Stewart, H.B.

Stewart, H. B.

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•Dezember 1974Institut für Neutronenphysik und ReaktortechnikProperties of Solutions and Eigenfunctions ofMultigroup Diffusion ProblemsH.B. StewartKF I< 2048Als Manuskript vervielfältigtFür diesen Bericht behalten wir uns alle Rechte vorGESELLSCHAFT FÜR KERNFORSCHUNG M. B. H.KARLSRUHE•KERNFORSCHUNGSZENTRUM KARLSRUHESeptember 1974 KFK 2048Institut für Neutronenphysik und Reaktortechnik"Properties of Solutions and Eigenfunctions ofMultigroup Diffusion Problems"byH. Bruce StewartGesellschaft für Kernforschung m.b.H., KarlsruheABSTRACTA variational formulation of the static multigroupdiffusion equations for heterogeneous assemblies providesa rigorous mathematical foundation for the solution ofthe time eigenvalue and the time-dependent initial valueproblems. Theorems on existence, uniqueness, regularity,and positivity are stated, and a proof that the gener-alized eigenfunctions are complete in L2 is indicated."Eigenschaften der Lösungen und der Eigenfunktionen derMehrgruppen-Diffusions-Probleme"KURZ FASSUNGDurch eine variationstheoretische Formulierung derstatischen Mehrgruppen-Diffusionsgleichungen für heterogeneAnordnungen wird die Lösung der Zeiteigenwert- und der zeit-abhängigen Anfangswertaufgabe mathematisch begründet.Sätze über Existenz, Eindeutigkeit, Regularität und Positivi-tät werden mitgeteilt; es wird ein Beweis angedeutet, daß dieverallgemeinerten Eigenfunktionen vollständig in L2 sind.September 1974We also choose to omit discussion of the multi--1-INTRODUCTIONA basic paper in the theory of the differentialequations for multigroup neutron diffusion has forsome time been that of HabetIer and Martino (1); theseauthors offered an approach to strict mathematicalanalysis of the time-dependent diffusion equations asweIl as of the effective multiplication rate eigenvalueproblem. For example, HabetIer and Martino showed thecompleteness of eigenfunctions (of the time rate eigen-value problem) for one-dimensional problems and forhomogeneous two- and three-dimensional problems. A mainresult of the present work is the completeness of eigen-functions for general heterogeneous problems in the higherdimensions.In addition to this extension of previous results, thepresent author has noticed important mathematical pointsin HabetIer and Martino's work which at least need furtherclarification. For this reason, a somewhat differentapproach in this paper - the weak or variational formulationof the static problem - will be developed systematicallyfrom the beginning. This approach employs functionalanalysis and much ofthe recent theory of elliptic boundaryvalue problems. We shall summarize rather completely theresults Of our analysis; full proofs will appear else-(_2)where-2-plication rate eigenvalue problem, and concentrate onthe time-dependent equations and the related time rateeigenvalue problem. Starting from the weak formulation,we establish the existence and uniqueness of solutionsto the initial value problem, the completeness of gener-alized eigenfunctions, and positivity properties ofsolutions.PROBLEMThe following assumptions suffice for all the resultsof this paper. The multigroup diffusion equations determinea neutron flux distribution .(x) = (.1(x) '.2(x) ""'.G(xt)in G energy groups within aspace domain n. Here nwill be a bounded connected open set in Euclidean n-space,n = 1,2,3, with boundary an made up of a finite numberof intersecting smooth (n-1)-dimensional surfaces. Tospecify smoothness we introduce the following definitions.A function f satisfies a Hölder condition with Hölderexponent ~ > 0 if If(x) -f(y) I ~ Mlx-yl~. The classCj'~(F) consists of functions on a set F whose partialderivatives up to order j. satisfy a Hölder conditionuniformly in F. An (n-1)-dimensional surface will becalled cj,~ if it is represented by local coordinatefunctions which, together with their inverses, are inCj'~(B), where B is a unit ball in n-space, and thesurface is locally the image of a plane in B • Using these-3-terms, the surfaces making up an must be c2,~ , andan as a whole must be co ,1 • (The latter conditionprevents the surfaces from joining in sharp cusps.)n mayaIso be subdivided by further c2,~ surfacesinto open subsets n , r = 1,2, ... R. These usuallyrcorrespond to different material regions. A point wheretwo of the smooth surfaces intersect is called a corner.(See Figure.)c ccß- .. ccFigure: A possible plane region n shbwing cornersindicated by C.-4-On n we have. group. diffusion coefficients dg(x),g = 1,2, .•• ,G, and. group transfer cross sections0gh(X), g,h = 1 ,2, ••• ,G. The latter include absorptionand scattering processes, and all fission processes asweIl. The time-dependent differential equations for multi-group diffusion would be= (1)where are inhomogeneous neutron sources; or,D<j> + S<j> = s. (2)The must be in for each subregionreal, positive, and everywhere greater than some positiveshallminimum valuebe ino. The neutron group velocities v gfor each r, and also positive withpositive minimum. The 0gh must be cO'~(nr)' real, andnon-positive for g ~ h. Note that we allow arbitrary up-and down-scattering and an unrestricted form of the fissionmatrix. The only additional restriction, for positivitytheorems only, will be transitivity of the 0gh.With equation (1) come certain boundary conditions andcontinuity conditions. At present, only the Dirichletboundary condition<j> = 0 on an (3)-5-will be considered. Continuity conditions arise atsurfaces an n an across which the coefficients haver sjump discontinuities. At points where anris smooth,and hence has a well-defined normal vector, one requirescfl g and the normal component ofd Vcf1 continuous across each interface.g g(4)Points where anris not smooth are the corners.The time-dependent equation (2) is closely relatedto the time rate eigenvalue equation-Dcf1 + Scf1 - -Lcf1 = Acf1. (5)The same conditions (3) and (4) go with this equation aswelle Neutron velocities may be included by replacing Lwith VL this remark applies throughout the sequel.A study of (5) yields valuable information about (2).WEAK SOLUTIONSLet L2 (n) be the space of complex measurable vectorfunctions cf1 on n whose norms11 cf1 11 = f I cf1 (x) 12 dxn-6-are finite; L2 (Q) is a Hilbert space with inner product($,$) = J~ $$ dx. Weak solutions will be functions in theHilbert space H~(Q): from the set of functions havingbounded continuous first partial derivatives and satis-fying the boundary condition (3), one takes the completionin the norm11 $ 11 1 = 11 $ 11 + J IV$ (x) 12 dx ,~that is, all possible limits of sequences converging in11· 11 1 , to obtain H~(~). Functions in H~(~) have firstpartial derivatives, belonging to L2 (n), in a generalizedsense; they need not be continuously differentiable in theusual sense Such functions also satisfy the zeroDirichlet boundary condition. in a generalized s~nse.The variational formulationof the static problem (5)amounts to replacing the differential equation (andcontinuity conditions) for $ by a condition on $expressed with abilinear form.For motivation, we recallthat the static diffusion equation (for one group) can bederived from the conditionJav j . \I dS = ~ (-0$ - s) dxfor all volumes V C~, together with the diffusion lawj = dV$. Now if Xv is a function which equals 1-7-inside V and zero outside, then VXV is a Diracdistribution for line integration along an outwardnormal across av ,so the left side above can be re-written- J d V ~ • VX V dxn= J (- 0 ~ - s ) X V dx ,na condition which we still require for all Xv withV c n. The variational formulation we are about to intro-duce is the same condition, but instead of all step functionswe use all ljJ E H1 (n) •oTo be precise, ~ E H1 (n)othe equationswill be a weak solution of-V'd V~9 9if+ = s 9(6 )B(~,ljJ) - J (L d V~ • V ljJgn 9 9 9(7)= = (s , ljJ)holds for all ljJ E H~ (n) •This weak formulation has become weIl known in themathematical theory of ellipti6 partial differential-8-equations, and a number of results can be borrowedfrom the established theory (1) ,(i) ,(~). For a givenbilinear form B, consider all pairs {~,s} such thatB(~,IjJ) = (s,ljJ) for all ljJ E H~(n). Let the set of all~ I S from such pairs be @, . which is a subset ofH~(n). It is not hard to show that any ~ E~ correspondsto exactly one sE L2 (n), which makes possible theDefinition: The mapping ~ + S is denoted by g, sothat a weak solution ~ of (6) is a solution of theoperator equation g~ = s.In fact, g is a closed linear operator in L2 (n),and one establishes the following apriori estimate:(8 )for some Ao> 0 and all ~ E~. One also discovers,using the Lax-Milgram theorem (1) that every sE L2 (n)has a solution of (g + Ao)~ = s; by (8) this solution ~is unique, so ~ + Aohas an inverse, which maps L2 (n)back onto @ C H1 (n). Rellich I s theorem (1) states thatoH~(n) is a compact subspace of L2 (n), so (g + Ao)-1 isa compact operator. Applying the Riesz-Schauder spectraltheory of compact operators (~) one can reach'l'heorem 1: The spectrum of L consists ofa countable=sequence of isolated eigenvalues of finite mUltiplicitywhich tend to infinity.Presently we will locate the eigenvalues more precisely.-9-REGULARITYThe operator L has been defined from the weak=formulation (7), but it is perhaps still unclear whywe say that a solution of ~~ = s is called a weaksolution of the differential equations (6). Weak solutionsneed only have partial derivatives of first order, andthese in a generalized sense. We would like to know thatweak solutions are more regular: that they actually havecontinuous partial derivatives of second order insiden which satisfy (6).rTheorem 2 Let !1,~ = s, where s € co, lJ in each nrThen for some lJl > 0, ~ f CO,lJl (n) and takes the valuezero continuously at an. Any point inside nrhas aneighborhood in which ~ t C2 ,lJl and satisfies thedifferential equations (6). At any point in anrwherethe interface is c 2 ,lJ, ~ € c 2 ,lJ 1 up to the interface frorninside nrand from inside the neighboringsatisfies the continuity conditions (4).n ,sandRemarks: The proof draws on results well known in thetheory of elliptic boundary value problems. The hard partis proving the regularity (i.e. differentiability) of ~;once that is known, the differential equations and thecontinuity conditions can be derived by integrating thebilinear form'by parts and using the arbitrary nature of ~.-10-We emphasize that Theorem 2 does not claim thesecond derivatives of ~ are bounded up to a cornerpoint. In fact, recent work (2) ,(~) has shown exampleswhere this is false. This raises substantial objectionto the analysis of HabetIer and Martino, who seem to claimthat second derivatives are bounded at corners. For example,one may only integrate by parts over a region away fromcorner singularities. The weak formulation has the advan-tage of defining a uniquely solvable problem and avoidsmany difficulties related to corner singularities. Also,as we tried to show inmotivating (7), the weak formulationhas some physical sense.TIME-DEPENDENT EQUATIONA basic tool for mathematical analysis of the time-dependent diffusion equations (1) is an estimate for the-1 .resolvent of -L, R(z.i-g) = (-~-z) • In Hilbert spacethis estimate follows without great difficulty if thespectrum of -g be shown to lie near the negative realaxis in the complex plane. Our multigroup diffusionoperator -L=consists of a self-adjoint operator -D=with spectrum on the negative real axis, perturbed by abounded operator g. One finds that the spectrum of-~ (which by Theorem 1 consists only of eigenvalues) thusfalls .within a semi-infinite band of fixed width aboutthe negative real axis.-11-Theorem 3: For any E > 0 there are constants X, C > °such that11 (}4 + z)-1 11 < c/izifor all complex z with Izi > X and larg zl~ TI - E.(9)This estimate represents the main condition underwhich -}4 generates an analytic semigroup (2) ofoperators U(t) for t > O. Recall that this semigroup,also written exp(-t}4), has the properties exp(-(t+s)g) =exp(-t}4)· exp(-sg) and (d/dt) exp(-tg) = -}4 exp(-t}4).According to the Hille-Yosida semigroup theory, Theorem 3further implies U(t) is analytic in t for complex twith Re t > 0, and tends strongly in L2 (Q) to theidentity operator as t + O. One can construct U(t) fromg by a modified inverse Laplace transform oftransformation integral converges due to (9).L·='theThe semigroup U(t) gives solutions to an abstractinitial yalue problem. Let and definecl> (t) =Then cl>(t) is a function of t with values in L2 (Q).By the properties above, cl>(t) is a solution of theabstract Cauchy problem-12-dcjl/dt + gcjlcjl (t) + .cjlo in= °L2 (n) as t + 0.Here the derivative is taken abstractly, in the senseof functions with values in a Banach space; it doesnot mean a partial derivative of cjl(x,t) with respectto t. However, using a more refined analysis of ourparticular ~, and estimates in the supremum norm onn, the existence of the partial derivative in t canbe established.Theorem 4: Let cjlo(x) f L2 (n). There exists a uniquefunction cjl(x,t) continuous in x and analytic in ton n x (O,T] and zero on an x (O,T] , twice continuouslydifferentiable in x in each n x (O,T] and satisfyingrthe time-dependent diffusion equation= ° (10)there, and satisfying the continuity condition (4) onsmooth parts of an n an for t > 0, and cjl(x,t) + cjlo(x)r sin L2 (n) as t + 0.Duhamel's principle yields solutions of the inhomo-geneous equations (1) as weIl (~) .The resolvent estimate of Theorem 3 above can alsobe used in proving that the generalized eigenfunctions ofgare complete in L2 (n). We say a non-zero function cjl-13-is a generalized eigenfunction of ~ correspondingto the eigenvalue A. if for some positive integer k,J(~ - Aj)k~ = o. Using (9) and arguments adapted from Agmon(lQ) and Dunford and Schwartz (ll), we get:Theorem ~: The generalized eigenfunctions of ~ arecomplete in L2 (Q). Every ~o E L2 {Q) has abiorthogonalexpansion in these generalized eigenfunctions convergingin L2 {Q), and the corresponding expansion for ~(x,t)converges in L2 {Q) uniformlyon [O,T].Remarks: By abiorthogonal expansion we mean an expansionin eigenfunctions of ~,the coefficients of the expansionbeing found by inner products with corresponding eigen-functions of the adjoint of ~ (which is an adjoint inthe strict mathematical as well as the formal sense) .Formulas may be found for example in Habetler andMartino The convergence of the expansion for ~ (x, t)depends on the fact that -~ generates a semigroup ofbounded operators in L2 (Q). Other partial results aboutcompleteness and convergence in supremum norm and in11· 11 1 are possible.We repeat that neutron group velocities may alwaysbe included (occasionally with extra effort) by replacingL with V L.= == =-14-POSITIVITYFor physical reasons one expects that a positiveinitial neutron distribution will result in a flux whichis positive for all later time. This expectation isconfirmed mathematically for the time-dependent diffusionequations (10). For the next two theorems we need to assumethe transitivity of the 0gh ' i.e. for any two groups g1and gm there is a transitive chain of groups g1 ,g2'" .gmsuch that ° is not everywhere zero.gi,gi+1Theorem 6: If ~o is non-negative and not identicallyzero, then ~(x,t) of Theorem 4 is non-negative, andstrictly positive except on aQ and possibly at corners.The proof that ~(x,t) is non-negative seemssurprisingly difficult; but 'once ~ is known to be non-negative, the strict positivity follows using classicalmaximum principles applied in each subregion Qr'Aside from its intrinsic interest, the positivity ofsolutions of (10) also helps prove the existence of adominant mode.Theorem 7: There is a dominant mode for the time-dependentequations (10) which is positive and unique up to a constantfactor, corresponding to a simple real eigenvalue oflarger than the real part of any other eigenvalue.L=One proves this theorem by applying results of Kreinand Rutman(ll)to the operator 'U(t), which is a positivecompact operator in L2 (Q) for any particular t > O.-15-CONCLUSIONSThe weak or variational formulation offers a usefulstarting point for rigorous mathematical analysis,especially as it allows one to bypass difficulties causedby singular solutions at corner points. From this basis,the existence anduniqueness of generalized solutions tothe time-dependent initial value problem for multigroupdiffusion can be established under very few restrictions.These generalized solutions are almost solutions in aclassical sense; it remains only to show that initialvalues are taken on continuously. (We showed they aretaken on in thesquare integral sense.) Solutions of thetime-dependent problem are positive for positive time ifthe initial values are non-negative, and there is a positivedominant mode. The generalized eigenfunctions are completein L2 , permitting biorthogonal expansions of arbitraryinitial values and of .the corresponding solutions forlater time.-16-ACKNOWLEDGEMENTSThe author thanks Drs. R. Froehlich and C. Guentherfor a number of suggestions which considerablyimproved the manuscript.-17-REFERENCES1 .G.J. HABETLER and M.A. MARTINO in Nuclear ReactorTheory , G. Birkhoff and E.P. Wigner, Eds., pp. 127 - 139,Arner. Math. Soc., Providence (1961).2 H.B. STEWART, to be published in ~. Math. Anal. ~.3 S. AGMON, Lectures 2n Elliptic Boundary Value Problems,Van Nostrand, Princeton (1965).4 C.B. MORREY, JR., Multiple Integrals in the Calculusof Variations, Springer, New York (1966).5 J. NECAS, Les Methodes Directes en Theorie desEguations Elliptigues, Masson, Paris (1967).6 T.KATO,·Perturbation Theory for Linear Operators,pp. 185ff., Springer New York (1966).7 R.B. KELLOGG in Numerical Solution of PartialDifferential Eguations 11 , pp. 351-400, B. Hubbard, Ed.,Academic Press, New York (1971).8 I. BABUSKA and R.B. KELLOGG in Numerical ReactorCalculations , pp. 473-486, J.W. Weil and C.N. Welsh, Eds.,lAEA, Vienna (1972).9 Ref. 6, pp. 487ff.10 S.AGMON, ~. Pure ~. Math. 15 , 119-147 (1962)11 N. DUNFORD and J.T. SCHWARTZ, Linear Operators,Part !!' pp. 1088-1119. Interscience, New York (1963)12 .M.G. KREIN and M.A. RUTMANi Uspehi Mat.Nauk ~,3-95 (1948) i Am. Math. Soc. Translations No. 26 (1950).

Properties of Solutions and Eigenfunctions of Multigroup Diffusion Problems

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Properties of Solutions and Eigenfunctions of Multigroup Diffusion Problems

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