .~+(~ +infl),
(64)
m-1 tin'=0
where we have defined
(65)
a ( ~ ) = 2 - ' a:(~) • C.
For ~ not in the E neighborhood of the points - iZ s, g = 1. . . . . G, there exists by equation (52) a large enough positive constant Ns, such that the following estimate holds for the matrix elements H,,(~):
Insi(¢)l ~
(66)
Next, for ~ in the E neighborhood of the point - iZ s, there exists a large enough positive constant Ms, such that the following estimate holds I$+s(~)l ~< M~,logl¢ + ix:sl -~,
for
14 + iXsl ~< e < 1.
(67)
Muitigroup linear transport--I
93
Consequently, using equations (67), (66), (57) and (59) in equation (64) and letting ~ belong to the E neighborhood of C,,- -iT, s - in[3, n ffi 1. . . . . oo, we obtain the following estimate
'{( )( ) ( ) t < ,.._,._, ( ).-,
I~+,(C)I ~< (Bs,/[CI) + ~
sup Ns~
sup Bg~ G m f i [C +/m'fll
--I
m'--O
]I I~ + im'fll(G - 1) supAg, I~ + infll-'
.-0
+
l-I I~+im'/3l-'Ms, logl~+iY~g+inBl-',
supN~
m'~0
I¢ - Cn~l,.< E < 1, n = 1 , . . . , ~ .
(68)
The summation term on the fight hand side of equation (68) can be easily shown to be bounded by R, ICI-~ where R, is a large enough positive constant. It is also a trivial argument to show that the third and fourth terms on the fight hand side of equation (68) are bounded by Z, ICI-I and V~ ICI-~ logic - ~ngl-~ respectively, with Z, and Vs, being again large enough positive constants. It follows then that I~+~(C)I satisfies the following estimate
I~+g(~)l~
n = l , 2 . . . . . oo,
(69)
where Wg,= R, + Z, + Bg,. From equations (67) and (69), it easily follows that for E < e-~ we have the estimate I~+g(~)l ~< T,,ICI -t logl~ - ~,,I-', n = 0, 1 , . . . ,
IC - ~,~t ~<~ < e x p ( - 1),
(70)
oo,
where T,, = Wg, + Vg, + Mg," (~s + ~). In a similar manner, it can be shown that the following estimate holds for I~+r (~)1 for all g' ~ g:
I~+r(~)l ~ Tgr, ICI-~ logic - ~n,I-~, I~ - ~,~1 <~~ < e x p ( - 1),
(71)
n ----1 , . . . , o 0 . Consequently, I~+~(~)1 satisfies the following estimate
, (72) for all ~ in the lower-half plane (Ira ~ < 0). We shall now use this estimate to prove that lAg(C)1 is indeed integrable on [ - i o o , - i Y , ] as we set out to prove earlier in this subsection. First, we note that by a repetition of a proof in the Appendix of Ref. [4], we have the following estimate for Gs(~) G , ( ~ ) = 8ni~-~ I' ~bs(0,#)d # +0(C-21og~), d- l
]CI -* ~ .
(73)
From equation (48) and the estimate in equations (72) and (52), it follows that lAg(l)] satisfies the estimate
,A,(~)I <...K~I~ I-2 + K,, I~I-21og I~I + B,,ICI-' (~=~ C,j)
g'~g
where
K@ and Kg~ are large enough positive constants.
94
R. DANIEL a n d G. C. POMRANING
Now, by a repetition of an argument in Ref. [4], we can assert that the estimate in equation (74) implies that IAg(~)l is integrable.on [-i,~o,,-iY~g] and that IeAg(~)l is square integrable on this same cut. The inversion integral in equation (49) therefore exists and through a change of variable, ~bg(x) can be written as
I
~bg(x) = 2n
~:k-I
(75)
bg(v)exp(-x/v) dv,
dO
where
(2~v)-~a,(- iv - ' ) .
b,(v) -
(76)
From equation (76), the function bs(v) has logarithmic singularities at the points vg. = - i ¢ ~ ~, n = 0 , 1 . . . . . ~ , and the points v t , . = - i ~ . ~ , , g ' ~ G , n = l . . . . ; ~ ; however, bg(v ) is H61der continuous on every subinterval of [0, Z f j] not containing the points vg, and vg,,, and is square integrable on [0, I f ' ] by virtue of the square integrability of 1~Ag(~)[ on [ - i ~ , -iEg]. It is trivial that ~bg(x) as given by equation (37) is continuous on [0, ~]. Moreover, by direct application of the Schwarz inequality and the fact that bg(v) E L2[0, ~ f ~], it is easy to show that ~bg(X) e L2[0, ~]. This then implies that the desired solution for ¢~(x) is indeed as represented in equation (37) with the functions bj(v) having the properties described above. The next task in the completeness proof is to show that the functions bj(v) as given by equation (76) are the unique solution for the expansion coefficients aj(v) of the set of singular integral equations f(p) =
fo'
dye,(0,/~),
0 < # ~< 1,
(77)
which arise upon imposing the boundary condition, equation (28), on the expansion in equation (48) or equivalently equation (49). The unique solvability is understood to be restricted to functions aj(v) which through the superposition in equation (36) yield solutions for 4~(x) of equation (31) which are continuous and square integrable on [0, ~ ] and to functions/~fg(/~) which are H61der continuous on [0, 1]. 3.3. Unique solvability of equation (77) We insert the solution for ¢~(x) from equation (75) into equation (31), and carry the integration over x' to obtain, after some rearrangements,
;
f0f
dv exp(-x/v)b(v) = 2 -I
dv
d~-I
R(x, ~).Z(x, v, ~)" C" b(v)
0
+
f0
d/a R(x,/~) • f(/~) -
f0 i
i~-~R(x,p).T(x,v,l~).C.b(v)d#,
dr2 -~
(78)
I
where b(v) is a column vector with elements Bg(v); T(x, v, #) and Z(x, v, I~) are diagonal matrix valued functions with entries
Zgg(x, v,/~) = co#(o~ Zg - #)-~ {exp[x(Zg # -~ - co -~)] - 1},
(79)
T , , ( x , v, ~,) = 1 - Z , , ( x , v, u ) .
(80)
and
Upon carrying the matrix products R • Z and R • T, equation (78) can be written as
fo'
;of fo'r'
dv exp(-x/v)b(v) = --
d/~co2-1 (co~" --/In) -1. C. b(v)exp(-x/og)
dv
d~
I
dvco2-t(wY-l~a)-~'R(x, lO'C'b(v)
dO
+
l
fo
d/~R(x, ~)" f(#).
(81)
Multigroup finear transport--I
95
By equating the coefficients of the linearly independent spatial modes (exponential modes) and assuming equation (18) to hold, we obtain,the vector.set-of singular integral equations ~ d v b ( v ) ~ , v ) = f~ dvco2-'(coY-/~0)-'" C" b(v) -['dv6(/~,co)co2-'[" d/z'(coY-#'U)-"C'b(v)+f(~), 0 < / ~ < 1 , (82) J0 J- 1 where ~(~, v) and 6(/~, co) are as defined in Section 2, and b(v) is a diagonal matrix with elements
b,(v). By replacing for #v(0,/z) in equation (77), we can readily see that we recover equation (82) which is solved by the functions bx(v ). A solution of equation (77) for the functions ag(v) therefore exists and is given by bg(v). The uniqueness can be shown in a simple way as follows. Suppose another set of functions hg(v) solves equation (77). By multiplying equation (77) by e x p ( - x / # ) and integrating over/~ from 0 to 1, we obtain equation (81) with hz(v ) replacing bs(v ). Since this equation was obtained from equation (78) through a reversible sequence of operations, we conclude that
2r~
;o'
h(v)exp(-x/v) dv
must be a solution of equation (31). If we require that the functions hs(v ) yield solutions ~bs(x) e L2[0, oo] n C[0, oo] (a sufficient condition being hs(v)e L2[0, Y~-']), then necessarily by the unique solvability of equation (31) we must have
fo' h(v)exp(- x /v ) dv = f ] b(v )exp(- x /v ) dv
(83)
hg(v) = bg (v), g = 1. . . . . G,
(84)
which clearly implies
hence proving uniqueness of the desired solution of equation (77) and terminating the completeness proof. 4. SUMMARY AND DISCUSSION We conclude this article with a summary of the important results obtained, and some recommendation for future investigations. We have shown that under the condition in equation (18), a continuum degenerate set of eigenfunctions of equation (4) is constructable. The ansatz is equation (5) and the explicit form of the eigenfunctions is given by equations (9), (10), (11), (15), (16) and (17). This set of eigenfunctions is complete for expanding the desired solution of the half-space exit distribution problem ("desired", in the sense discussed at the outset of Section 3). The completeness property removes the degeneracy of the set, and was shown to hold for essentially H61der continuous incident angular currents with the expansion coefficients having logarithmic singularities at an infinite set of points which becomes dense near zero, being otherwise H61der continuous, and satisfying the property of square integrability. Whether eigenfunctions can be constructed without the restrictive condition in equation (18) still requires further investigation. One likely ansatz is that given by equation (27). One interesting property of this half-range complete set of eigenfunctions is that in the homogeneous media limiting case of s = ~ we do not recover the discrete eigenmodes which are known to exist for a large class of matrices C. This peculiar discontinuity in the spectrum of v as s --. oo will be seen to be a source of numerical difficulties in the solution for the exit distributions via collocation schemes, n The fact that the logarithmic singularities in the expansion coefficients as(v ) becomes dense on the entire interval [0, ~-1] is conjectured to be a compensation for the non-recovery of discrete modes which carry the asymptotic solution. The technicalities of such a compensation are not understood yet. Finally, we conclude by posing the open problem of eigenfunction construction for the purpose of solving the finite slab reflection-transmission problem. Whether a complete set can be
96
R. DANIEL and (3. C. POMRA~NG
constructed for this purpose is doubtful if done with a simple ansatz as in equation (5); our experience with the one speed case ~5 suggests that an ansatz as in equation (27) with N = ~ is needed. Acknowledgement--This work was partially supported by the National Science Foundation. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
B. J. Martin, J. appl. Math. 20, 703 (1971). E. W. Larsen, Prog. nucl. Energy 8, 203 (1981). T. W. Mullikin and C. E. Siewert, Ann. nucl. Energy 7, 205 (1980). E. W. Larscn and T. W. Mullikin, J. Math. Phys. 22, 866 (1981). N. I. Muskhelishvili, Singular Integral Equations. Nordhoff, Groningen (1953). C. E. Siewert and P. Bcnoist, Nucl. Sci. Engng 69, 156 (1979). P. (3randjcan and C. E. Siewert, Nucl. Sci. Engng 69, 161 (1979). R. D. M. (3arcia and C. E. Siewert, JQSRT 25, 277 (1981). W. W. Engle Jr, The User's Manual for ANISN: A One Dimensional Discrete Ordinate Code with Anisotropic Scattering, K-1693, Oak Ridge Gaseous Diffusion Plant (1967), available from Radiation Shielding Information Center, Oak Ridge National Laboratory, Oak Ridge, Tenn. as CCC 82/ANISN (March, 1967). C. T. Kelley and T. W. Mullikin, J. int. Eqs. 4, 77 (1982). T. W. Mullikin, Some Singular Integral Equations in Linear Transport Theory, Sandia National Laboratory Report SAND-80-1069, Albuquerque, N.M. (1980). R. Daniel and (3. C. Pomraning, JQSRT 37, 97 (1987). C. T. Kelley, J. int. Eqs. 3, 261 (1981). Carrier, Krook and Pearson, Functions of a Complex Variable. Mc(3raw-Hill, New York (1966). R. Daniel, (3. C. Pomraning and E. W. Larsen, J. Trans. Theory Statist. Phys. 11(2), 75 (1982).
0022-4073/87 $3.00+ 0.00 Copyright © 1987 Pergamon Journals Ltd
MULTIGROUP LINEAR TRANSPORT IN THE EXPONENTIAL HALF-SPACE--I R. DANIEL* and G. C. POMRANING School of Engineering and Applied Science, University of California, Los Angelvs, CA 90024, U.S.A. (Received 18 July 1984) Abstraet--A continuum set of singular eigenfunctions is constructed for the multigroup linear transport equation in the exponential atmosphere. This eigenfunction set is then shown to be complete for solving the subcritical half-space exit distribution problem.
1. I N T R O D U C T I O N
The exponential atmosphere t has received major attention in the continuing: development of transport theory methods for the solution of linear transport theory problems in media where the single scatter albedo is a continuous function of the optical depth variable. Particular emphasis was placed on the development of the method of singular eigenfunetion expansions. The following form of the one speed linear transport equation with isotropic scattering was considered:
#(a/Ox)~/(x, I~) + O(x, #) = (co/2)exp(-x/s) I ~ #(x, ~) dl~,
(1)
3- l where co and s are both positive constants and characterize the exponential media in question. A continuum set of singular eigenfunetions was constructed 3 and shown to be complete4 for solving the half-space exit distribution problem consisting of the transport equation, equation (1) subject to the boundary conditions ~(0, #) = f ~ ) , lim
O
(2)
~(x,#)=O, - 1 ~
(3)
X~oO
Through an expansion of the solution in these eigenfunetions and the use of an orthogonality relation, a singular integral equation was derived 3 for the exit distribution ~b(0, - p ) , 0
#(O/c~x)d/(x, I~) + ~" d/(x, p) = {[exp(-x/s)]/2}C. I t ¢/(x, g) alp, ,/- t
(4)
where #(x, g) is a column vector with its gth element #s(x,/~) representing the angular flux in group g; Ir is a diagonal G x G matrix with the gth diagonal entry Yz corresponding to the total *Present address: 10535 Wilshire Boulevard, No. 401, Los Angeles, CA 90024, U.S.A. 85
R. DANIEL and G. C. POblRANING
86
collision cross section in group g; C is a general G x G matrix with its entries C o corresponding to the scattering cross section from group j to i. We shall assume throughout that the groups are numbered in the following order: E~ ~< E~_ ~. . . . . ~< E~. We shall also assume that the optical depth transformation has been carried in such a way that E c = 1. In Section 2 we shall construct eigenfunctions for equation (4) which vanish as x --, ~ . In Section 3, we shall consider the half-space exit distribution problem and prove the completeness of the eigenfunction set of Section 1 for its solution. Finally in Section 4, we make some concluding remarks and recommendations for further investigation. In the companion article, ~2 we consider the problem of obtaining the exiting group fluxes from the half space.
2. C O N S T R U C T I O N
OF E I G E N F U N C T I O N S
We consider the ansatz for elementary solutions: 0r(x,/~) = ¢ ~ ( u ) e x p ( - x / v ) + X , ~ ) e x p ( - x/og),
(5)
where o9 - 1 = v- ] + s - ~. Inserting into equation (4) and equating the vector coefficients of the exponential linearly independent spatial modes, we obtain the vector equations O- - / ~ v -l 0 • Ikv~) = 0,
(6)
(~- --/m)-l 0) • Xv(/~) = ( C • ~ 1 ¢~(/t) d/~)/2,
O=(C.
(7) (8)
Before we present the solution of equations (6)-(8), we first define the interval ~ to be that segment of the real line contained between the points zero and ~-~. Clearly ?x c (0, 1). We also define the interval ~rg to be the segment contained between Y.g-_J~ and Y.~-~. Then for v E ~s, for some g between 1 and G, equation (6) has the distributional vector solution 0,(U) = a(v). 6 ~ , v),
(9)
where #(/~, v) is a vector distribution with elements 6j(la, v) = 6(U - v ~ j ) ,
j = 1, 2 . . . . .
G,
(10)
and where a(v) is a G x G diagonal matrix valued function with its entries given by au(v ) = as(v)ru,
= 0,
j = g, g + 1. . . . .
G
otherwise
(11)
his here is the familiar Kronecker delta, and as(v ) is an arbitrary test function. Inserting for ~,(/~) from equation (9) into equation (7), we obtain X,(#) = 2 - ' 0 - -
UOg-' 0)-'" C" a(v). f " d/~ 6(U, v). j- I
(12)
Since, in general, we may assume that det C # 0, equation (8) implies then the condition ~
X , ( u ) d u = O.
(13)
I
Inserting for X ~ )
from equation (12) we obtain the requirement 0--
2-,
du'(Z
-
• C • a(v)
•
d/~ ~(~, v).
(14)
For a given v e %, equation (14) represents G homogeneous equations for the ( G - g ) unknowns G. In general, it may be assumed that these reduced rows of C are linearly independent. This then implies that the only
a j ( v ) , j = g , g + 1. . . . . G with the coefficients being Cu, i = 1. . . . . G , j = g . . . . .
Multii~'oup linear transport--I
87
solution for the unknowns is the trivial one leading to trivial eigvnfunctions. Avoiding this construction, we rewrite Xv00 as Xv(p) = (co/2) (co~" - / ~ l ) -=. C. a(v). I ~ d/~ &(p, v) - &(p, co). ~(v), j- I
(15)
where 6(p, co) is a diagonal matrix with its entries given by
~ u 0 ~ , c o ) = ~ -co~:j)~,~, j = 1,...,G,
(16)
and Z(v) is a vector valued function to be determined from the requirement in equation (13). If coEl <1
for j = l . . . . ,G
and all v e U y g ,
then given a(v), equation (13) is satisfied by a unique choice of Z(v). Inserting for X~(/~) from equation (15) into equation (13), we obtain the solution for ~(v) as
J,(v)=(co/2)f~,
• c
• a(v)
A necessary and sufficient condition to guarantee that co Y./< 1 for j = 1. . . . . G and all
•
d/u~(/~, v).
(17)
v ~ U 7g
[i.e. all v E (0, 1)] is that
[s/(s + I)]Y~, < 1. (18) In summary, we have constructed a degenerate continuum set of eigenfunctions given by equations (5), (9), (15) and (17), with the spectral parameter v covering the interval (0, 1), provided the inequality in equation (18) is satisfied. The degeneracy of this set is due to the arbitrariness in the choice of the matrix elements in a(v). We can express this degeneracy in a more convenient way by rewriting q,,(/~) for v ~ Ys as G
0,~) = ~ aj(v)OJO, v),
(19)
J-s
where the elements of 0JO, v) are all zeroes except for the jth one which is given by ~b~(~, v) = 6 (/.= - v Y-j).
(20)
Each of the vectors #JO, v) generates an eigenvector solution of the transport equation for v ~ yj given by ~,(x, ~) = 0'(~, v)exp(-x/v) + XJ(~, v)exp(-x/co) (21) with XJ(#, v) = (co/2)(co~ -/~ ~)-'' C. e~- 6Q~, co). U(v), (22) where e / is a unit vector with only the jth element different from zero and equal to one, and is uniquely determined from f~ 0X~(~,, v)d~ =
ZJ(v) (23)
aS
/,i
~U(v) = (co/2) | (cox - #0)-' d/~- C. ~ (24) J- I provided equation (18) is satisfied. The general solution of the multigroup transport equation can now be written in the two equivalent forms #,(x,/~) ffi
fo'
dv
~v(x,/~)
(25)
or
¢/(x,#)--j.~f~jdvaj(v)#~(x,/,t).
(26)
88
R. DANIEL and G. C. POMRANING
The condition in equation (18) is certainly restrictive with regard to the exponential decay parameter s. A different ansatz is needed to construct eigenfunctions if it is violated. This may be achieved by adding more spatial modes, such as the ansatz N
dpv(X,p)= Z X , , ( v , p ) e x p [ - x ( v
l+ns
')]
(27)
n=0
where N is to be determined depending on s. We do not have, however, any explicit regarding it. In the next section, we define the half-space exit distribution problem, discuss its solvability, and prove that for a unique choice of ~(v) the superposition in equation equivalently equation (26) solves this problem. This property of the eigenfunction set is referred to as half-range completeness.
results unique (25) or usually
3. COMPLETENESS OF THE EIGENFUNCTIONS SET FOR SOLVING THE HALF-SPACE DISTRIBUTION PROBLEM The half-space exit distribution problem consists of solving equation (4) subject to the boundary conditions ~ ( 0 , p) = f(p),
limO(x,p)=0,
x~oo
0 < p .< J.
(28)
-l~
(29)
We seek a solution ~k(x, p) which is differentiable in x and integrable in p. The boundary value problem can be converted into a set of integral equations for the scalar group fluxes qbg(X), g = 1. . . . , G defined by q~g(X) = 2~
dp Og(X, p).
(30)
1
In matrix notation, this set of integral equations can be written as
dp(x) = 2 -j
f0
dx' e x p ( - x ' / s ) M
(Ix - x'l)" C. ¢(x') + 2~
;
dp I~(x, p) • f(p),
(31)
where ~ ( I x - x'l) and I~(x, p) are diagonal matrix valued functions with entries MggQX -- x'[) =
E I ( ~g
IX - - X" I ) ,
(32)
Rgg(X, IA) --~ exp ( - Z e x /p ),
(33)
and E, (0) is the familiar first order exponential integral function of argument 0. A unique solution for ¢(x) is known to exist 's with qbg(x)eL2[O, oo]~ C[0, oo], l~
(2)fg(p)eL'[0,1],
g = 1. . . . . G.
The corresponding unique solution '3 of the boundary value problem for ~,(x, p) is then given in terms of ¢(x) by @(x,p)=(4~#)-'
dx'exp(-x'/s)N(x-x',p).C.~(x')+R(x,p).f(p),
@(x,p)= --(4rip) l f ; d x ' e x p ( - x ' / s ) i ( x
-x',p).C.¢(x'),
O
(34) (35)
Clearly, the superposition in equation (25) is a solution of equation (4) which is differentiable in x and integrable in p. Moreover it satisfies the boundary condition in equation (21). Thus, if for a unique choice of a(v), this superposition satisfies the boundary condition at x = 0, given by
Multigroup linear t r a n s p o r t ~ I
89
equation (28), and if for this same choice of a(v), the corresponding superposition for the scalar flux vector O(x) given by ~(x) = 2n
dft
;0'
dv~,(x, p)
(36)
is the unique solution of equation (31) with its elements (~e(x)~ L2[0, oo] f)C[0, oo], then the superposition in equations (25), or equivalently (26), represents the solution in equation (34) and (35), and the eigenfunctions set is, in this sense, half-range complete. We shall prove this half-range completeness property by first taking the half-space Fourier transform of equation (31) (essentially, it is a Laplace transform). Second, through analysis in the transform plane, we shall show that provided the inequality in equation (18) is satisfied then the desired solution for ~(x) can be written from the inversion integral as q)(x) =
2~zJ=,~f,j dvb,(v)exp(-x/v)e/,
(37)
where bj(v) are uniquely determined functions of v on ?j, forj = 1. . . . , G. Moreover, we show that these functions bj(v) are the unique solutions of the set of singular integral equations that arise for the expansion coefficients aj(v) when we require that the superposition in equation (26), or equivalently (25), satisfies the boundary condition in equation (28). Since the corresponding superposition for the scalar flux factor from equation (26) [or equation (25)] has precisely the same form as equation (37) with bj(v) replaced by aj(v), it then becomes clear that the half-range completeness property is guaranteed for a unique choice of the expansion coefficient given by
aj(v) bj(v). =
(38)
Our proof will be a generalization of a proof already carried out for the one speed (one group) isotropic scattering case. 4 We now define extensions of ~,(x, #) and ~(x) to all x by
fV,(x, #),
x >1 o, x < o,
(39)
~b+(x) - 27t I' q,+ (x, ~) d~. d- I We also define their half-space Fourier transforms as
(40)
~+ (x, p) - \ 0 ,
~;+ (~' P) = I ~ (Ix exp(i~x) g,+ (x, p),
(41)
~+ (~) = f~oo (ix exp(i~x) 4+ (x).
(42)
Multiplying equation (4) by exp(i~x), integrating over x from 0 to oo and using the definitions in equations (39)-(42) we obtain (~ -- i/t~ U)~; + (~, p) = # ~, (0, g) + (47t)-' C . ~ + (~ + ifl),
(43)
where p =
s-1
(44)
Multiplying by 2~z(~ - i#~ 0)-I, integrating over p from - 1 to 1 and using equations (40) and (42), we obtain the half-space Fourier transform of equation (31), ~+ (() -- G(() + 2 -I F(~). C. ~+ (~ + ifl),
(45)
where we have defined the column vector G(q) by G(~) = 2~t
I'
-I
~t(~: - i~# 0)-I. O (0, #) d#
(46)
90
R. DANIEL and G. C. P o ~ N ( }
and the diagonal matrix 0=(~) by ~:(~) =
f,
d#(Ii - i~# a)-'.
(47)
--1
Since q~÷~(x) s L210, 0o] N C[0, 0o], then in the upper half plane, its half-space Fourier transform ~+g(~) is analytic, vanishes as [~1 ~ 0o and is therefore uniformly bounded. To analytically continue ~+ (~) to the lower half plane we shall make use of equation (45). To this end, we need to understand the properties of G(~) and g:(~). We shall assume that ~f~(~), g = 1. . . . . G, is a H61der function on [0, 1]. It then follows that G~(~) is analytic in the entire lower half plane cut from ( - i oo, - i Z s ] and has in general a logarithmic singularity at ~ = - i Z g . (The only exception would be if ~fz(~) vanishes as p ~ 1.) Next, the function F~g(~) is analytic in the lower half plane off the cut ( - i oe, - i Z ~ ] and has a logarithmic singularity at ~ = - i Z v With the domain of analyticity of G~(~) and Fgg(~) identified in the lower half plane (Ira ~ < 0) for all g, we can, through equation (45), determine the domain of analyticity in the lower half plane of the analytic continuation of each component of ~÷ (~). We have G
r~+~(~) = Gg(~) + 2-' Fu(~) ~ C~r~ ÷j(~ + ifl),
g = 1. . . . . G.
(48)
j=l
From the analyticity of ~÷j(~) in the upper half plane for all j and the analyticity of G~(~) and Fg~(~) for ~ ¢ ( - i o o , - i ) , it is possible to construct, through a recursive argument using the formula in equation (48), the analytic continuation of ~÷g(~) to the lower half plane cut from (-ioo, -il. Moreover, it is clear from equation (48) that q~÷g(~) is discontinuous across the cut ( - J o e , -iZ~). It remains to identify whether the segment [2-iZg, - i ] belongs to its domain of analyticity. Clearly, from equation (48), if Z I - 1 < fl then 4~÷g(~) is analytic for ~ ~ [-iY.~, - i ] since ~ + ifl belongs to the domain of analyticity of q~+j(~) for allj. This condition on X~ and fl is precisely what we had imposed earlier in the construction of eigenfunctions in equation (18). To summarize our results up to this point: q~+,(~) is analytic in the entire lower half plane cut from (--io0, -- iX~]. Moreover, from equation (48), $+~(~) has a logarithmic singularity at ~ = - i Z ~ , and by recursion at all the points ~ = - i X ~ - infl, n = 1. . . . , oo, and ~m~.= - i Z ~ - imfl, m = l , . . . , oo, for all g' # g . Our aim is to deform the path of the inversion integral of O+~(~) along the real axis, into an integral around the cut ( - i oo, -iX~]. To justify this deformation we need to establish the following two facts: (1) That the inversion integral along the connecting contour, that is the semi-circle with radius R and center at the origin, vanishes as R ~ oo. By Jordan's Lemma, ~4 a sufficient condition for this is that I~+~(~)1 be bounded for R large enough by a function of I~1 which vanishes as Ill ~ ~ . (2) Denoting by q~+~s(~) the limiting values of q~+~(~) as ~ approaches the cut ( - i ~ , - iZ~] from the right and the left, respectively, we need to show that the inversion integral around the cut, i.e. /" - iI:~ (2hi) -~ J-,o~ d~ [$ ++,(~) - q~~.,(~)]exp(- i~x)
(49)
exists for all x >t 0. We shall now devote two subsections, 3.1 and 3.2, for proving facts (1) and (2) respectively. 3.1. A uniform bound on I$+,(¢)1 We shall establish fact (1) by proving that I$+,(~)1 < As,/l~ I,
(50)
As, being a large enough positive constant and ~ being anywhere in the lower half-plane not belonging to some ~ neighborhoods of the points ~g, n = 0, 1. . . . . ~ , and ~,~g,, m = 1, . . . , ~ , all g' :# g. To prove equation (50), we shall use equation (48) and some properties of Gs (¢) and Fs~ (~). For the case g = G, it has been shown ~ that IG~(£)I and Ig~(~) I are uniformly bounded in the lower
Multigroup linear transport~I
91
half plane cut from the ~ neighborhood of the point - i by B,/I~I, with B, a large enough positive constant. Through a slight modification of their proof, it can be shown that for any g and any such that I~ + i ~ l > ~, IG~(~)I < Bs,/l~l,
(51)
IF,~(~)I < B,,/I~I,
(52)
where B~, are large enough positive constants. Inserting these estimates into equation (48), we have, for all ~ outside the e neighborhood of - i ~ , the following inequality G
Iq~+~(~)l ~< (B~,/I~I) + (2-~B~/I~I) ~ C~l~+s(~ + i[j)l,
g - 1. . . . . G.
(53)
To proceed with the proof, we subdivide the lower half plane, cut from the E-neighborhoods of the points ~,g = -iY.g - in[J, g = 1 , . . . , G, n = 0, 1. . . . , ~ , into an infinite number of strips Sin., of width [j with Sin,, = {~: - ( m + 1)[j ~< Im~ ~< -re[j}, m = 0, 1. . . . . oo. Since for all g, I~+g(~)l is uniformly bounded in the upper half plane by some large enough positive constant M, we have from equation (53), for ~ ~ So.,, G
I~+s(~)l <~ (B~c/l~l) + 2-~M(Bs,/I~I) ~, C~,
g -- 1. . . . . G.
(54)
j=l G
If we define Dg,0 = Be, + 2 -~ M B ~ ~ C~, then, for ~ E So,, we have from equation (54). j=l
I~+~(¢)1
~Ds,o/l~l,
g = 1. . . . .
G.
(55)
Similarly, using this estimate and equation (53), we can easily deduce that for ~ ¢ Su the following estimate holds:
I~+~(~)1 ~
g = I,...,G,
(56)
where Ds, t is some large enough positive constant. Continuing in this way, we can show that, for any m and ~ e S,,,
I~+,(¢)1 ~
=
1.... ,G.
(57)
where, again Dg,m is a large enough positive constant. If we can show that the sequence {Dg,m}~.t can be taken to be bounded, then clearly for ~ UmS,,~ we would have the result
1~+8(~)1 ~
(58)
Dr, m .
ra
Since q~+~(¢) is bounded in the ~ neighborhoods of -i~.s., g' # g, we conclude then that there exists a large enough positive constant Az, such that I~+~(~)1 ~< As,/l~l,
(59)
for all ¢ not in the E neighborhoods of the points l , t - - - - i E g - i n [ j , n = 0 , 1 , . . . , ~ , and ~,,,g. = - i Y . e -ira[j, m = 1. . . . . ~ , g' ~ g. This is precisely the desired result in equation (50). To prove that the sequence {Dg,m }~. ~can be taken as bounded, we return to equation (57) which implies that I~+~(~)1 is bounded in every strip S,~, m >I 1. This then allows us to define Us,,. - sup I$+s(~)l, ~s.~
g = 1. . . . . G;
m >t 1
(60)
which, from equation (53) satisfies G
U,o,, <~(B,,/rn[j) + 2-'fBs,/m[j) ~ CnUs,,,,_ ,, j-I
m >1 1, g = 1. . . . . G.
(61)
92
R. D ~ L and G. C. POMRANING
Next, recalling Co = s u p g,)
and defining U,,. = sup U~,r., B, = sup Bs,, g
g
we obtain from equation (61) the inequality
Uo. <~(B,/m~) + CoG& U<,._,/2m~,
m i> 1.
(62)
A recursive inequality of this form, was shown 4 to imply lira U~m= 0.
(63)
From equation (62), this implies that (-1o, vanishes at least as fast as m - '. The same result necessarily holds regarding ]q~+s(~)[ for ~ E S,,~. Consequently, since for ~ s Sin,, ]~l-~= O(m-z) as m ~ ~ uniformly in Sin,, we conclude from equation (57) that the sequence Dg,,, is indeed bounded. Thus, proving the estimate in equation (59) or (50). An alternate way to derive this same result is to realize that, from equation (63), we have the uniform boundedness by some constant of Iq~+g(~)[ for
eEu&, for all g. Using this information in equation (53), we readily deduce the result in equation (59). We have therefore shown that the inversion integral vanishes around the infinite semi-circle. The next subsection is devoted to proving that the inversion integral around the cut ( - i o o , -iY.e] exists. 3.2. Convergence of the inversion integral around the cut We consider the inversion integral of q~+g(¢) around the cut ( - i v , -iY~g] in equation (49). We first recall that q~+g(~) has logarithmic singularities at the points ~g. and ~s',.. Moreover, Gg(~) and F~(~) are H61der continuous on the open interval ( - i ~ , -iZs). Using equation (48), it can be easily shown through a recursive argument, similar to previous ones, that this implies that q~_g(¢) is Hflder continuous on every closed interval of ( - i v , - i Y . s ) not containing the logarithmic singularity points ep and ¢s',." Therefore, the discontinuity Ag(~) = q~+g(~) - q~+s(~), has only logarithmic singularities on the cut ( - i v , - i Y s ] at the points ~s. and ~s',., and is otherwise H61der continuous. It follows that As(~ ) is integrable and square integrable on [ - Jr, - iY.s] for any finite r > x:s" The existence of the inversion integral in equation (49) can be easily shown to be guaranteed if IAs(~) I is integrable on [ - i oo, -iX:g]. To prove this, we first obtain bounds on Iq~+s(~)l in the neighborhoods of its logarithmic singularities. Applying equation (45) recursively n times, we easily obtain the relation
H(¢ +im'fl)
#J+(~)=G(~)+
.G(~ +im[3)+
H(~ +im'[3) .~+(~ +infl),
(64)
m-1 tin'=0
where we have defined
(65)
a ( ~ ) = 2 - ' a:(~) • C.
For ~ not in the E neighborhood of the points - iZ s, g = 1. . . . . G, there exists by equation (52) a large enough positive constant Ns, such that the following estimate holds for the matrix elements H,,(~):
Insi(¢)l ~
(66)
Next, for ~ in the E neighborhood of the point - iZ s, there exists a large enough positive constant Ms, such that the following estimate holds I$+s(~)l ~< M~,logl¢ + ix:sl -~,
for
14 + iXsl ~< e < 1.
(67)
Muitigroup linear transport--I
93
Consequently, using equations (67), (66), (57) and (59) in equation (64) and letting ~ belong to the E neighborhood of C,,- -iT, s - in[3, n ffi 1. . . . . oo, we obtain the following estimate
'{( )( ) ( ) t < ,.._,._, ( ).-,
I~+,(C)I ~< (Bs,/[CI) + ~
sup Ns~
sup Bg~ G m f i [C +/m'fll
--I
m'--O
]I I~ + im'fll(G - 1) supAg, I~ + infll-'
.-0
+
l-I I~+im'/3l-'Ms, logl~+iY~g+inBl-',
supN~
m'~0
I¢ - Cn~l,.< E < 1, n = 1 , . . . , ~ .
(68)
The summation term on the fight hand side of equation (68) can be easily shown to be bounded by R, ICI-~ where R, is a large enough positive constant. It is also a trivial argument to show that the third and fourth terms on the fight hand side of equation (68) are bounded by Z, ICI-I and V~ ICI-~ logic - ~ngl-~ respectively, with Z, and Vs, being again large enough positive constants. It follows then that I~+~(C)I satisfies the following estimate
I~+g(~)l~
n = l , 2 . . . . . oo,
(69)
where Wg,= R, + Z, + Bg,. From equations (67) and (69), it easily follows that for E < e-~ we have the estimate I~+g(~)l ~< T,,ICI -t logl~ - ~,,I-', n = 0, 1 , . . . ,
IC - ~,~t ~<~ < e x p ( - 1),
(70)
oo,
where T,, = Wg, + Vg, + Mg," (~s + ~). In a similar manner, it can be shown that the following estimate holds for I~+r (~)1 for all g' ~ g:
I~+r(~)l ~ Tgr, ICI-~ logic - ~n,I-~, I~ - ~,~1 <~~ < e x p ( - 1),
(71)
n ----1 , . . . , o 0 . Consequently, I~+~(~)1 satisfies the following estimate
, (72) for all ~ in the lower-half plane (Ira ~ < 0). We shall now use this estimate to prove that lAg(C)1 is indeed integrable on [ - i o o , - i Y , ] as we set out to prove earlier in this subsection. First, we note that by a repetition of a proof in the Appendix of Ref. [4], we have the following estimate for Gs(~) G , ( ~ ) = 8ni~-~ I' ~bs(0,#)d # +0(C-21og~), d- l
]CI -* ~ .
(73)
From equation (48) and the estimate in equations (72) and (52), it follows that lAg(l)] satisfies the estimate
,A,(~)I <...K~I~ I-2 + K,, I~I-21og I~I + B,,ICI-' (~=~ C,j)
g'~g
where
K@ and Kg~ are large enough positive constants.
94
R. DANIEL a n d G. C. POMRANING
Now, by a repetition of an argument in Ref. [4], we can assert that the estimate in equation (74) implies that IAg(~)l is integrable.on [-i,~o,,-iY~g] and that IeAg(~)l is square integrable on this same cut. The inversion integral in equation (49) therefore exists and through a change of variable, ~bg(x) can be written as
I
~bg(x) = 2n
~:k-I
(75)
bg(v)exp(-x/v) dv,
dO
where
(2~v)-~a,(- iv - ' ) .
b,(v) -
(76)
From equation (76), the function bs(v) has logarithmic singularities at the points vg. = - i ¢ ~ ~, n = 0 , 1 . . . . . ~ , and the points v t , . = - i ~ . ~ , , g ' ~ G , n = l . . . . ; ~ ; however, bg(v ) is H61der continuous on every subinterval of [0, Z f j] not containing the points vg, and vg,,, and is square integrable on [0, I f ' ] by virtue of the square integrability of 1~Ag(~)[ on [ - i ~ , -iEg]. It is trivial that ~bg(x) as given by equation (37) is continuous on [0, ~]. Moreover, by direct application of the Schwarz inequality and the fact that bg(v) E L2[0, ~ f ~], it is easy to show that ~bg(X) e L2[0, ~]. This then implies that the desired solution for ¢~(x) is indeed as represented in equation (37) with the functions bj(v) having the properties described above. The next task in the completeness proof is to show that the functions bj(v) as given by equation (76) are the unique solution for the expansion coefficients aj(v) of the set of singular integral equations f(p) =
fo'
dye,(0,/~),
0 < # ~< 1,
(77)
which arise upon imposing the boundary condition, equation (28), on the expansion in equation (48) or equivalently equation (49). The unique solvability is understood to be restricted to functions aj(v) which through the superposition in equation (36) yield solutions for 4~(x) of equation (31) which are continuous and square integrable on [0, ~ ] and to functions/~fg(/~) which are H61der continuous on [0, 1]. 3.3. Unique solvability of equation (77) We insert the solution for ¢~(x) from equation (75) into equation (31), and carry the integration over x' to obtain, after some rearrangements,
;
f0f
dv exp(-x/v)b(v) = 2 -I
dv
d~-I
R(x, ~).Z(x, v, ~)" C" b(v)
0
+
f0
d/a R(x,/~) • f(/~) -
f0 i
i~-~R(x,p).T(x,v,l~).C.b(v)d#,
dr2 -~
(78)
I
where b(v) is a column vector with elements Bg(v); T(x, v, #) and Z(x, v, I~) are diagonal matrix valued functions with entries
Zgg(x, v,/~) = co#(o~ Zg - #)-~ {exp[x(Zg # -~ - co -~)] - 1},
(79)
T , , ( x , v, ~,) = 1 - Z , , ( x , v, u ) .
(80)
and
Upon carrying the matrix products R • Z and R • T, equation (78) can be written as
fo'
;of fo'r'
dv exp(-x/v)b(v) = --
d/~co2-1 (co~" --/In) -1. C. b(v)exp(-x/og)
dv
d~
I
dvco2-t(wY-l~a)-~'R(x, lO'C'b(v)
dO
+
l
fo
d/~R(x, ~)" f(#).
(81)
Multigroup finear transport--I
95
By equating the coefficients of the linearly independent spatial modes (exponential modes) and assuming equation (18) to hold, we obtain,the vector.set-of singular integral equations ~ d v b ( v ) ~ , v ) = f~ dvco2-'(coY-/~0)-'" C" b(v) -['dv6(/~,co)co2-'[" d/z'(coY-#'U)-"C'b(v)+f(~), 0 < / ~ < 1 , (82) J0 J- 1 where ~(~, v) and 6(/~, co) are as defined in Section 2, and b(v) is a diagonal matrix with elements
b,(v). By replacing for #v(0,/z) in equation (77), we can readily see that we recover equation (82) which is solved by the functions bx(v ). A solution of equation (77) for the functions ag(v) therefore exists and is given by bg(v). The uniqueness can be shown in a simple way as follows. Suppose another set of functions hg(v) solves equation (77). By multiplying equation (77) by e x p ( - x / # ) and integrating over/~ from 0 to 1, we obtain equation (81) with hz(v ) replacing bs(v ). Since this equation was obtained from equation (78) through a reversible sequence of operations, we conclude that
2r~
;o'
h(v)exp(-x/v) dv
must be a solution of equation (31). If we require that the functions hs(v ) yield solutions ~bs(x) e L2[0, oo] n C[0, oo] (a sufficient condition being hs(v)e L2[0, Y~-']), then necessarily by the unique solvability of equation (31) we must have
fo' h(v)exp(- x /v ) dv = f ] b(v )exp(- x /v ) dv
(83)
hg(v) = bg (v), g = 1. . . . . G,
(84)
which clearly implies
hence proving uniqueness of the desired solution of equation (77) and terminating the completeness proof. 4. SUMMARY AND DISCUSSION We conclude this article with a summary of the important results obtained, and some recommendation for future investigations. We have shown that under the condition in equation (18), a continuum degenerate set of eigenfunctions of equation (4) is constructable. The ansatz is equation (5) and the explicit form of the eigenfunctions is given by equations (9), (10), (11), (15), (16) and (17). This set of eigenfunctions is complete for expanding the desired solution of the half-space exit distribution problem ("desired", in the sense discussed at the outset of Section 3). The completeness property removes the degeneracy of the set, and was shown to hold for essentially H61der continuous incident angular currents with the expansion coefficients having logarithmic singularities at an infinite set of points which becomes dense near zero, being otherwise H61der continuous, and satisfying the property of square integrability. Whether eigenfunctions can be constructed without the restrictive condition in equation (18) still requires further investigation. One likely ansatz is that given by equation (27). One interesting property of this half-range complete set of eigenfunctions is that in the homogeneous media limiting case of s = ~ we do not recover the discrete eigenmodes which are known to exist for a large class of matrices C. This peculiar discontinuity in the spectrum of v as s --. oo will be seen to be a source of numerical difficulties in the solution for the exit distributions via collocation schemes, n The fact that the logarithmic singularities in the expansion coefficients as(v ) becomes dense on the entire interval [0, ~-1] is conjectured to be a compensation for the non-recovery of discrete modes which carry the asymptotic solution. The technicalities of such a compensation are not understood yet. Finally, we conclude by posing the open problem of eigenfunction construction for the purpose of solving the finite slab reflection-transmission problem. Whether a complete set can be
96
R. DANIEL and (3. C. POMRA~NG
constructed for this purpose is doubtful if done with a simple ansatz as in equation (5); our experience with the one speed case ~5 suggests that an ansatz as in equation (27) with N = ~ is needed. Acknowledgement--This work was partially supported by the National Science Foundation. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
B. J. Martin, J. appl. Math. 20, 703 (1971). E. W. Larsen, Prog. nucl. Energy 8, 203 (1981). T. W. Mullikin and C. E. Siewert, Ann. nucl. Energy 7, 205 (1980). E. W. Larscn and T. W. Mullikin, J. Math. Phys. 22, 866 (1981). N. I. Muskhelishvili, Singular Integral Equations. Nordhoff, Groningen (1953). C. E. Siewert and P. Bcnoist, Nucl. Sci. Engng 69, 156 (1979). P. (3randjcan and C. E. Siewert, Nucl. Sci. Engng 69, 161 (1979). R. D. M. (3arcia and C. E. Siewert, JQSRT 25, 277 (1981). W. W. Engle Jr, The User's Manual for ANISN: A One Dimensional Discrete Ordinate Code with Anisotropic Scattering, K-1693, Oak Ridge Gaseous Diffusion Plant (1967), available from Radiation Shielding Information Center, Oak Ridge National Laboratory, Oak Ridge, Tenn. as CCC 82/ANISN (March, 1967). C. T. Kelley and T. W. Mullikin, J. int. Eqs. 4, 77 (1982). T. W. Mullikin, Some Singular Integral Equations in Linear Transport Theory, Sandia National Laboratory Report SAND-80-1069, Albuquerque, N.M. (1980). R. Daniel and (3. C. Pomraning, JQSRT 37, 97 (1987). C. T. Kelley, J. int. Eqs. 3, 261 (1981). Carrier, Krook and Pearson, Functions of a Complex Variable. Mc(3raw-Hill, New York (1966). R. Daniel, (3. C. Pomraning and E. W. Larsen, J. Trans. Theory Statist. Phys. 11(2), 75 (1982).