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Multigroup linear transport in the exponential half-space—II
J. Quant. Spectrosc. Radiat. Transfer Vol. 37, No. 1, pp. 97-105, 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--II R . DANIEL,* V. C. BADHAM a n d G . C. POMRANING School of Engineering and Applied Science, University of California, Los Angeles, CA 90024, U.S.A.
(Received 18 July 1984) AINtract--The multigroup half-space exit distribution problem in an exponential atmosphere is considered by two separate approaches. The first approach consists of an expansion of the multigroup angular flux solution in a half-range complete set of eigenfunctions. Then, through an orthogouality relation between these eigenfunctions and those of the adjoint transport equation, a set of singular integral equations is derived for the multigroup exit distributions. The half-range completeness of the eigenfunctions is then shown to imply that these singular integral equations possess a unique solution. The second approach consists of deriving this same set of singular integral equations by Laplace transforming the set of integral equations for the scalar group fluxes. Uniqueness of the solution is also shown in an appropriate function space. A multigroup version of the F-N method is then formulated and applied to the solution of the set of of singular integral equations. Convergence is proven and the numerical results for the two group albedo is seen to be in good agreement with those obtained from the standard S-N method calculations.
1. I N T R O D U C T I O N
This article is a continuation of companion paper ~ where we have considered the purely mathematical problem of constructing eigenfunctions of the multigroup linear transport equation in an exponential atmosphere and proven their completeness for solving the half-space exist distribution problem. Here we shall use these mathematical results for the purpose of solving the half-space exit distribution problem (already defined in Ref. [1]) and obtaining numerical results. The following is an outline of this article. Sections 2 and 3 are devoted to a derivation of a set of singular integral equations for the multigroup exist distributions and to a proof of their unique solvability. In Section 4, we derive this same set of singular integral equations in an alternate way, prove its unique solvability and present the theoretical background which is necessary for proving convergence of a multigroup F-N scheme which we formulate in Section 5 and apply to its solution. In Section 6, we present an F-N calculation for the algebraically tractable two group case and compare it to the results found from the well tested standard S-N method embodied in the ANISN code. 2 Finally, Section 7 is devoted to conclusions, a discussion of the results and recommendations for future work. 2. S I N G U L A R
INTEGRAL EQUATIONS VIA EIGENFUNCTION
FOR THE EXIT EXPANSION
DISTRIBUTIONS
The half-space completeness proof presented in the previous article t allows for the expansion of the solution ~,(x, g) as in equation (26) of Ref. [1], i.e.
f,
(l)
where all the necessary definitions have been made in Ref. [1]. In order to derive from equation (1) a set of singular integral equations for the exiting group fluxes we shall first establish an orthogonality relation between the eigenfunctions constructed in Ref. [1] and those of the adjoint multigroup transport equation
•_.,*(x,l~)={texp(-x/s)]/2}C+'f"d-
~*(x,/~') dt~', i
where C + is the transpose of C. *Present address: 10535 Wilshire Boulevard, No. 401, Los Angles, CA 90024, U.S.A. Q.S.R.T.37/~--G
97
(2)
98
R. DANIELet
aL
To construct the adjoint eigenfunctions, we rewrite equation (2) with a simple change of independent variable as
q,*(x, -it') dit'.
It(~/Ox)O*(x, - i t ) + r-.O*(x, - i t ) = { [ e x p ( - x / s ) ] / 2 } C + •
(3)
I
Assuming that equation (18) of Ref. [1] is satisfied, then eigenfunctions of equation (3) are readily obtained from the eigenfunctions constructed in Ref. [1] by simply replacing C by C +. If, in addition, we replace It by - i t in these eigenfunctions we obtain the desired eigenfunctions of the adjoint equation, equation (2). Explicitly then, we have the eigenfunctions dp*J(x, It) = O *J(it, v) e x p ( - x /v ) + X*J(Ia, v) e x p ( - x / c o ) ,
(4)
where the j t h element of ~*J(it, v) is given by
(5)
¢,?J(it, v) = ,~(it + vr.j)
and where X*J(it, v) = (co/2) (co Y +
ItO) -1
"C + .e j -- ~(--it, co)./t*J(v)
(6)
with 2*J(v) = (~o/2)
t,l
! dit(,o+,u)-'.C+.e,; ,/-
(7)
I
the degeneracy of this set is the same as the degeneracy of the set constructed in Ref. [I], i.e. for j = I..... G, #~*:(x,It) is an eigenfunction in the distributional sense for v ~ y:. The orthogonality relation can n o w be easily derived as follows. W e consider the transport equation with ~i(x,It) as a distribution solution for v e Yi, i.e.
It(~3/Ox)gp~(x, It) + ~_. ~',(x, It) = { [ e x p ( - x / s ) ] / 2 } C .
dp',(x, It') dit';
(8)
I
and its adjoint with 4~*)(x, It) as a distributional solution for v ' e ~,~, i.e. - i t ( O / O x ) O y J ( x , It) + I . . ~ J ( x ,
It) = { [ e x p ( - x / s ) l / 2 } C
÷.
:,
~,*J(x, It') dit'.
(9)
-I
We multiply equation (8) by [ ~ J ( x , It)lT and equation (9) by [4~(x, It)lT, subtract the two results and integrate over It from - 1 to 1. Using simple matrix identities, we obtained the distributional relation ditit [4,'v(x, It)]v. q~*,J(x, it) = 0,
(OlOx)
(10)
I
where v ET~ and v'eT: for any i , j = 1. . . . . G. Integration equation (10) over x from 0 to m, and using the fact that at m both ~ ( x , It) and ~*J(x, It) vanish, we obtain
f'
ditit [~iv(0, It)]T'~*J(0, It) = 0,
V e ~i,
V' E 7j.
(] 1)
-I
Equation (11) is the desired'orthogonality relation and holds in a distributional sense. We now multiply equation (11) by the expansion coefficient ai(v) (uniquely determined from the half-range completeness proof) integrate over v on 7~ and sum over i from 1 to G. Interchanging the order of integration and using equation (1) with x = 0, we obtain the set of equations ditit[0(0, It)]T'~J(0, It) = O,
j = 1 . . . . , G,
v" ~ 7j.
(12)
I
Upon replacing the explicit form for ~*J in equation (12), it becomes clear that it is a coupled set of singular integral equations for the multigroup exit distributions It0g(0, It), It < 0, g = 1. . . . . G. It can be rewritten in a more convenient form by replacing It by - i t in the integral from - 1 to
Multigroup linear transport--II
99
0, and ¢(0, #) by the incident flux vector f(#) for u > 0, thus yielding the set of equations d##[#,(0,-#)]r.~by(0,-#)= d##[f(#)]T.Oy(0,#), v'eTj, j = l . . . . . G. (13) fo' fo' Clearly, the fight hand side of equation (13) is a known term, whereas the left hand side represents singular integrals over the unknown exit group distributions. The question of existence and uniqueness of solutions to this set is easily answered in the next section as a generalization of an argument already presented for the one group (one speed) case by Larsen and Mullikin) 3. UNIQUE SOLVABILITY OF THE SET OF SINGULAR INTEGRAL EQUATIONS We shall exploit the half-range completeness property of the eigenfunction set ~i~(x, #) to show that a unique solution exists to equation (13) in a class of functions which we shall determine. Existence of a solution is insured since the superposition in equation (1) which by half-range completeness represents the desired solution of the half-space exit distribution problem, also necessarily satisfies, when taken at x = 0 and # < 0, the set of equations in equation (13). Moreover, from the integral relation between the exit distribution vector and the scalar flux {see equation (35) in Ref. [1]} such a vector solution is dearly analytic in all its elements for # ~ [ - l, 0). Uniqueness of this solution can be shown within the wider class of H61der continuous vector functions as follows. We assume another solution, ~'(0, - # ) , exists and is H61der continuous in all elements on (0, 1). The difference A(0, #) between ~ (0, - #) and ~'(0, - #) is H61der continuous and satisfies the homogeneous set of singular integral equations
~0
' d## [A(0' #)]T'~b*J(0' - # ) =0,
J = 1,... ,G.
(14)
Now ~*J(0, - # ) is obtained from ~{,(0, #) by simply replacing C by C +. Thus if sup i,j
C,j
is small enough then the eigenfunction set ~*J(0, - # ) is half-range complete for expanding the H61der continuous vector function A(0, #). Assuming the expansion coefficients are the functions bj(v'), we can then multiply equation (14) by bj(v'), integrate over v' on ~j, sum overj from 1 to G, interchange the order of the # and v" integrals and use the half-range expansion for A(0, #) to obtain the scalar equation
f0
1d## [A(0, # )]T[A(0, # )] = 0,
(15)
which clearly shows that A(0, # ) = 0, and completes the uniqueness proof. We now summarize our results in Sections 2 and 3. We considered the adjoint multigroup transport equation. Under the condition in equation (28) of Ref. [1], the adjoint eigenfunctions $*J(x, #) are obtained from the eigenfunctions of the transport equation ~#{(x,#) by simply replacing C by C + and # by - # . An orthogonality relation among these eigenfunctions was established and is given by equation (11). This orthogonality relation together with the expansion in equation (1) were used to derive a set of coupled singular integral equations for the exiting group distributions as in equation (13). A solution to this set of equations is guaranteed to exist since the eigenfunction set ~{(x,#) is complete for expanding the solution to the half-space exit distribution problem under the conditions already discussed in Ref. [1]. This solution to equation (13) is given by the expansion in equation (1) taken at x = 0 and # < 0 and is analytic in # on (0, 1). Uniqueness of this solution is guaranteed in the wider class of H61der continuous functions on (0, 1] (analytic functions are a subclass of it) provided the conditions for half-range completeness of the eigenfunction set ~{(x, #) are satisfied when C is replaced by C +. Exact solutions to equation (13) are not known to exist to date and, just as in the one speed (one group) case, are too difficult to obtain. We shall instead be seeking approximate solutions
100
R. DANIEL et al.
through a multigroup version of the F-N method. The theoretical foundation necessary to prove convergence of the F-N collocation scheme which we will be using is provided by following an alternate approach to deriving singular integral equations for the exiting distributions, as was first demonstrated for the one speed case by Mullikin. 5 This is the subject of the next section.
4. A L T E R N A T E D E R I V A T I O N OF THE S I N G U L A R I N T E G R A L EQUATIONS AND T H E I R U N I Q U E SOLVABILITY We consider equation (35) of Ref. [1] in which we set x = 0 and change variable from/~ to - p to obtain
dx'exp(-x'/s)R(-x',
/ ~ ( 0 , --/t)=(1/47t)
-/t)'C,ib(x'),
0 < # ~< 1,
(16)
where the elements of the diagonal matrix R ( - x ' , - / t ) have been defined ~ as
Rgg(-X', -kt) = e x p ( - x ' X g / # ) .
(17)
Defining the scalar Ig(x') by G
Ig(x') = (1/4n) exp(--x'/s) ~ CgSpj(x'),
(18)
j=J
it is then clear that/~Og(0, -/~) is the Laplace transform of Ig(x') with the tramsform variable being Zg//~. By defining the vector R(z) from R(z)=
f:
exp(-x'z)I(x')dx',
g = 1. . . . ,G,
(19)
where z is allowed to be complex, we obtain the relation
Rg(~.g/lt) =/~kg(0, -/~),
0 < U ~< 1, g = 1. . . . . G.
(20)
Now, from equation (31) of Ref. [1] and the definition in equation (18), it is easy to show that l(x) satisfies a set of coupled integral equations given by
f
l(x) = (1/2) e x p ( - x / s ) C ,
oo
dx' ~ ( J x - x' I)'I(x')
do
+(l/2)exp(-x/s)C.
f,
d~ R(x,~)-f(~),
(21)
do
where the diagonal matrices ~ and ~ have been defined in Ref. [1]. The definition of equation (19) can then be used to transform equation (21) into a set of coupled singular integral equations for the elements of R(z). Thus, by multiplying equation (21) by e x p ( - xz), integrating over x from 0 to ~ , using the definition of the first order exponential integral function in the elements of M, interchanging the order of the resulting integrals and using the definition in equation (19) we obtain the set of equations Rg(z) + (1/2)
j=l
Cgj
dl.t' Rj(Ej/I.t')/[Ej - I.t'(z + l/s)]
-- (1/2) ~,, CgjRj(z + 1/s) j=l
= (1/2)
~01
dla'/[Y.j- p'(z + l/s)] I
G
d u ' Z Cgjlt'fj(l~')/[~.j+ lz'(z + l/s)], j=l
g = 1. . . . . G.
(22)
By letting z = Y~g//~in the gth equation and using equation (20), we obtain a set of coupled singular
Multigroup linear transport--II
101
integral equations for the exiting group angular distributions #$g(0, - g ) , g = 1. . . . . G: °
#~O,(O, --#) + (1/2) ~..=C~
-- (1/2)
=(1/2)
j=l
d#'#'~Oj(O,-g')/[Zj - #'(Z,/# + I/s)]
CsjZ~qgqJj(O,-Y.,jqg)
d#'/[Ej - U'(X, Iu + l/s)]
1
fO dg' G j=l
0<~<1,
g = l . . . . . G,
(23)
where the function qg is defined from
(24)
1/q, = Y.,gl# + lls.
In equation (23), the function Xyqs may exceed unity; in this case the function Ejqz~j(0, -Xjqs) is defined through an extension of equation (16) to values of the argument # outside the physical range (0, 1]. In the gth equation of the set in equation (23), we make a convenient change of notation from # to v~z, 0 < v ~< 1/Y,s, to obtain vZ, t#~(O, - r E , ) + ( 1 / 2 )j=t E~ C,j f0l dv'#'$j(O, -#')/(Zj - #'/oJ)
=
(1/2)
~01d # ' E G j=l
+ ~'/o~),
0 < v .< l/X,,
, = 1. . . . . G.
(25)
Now, if the restriction on the eigenfunction construction in Ref. [1], equation (18), is satisfied then it is a very simple exercise to check that equation (25) is identical, apart from dummy variables and indices, to equation (17) derived via eigenfunction expansion and the use of the orthogonality relation. The desired solution to equation (23) {or, if equation (18) in Ref. [1] is satisfied, then also equivalently to equation (17)} is that which is given by equation (16) in terms of ~(x). Here, ~(x) is that solution of equation (31) of Ref. [1] whose existence and uniqueness is guaranteed, with elements ~b~(x) continuous and square integrable on [0, oo], provided sup C U i,j
is small enough, t As we mentioned earlier in Section 3, a closed form is not known to exist for this solution for #~(0, - # ) . For the purpose of analyzing convergence of a numerical solution to this desirable solution it will be useful to prove its uniqueness in an appropriate function space. Such an analysis was first performed by Mull/kin 5 and in a subsequent article by Kelley and Mull/kin6 for the one group (one speed) case. Here we shall simply generalize their arguments to our muir/group case. We note that, from our earlier assumption in Ref. [1] that det C # 0 and from the use of the Schwarz inequality, it is clear that equation (21) has a unique solution for I(x) with Is(x ) ~ L2[0, oo] N C[0, co] whenever equation (31) of Ref. [1] has a unique solution for ~(x) with Sg(x) satisfying the same properties. Consequently, since R(z) is the Laplace transform of/(x) and since equation (22) is the Laplace transform of equation (21) then equation (22) has a unique solution for R(z) with Rg(z) in the Hardy space H 2. This space is the Hilbert space of functions which are Laplace transforms of functions in L2[0, oo]. Every function in H 2 is analytic in the right half plane and the inner product between any two functions F(z) and G(z) in H 2 is defined as
{F, G)m = (1/2~) f
o F(it)G(it) dt.
(26)
R. DANIELet al.
102
It follows then that the desired solution of equations (23) and (16), is the only solution which is analytic on (0, 1], which can be analytically continued from equation (20) to the right half-plane of Eg//~ and be in H ~. Based on the foregoing observations and facts we shall, in the next section, formulate a multigroup F-N scheme which when applied to equation (23) or equivalently to equation (13) {if equation (18) in Ref. [1] is satisfied} would converge to this desired solution f o r / t o (0, -/~).
5. THE M U L T I G R O U P F-N C O L L O C A T I O N SCHEME We shall first begin by presenting some mathematical preliminaries necessary for proving convergence of the numerical F-N scheme which we present later in this section. The sequence of functions ( 0 . ( z ) = (z + z.)-~}~=l, where the set of points {zn}~ belongs to the positive real axis (0, oo) and has at least one accumulation point not at zero or infinity, was shown 5'6 to form a basis for a 2. We now consider a Hilbert space H of column vectors with G elements in H 2 and the inner product between any two vectors F(z)=[Fl(z) . . . . . F~(z)] T and G(z) = [G,(z) . . . . . G~(z)] r in H defined by G
(F, G ) . = ~ (Fg, Gg)n:.
(27)
g=l
Next, we define a sequence of vector functions {0.i( Z )}.oo,G = ,. i=, to be a sequence of column vector functions with elements zero except for the ith element which is 0.(z). It then follows that the o~,G sequence {0.i( z )}.= 1,i=1 forms a basis for H. We now consider equation (22) and write it symbolically as R(z) + 0_.R(z) = F(z).
(28)
where 0_ is a G × G matrix operator with a general element Ln corresponding to the integral operator on Rj(z) in the g t h equation, and where F (z) is a known column vector with the gth element Fg(z) corresponding to the known term on the right hand side of equation (22). The integral operators L~. are known to be Hilbert-Schmidt on H 2 by a simple generalization of an earlier argument 6 relevant to the one group case. Consequently the matrix operator 0_ is Hilbert-Schmidt on H. A well known procedure for obtaining approximate solutions to equation (28) is the Galerkin 7 procedure. This consists of inserting the expansion G
Ru(z) = ~
N
~ a, AO,i(z),
N integer 1> 1,
(29)
i=1 n = l
into equation (28) and evaluate the scalar coefficients ani by matrix inversion of the set of N × G algebraic equations: ( R N , 0 , . j ) / ~ + ( 0 _ . R N , 0 , . j ) n = ( F , 0,.j)H, j = l
. . . . . G, m = l , . . . , N ,
(30)
where the sequence of vector functions 0m:(z) forms any basis for H generated by a set of points Zr. accumulating on (0, oo). With 0_ being Hilbert-Schmidt on H, sufficient conditions are known to exist 7 to guarantee that the expansion in equation (29) would converge, as N ~ 0% to the unique solution in H of equation (28). The key observation here was made by Mullikin. 5 He proved that evaluating an arbitrary Hardy space function F(z) at a point zn is exactly the same as evaluating the inner product in H 2 of F(z) and 0,(z), that is
F(z,) = (r, O,)u2.
(31)
It is then obvious that evaluating the ith component of a vector function F e H at the point z, is the same as taking the inner product in H of F(z) with 0n;(z), i.e. Vi(zn) = (F, Oni)I'l"
(32)
Consequently, the Galerkin procedure described above is the same as a collocation scheme whereby the expansion (approximation) in equation (29) is inserted into equation (28) and the expansion
Multigroup linear transport~II
103
coefficients a,i are then evaluated by direct matrix inversion following collocation of the resulting vector equation for RAtat the points zm, m ffi 1. . . . . N. Such a collocation scheme is thus guaranteed to converge, as N ~ 0% to the unique solution in H of equation (28). With this collocation method established as a procedure for solving equation (22), we could directly deduce from it an F-N scheme which when applied to equation (23) or equivalently to equation (13) {if equation (18) of Ref. [1] is satisfied} would converge to the desired solution for # ~ (0, - # ) ; however, we shall instead follow a somewhat reverse direction. We shall generalize an F-N scheme first used with empirical success .by Garcia and Siewert s for the one group (one speed) case, the so-called "natural basis" expansion. We will then identify it as a special version of our collocation scheme, thus proving Convergence. We consider the approximation G
N
¢/N(O, - I z ) = ~ j=l
~ aj, gp~,(O, -/~),
0 < # ~< 1,
(33)
i=l
where {vi}~u=lis a set of points on (0, Zi-I) which is assumed to have at least one accumulation point as N ~ ~ , ~ ( x , / ~ ) , withj = 1 , . . . , G, is the degenerate eigenfunction set constructed in Ref. [1], and aj, are unknown expansion coefficients to be determined. If equation (18) of Ref. [1] is satisfied and if the conditions for half-range completeness of the eigenfunctions are present then the approximation in equation (33) may be referred to as an expansion in the natural basis, since, as it was first mentioned in the one speed case, 9 it is motivated by the expansion of the solution #(0, - # ) in the eigenfunction set. However, it should be mentioned, that we are adopting this approximation even if equation (18) of Ref. [1] is not satisfied, in which case the term "natural basis" would seem irrelevant. We proceed by replacing ~ , ( 0 , - # ) by its explicit form from equations (21), (22) and (24) in Ref. [1]. We then obtain for the gth component G
At
#~Og~(0,--#)=~ ~aj,/tCgj/2(o~,l~s+/t), j=l
0
g=l,...,G,
(34)
i~l
where o~, = vis/(v, + s). We then change variables from t~ to v~g in equation (34) and insert this expression into equation (24) or equivalently equation (13) {if equation (18) of Ref. [1] is satisfied}. Next we collocate the resulting vector equation by evaluating the gth component at the set of points v,., i = 1. . . . . N. This leads to a set of G x N algebraic equations for the G x N unknowns a~. To prove convergence of this F-N scheme to the desired solution, we replace l~g//t in equation (34) by z. Then, through simple algebraic manipulations it becomes clear that the expansion in equation (34) is equivalent to the expansion for Rg(z) G
N
RgAt(z) = ~, ~ (aj, C,j/2o~,)(z + co7')-'. j-t
(35)
:=l
Upon interchanging the order of summation in equation (35) and carrying the summation over j we obtain the expansion N
RgAt(z) = ~ A,g(z + co7')-',
g = 1. . . . . G
(36)
i=l
where the coefficients A~8 are defined as G
A~g = ~ aj~C~s/2oJ,,
g = 1. . . . . G,
i = 1 . . . . . N.
(37)
j--I
In vector form equation (36) can be rewritten as G
RAt(z) = ~
N
~ A,,~%(z)
(38)
g ~ l i--I
where the vector functions ~ ( z ) are as defined earlier with the points z~ in this case equal to o~-~. Since the points vi, i = 1, . . . , N, have an accumulation point on (0, ]~-t) as N ~ ~ , then clearly the points co~have an accumulation point on (s -t + ~t, ~). It follows then that the expansion in equation (38) is in a basis for H.
104
R. DANIEL
et al.
Next, insertion of the expansion in equation (34) into equation (24) or equation (13) {if equation (18) of Ref. [1] is satisfied}, changing variables in the gth component of the resulting vector equation from 11 to VEg, and evaluating this gth component at the set of points vr, is equivalent to our previously established collocation method with the collocation points here being z m = vm ~, m = 1. . . . . N, and accumulating on (£1, ~ ) . Our multigroup F-N scheme is thus guaranteed to converge as N ~ ~ to the desired solution. In the next section we present an algebraically simple application of this F-N scheme to the two group case and display the numerical results which we have obtained on the digital computer vs two group calculations using the standard ANISN code? 6. TWO G R O U P F-N C A L C U L A T I O N S We consider the algebraically simplest multigroup case: the two group case with G = 2. We then have the expansions N
N
111/11N(0' --11) = E a~il~Ct,/2(c°iEJ + 11) + ~. a2~11C~2/2(coiEt + 11). i=1 N /~t//2N(0' --11) =
E i=1
(39)
i=l N
ali11C21/2(oJi '1- 11) 'F E a2i#C22/2(t°i + 11) .
(40)
i=l
These expansions are now inserted into equation (23) or equation (13) [if s (s + 1)-~E~ < 1] and the integrals on the left hand side of the resulting two equations are then evaluated analytically. On the right hand side of these two equations we have assumed that the incident fluxes f~ (11) and f2 (#) are isotropic and given by fj (11) = 1,
(41)
£(11) = O.
(42)
The integrals on the right hand side are also evaluated analytically. Next, in the first equation # is replaced by vE~, and in the second, 11 is replaced by v. The system of two equations is then collocated at the set of points v~, chosen equally spaced on (0, E/~ ~) for each choice of N, and the resulting algebraic system of 2N equations is then solved by a matrix inversion routine to yield the expansion coefficients a , and a2i , i = l . . . . . N . The two group albedos defined by ALB1 = and ALB2
=
[fo' [fo'
]/;o' ]/;o'
11~'j~v(0,-11) d11
11 d11
11~2N(0,--11) d11
# d11
can then be computed by carrying out the integrals analytically. The convergent results for the quantities ALB1 and ALB2 are presented in Table 1 together with the ANISN code results for several values of the exponential decay parameter s. The special data for the various parameters was: E~ = 1.005, C~ =0.5, G2 =0.3, C2~ = 0 . I , 6'22 = 0.4, convergence criterion E = 10 -4. Note that the condition on the applicability of our scheme to equation (13) is satisfied for all values of s which are reported. Table 1. A comparison between F-N results and ANISN results s
F-N ALB1
ANISN ALBI
F-N ALB2
ANISN ALB2
0.5 1.0 5.0 10.0 50.0 100.0 300.0
0.08057 0.10304 0.13959 0.14758 0.15538 0.15649 0.15724
0.08050 0.10318 0.13986 0.14784 0.15564 0.15674 0.15749
0.01798 0.02381 0.03440 0.03699 0.03958 0.03995 0.04021
0.01735 0.02384 0.03449 0.03704 0.03962 0.03999 0.04025
Multigroup linear transport--II
105
We note that the F-N results are in good agreement with the ANISN code. However, for cases not reported with s near 103 the F-N method results display numerical instability problems as N increases. These difficulties may be purely numerical, such as error growth in the matrix inversion routine as N increases and s is very large. They may also be attributed to the choice of collocation points being uniformly spaced and the expansion basis varying with N. Finally, they may be attributed to the fact that at s = oo, the constraints to which the singular integral equations are subjected to guarantee unique solvability are not reproduced in a uniform manner thus perhaps rendering the convergence rate of the solution too slow as s increases. In summary, these difficulties are not well understood yet. 7. C O N C L U S I O N S
AND
RECOMMENDATIONS
FOR
FURTHER
STUDY
From the results in this article and in the companion publication t we may conclude that the multigroup treatment of the exit distribution problem from a half-space is fairly well understood. Future study ought to deal with the finite slab problem through extensions of an analysis presented previously for the one speed case. 4,9 Acknowledgement--This work was partially supported by the National Science Foundation. REFERENCES 1. R. Daniel and G. C. Pomraning, JQSRT 37, 85 (1987). 2. W. W. Engle Jr, The User's Manual for ANISN: A One Dimensional Discrete Ordinates 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). 3. E. W. Larsen and T. W. Mullikin, J. Math. Phys. 22, 866 (1981). 4. T. W. Mullikin and C. E. Siewert, Ann. nucl. Energy 7, 205 (1980). 5. T. W. MuUikin, Some Singular Integral Equations in Linear Transport Theory, Sandia National Laboratory Report SAND-80-1069, Albuquerque, N.M. (1980). 6. C. T. Kelley and T. W. Mullikin, J. /nt. Eqs. 4, 77 (1982). 7. G. I. Marchuk, Methods of Numerical Mathematics. Springer, New York (1975). 8. R. D. M. Gareia and C. E. Siewert, JQSRT 25, 277 (1981). 9. R. Daniel and G. C. Pomraning, J. Math. Phys. In press.
0022-4073/87 $3.00+0.00 Copyright © 1987 Pergamon Journals Ltd
MULTIGROUP LINEAR TRANSPORT IN THE EXPONENTIAL HALF-SPACE--II R . DANIEL,* V. C. BADHAM a n d G . C. POMRANING School of Engineering and Applied Science, University of California, Los Angeles, CA 90024, U.S.A.
(Received 18 July 1984) AINtract--The multigroup half-space exit distribution problem in an exponential atmosphere is considered by two separate approaches. The first approach consists of an expansion of the multigroup angular flux solution in a half-range complete set of eigenfunctions. Then, through an orthogouality relation between these eigenfunctions and those of the adjoint transport equation, a set of singular integral equations is derived for the multigroup exit distributions. The half-range completeness of the eigenfunctions is then shown to imply that these singular integral equations possess a unique solution. The second approach consists of deriving this same set of singular integral equations by Laplace transforming the set of integral equations for the scalar group fluxes. Uniqueness of the solution is also shown in an appropriate function space. A multigroup version of the F-N method is then formulated and applied to the solution of the set of of singular integral equations. Convergence is proven and the numerical results for the two group albedo is seen to be in good agreement with those obtained from the standard S-N method calculations.
1. I N T R O D U C T I O N
This article is a continuation of companion paper ~ where we have considered the purely mathematical problem of constructing eigenfunctions of the multigroup linear transport equation in an exponential atmosphere and proven their completeness for solving the half-space exist distribution problem. Here we shall use these mathematical results for the purpose of solving the half-space exit distribution problem (already defined in Ref. [1]) and obtaining numerical results. The following is an outline of this article. Sections 2 and 3 are devoted to a derivation of a set of singular integral equations for the multigroup exist distributions and to a proof of their unique solvability. In Section 4, we derive this same set of singular integral equations in an alternate way, prove its unique solvability and present the theoretical background which is necessary for proving convergence of a multigroup F-N scheme which we formulate in Section 5 and apply to its solution. In Section 6, we present an F-N calculation for the algebraically tractable two group case and compare it to the results found from the well tested standard S-N method embodied in the ANISN code. 2 Finally, Section 7 is devoted to conclusions, a discussion of the results and recommendations for future work. 2. S I N G U L A R
INTEGRAL EQUATIONS VIA EIGENFUNCTION
FOR THE EXIT EXPANSION
DISTRIBUTIONS
The half-space completeness proof presented in the previous article t allows for the expansion of the solution ~,(x, g) as in equation (26) of Ref. [1], i.e.
f,
(l)
where all the necessary definitions have been made in Ref. [1]. In order to derive from equation (1) a set of singular integral equations for the exiting group fluxes we shall first establish an orthogonality relation between the eigenfunctions constructed in Ref. [1] and those of the adjoint multigroup transport equation
•_.,*(x,l~)={texp(-x/s)]/2}C+'f"d-
~*(x,/~') dt~', i
where C + is the transpose of C. *Present address: 10535 Wilshire Boulevard, No. 401, Los Angles, CA 90024, U.S.A. Q.S.R.T.37/~--G
97
(2)
98
R. DANIELet
aL
To construct the adjoint eigenfunctions, we rewrite equation (2) with a simple change of independent variable as
q,*(x, -it') dit'.
It(~/Ox)O*(x, - i t ) + r-.O*(x, - i t ) = { [ e x p ( - x / s ) ] / 2 } C + •
(3)
I
Assuming that equation (18) of Ref. [1] is satisfied, then eigenfunctions of equation (3) are readily obtained from the eigenfunctions constructed in Ref. [1] by simply replacing C by C +. If, in addition, we replace It by - i t in these eigenfunctions we obtain the desired eigenfunctions of the adjoint equation, equation (2). Explicitly then, we have the eigenfunctions dp*J(x, It) = O *J(it, v) e x p ( - x /v ) + X*J(Ia, v) e x p ( - x / c o ) ,
(4)
where the j t h element of ~*J(it, v) is given by
(5)
¢,?J(it, v) = ,~(it + vr.j)
and where X*J(it, v) = (co/2) (co Y +
ItO) -1
"C + .e j -- ~(--it, co)./t*J(v)
(6)
with 2*J(v) = (~o/2)
t,l
! dit(,o+,u)-'.C+.e,; ,/-
(7)
I
the degeneracy of this set is the same as the degeneracy of the set constructed in Ref. [I], i.e. for j = I..... G, #~*:(x,It) is an eigenfunction in the distributional sense for v ~ y:. The orthogonality relation can n o w be easily derived as follows. W e consider the transport equation with ~i(x,It) as a distribution solution for v e Yi, i.e.
It(~3/Ox)gp~(x, It) + ~_. ~',(x, It) = { [ e x p ( - x / s ) ] / 2 } C .
dp',(x, It') dit';
(8)
I
and its adjoint with 4~*)(x, It) as a distributional solution for v ' e ~,~, i.e. - i t ( O / O x ) O y J ( x , It) + I . . ~ J ( x ,
It) = { [ e x p ( - x / s ) l / 2 } C
÷.
:,
~,*J(x, It') dit'.
(9)
-I
We multiply equation (8) by [ ~ J ( x , It)lT and equation (9) by [4~(x, It)lT, subtract the two results and integrate over It from - 1 to 1. Using simple matrix identities, we obtained the distributional relation ditit [4,'v(x, It)]v. q~*,J(x, it) = 0,
(OlOx)
(10)
I
where v ET~ and v'eT: for any i , j = 1. . . . . G. Integration equation (10) over x from 0 to m, and using the fact that at m both ~ ( x , It) and ~*J(x, It) vanish, we obtain
f'
ditit [~iv(0, It)]T'~*J(0, It) = 0,
V e ~i,
V' E 7j.
(] 1)
-I
Equation (11) is the desired'orthogonality relation and holds in a distributional sense. We now multiply equation (11) by the expansion coefficient ai(v) (uniquely determined from the half-range completeness proof) integrate over v on 7~ and sum over i from 1 to G. Interchanging the order of integration and using equation (1) with x = 0, we obtain the set of equations ditit[0(0, It)]T'~J(0, It) = O,
j = 1 . . . . , G,
v" ~ 7j.
(12)
I
Upon replacing the explicit form for ~*J in equation (12), it becomes clear that it is a coupled set of singular integral equations for the multigroup exit distributions It0g(0, It), It < 0, g = 1. . . . . G. It can be rewritten in a more convenient form by replacing It by - i t in the integral from - 1 to
Multigroup linear transport--II
99
0, and ¢(0, #) by the incident flux vector f(#) for u > 0, thus yielding the set of equations d##[#,(0,-#)]r.~by(0,-#)= d##[f(#)]T.Oy(0,#), v'eTj, j = l . . . . . G. (13) fo' fo' Clearly, the fight hand side of equation (13) is a known term, whereas the left hand side represents singular integrals over the unknown exit group distributions. The question of existence and uniqueness of solutions to this set is easily answered in the next section as a generalization of an argument already presented for the one group (one speed) case by Larsen and Mullikin) 3. UNIQUE SOLVABILITY OF THE SET OF SINGULAR INTEGRAL EQUATIONS We shall exploit the half-range completeness property of the eigenfunction set ~i~(x, #) to show that a unique solution exists to equation (13) in a class of functions which we shall determine. Existence of a solution is insured since the superposition in equation (1) which by half-range completeness represents the desired solution of the half-space exit distribution problem, also necessarily satisfies, when taken at x = 0 and # < 0, the set of equations in equation (13). Moreover, from the integral relation between the exit distribution vector and the scalar flux {see equation (35) in Ref. [1]} such a vector solution is dearly analytic in all its elements for # ~ [ - l, 0). Uniqueness of this solution can be shown within the wider class of H61der continuous vector functions as follows. We assume another solution, ~'(0, - # ) , exists and is H61der continuous in all elements on (0, 1). The difference A(0, #) between ~ (0, - #) and ~'(0, - #) is H61der continuous and satisfies the homogeneous set of singular integral equations
~0
' d## [A(0' #)]T'~b*J(0' - # ) =0,
J = 1,... ,G.
(14)
Now ~*J(0, - # ) is obtained from ~{,(0, #) by simply replacing C by C +. Thus if sup i,j
C,j
is small enough then the eigenfunction set ~*J(0, - # ) is half-range complete for expanding the H61der continuous vector function A(0, #). Assuming the expansion coefficients are the functions bj(v'), we can then multiply equation (14) by bj(v'), integrate over v' on ~j, sum overj from 1 to G, interchange the order of the # and v" integrals and use the half-range expansion for A(0, #) to obtain the scalar equation
f0
1d## [A(0, # )]T[A(0, # )] = 0,
(15)
which clearly shows that A(0, # ) = 0, and completes the uniqueness proof. We now summarize our results in Sections 2 and 3. We considered the adjoint multigroup transport equation. Under the condition in equation (28) of Ref. [1], the adjoint eigenfunctions $*J(x, #) are obtained from the eigenfunctions of the transport equation ~#{(x,#) by simply replacing C by C + and # by - # . An orthogonality relation among these eigenfunctions was established and is given by equation (11). This orthogonality relation together with the expansion in equation (1) were used to derive a set of coupled singular integral equations for the exiting group distributions as in equation (13). A solution to this set of equations is guaranteed to exist since the eigenfunction set ~{(x,#) is complete for expanding the solution to the half-space exit distribution problem under the conditions already discussed in Ref. [1]. This solution to equation (13) is given by the expansion in equation (1) taken at x = 0 and # < 0 and is analytic in # on (0, 1). Uniqueness of this solution is guaranteed in the wider class of H61der continuous functions on (0, 1] (analytic functions are a subclass of it) provided the conditions for half-range completeness of the eigenfunction set ~{(x, #) are satisfied when C is replaced by C +. Exact solutions to equation (13) are not known to exist to date and, just as in the one speed (one group) case, are too difficult to obtain. We shall instead be seeking approximate solutions
100
R. DANIEL et al.
through a multigroup version of the F-N method. The theoretical foundation necessary to prove convergence of the F-N collocation scheme which we will be using is provided by following an alternate approach to deriving singular integral equations for the exiting distributions, as was first demonstrated for the one speed case by Mullikin. 5 This is the subject of the next section.
4. A L T E R N A T E D E R I V A T I O N OF THE S I N G U L A R I N T E G R A L EQUATIONS AND T H E I R U N I Q U E SOLVABILITY We consider equation (35) of Ref. [1] in which we set x = 0 and change variable from/~ to - p to obtain
dx'exp(-x'/s)R(-x',
/ ~ ( 0 , --/t)=(1/47t)
-/t)'C,ib(x'),
0 < # ~< 1,
(16)
where the elements of the diagonal matrix R ( - x ' , - / t ) have been defined ~ as
Rgg(-X', -kt) = e x p ( - x ' X g / # ) .
(17)
Defining the scalar Ig(x') by G
Ig(x') = (1/4n) exp(--x'/s) ~ CgSpj(x'),
(18)
j=J
it is then clear that/~Og(0, -/~) is the Laplace transform of Ig(x') with the tramsform variable being Zg//~. By defining the vector R(z) from R(z)=
f:
exp(-x'z)I(x')dx',
g = 1. . . . ,G,
(19)
where z is allowed to be complex, we obtain the relation
Rg(~.g/lt) =/~kg(0, -/~),
0 < U ~< 1, g = 1. . . . . G.
(20)
Now, from equation (31) of Ref. [1] and the definition in equation (18), it is easy to show that l(x) satisfies a set of coupled integral equations given by
f
l(x) = (1/2) e x p ( - x / s ) C ,
oo
dx' ~ ( J x - x' I)'I(x')
do
+(l/2)exp(-x/s)C.
f,
d~ R(x,~)-f(~),
(21)
do
where the diagonal matrices ~ and ~ have been defined in Ref. [1]. The definition of equation (19) can then be used to transform equation (21) into a set of coupled singular integral equations for the elements of R(z). Thus, by multiplying equation (21) by e x p ( - xz), integrating over x from 0 to ~ , using the definition of the first order exponential integral function in the elements of M, interchanging the order of the resulting integrals and using the definition in equation (19) we obtain the set of equations Rg(z) + (1/2)
j=l
Cgj
dl.t' Rj(Ej/I.t')/[Ej - I.t'(z + l/s)]
-- (1/2) ~,, CgjRj(z + 1/s) j=l
= (1/2)
~01
dla'/[Y.j- p'(z + l/s)] I
G
d u ' Z Cgjlt'fj(l~')/[~.j+ lz'(z + l/s)], j=l
g = 1. . . . . G.
(22)
By letting z = Y~g//~in the gth equation and using equation (20), we obtain a set of coupled singular
Multigroup linear transport--II
101
integral equations for the exiting group angular distributions #$g(0, - g ) , g = 1. . . . . G: °
#~O,(O, --#) + (1/2) ~..=C~
-- (1/2)
=(1/2)
j=l
d#'#'~Oj(O,-g')/[Zj - #'(Z,/# + I/s)]
CsjZ~qgqJj(O,-Y.,jqg)
d#'/[Ej - U'(X, Iu + l/s)]
1
fO dg' G j=l
0<~<1,
g = l . . . . . G,
(23)
where the function qg is defined from
(24)
1/q, = Y.,gl# + lls.
In equation (23), the function Xyqs may exceed unity; in this case the function Ejqz~j(0, -Xjqs) is defined through an extension of equation (16) to values of the argument # outside the physical range (0, 1]. In the gth equation of the set in equation (23), we make a convenient change of notation from # to v~z, 0 < v ~< 1/Y,s, to obtain vZ, t#~(O, - r E , ) + ( 1 / 2 )j=t E~ C,j f0l dv'#'$j(O, -#')/(Zj - #'/oJ)
=
(1/2)
~01d # ' E G j=l
+ ~'/o~),
0 < v .< l/X,,
, = 1. . . . . G.
(25)
Now, if the restriction on the eigenfunction construction in Ref. [1], equation (18), is satisfied then it is a very simple exercise to check that equation (25) is identical, apart from dummy variables and indices, to equation (17) derived via eigenfunction expansion and the use of the orthogonality relation. The desired solution to equation (23) {or, if equation (18) in Ref. [1] is satisfied, then also equivalently to equation (17)} is that which is given by equation (16) in terms of ~(x). Here, ~(x) is that solution of equation (31) of Ref. [1] whose existence and uniqueness is guaranteed, with elements ~b~(x) continuous and square integrable on [0, oo], provided sup C U i,j
is small enough, t As we mentioned earlier in Section 3, a closed form is not known to exist for this solution for #~(0, - # ) . For the purpose of analyzing convergence of a numerical solution to this desirable solution it will be useful to prove its uniqueness in an appropriate function space. Such an analysis was first performed by Mull/kin 5 and in a subsequent article by Kelley and Mull/kin6 for the one group (one speed) case. Here we shall simply generalize their arguments to our muir/group case. We note that, from our earlier assumption in Ref. [1] that det C # 0 and from the use of the Schwarz inequality, it is clear that equation (21) has a unique solution for I(x) with Is(x ) ~ L2[0, oo] N C[0, co] whenever equation (31) of Ref. [1] has a unique solution for ~(x) with Sg(x) satisfying the same properties. Consequently, since R(z) is the Laplace transform of/(x) and since equation (22) is the Laplace transform of equation (21) then equation (22) has a unique solution for R(z) with Rg(z) in the Hardy space H 2. This space is the Hilbert space of functions which are Laplace transforms of functions in L2[0, oo]. Every function in H 2 is analytic in the right half plane and the inner product between any two functions F(z) and G(z) in H 2 is defined as
{F, G)m = (1/2~) f
o F(it)G(it) dt.
(26)
R. DANIELet al.
102
It follows then that the desired solution of equations (23) and (16), is the only solution which is analytic on (0, 1], which can be analytically continued from equation (20) to the right half-plane of Eg//~ and be in H ~. Based on the foregoing observations and facts we shall, in the next section, formulate a multigroup F-N scheme which when applied to equation (23) or equivalently to equation (13) {if equation (18) in Ref. [1] is satisfied} would converge to this desired solution f o r / t o (0, -/~).
5. THE M U L T I G R O U P F-N C O L L O C A T I O N SCHEME We shall first begin by presenting some mathematical preliminaries necessary for proving convergence of the numerical F-N scheme which we present later in this section. The sequence of functions ( 0 . ( z ) = (z + z.)-~}~=l, where the set of points {zn}~ belongs to the positive real axis (0, oo) and has at least one accumulation point not at zero or infinity, was shown 5'6 to form a basis for a 2. We now consider a Hilbert space H of column vectors with G elements in H 2 and the inner product between any two vectors F(z)=[Fl(z) . . . . . F~(z)] T and G(z) = [G,(z) . . . . . G~(z)] r in H defined by G
(F, G ) . = ~ (Fg, Gg)n:.
(27)
g=l
Next, we define a sequence of vector functions {0.i( Z )}.oo,G = ,. i=, to be a sequence of column vector functions with elements zero except for the ith element which is 0.(z). It then follows that the o~,G sequence {0.i( z )}.= 1,i=1 forms a basis for H. We now consider equation (22) and write it symbolically as R(z) + 0_.R(z) = F(z).
(28)
where 0_ is a G × G matrix operator with a general element Ln corresponding to the integral operator on Rj(z) in the g t h equation, and where F (z) is a known column vector with the gth element Fg(z) corresponding to the known term on the right hand side of equation (22). The integral operators L~. are known to be Hilbert-Schmidt on H 2 by a simple generalization of an earlier argument 6 relevant to the one group case. Consequently the matrix operator 0_ is Hilbert-Schmidt on H. A well known procedure for obtaining approximate solutions to equation (28) is the Galerkin 7 procedure. This consists of inserting the expansion G
Ru(z) = ~
N
~ a, AO,i(z),
N integer 1> 1,
(29)
i=1 n = l
into equation (28) and evaluate the scalar coefficients ani by matrix inversion of the set of N × G algebraic equations: ( R N , 0 , . j ) / ~ + ( 0 _ . R N , 0 , . j ) n = ( F , 0,.j)H, j = l
. . . . . G, m = l , . . . , N ,
(30)
where the sequence of vector functions 0m:(z) forms any basis for H generated by a set of points Zr. accumulating on (0, oo). With 0_ being Hilbert-Schmidt on H, sufficient conditions are known to exist 7 to guarantee that the expansion in equation (29) would converge, as N ~ 0% to the unique solution in H of equation (28). The key observation here was made by Mullikin. 5 He proved that evaluating an arbitrary Hardy space function F(z) at a point zn is exactly the same as evaluating the inner product in H 2 of F(z) and 0,(z), that is
F(z,) = (r, O,)u2.
(31)
It is then obvious that evaluating the ith component of a vector function F e H at the point z, is the same as taking the inner product in H of F(z) with 0n;(z), i.e. Vi(zn) = (F, Oni)I'l"
(32)
Consequently, the Galerkin procedure described above is the same as a collocation scheme whereby the expansion (approximation) in equation (29) is inserted into equation (28) and the expansion
Multigroup linear transport~II
103
coefficients a,i are then evaluated by direct matrix inversion following collocation of the resulting vector equation for RAtat the points zm, m ffi 1. . . . . N. Such a collocation scheme is thus guaranteed to converge, as N ~ 0% to the unique solution in H of equation (28). With this collocation method established as a procedure for solving equation (22), we could directly deduce from it an F-N scheme which when applied to equation (23) or equivalently to equation (13) {if equation (18) of Ref. [1] is satisfied} would converge to the desired solution for # ~ (0, - # ) ; however, we shall instead follow a somewhat reverse direction. We shall generalize an F-N scheme first used with empirical success .by Garcia and Siewert s for the one group (one speed) case, the so-called "natural basis" expansion. We will then identify it as a special version of our collocation scheme, thus proving Convergence. We consider the approximation G
N
¢/N(O, - I z ) = ~ j=l
~ aj, gp~,(O, -/~),
0 < # ~< 1,
(33)
i=l
where {vi}~u=lis a set of points on (0, Zi-I) which is assumed to have at least one accumulation point as N ~ ~ , ~ ( x , / ~ ) , withj = 1 , . . . , G, is the degenerate eigenfunction set constructed in Ref. [1], and aj, are unknown expansion coefficients to be determined. If equation (18) of Ref. [1] is satisfied and if the conditions for half-range completeness of the eigenfunctions are present then the approximation in equation (33) may be referred to as an expansion in the natural basis, since, as it was first mentioned in the one speed case, 9 it is motivated by the expansion of the solution #(0, - # ) in the eigenfunction set. However, it should be mentioned, that we are adopting this approximation even if equation (18) of Ref. [1] is not satisfied, in which case the term "natural basis" would seem irrelevant. We proceed by replacing ~ , ( 0 , - # ) by its explicit form from equations (21), (22) and (24) in Ref. [1]. We then obtain for the gth component G
At
#~Og~(0,--#)=~ ~aj,/tCgj/2(o~,l~s+/t), j=l
0
g=l,...,G,
(34)
i~l
where o~, = vis/(v, + s). We then change variables from t~ to v~g in equation (34) and insert this expression into equation (24) or equivalently equation (13) {if equation (18) of Ref. [1] is satisfied}. Next we collocate the resulting vector equation by evaluating the gth component at the set of points v,., i = 1. . . . . N. This leads to a set of G x N algebraic equations for the G x N unknowns a~. To prove convergence of this F-N scheme to the desired solution, we replace l~g//t in equation (34) by z. Then, through simple algebraic manipulations it becomes clear that the expansion in equation (34) is equivalent to the expansion for Rg(z) G
N
RgAt(z) = ~, ~ (aj, C,j/2o~,)(z + co7')-'. j-t
(35)
:=l
Upon interchanging the order of summation in equation (35) and carrying the summation over j we obtain the expansion N
RgAt(z) = ~ A,g(z + co7')-',
g = 1. . . . . G
(36)
i=l
where the coefficients A~8 are defined as G
A~g = ~ aj~C~s/2oJ,,
g = 1. . . . . G,
i = 1 . . . . . N.
(37)
j--I
In vector form equation (36) can be rewritten as G
RAt(z) = ~
N
~ A,,~%(z)
(38)
g ~ l i--I
where the vector functions ~ ( z ) are as defined earlier with the points z~ in this case equal to o~-~. Since the points vi, i = 1, . . . , N, have an accumulation point on (0, ]~-t) as N ~ ~ , then clearly the points co~have an accumulation point on (s -t + ~t, ~). It follows then that the expansion in equation (38) is in a basis for H.
104
R. DANIEL
et al.
Next, insertion of the expansion in equation (34) into equation (24) or equation (13) {if equation (18) of Ref. [1] is satisfied}, changing variables in the gth component of the resulting vector equation from 11 to VEg, and evaluating this gth component at the set of points vr, is equivalent to our previously established collocation method with the collocation points here being z m = vm ~, m = 1. . . . . N, and accumulating on (£1, ~ ) . Our multigroup F-N scheme is thus guaranteed to converge as N ~ ~ to the desired solution. In the next section we present an algebraically simple application of this F-N scheme to the two group case and display the numerical results which we have obtained on the digital computer vs two group calculations using the standard ANISN code? 6. TWO G R O U P F-N C A L C U L A T I O N S We consider the algebraically simplest multigroup case: the two group case with G = 2. We then have the expansions N
N
111/11N(0' --11) = E a~il~Ct,/2(c°iEJ + 11) + ~. a2~11C~2/2(coiEt + 11). i=1 N /~t//2N(0' --11) =
E i=1
(39)
i=l N
ali11C21/2(oJi '1- 11) 'F E a2i#C22/2(t°i + 11) .
(40)
i=l
These expansions are now inserted into equation (23) or equation (13) [if s (s + 1)-~E~ < 1] and the integrals on the left hand side of the resulting two equations are then evaluated analytically. On the right hand side of these two equations we have assumed that the incident fluxes f~ (11) and f2 (#) are isotropic and given by fj (11) = 1,
(41)
£(11) = O.
(42)
The integrals on the right hand side are also evaluated analytically. Next, in the first equation # is replaced by vE~, and in the second, 11 is replaced by v. The system of two equations is then collocated at the set of points v~, chosen equally spaced on (0, E/~ ~) for each choice of N, and the resulting algebraic system of 2N equations is then solved by a matrix inversion routine to yield the expansion coefficients a , and a2i , i = l . . . . . N . The two group albedos defined by ALB1 = and ALB2
=
[fo' [fo'
]/;o' ]/;o'
11~'j~v(0,-11) d11
11 d11
11~2N(0,--11) d11
# d11
can then be computed by carrying out the integrals analytically. The convergent results for the quantities ALB1 and ALB2 are presented in Table 1 together with the ANISN code results for several values of the exponential decay parameter s. The special data for the various parameters was: E~ = 1.005, C~ =0.5, G2 =0.3, C2~ = 0 . I , 6'22 = 0.4, convergence criterion E = 10 -4. Note that the condition on the applicability of our scheme to equation (13) is satisfied for all values of s which are reported. Table 1. A comparison between F-N results and ANISN results s
F-N ALB1
ANISN ALBI
F-N ALB2
ANISN ALB2
0.5 1.0 5.0 10.0 50.0 100.0 300.0
0.08057 0.10304 0.13959 0.14758 0.15538 0.15649 0.15724
0.08050 0.10318 0.13986 0.14784 0.15564 0.15674 0.15749
0.01798 0.02381 0.03440 0.03699 0.03958 0.03995 0.04021
0.01735 0.02384 0.03449 0.03704 0.03962 0.03999 0.04025
Multigroup linear transport--II
105
We note that the F-N results are in good agreement with the ANISN code. However, for cases not reported with s near 103 the F-N method results display numerical instability problems as N increases. These difficulties may be purely numerical, such as error growth in the matrix inversion routine as N increases and s is very large. They may also be attributed to the choice of collocation points being uniformly spaced and the expansion basis varying with N. Finally, they may be attributed to the fact that at s = oo, the constraints to which the singular integral equations are subjected to guarantee unique solvability are not reproduced in a uniform manner thus perhaps rendering the convergence rate of the solution too slow as s increases. In summary, these difficulties are not well understood yet. 7. C O N C L U S I O N S
AND
RECOMMENDATIONS
FOR
FURTHER
STUDY
From the results in this article and in the companion publication t we may conclude that the multigroup treatment of the exit distribution problem from a half-space is fairly well understood. Future study ought to deal with the finite slab problem through extensions of an analysis presented previously for the one speed case. 4,9 Acknowledgement--This work was partially supported by the National Science Foundation. REFERENCES 1. R. Daniel and G. C. Pomraning, JQSRT 37, 85 (1987). 2. W. W. Engle Jr, The User's Manual for ANISN: A One Dimensional Discrete Ordinates 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). 3. E. W. Larsen and T. W. Mullikin, J. Math. Phys. 22, 866 (1981). 4. T. W. Mullikin and C. E. Siewert, Ann. nucl. Energy 7, 205 (1980). 5. T. W. MuUikin, Some Singular Integral Equations in Linear Transport Theory, Sandia National Laboratory Report SAND-80-1069, Albuquerque, N.M. (1980). 6. C. T. Kelley and T. W. Mullikin, J. /nt. Eqs. 4, 77 (1982). 7. G. I. Marchuk, Methods of Numerical Mathematics. Springer, New York (1975). 8. R. D. M. Gareia and C. E. Siewert, JQSRT 25, 277 (1981). 9. R. Daniel and G. C. Pomraning, J. Math. Phys. In press.