~ Pergamon Progress #~ Nuclear Energy, Voh 34, No. h pp. 77-86, 1999

1998 Published by Elsevier Science Ltd All rights reserved. Printed in Great Britain

P lh S0149-1970(97)00112-1 0149-1970/99/S--see front matter

CONVERGENCE OF THE LTSN METHOD: APPROACH OF Co SEMIGROUPS

l~ul~n Panta Pazos * Marco Tullio de Vilhena **

' Prosr~ms de P6e.Ontdut~,io em Bapmhuls do ]din-,, Me~drSlc~, de M¢,~'im, UPROS

Av Oswsldo Anmbs, 20, n h $1~, CliP 000~190, Porto Ale65~. RS - BRAZIL

" IMtltuto de M,aem/ti~. DMPA - D'PROS - C.smpus de Vd~

Av.Beoto Oot~dv,., ~ 0 , U"~P ~1~,0~-900, Porto Abs'm. R$ - BP, AZil

April, 1997

Abstract

h thh work we repo..-'t the conwergence of the LTS.~ solution for the steady st, ste ~ the Co sen~l~'ou p approach fast condde~mg that the total e~u :eaion and ,eatteeinf ke~nei are both independent with respect to the spatial variable. Thin we comider that the tot~ e~o. ~eetion sad ~eatteri.~ kem,.A hinge on the spstisl varisble and apply the procedure d P~y and 'l%asbe [Pa~y 1983] and ['1%nabe 1979]. © 1998 Published by Elsevier Science Ltd. All rights reserved.

X~YW0aDS Co memi~oup, LTSN Maho4, $~eedng Kernel.

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

Much of the ~orts in the t~aiport theory is devoted on N a r ~ of methods that 8~aerste ~cumte results. One d the &ter-,;ni-fic n~tho& is the LTSN method which estsb]khes an An aly~cal f ~ o n for the so- ]ufion d slab- Seometry discrete m4h~ates problmns considering auisotroplc sestterkg and m,dtlsroup too- d,l [s,m,ic~no and ~ I~IA], [Barmch4/o mad V'~,na m91B], [B,~ch,no ,~d VilhQ ImlC],

77

78 R.P. Pazos and M. T. de Vilhena

[Bamichello sad Vilhena 1993], [Vilhena and Barichello 1991], [Vilheua and Barkhello 1995]. The step, emboddied in the idea of this method are now briefly dimmed. The Leplace ~andorm is applied to the set of discrete ordinates equa~ons, rewriting in an e~gehmic linear system for the trtnd~rmed angular flux, which is solved analytically exploiting the structure of the associated matrix. The angular flux is then analy- ticaHy restored udug the Heavidde expansion technique. This procedure leads to an analytical formnktion for the angular flex in terms of its initial value (at z = 0). Since only the components of the incoming flux at z = 0 are known, the oulzo~ing components at z = 0 are determined applying the boundary condition at ==l ,

In this work we apply the strong continuous semigroup approsch which results sucesdully to study the Linear Transport Equation, [Dautray and Lions 1984-1985], [Nagel et al. 1986]. First we consider the Steady State for the slab - geometry problem with the total crou section and the scatteeinf l~evnel both independent d the spatial variable = in all the interval with non. homogeneous boundary conditions. We apply s ¢uadrature scheme sad obtain a coupled system d ];near equ~ons. The most important part con.;.ts in applying the Approzimation Theorem of the Co mmigronp theory for the convergence of the ~ear .ystem d e q ~ a twal t~ d the LT$s method.

When the total crou s~tion sad the scatteeing tcvnel h; ,~ on the spatial variable we establish spatial meshs, such that in esch subinterval both functions are constants, and is possible to apply the Laplace Tr~dorm and we obtain the standard solution [BerrlchelIo 1992] for each subinterval. The next step follows the approach of Pazy - Tavabe to establish the convergence of the linear system of equations.

In both cases the convergence is resulting of an ~ quadrature scheme sekction. h the Section 2 we give a complete revision concerning derivation of the LT$~ method. Section 3

contains the semigronp approach for the Steady State of 1 - D transport problem if the total cross section and scattevin9 kernel ere independent of the spatial variable. Finally the Section 4 enaliw the case if both functions h ; .~ on the spatial variable.

2 T H E LTSlv M E T H O D

Let us consider the following one dimensional $~v problem in a hmnogeneons dab:

I N ( )+ = "~Wk'Pk(=) + Q.(.) with • ( [o, l]

(a) V',.(O)=/(..),

(b) @re(l)= g(~m),

fro" ~m > 0 )

for/Jm < 0 boundary conditionJ

where:

(c) ,~,,,(.) is the a n ~ ~.x on the ~,,,, ah-e~on

(e) (~.(=)= q ( = ' ' ) ,o~e ~ t i o n /;m

(1)

Convergence of the LTSN method 79

Hm i~ =iopt~d the m a d ~ noutti~ To obtain the £TSs mluti~m our fu~t pea i= apply the Laplace trm~orm to the linur tremport

equation, ~ the mv, t,,~ ~ m t i c ~ e t r m~te=n:

M,,(,),7,(,) = ,;(o) + ~(,) (2) where the cim~mfl~ ace /dmotes the Laplace tnadorm d the correspondent vector function.

The mstrix Ms(*) , is expremd = Ms(s) =~ I+ A (3)

where the entries of the matrix A are:

A~j =

- - N i = j

(s=)

where

B(=) = ~-z [B(,)] = ~ P/'e"' = k=l

rk are the root~ of the detennina= of the pol.vnm~ of M(J), pt m the codBcimt m a i m rwnlting fi'un the Laplace inversion amd the term H(=~) k written = a convolution

= S(.) , ~(.) = Jo" B( , - ~)~(~)~. (6b) #(=)

Since only N/2 cmnpone~ts of the yet't= ~(0) ¢ortesp~di~ to the incom~-$ azq~oIsr flux I t the boundary :: = 0 are known, theu the solution 8ivez by the equations (5) is not utterly determlu=d. Neverthek=, m - , =nalyti~ formlaion wu e~tabli~ed, the N/2 unknow mnponem of the vector ~(0) =m be reed~ obtained by applying the boundary condition (lb) st ~ = i end solving the resulting linear system. To this end, the equation (5) is recast as:

,(0 ~,,(t) ~,,(t) n=(l)

,,her, th. Jr/2 ¢=npo~ent. of th. ,~to~ ¢~(0) = . ~=o.n. , ,hm= ~ =t~. . o~ ~ vector ~(0). =r. . . l~,wn. Con=quently from the equation (6), the vector i~ obtained u:

" l l " - = B~(),~(o) ~;~(o) ~(~) [~() - ~=(~)] (~)

(5=)

_di% x i ~ j.

Bari~ello achieved the foIlowh~ solution for the equation ( 2),

~(,) = B(,)~(o) + n(,)~(,) (4)

The m t r i ~ / ~ for k = 0 : N - 1 were abo detw-!,,4d emlytically [Berrichello l ~ ] . Noticing that each eleme~ of the M~rl(J) nmtrJx is a r~tionsl function, the= the Lspkce inversion k

executed by the Huviaide expanR!on technique, yielding

@(=) = B(=)¢(O)+/7(¢) (5)

80 R.P. Pazos and M. T. de Vilhena

3 STEADY STATE WITH THE TOTAL CROSS SECTION AND SCATTERING KERNEL NOT DEPENDING ON THE SPATIAL VARIABLE

Let us co=ider the fogow~ one~;=,,-,,;oml one-sro.p ] inm traa~ort eqtmi~ Let [0,/] be sn inter~l reprm.zt~ the ~ b geometry; e i. the angle between the tmjector~ of a particle and the ~-*-;., ~ = coa(8), We denote with:

k(tz',/~) scattering kin'rid

q(~, #) source term (s)

~ total ~om section

~(x,v) dmt~ of n.utro-.

The one-dimensional linear transport problem is written as:

p].

t~(O,.) = A(p) for p > 0 I bounda~t ¢onditiona

¢(t,~) = s(~) for ~ < o j

(g)

Here the notation is standard and the base spac. is, of course E = L a ([0,/~ x [-I, I D. The $,v method comists in .electing s finite n,,,,,her of dimzte dir~tiom Pro, in the interval [-I, I] and spplyi~ s guad~tu~ acherne

(i)

iffil conver~ to the ~egml

fl .u~ thm¢ for all I~ /(~) m

~ , # O, with i = I..N.,;

(10)

(b) ~ , > O, with i = I..N.,~

(c) .up., E~'~ ~.,, < ~;

(d) If~t.~= ~ l#,~, [. then <(l+log#t~.,,l)

(11)

for a universal co.taut e.

Convergence of the LTSv method 81

We d ~ the follo~q m ~ :

r . = L~(~,~)]

~,.(~) = [q(=, ~,)]

~.(=) = [~,(=)]

w, = [,,,~,~]

smocked scsttering mstrlx

discrete ordlnate, msUi~

~. ,ource vecto~

wocisted collision mstrix

moci=ed w e i ~ ~ i x ,

(n)

with /~(=, /~) = k(/~,/~') ~(=, #') d/~' !

(lr)

We now obte~n the following system of equ~om, with the correq),~dh~ boundary condit3om

~.~'(=) + Z,,~,.(,,)= K~W.t.(,.)+ ~,,C ~)

,I,,,(o) = A,, (lS)

~,,,(t) = .s.

The functional space considered is E, = ~=l Ll([O,l]), for er~mple, l~t is po~'ble to take its sul~pace of n-upks of piecewise continuous functi~s with discontinuity points st the interph=te pointa, i.e. points such thag Z. and K,~ are not tortuous ( o/ ~vat type).

We spply now the Laphge '~mmf=m with re~:cct to the spatial vari~le md obtain:

[o.,, .,. M:, Co.,,,, - ,~,.)] ~.l,)-- ~;"~,<o),,,,co). I - )

The ,equemce of t~-~mformed molu~om is

T -1 -1 ^ ~.c,l=[,,.,~,;'I~,,.-K~.l] (M~ ,.I,l÷,.iol). I~l the inverse mmd'orm of (I~) we obUdn:

~ . ( . ) = , - r ' % ( o ) + / o ' , -r'c'-¢l ~ ; ' 0 . ( ~ ) ~ (161

We consider the ~ 1. The spaces ~ spprosdmAtes E, and the LTSs transport operators T~ sppro- ximste the ori~al ~mport operator T given by the equstion (9). In this cue the ,olution of (9) i, given in the operstor form

82 R.P. Pazos and M. T. de Vilhena

7" E ' E B b w space

T. Es spproximste space

Figure 1: Apprc~mstion clhgrsm.

h an element~ form the ,emigroups generated by the operators ~ sstidy the atabili~ condition, L e.

II~.ll-< ~ " , 08)

whe~ G~. = e -7'' , M = 1 became these are cont~actlon semigroups. In fact,

{ IIT.II < II~xll Ilqd, - K~.W, II (19)

< ~.~{~ +l,j} - miaH~d

because we select one qmutmtt=e scheme .ati.~/ing ( 11 (c)), ~ we can nor,-,d;q the weights. ~rthm'more m;.~ {#/} exists by (11 (a)). Finally the Approximation Theorem of Co-semiiroul~, [Ooldstein 1985], asserts that G~ c=zm'ps to the L'o-smilFoup ~ t e d by Tfo¢ = in ~ s c t mbset, dR +. The convergence is obtsin~.

4 STEADY STATE WITH THE TOTAL CROSS SECTION AND SCATTERING KERNEL DEPENDING OF THE SPA- TIAL VARIABLE.

Let us considm" the same oz~;,,,,,-,;~,,d one-group linear trmmport equation, but now the total erou aeetion and amtteeing ~emel h; .~ on the spatial variable z. The oae-&mmmud finest Uamport problem is written as:

# ' 8~.(=,#) + ¢(=, /J)¢(=, #) = f_l k(=, , ' , #),(z, #') d/] + q(=,#)

(~o) ~b(O, #) - A(9) for ~ > 0 I boundo~ ¢onditio.a

,PO, t,) = BO,) f~ ~, < o J The notation is the same that in the last section. We select a finite number of discrete direction, ~ in the interml [~.1,1] And take a fuadmtuve #¢hemeln the integral term of equation (20), namely we use (10) and the co=Utio= (Xl).

Convergence of the LTSN method 83

0 I I

:~1~ "~!9

z~ i~ the middle point of the cell Cj

1 ' I I I I

~-~ ~j+~ :~..+]~

2: Mesh ~or spatial variable.

The follo~.~ anays:

depend d the ~ a l vL-'iable ~. We now obtain the followi~ system of equstiom, w/th the correspondh~E boundary cond/tiom

M.~'(,) + E.(,)~.(,) = K~(,)W.~.(,) + 0.(,)

(21)

~,(0) = A. (22)

~,(0 = e .

The ftm~omd space comddered is 1~ - I'[~=1 LI([O,/]), for example, but is pomdble to take its ~ of n-uplm of piecewke continuom functions with disconthnfity points at the inter~hase points, i,e. polnts such that E, and K, ea'e not continuom (o/fira ~]pe).

We e6tsblish now a ~,~tial mesh

0== t <z ! < : t < ... < =~,,,,~½ =1 (~)

su~ that we c~ conaidea" o'j constemZ in the hlterval [.'i_~,X~+~], ~ the geu~le for the scatter~ matr~

kernel K~. We obtain the following sequence of systems of coupled ~erenfial equations:

%(, ) = ~;1 [-,.~z. + K'~jw.] ~.~(,) + M;'q.~(,)

~..,(o) = A .

for j-- 2-..jm~ (u)

~,~...(z) = B,

84 R.P. Pazos and M. T. de Vilhena

The next step combts in applying the Laplace t n ~ with respect to the q)afial ~u'iahte ~S = z - z . ½,

obtaining for the operator r ~ - M~'I[,~..j]~. - K~W,~]

and the mquence d trandormed eolufiom are ['Benidse]]o 1992]

= u;~Q,~(,s) + ~.a(', ~)

Ta]~ng the invene on (26) it turns out

~,~(~s)=, "'~'' ~,~-1(~,.~)+ ~ M; q ,~ (~ j )d~ j (~r) ~-½

In the same way M [Pazy 1983] and ~msbe 1979] we defme the stable family o.f operators, noting that ~(z) is the generator of e-'r.( '),

Definition 4.1 {~(z) ~ £(r~,,) : 0 <_ z <_ l} it said to be ruble for (t!£) (~ e,~U ~al constant~ M > 1 and ~ ~ It such that

,,(~(~)) ~ (~,~) we[o,t ]

# - M (2s) II ~ R(x,~a)ll < w >

#=1 - (k-.,)~'" The ~h..vs~e~zafion proposition estalxlJth that if ~(=) 81metal;e8 e -'T'('), theu the fol low~ statements

neewa~ and .~eient conditao,,, for the . t~ i l i ~ d the family of opmto , {~(x) : 0 __. x _< I}:

J - "E::", II IX ,-°JT"#II < u , f o r ' s -> o (~)

./--1

S.,, M II ]'[ [As - ~a] -1 II < for As > w (3o)

s:~ - l]i:; '(xs - ~) We now de~e

u

Y;- z.nIIc, Cto, ;R+) (sl)

r . = W. e Y " v.' e s . ) Now it is lxmble to apply Theorem 3.1 of [Puy 1983] ,rod results the ex~te~ce and m c i ~ d solution for the (22) for the case Q.(z) = 0 (hom.en.ou. quation), lucia that for all 0 _< U -< ar _< i, we have

r l ~ , ' I f _< M , ' ( ' )

a~w:,,v.l.~, = ~(~)v. (s2)

Convergence of the L TSN method 85

,,h=~ ~ " r, pr,==t, t~ ¢=r=p~dm ,~o~=tio, op,,mo,. F = ~ proof a,ta~ty ~ {r4=)l = ~ [o.q} we hsve only (29) two steps:

i= l

lit.all _< IIMCZll [,~ + IIK.~aw.IIz] (33)

but ,{M~"[, ,fists becsuse/~,~,, sad {[K~jW, t[[~ =m~. {~ki,{~,[}. In genersl it i. easy to see that

=. No,, ,,ith M = ~ =d ~, = [,ff=. (,, + ~,~)]/,.,{-{..,.,}, it i. , . ~ (*), ~, ~ , ~ = ==a~o. of

.tsb'flJty. The bridge for the i~ornog~neou~ ~luafion is in [Puy 1983]~n Theorem 4.2, sad [Tansbe 1979], that is to ssy, e=sts sY,,-vslued solution of (2=) for On(z) E ¢ ([a, hi, It+) :

~.C=) = ~ . 'A . + [" W'~0.(0 d~ (34)

R~MARK l. ~e concUtion (U (¢)) d the ,~q~,-ce d q=dram nd= g=rmt~, th= we ~me

t~e mesh ,o that m~. I,~+]r - z~.,I is szm~er. P J ~ 2. The convergence of S= appr~, , t ion to (~0) is by meam of ~tte~-Yo=ida-Nev~u

Theorem [GoldsteJn 1988], or by the Bia¢rete Schem~- Conue~enc¢ Theorem ~ 108~].

5 C O N C L U S I O N S

The use d Co-Jemigvoup approa¢/= for the LJnesr Transport Equstion is essential for st~uJ7 the e~tence and un/q~eness of the solution ~)sutnsy =rod Lions 1984-1985], ~=~F,l et ~, 1986]. We hsve used the Ap, pmzimafion Theorem to prove the ¢om, ergence when the tot=] crwl meciion sad the mcattevin$ /~emd are independent of the spatial va~-J=~ble. But such approsch is also spproplste for the di#eeete, ordinate# #ystem of equstions when both a(z) sad k(::,l~,l~') ]tinge in ~. The ftmdsmenta] ,l~ce ¢on~i&red is now the c&tesisu product of certain spaces £~- type. In genera] for the Stesdy State Problm

~.(0)=A, ~.(l) = e .

(3s)

.h,,. T. ;, the S.-t..,po~ op,.to, d~,d by [-~.(=) + K~(=)W.] *.(=), ,=d th, th.,~=, ~ - - to the ~ . spon ope,~tor m .pplied in the ,e~e of CP'y 19S.~] *,,cl [ ~ 19~9]. ~ the ~ of g e o = ~ . depending of two or more spatial ~rJ~b~ the t=e of the strong cont/nuom sendgroup sppronch will be a imports= tool to obtain importsat result, in the c o n v e n e of mtmed~ methods of Trmsport Theory.

86 R.P. Pazos and M. T. de Vilhena

6 ACKNOWLEDGMENT

The tint and the second authors are respectively indebted to FAPERGS (~ndagdo do Amparo t~ Puguiaa do Rio Gmnde do Su/) end CNPq ( Conadho Naaional de Deaenvolvimento ~ienttfi¢o e Tecnoldgi¢o) for the portal support d tim work.

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1992] BARRICHELLO Liliene : Fomula~do Analtfiea ~re $oIwdo do Problm,~ de Odenada Diaereta Unidimensiona~ Tree de Doutorsdo, UFRGS, 1992.

[Bsrrrk:he]]o sad Vilhons 1991A] BAEEICHELLO Lilisue end VILHENA M.T.: An Ands, tirol Solution/or the One-Dimensional Discrete Odinateo Problema Udna Laplace 7bans~ore, X IV e' B ~ Congrw on Applied and Computational Mathematics, Rio, 1991.

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[ ~ 1979] TANABE Hiroki : Evolution E~uations, Pitman, 1979.

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[Vi]]xens end Bsrichello 1991] VILHENA M.T. sad BARICHELLO I£fisne : A ?;e~ Analogical Approach to Solve the Neutron Tmnspor~ ~quation Kemteelmib; M(~), pp &34-&~, 1991.

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