P e r g a m o n Progress in Nuclear Energy, Vol. 33. No. 4. pp. 393--401. 1998

O 199.8 Elsevter Science Lid All rights .:served. Printed in Great Britain

0149--1970/98 $19.00 ÷ 0.00

PII : S 0 1 4 9 - 1 9 7 0 ( 9 6 ) 0 0 0 2 4 - 8

A RECURSIVE METHOD TO INVERT THE LTSN MATRIX

Jacques Branche~, Augusto Cardona 2 and Marco T~llio de Vilhena z

'PPGE~- UFRGS

Av. Osvaldo Aranha, 99- ~ andar

90046-900, Porto Alegre - Brazil

E-Mail jacques@cesup, ufrgs, br

zIM~ - PUCRS

Av. Ipiranga, 6681 - pr~dio 15

90619-900, Porto Alegre - Brazil

E-Mail avcardona@music, pucrs, br

Abstract

In this work it is presented a recursive method to invert the LTS~ matrix. Numerical simulations are presented.

0 1 9 9 8 E l s e v i e r S c i e n c e L t d . A l l r i g h t s r e s e r v e d .

l.Introduction

In the last years, Vilhena and co-workers proposed an analytical method, coined LTSN method, to solve the multidimensional discrete ordinates problem (S, problem) [I-II]. For the one-dimensional case, its main feature encompasses the following steps: application of the Laplace transform to the set of SN equations, analytical solution of the resulting algebraic system for the transformed angular flux and reconstruction of the angular flux employing the Heaviside expansion technique. On the other hand, the idea involved on the solution of multidimensional SN problem, relies on its transformation into a set of one- dimensional S~ problems by transverse integration. This procedure allows the establishment of analytical solutions for the average angular fluxes using the one-dimensional approach. In the foregone works, the analytical solution of the linear algebraic system was achieved recasting the system into matrix form and carrying out the calculation of the determinant and adjoint matrix, for an arbitrary N, of the so called LTSN matrix. The feasibility of this procedure is due to the structure of the LTS, matrix [I]. It is relevant to stress the fact that the number of operations required by this method to solve the algebraic system is proportional to N!

393

394 J. Brancher e t a l .

(read N factorial), consequentely this method is inappropriate, under the computacional point of view, to solve SN problems with large N, which occur for instance in radiative transfer problems.

Therefore, in this work, it is depicted a new approach to solve the algebraic linear system in order to improve the capability of the LTSN method to solve SN problems with large N. To this end, this idea is exemplified for isotropic problem. Indeed, the LTSN matrix, for an arbitrary N, is appropriately modified through elementary operations into a form such that the resulting matrix posses a special structure, when seeing as block matrix, because its first entry, corresponding to the first row and column, is the matrix LTSn-I. From this result, stems a relationship between the modified LTSn and LTSn-I matrices, for n ranging from 2 to N. Now starting with n = 2, proceeding recursively, the inverse of the LTSN is obtained evaluating the matrix LTSn (for n = 2 : N) in terms of the matrix LTSn-I, through the application of the partitioning method [12].

The paper is outlined as follow: in section 2, is reported how the LTS, matrix is inverted by the recursive method and in section 3, are presented numerical simulations for the scalar flux by the LTSN formulation with N ranging from 4 to 120 as well discussion of the results achieved.

2. The recursive method

In order to elucidate the idea embodied by the recursive method, let us consider the following isotropic SN problem [i]:

N d~n (x) + ~n(X) = c ~n 7 ~k~Pk(X)' n = I:N. (i) dx

k=l

which has the well known LTSN solution [1,2]:

~_(x) = exp (SnX) ~ QN(s n) . q0(0) , in=0 = --

(2)

where the column vector <0(x) is defined as [q00(x)...q)N(X)] T and dN(S),

QN(s) and the coefficients s n denote respectively the determinant,

the adjoint matrix and the eigenvalues of the matrix AN(S) written

as:

Recursive method to invert the LTS N matrix 395

AN(S} =

s + I c (o1 c (o2

C (O1 I C (O2 S +

2 J/2 J/2 2 ~2

:

C(O N

2 ~I

C(O N

2 ~2

c C°l c (0 2 i c (o N ... s +

2 BN 2 ~N BN 2 BN

(3)

Here was used the standard notation and prime stands for derivative respect to s.

In order to apply the partitioning method [12], let us

perform some elementary operations over the matrix ~N~), namely

subtraction of rows. It turns out the modified LTSN matrix, denoted

A~(s), whose entries are expressed as:

a~,j =

c (I + s~, t) 6j, 1 - ~(oj , se i = 1

(I + s#i), se i ~ i e i = j

-(I + S#j), se i ~ 1 e i = j + I

, (4)

0, em outro caso

where ~i,j stands for the Kronecker's delta.

To calculate the inverse of the matrix A'(s) recursively, we --N

start defining the following matrices:

396 J. Branchcr et al .

B (s) =

c _ ~ COn.

2

Bn - 1 (s)

0 .-- 0 -g + SBn_ I) ~% + S~n'~

(5)

C for n = 2 : N and . Bl(s)= (I + S~l)- ~I. It is important to notice

t h a t B (s) h a s a s u i t a b l e s t r u c t u r e t o be i n v e r t e d b y t h e

partitioning method [121. Following this procedure it comes out

t h e f o l l o w i n g f o r m u l a f o r t h e i n v e r s e o f t h e m a t r i x B (s):

B_I(s) = __I pn(s) , with n = 2 : N, (6) =n D n (s) =

where Dn(s ) denotes the determinant of the matrix B (s) and pn(s) is m n

t h e a d j o i n t m a t r i x o f B (s), whose e l e m e n t s a r e d e f i n e d a s :

n-I pnn, j(s) = (i + S~/n_l). P~_I,j(s) , (7)

if j = 1 : (n - i), or:

p~, n (S) C pn-l. • =---C°n- i,l IS), (7a)

2

if i = 1 : (n - I), or:

pn, n(S) = Dn_l(S) , (7b)

or:

P~,~1(s). D~(s) + c P~_1,j(s). £ + S~n-l)

~, = , (7C) Dn- i (s)

Recursive method to invert the LTS N matrix

if Dn_l(S)~ 0 and i, j = 1 : (n-l), or:

2 2 (on-l" (I + S~n) + (o n . ~ + S~n_l)

p~,3(s ) = c. (I + S}/n_2) • 2 (on-l- Dn-2(s)

• Pn-2,j(S) + Dn_2(S ) '

if Dn_l(S) = 0 and i, j = 1 : (n - 2), or:

n-I Dnn_l,j(S) = (I + S~n). Dn_l,j(s) ,

if Dn_l(s) = 0 and j = 1 : (n - 2), or:

2 2 pnn_l(S) c (on-l" (I + S~n) + (on" (i + S~n_ I) n-2

• = --- • Pi,l (s) ,

2 (on-I

if Dn_l(s) = 0 and i = 1 : (n - 2), or:

~ n - 2 . . • i ,1 i S ) .

397

(7d)

(7e

(7f

pnn_l,n_l(s) = (I + S~n). Dn_2(S) , (7g

if Dn_l(s) = 0, with n = 3 : N. Here for n = 2 we have:

C p21(S) = I + SZI and p222(s) = I + s~ l - ~ col. P~l(s) = I + s~2; P22(s) = ~ (o2; c

Exploiting the structure of the matrix B (s), the following -n

recurrence formula for the determinant is readily obtained using its definition and is quoted as:

n-I

Dn(S) (I + S~n). Dn_l(s) C H = - --.(0 n . (I + S~k) , n = 3:N, 2 k=l

(8

C where D2(s) = ~ + s~l)~ + s~2)- ~[(ol~ + s~2)+ (o2~ + S~l)]. Now,

replacing (8), for decreasing n, sucessively in the RHS of equation (8), it turns out:

n n n C

k=l m=l k=l k~m

(9)

398 iBranchcre ta l .

Finally, observing that ~(s) = ~N(S), the LTS, solution is

then encountered and expressed as:

q)(x) = e x p (SnX) - pN(s n) . [ " , - - n = 0 D~l{Sn) = ' - -

(10)

where the eigenvalues s n are the zeros of DN(S), which are evaluated

using the LAPACK subroutines [13], and the vector [ is defined as

[ B I ~ I ( 0 ) B 2 ~ 2 ( 0 ) _ ~ 1 ~ 1 ( 0 ) . . . BN(PN(0) _ B N _ I ~ N _ I ( 0 ) I T .

As final remark, it is important to notice that the adjoint

matrix ~n(s), described by equations (7)-(7g), is well defined,

because the determinants Dn_l(S) and Dn_2(s) don't vanish

simultaneously when s = sk. These results can be easily verified given a closed look to equations (8) and (9).

3. Numerical Results and Conclusion

In order to check the LTSN formulation with this method of matrix inversion, let us consider the following SN problem in a homogeneous slab with the following parameters: slab tickness 40 mfp, c = 0.999; 0.950; 0.900 and 0.800 and boundary conditions:

(Pk(0) = I, for ~k > 0, and (Pk(40) = 0, for ~k < 0. The reason for

choosing this problem is the fact it has well known solution [14]. To ilustrate the numerical convergence of the LTSN method, in table 1 are depicted the results for the scalar flux at x = 20 m[p for N ranging from 4 to 120. Numerical comparisons weren't mading because the solution considered exact (S4B [14]) has only three significant digits.

Examining the results in table 1 is readily realized the convergence of the LTSN solution, with at least six significant digits for N = 120. Taking into account that in a forthcoming paper will be shown the convergence of the LTSN to the Case solutions [15], we believe that we may assert that the results achieved are correct with at least six significant digits. Furthermore, it is relevant to point out that the computational effort to run the LTSN solution with N = 120 in a CRAY Y-MP-2E machine was 3.13 seconds of CPU time with a performance of 112.25 Mflops by second of CPU. Since, we already devised a manner to apply the recursive method to invert the LTSN matrix for SN problems with anisotropic scattering, now being implemented, we are confident that the LTSN method will be usefull to generate benchmark problems and will become an atractive method to solve one-dimensional transport problems.

Rccursiv¢ m e t h ~ to invert the LTS s matrix

Table I. Numerical simulations for the scalar flux at a homogeneous slab using the LTSN formulation

399

N c = 0.999 c = 0.950 c = 0.900 c = 0.800

4

6 8 I0 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 5O 52 54 56 58 60 62 64 66 68 7O 72 74 76 78 80 82 84 86 88 9O 92 94 96 98 i00 102 104 106 I08 II0 112 114 116 118 120

0.583633304085E+00" 0.583377860066E+00 0.583299698224E+00 0.583265491093E+00 0.583247468793E+00 0.583236808471E+00 0.583229979875E+00 0.583225342748E+00 0 583222049818E+00 0 583219627248E+00 0 583217793116E+00 0 583216371102E+00 0 583215246326E÷00 0 583214341351E+00 0 583213602405E+00 0 583212991182E+00 0 583212479880E+00 0 583212047832E+00 0 583211679463E+00 0 583211362860E+00 0 583211088721E+00 0 583210849812E+00 0.583210640341E+00 0.583210455634E+00 0.583210291973E+00 0.583210146255E+00 0.583210015960E+00 0.583209898969E+00 0.583209793513E+00 0.583209698176E~00 0.583209611683£~00 0.583209532954E+00 0.583209461121E+00 0.583209395356E+00 0.583209335042£+00 0.583209279555E~00 0.583209228413E+00 0.583209181169E+00 0.583209137451E+00 0.583209096878£~00 0.583209059192E+00 0.583209024121E+00 0.583208991422E+00 0.583208960879E+00 0.583208932324E+00 0.583208905579£+00 0.583208880492E÷00 0.583208856911E÷00 0.583208834748E+00 0.583208813867E+00 0.583208794201E+00 0.583208775629E+00 0.583208758100E+00 0.583208741509E+00 0.583208725811E+00 0.583208710906E+00 0.583208696768E+00 0.583208683348E÷00 0.583208670647E+00

0.769020258529E-03

0.766531811775E-03 0.765730532133E-03 0.765375157798E-03 0.765186708226E-03 0.765074817275E-03 0.765002967823£-03 0.764954092593E-03 0.764919341113E-03 0.764893750037E-03 0.764874360065E-03 0.764859317543E-03 0.764847413247E-03 0.764837831027E-03 0.764830003842E-03 0.764823527772E-03 0.764818108777E-03 0.764813528674E-03 0.764809622812E-03 0.764806265047E-03 0.764803357404E-03 0.764800822876E-03 0.764798600250E-03 0.764796640352E-03 0.764794903360E-03 0.764793356695E-03 0.764791973534E-03 0.764790731621E-03 0.764789612344E-03 0.764788600089£-03 0.764787681615£-03 0.764786845699E-03 0.764786082730E-03 0.764785384454E-03 0.764784743769E-03 0.764784154500E-03 0.764783611301E-03 0.764783109482E-03 0.764782644953E-03 0.764782214096E-03 0.764781813743E-03 0.764781441079E-03 0.764781093597E-03 0.764780769108E-03 0.764780465595E-03 0.764780181301E-03 0.764779914650E-03 0.764779664188E-03 0.764779428635E-03 0.764779206846£-03 0.764778997762E-03 0.764778800420E-03 0.764778613976E-03 0.764778437634E-03 0.764778270672E-03 0.764778112438E-03 0.764777962324E-03 0.764777819794E-03 0.764777684378E-03

0.365535707222E-04

0.364400929874E-04 0.363918426022E-04 0.363702066811E-04 0.363586635839E-04 0.363517848465E-04 0.363473571019E-04 0.363443400266E-04 0.363421921062E-04 0.363406088369E-04 0.363394082949E-04 0.363384763469E-04 0.363377384447E-04 0.363371442219E-04 0.363366586552E-04 0.363362567785E-04 0.363359204063E-04 0.363356360375E-04 0.363353934788E-04 0.363351849174E-04 0.363350042838£-04 0.363348468052E-04 0.363347086863E-04 0.363345868790E-04 0.363344789121E-04 0.363343827655E-04 0.363342967746E-04 0.363342195578E-04 0.363341499603E-04 0.363340870121E-04 0.363340298922E-04 0.363339779028£-04 0.363339304472£-04 0.363338870130£-04 0.363338471588E-04 0.363338105011E-04 0.363337767072£-04 0.363337454868£-04 0.363337165847E-04 0.363336897767E-04 0.363336648655£-04 0.363336416765E-04 0.363336200542E-04 0.363335998603E-04 0.363335809727E-04 0.363335632801E-04 0.363335466849E-04 0.363335310967E-04 0.363335164366E-04 0.363335026322E-04 0.363334896179£-04 0.363334773349E-04 0.363334657297E-04 0.363334547524E-04 0.363334443593E-04 0.363334345092E-04 0.363334251654E-04 0.363334162935E-04 0.363334078619E-04

0 719560926422E-06

0 726222018356E-06 0 725197004862E-06 0 724698810637E-06 0 724431249924E-06 0 724270720916E-06 0 724166940102E-06 0 724096001634E-06 0 724045380463E-06 0 724007998799E-06 0 723979612374£-06 0.723957550713E-06 0.723940065489E-06 0.723925973221E-06 0.723914449645E-06 0.723904906400E-06 0.723896914422E-06 0.723890154847E-06 0.723884386721E-06 0.723879425228E-06 0.723875126684E-06 0.723871378033E-06 0.723868089332E-06 0.723865188299E-06 0.723862616316E-06 0.723860325438E-06 0.723858276145E-06 0.723856435621E-06 0.723854776446E-06 0.723853275556E-06 0.723851913433E-06 0.723850673488E-06 0.723849541546E-06 0.723848505391E-06 0.723847554539E-06 0.723846679858E-06 0.723845873427£-06 0.723845128353E-06 0.723844438527E-06 0.723843798642E-06 0.723843203985E-06 0.723842650391E-06 0.723842134159£-06 0.723841652005E-06 0.723841201016E-06 0.723840778534E-06 0.723840382194£-06 0.723840009922E-06 0.723839659808E-06 0.723839330076£-06 0.723839019234E-06 0.723838725833E-06 0.723838448593E-06 0.723838186369£-06 0.723837938082E-06 0.723837702752E-06 0.723837479470E-06 0.723837267523E-06 0.723837066076E-06

Read as 0.583633304085 x i0 °

400 J. Brancher et al.

Acknowledgements.

The first, second and third authors are respectively indebted to CAPES (Coordenag~o de Aperfeigoamento de Pessoal de Nivel Superior), CNEN (Comiss~o Nacional de Energia Nuclear) and CNPq (Conselho Nacional de Desenvolvimento Tecnol6gico e Cientifico) for partially support this work. The computational work was developed at CESUP (Centro Nacional de Supercomputag~o da Universidade Federal do Rio Grande do Sul).

References.

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[2]BARICHELLO, L. B.; VILHENA, M. T.: A General Approach to One Group One Dimensional Transport Equation. Kerntechnik, v. 58, n. 3, p. 182-184, 1993.

[3]OLIVEIRA, J. V.; AGOSTINI, M. N.; BARICHELLO, L. B.; VILHENA, M. T.: Analytical Formulation to the Solution of the One- Dimensional Discrete Ordinates Neutron Transport Problem with Anisotropic Scattering. Proccedings of the 9th Brazilian Meeting on Reactor Physics and Termal Hydraulics, Caxambd, Brazil, p. 72-77, 1993.

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[5]BARICHELLO, L. B.; VILHENA, M. T.: An Inverse Problem at Neutron and Radiation Transport. Proccedings of the 9th Brazilian Meeting on Reactor Physics and Termal Hydraulics, Caxambd, Brazil, p. 22-24, 1993.

[6]SEGATTO, C. F.; VILHENA, M. T.: Extension of the LTSN Formulation for Discrete Ordinates Problem Without Azimuthal Symmetry. Annals of Nuclear Energy, v. 21, n. II, p. 701-710, 1994.

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[8]VILHENA, M. T.; SEGATTO, C. F.: A New Iterative Method to Solve the Radiative Transfer Equation. Journal of Quantitative Spectroscopy and Radiative Transfer. In press.

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Recursive method to invert the LTS s matrix 401

Proccedings of the 9th Brazilian Meeting on Reactor Physics and Termal Hydraulics, Caxamb~, Brazil, p. 90-92, 1993.

[10]ZABADAL, J.; VILHENA, M. T.; BARICHELLO, L. B.: Solution of the Tree-Dimensional One Group Discrete Ordinates Problem by the LTSN Method. Annals of Nuclear Energy, v. 22, n. 2, p. 131- 134, 1995.

[II]ZABADAL, J.; VILHENA, M. T.; BARICHELLO, L. B.: An Analytical Solution for the Two-Dimensional Discrete Ordinates Problem In a Convex Domain. Progress in Nuclear Energy. In press.

[12]DEMIDOVICH, B. P.; M_ARON, I. A.: Co~utational Math-m~tics. MIR Publishers, Moscow, 1987.

[13]ANDERSON E.; ET ALLI: LAPACK, Users'Guide. Society for Industrial and Applied Mathematics - SIAM, Philadelphia, 1992.

[14]RULKO, R. P. & LARSEN, E. W.: The PN Theory as an Asymptotic Limit of Transport Theory in Planar Geometry - II: Numerical Results. Nuclear Science and Engineering, v. 109, n. I, p. 76- 85, 1991.

[15]PAZOS, R. P.; VILHENA, M. T. : Convergence of the LTS. Method: An Approach of C0-Semigroup. To appear.