Kyncl, J.

Marek, I.

English

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 4Jan Kyncl; Ivo MarekSome problems in neutron transport theoryIn: Jiří Fábera (ed.): Equadiff IV, Czechoslovak Conference on Differential Equations and TheirApplications. Proceedings, Prague, August 22-26, 1977. Springer-Verlag, Berlin, 1979. LectureNotes in Mathematics, 703. pp. [224]--232.Persistent URL: http://dml.cz/dmlcz/702223Terms of use:© Springer-Verlag, 1979Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czSOME PROBLEMS IN NEUTRON TRANSPORT THEORY +) J.Kyncl, I.Marek, Praha 1. Introduction. Formulation of the problem. Today when many nuclear power stations produce a significant portion of the world's electricity, it may be assumed that most problems in reactor physics have been solved. Indeed, there are mathematical models which function well in the sense that they are in good agreement with experimental models of both qualitative and quantitative level. However, despite this fact, there remain some unsolved problems of both theoretical and practical nature. We will show this claim on two basic problems which frequently occur in re-actor calculations. Problem 1. To determine the critical value irQ £ I c[_0f+*>) of the fuel enrichment and the corresponding neutron density N in reactor body G ; to investigate the numerical aspects. Problem 2. To examine the time behaviour of the neutron density in particular for t -» oo # By the reactor we mean a convex body G where absorption, sca-ttering and creation of neutron take place. The neutron density N is a function of velocity z = (v„t v . v ) , space variables r = / % x y z — = (x,y,z; and time t • Since any interaction between neutrons themselves is negligib-le, the models of reactor physics must be linear. Obviously, such models apply in other fields too when the particles under conside-ration do not interact as in the case of neutrons. From physical point of view the most rigorous model is defined by the Boltzmann linearized operator, that is N satisfies d (1.1) . -jp£ N = LN + SN + FN and (1.2) N(r,x,t) = 0 for r e c?G and (n,^) < 0 where n i s the outer normal, and (1.3) LN 2 -v grad N - v jr(z:,v)N , (1.4) SN S Jd£'z ZS(^->Z,£) N(£ fv ft) , ) Delivered by the second author. 225 (1.5) FN 5- J d ^ V V (v') If(v') 7(^z)N(llfi,t) and where £ , 4-3, -£-f, ^ and ")( are characteristics of the reactor medium with respect to collisions with neutrons; v deno-tes the length of the vector v . Problem 1 is connected with the stationary case of (1.1) in which the characteristics £,£ .etc. have to be arranged in such a way that the problem (1.6) 0 = (L + S + F)NQ, N0(£,£) = 0 for £ € d G and (&,£) < ° > has a positive solution for fixed geometrical configuration of G. More precisely, the stationary state depends on some (criticality) parameter: L = L(jr), S = S(y),F=F(j) and one has to find a 0"0 such that the problem (1.7) [L(r> + S(/0 + ^ F ( / ) ] N o = 0 > NQ(r,£) =0 for £ € 5>G, (£,n) < 0 has a nonnegative nontrivial solution N corresponding to ^(7^)= = 1 (criticality condition). From theoretical point of view the Problem 1 was more or less completely solved (see C23t[33t[53>[203). Concerning the calculations there are difficulties connected with the fact that the data of most of the models change rapidly with respect to space variables. A real reactor situation is such that domain G consists of periodically changing subdomains GL each of which contains cells with fuel kernel and several shells having various purposes. The cells are rather small in the sense that when any variant of the finite element method is used the majority of elements are then determined by discontinuous data. However, in practice we are not interested in the detailed behaviour of the differential neutron density; only some global characteristics of the solution are needed such as integral neutron flux, parameter of criticality etc. In order to be able to obtain such characteris-tics some appropriate procedures and simplifications have to be u-sed. A very efficient procedure how to overcome difficulties with rapidly oscillating data a homogenization method has been invented. The homogenization procedure is an approximate method by which the original problem is approximated by another one (usually with res-pect to space variables) in which the data do not change so rapid-ly as the original data; the data are in a special way averaged, or, as we say homogenized. 226 Some mathematical and numerical aspects of the problem of cri-ticality and homogenization will be considered in a subsequent pa-per. In this paper Problem 2 will be considered in detail. 2. Fundamental decay mode and asymptotic behaviour in time. In order to investigate the asymptotic time-behaviour of any particle distribution N , we examine in more detail the Boltzmann operator A == L + S + F defined by (1.2) - (1.5)« We remark that F = 0 if the medium under consideration is nonmultiplying (a mode-rator). Let Y be any Ir°(G x R ,w) for p € (!,+ <*>), where w ^ 0 is a suitable weight function. It is known that L is an infinitesimal generator of a semi-group T(t;L) of class (CQ) and that ||T(t;L)|f ^ a"*** , where A* = inf {v £(r,v) : v €R 3, £ £R 3, v = 1*1 }. Since both of the operators S and F are bounded, we see that A is an infinitesimal generator of a semigroup of operators T(t;A) of class (CQ) (C6 , p.403, Th.l3«2.l]) . The semigroup T(t;L) can be written explicitly by integrating the corresponding first-order differential equation, and we can conclude that T(t;L) is a semigroup of positive operators, that is, T(t;L) leaves inva-riant the cone of elements in Y with nonnegative representatives. Since the operators S and F are defined by nonnegative kernels, the semigroup T(t;A) is also nonnegative ([6 , p.418, Corolla-ry 4]). Moreover, since the kernel of S is positive almost everywhe-re, we may conclude that the semigroup T(t;A) is primitive for sufficiently large t > 0 , that is, for every nonnegative u € Y, u /-* 0 , there is a T u > 0 such that v = T(t;A)u is positive almost everywhere for t > Vy' • The spectrum &(k) has the following structure: Every A for which Re A ~ - A = 0 belongs to the continuous spectrum, i.e. CeT(A) o { A :Re A = - X * } . On the other hand, A for which Re A >- A*+ ||S + F || belongs to the resolvent set ffl (A) . If the body G is sufficiently small, there are no further points in & (A) except those in { A : ReAi- A*} (U.JK Hence, we must assume that the strip - A * < R e A s -A*+ II S + F l| has 227 a nonempty intersection with o (A) • Let A be such that any ^ for which Re A > A Q belongs to the resolvent set p(A), while there exists a A.. £ o'(A) with Re A. = A # i 1 o It is known that N(t) = T(t;A)NQ , where N 6 Y , is a unique solution of the problem ([6, p#359, Theorem 11.5.3]) ££ N = AN , N(0) = NQ . We assume that N is nonnegative almost everywhere and now inves-tigate the behaviour of N as T -» + oo # A standard procedure [19, p»210-213J consists in estimating the semigroup operator by using the resolvent inversion formula [6, The-orem 11.6.1, p.363] CC++&) T(t;A)NQ = li* 2J*T ( e M R ( A > A ) N 0d A » <**»** (Oi *-0> t U-AJQ where A Q = Re AQ is such that Re ^ ^ ^ 0 imPlies ^ ^ JO (A) . For such a procedure we must have complete information about that part of the spectrum of A in the region Re A > - ^ • We propose a more direct and much simpler approach. We formulate it in an abstract way. Let Y be a real Banach space, X = Y © iY its complexifica-tion. Let K C Y be a generating and normal cone. We say that a linear bounded operator T €. B(Y) = (Y -> Y) is K-positive if Tx€K whenever x € K . We also have a partial ordering in Y defined as follows x 6 y4==>y - y € K . Similarly T £ S<=>(S-T)K C K . We call an element y € K quasiinterior if x'(y) > 0 for all linear functionals 0 ? x'€Y' such that x'(x) =-0 for all x € K ; here Y' is the dual space of Y • If the cone K is such that the partial ordering of Y gene-rated by K is a lattice order, that is, sup {x,y] and inf £x,y] exist for every pair of elements x and y in Y , we call Y a Banach lattice. In the following theorems (Theorem 1-3) we assume that Y is a Banach lattice generated by a cone K . We shall apply some deep results due to F.Niiro and I.Sawashima [16] and H.H.Schaefer [18, P.328-333J. Theorem 1. Let A be an infinitesimal generator of a semigroup of operators T(t;A) of class (CQ) • Let ^ Q be such that (2.1) A 6 <^(A) =-> Re A ^ A 0 = Re A Q , 228 and let }\ be a Fredholm eigenvalue, i.e. an isolated pole of R(vK1,A) to which there corresponds a finite-dimensional eigenspace (2.2) 7CQ = [u 6 Y : (A - A Q D k u = 0 for some k = l,2,...}. Let the semigroup T(t;A) be K-positive for t ~ 0 . Then (2.3) S'(A) O { A : Re A = A0 } = [A0,..., A g} and s ^ t (2.4) T(t;A) = H e j [B. + Z.(t)]+ W(t) , 0=o J J where (2.5) B.l^ = B]£B.= ^.kB. , B.Z.(t) = Z.(t)B. = Z.(t) , (2.6) B.W(t) = W(t)B. = © , j,k = 0,...,s J J and (2.7) lim e~ ° lfw(t)// = 0 , lim t"q+1//Z .(t)// = 0 , where q is the order of A Q as a pole of B.(fvfk) . Moreover, Brt „ = lim (P- A )q R( P ,A) is K-positive and ' ofq p ^ A o > o > ' hence, if u 6 K, then v = B u j* 0 is an eigenvector of A in K : Av„ = A,v • o o o Theorem 2. If the semigroup T(t;A) in Theorem 1 is such that for every u £ K , u / 0 , there exists ^(ii) ̂ 0 and a po-sitive integer p = p(u) such that QT(t;A)Jpu is quasi-interior with respect to K for t > 17 (u)f then (a) s = 0 in (2.3) , B Q > q = BQ ; (b) 7C0 = BQY with dim ^fQ = 1 , and B0v is quasi-interior whenever v 6 K , v ^ 0 ; i f y £ K is any eigenvector of A then y = cuQ = BQv0 , vQ € K , vQ ± 0 . Furthermore, (.) T(t;A> - .*•* B 0 • « , ) with -A t (2.8) lim e ° //w(t)l/ = 0 . t-*°° Under the hypotheses of Theorem 2, we consider the Cauchy pro-blem 229 (2.9) ^ u(t) = Au , u(0) = u Q e K . By Theorem 2 we have the following representation of the solu-tion u(t ) = T(t;A)uQ = Aftt = e B0u0 + W(t)uQ • It follows from (2.8) that (2.10) lim e ° u(t) = B„u„ • t-^oo ° ° Thus we have Theorem !• The asymptotic behaviour of any solution u(t) of (2.9) is non-oscillatory. Remark• Let Mf0 be an eigenvalue of A , and M_ the corres-ponding eigenvector. We see that any M(t) of the form e MQ is always a solution of (2.9) with u(0) = M #• Such a solution is called a decay mode; a decay mode is called fundamental if M £ K, Mo * ° • It is easy to see that the normalized fundamental decay mode is unique if T(t;A) fulfils the hypotheses of Theorem 2. To apply our previous theory we have to show only that the point A 0 f the bound of the spectrum ff'(A), is an isolated pole of the resolvent operator R(^,A) • We emphasize this fact because a com-plete analysis of the existence of decay modes and the uniqueness of the fundamental decay mode can be made without any further in-formation about the spectrum of the operator A • This makes our approach different from the sort of analysis proposed by others. On the other hand, we describe only the peripheral part of the spect-rum of the semigroup T(t;A) • If we make assumptions involving compactness about ST(t;L)S [22] or other closely related assump-tions, we can give a complete description of ^(A) • Actually, un-der certain assumptions concerning compactness of T(t^;L)S ••• T(t£;L)S it has been shown that every AJJ 6 6^(T(t;A)) for which f ̂ f>e"" has the form ^W = eA , where A is an isolated pole of R(/^,A) with finite-dimensional invariant subspace Wh) = { u : (A - M ) k u = 0 for some k = 1,2,.~} , [21],[22], [19]# However, these assumptions are not fulfilled in general, e.g. for some models including the case of inelastic scattering in the 230 high-energy range £ll]• On the other hand, our theory does not cover the model exclu-ding the up-scattering. With some minor modifications this case can also be considered by our method and the main results, such as the final Theorem, remain valid in general. We already know that the semigroup T(t;A) is K-positive in Y , where K is the cone of elements of Y = Lp(GxR3,w), 1< p <+O0, with nonnegative representatives, w = 0 • It follows that R(a,A) is also K-posiiive, where a > max(0, ̂ Q) ([6 , Theorem 11.7.2]). Let us write A in the form A = L + S 1 + S2 + F , i n which S-. includes the elastic scattering and the inelastic scattering in high-enargy range and S2 the inelastic scattering in low-ener-gy range. Since S^ is bounded and S2 and F are compact opera-tors, we have that R(a,A) = RtajL+S-^ + R(a,A) (S2+F)R(a,L+S1) . A crucial assumption for the applicability of our theory is the fulfilment of the strict inequality (2.11) r = r(R(a,A)) > r(R(a,L+S1)) = r± , the relation r ~ r^ being trivial. Actually, we have Lemma. Under the assumption (3»ll) for some a > max (0,- A + + /[s+F/Y ) the point P^Q is a pole of the resolvent operator R(̂ c/,A) . Remark, We note that the validity of (2.11) follows from the compactness of (S+F)T(t;L)(S+F) and similar other assumptions, as we have mentioned above. The converse is obviously not neces-sarily true, as we have mentioned, in the case of inelastic scatte-ring in the high-energy range. Proof of the Lemma. The operator R(a,A) is an operator of Ra-don-Nikolskii type [14] whence it follows that its peripheral spec-trum consists of a finite set of Fredholm eigenvalues VQ,«.., V • Obviously, Vj = a- a. t where Aj £ & (A) , j = 0,...,s , | Vjl = r(R(a,A)) . We identify A 0 by setting r(R(a,A)) = J^ # 231 Let R(a,A)y. = V-y. 9 y . ± 0 and l e t b > a . We see that J 0 0 o o Thus, for all b > a we have that b - A I =- I b - A Q for every 7^ € & (A) . Since b can be arbitrarily large, we con-clude that A € <f (A) implies that Re A = AQ • Because r = r(R(a,A)) G ^(R(a,A)) , the spectral mapping theorem shows that 7\Q is a Predholm eigenvalue of A since r has this pro-perty with respect to R(a,A) • This completes the proof of the Lemma. The conclusion of the Lemma implies that Theorems 1-3 apply to those cases of neutron transport where the assumption (3*11) holds. In our opinion, this is the case in most of the models used until now. As a consequence we have the following final result. Theorem. If (̂ (A) ̂  {A : Re A > - A*} / 0 , then there exists exactly one normalized fundamental decay mode I A 0*-O and we ha-ve that for every solution N of §£ N = AN , N(0) = NQ £ 0 , lim f/e ° N(t) - cM0 '( = 0 , t->oo ° where c > 0 is a constant independent of t , More precisely, cMQ = PNQ , where P is the residue of the Laurent expansion of R( £f,A) about the point A Q • We remark that this last theorem gives a solution to Problem 10 of Kaper's Collection of problems in L7J. References [l] Albertoni S., Montagnini B.:On the spectrum of neutron transport equation in finite bodies. J.Math.Anal.Appl.Vol.13^1966), 19-48. [2] Birkhoff G.: Reactor criticality in transport theory. Proc.Nat. Acad.Sci. USA, 45(1958), 567-569. [3] Birkhoff G., Varga R.S.: Reactor criticality and nonnegative ma-trices. SIAM J.Appl.Math.Vol.6(1958), 354-377. 232 C 4] Borysiewicz M., Mika J.: Time behavior of thermal neutrons in moderating media. J.Math.Anal.Appl.Vol.26(1969), 461-478. L 5] Habetler G.J., Martino M.A.: The multigroup diffusion equations of reactor physics. Report KAPL-1886, July 1958. [6] Hille E., Phillips R.S.: Functional Analysis and Semigroups. Revised Edition. Providence 1957* Russian translation Izd.Ino-strannoj Lit.Moscow 1962. f" 7] Kaper H.G.:A collection of problems in transport theory. Trans-port Theory and Stat.Physics 4(3)(1975), 125-134. Q 8] Kato T.:Perturbation Theory for Linear Operators. Springer-Ver-lag, Berlin-Heidelberg-New York 1965. £ 9j Krein M.G., Rutman M.A.: Linear operators leaving a cone invari-ant in a Banach space. Uspekhi mat.nauk 111:1 (1948), 3-95. (Russian); Amer.Math.Soc.Translations no. 26(1950), 128 pp. [lo] Kyncl J., Marek I.: Relaxation lengths and nonnegative solutions in neutron transport. Apl.mat.22(1977), 1-13. [ill Larsen E.W., Zweifel P.F.:0n the spectrum of the linear trans-port operator. J.Math.Phys.Vol.15(1974),1987-1997. Ql2l Marek I.: IVobenius theory of positive operators. Comparison theorems. SIAM J. Appl.Math. 19(1970), 607-628. [13J Marek I.:On Fredholm points of compactly perturbed bounded li-near operators. Acta Univ.Carol.-Math.Phys.Vol.17(1976), No.l, 65-72. [14] Marek I.:On some spectral properties of Radon-Nikolskii opera-tors and their generalizations. Comment.Math.Univ.Carol. 3:1 (1962), 20-30. [15J Mika J.:Neutron transport with anisotropic scattering. Nucl.Sci. Engi.ll(1961), 415-427. [l6j Niiro F., Sawashima I.:On spectral properties of positive irre-ducible operators in an arbitrary Banach lattice and problems of H.H.Schaefer. Sci Papers College General Education Univ.To-kyo 16(1966), 145-183. [17J Sawashima I.:Spectral properties of some positive operators. Natur.Sci.Rep.Ochanomizu, Univ.l5(l964), 55-64. [l8j Schaefer H.H.IBanach Lattices and Positive Operators. Springer Verlag Berlin-Heidelberg-New York 1974. [19J Shikhov S.B.:Lectures in Mathematical Theory of Reactors. I. Linear Theory. Atomizdat, Moscow 1973. (Russian) [20J Varga R.S., Martino M.A.tThe theory for numerical solution of time-dependent and time-independent multigroup diffusion equa-tions. Proc.of the Second Intern. Conf. in the Peaceful Uses of A.E.XVI, 570-577. Ref.P/154, Geneve 1958. [2lJ Vidav I.:Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator. J.Math.Anal.Appl.22(1968T, 144-155. [22J Vidav I.:Spectra of perturbed semigroups with applications to transport theory. J.Math.Anal.Appl.30(1970), 264-279. Authors'address: Nuclear Research Institute, 25068 Se2 u Prahy, Czechoslovakia Caroline University, Faculty of Mathematics and Physics. Malostranskd n6m.25, 11800 Praha 1, Czechoslovakia 

Some problems in neutron transport theory

http://dml.cz/bitstream/handle/10338.dmlcz/702223/Equadiff_04-1979-1_32.pdf

Some problems in neutron transport theory

Abstract

Similar works

Full text

Available Versions

Institute of Mathematics AS CR, v. v. i.