Numerical Modeling of Neutron Transport

NUMERICAL MODELING OFNEUTRON TRANSPORT

Master’s Thesis

submitted by

Bc. Milan Hanus

completed at

Department of MathematicsFaculty of Applied Sciences

University of West Bohemia

under the supervision of

Ing. Marek Brandner, Ph.D.

Pilsen, August 2009

Declaration

I hereby declare that this Bachelor’s Thesis is the result of my own work andthat all external sources of information have been duly acknowledged.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Milan Hanus

Abstract in English

The aim of this thesis is to assess the possibilities of improving the diffusionbased method for steady-state neutronic analyses of nuclear reactors with he-xagonal assemblies via the transport theoretical model. Mathematical basisof neutron transport theory is established first, including the functional ana-lytic setting of the two main problems of steady-state neutron transport –the core criticality eigenvalue calculations and the fixed neutron source pro-blem. Discretization of the governing equation with respect to energy (themultigroup approximation), direction of motion (the SP3 approximation) andspace (the finite volume method) is described next. The core criticality pro-blem of finding the dominant eigenpair of the transport operator is solved bythe source iteration (a variant of the power method), which is also explainedin the text. Numerical experiments with both the diffusion and the transportapproximation are performed at the end and their results analyzed, includingthe most recent developments of the diffusion solver – namely the conformalmapping for the transverse integrated nodal method in the hex-z geometryand the homogenization of input data originating in thermal-hydraulics cal-culations performed by Skoda JS, a. s. (Skoda Nuclear Machinery).

Keywords:

reactor physics, neutron transport equation, fixed source problem, sphericalharmonics, Legendre polynomials, slab geometry, simplified spherical har-monics method, PN , SPN , neutron diffusion equation, albedo boundary con-ditions, multigroup approximation, reactor criticality, eigenvalue problem,source iteration, finite volume method, CMFD, FMFD, neutron flux, nodalmethod, transverse integration, hexagonal assembly, homogenization.

Abstrakt v cestine

Cılem teto prace je zhodnotit moznosti pouzitı transportnı teorie pro zpresnenıdifuznı metody, vyvinute pro urcovanı neutronove-fyzikalnıch charakteris-tik aktivnı zony reaktoru s sestihrannymi palivovymi kazetami. V uvoduje zpracovana matematicka teorie stacionarnıho transportu neutronu a de-finovany dve zakladnı ulohy, jez se s jejım vyuzitım resı – stanovenı kri-tickeho cısla reaktoru (uloha na vlastnı cısla) a stanovenı neutronoveho polevznikleho nemennym externım zdrojem neutronu. Pote je popsana diskreti-zace prıslusne rovnice, jız se rozlozenı neutronu v danem prostredı rıdı. Ener-geticka zavislost je vyjadrena mnohagrupovou aproximacı, smerova zavislosttzv. SP3 aproximacı a prostorova zavislost metodou konecnych objemu. Kri-ticke cıslo reaktoru (dominantnı vlastnı cıslo transportnıho operatoru) jehledano v textu zevrubne popsanou variantou mocninne metody. Na zaverjsou uvedeny a analyzovany numericke vysledky vypoctu s difuznı i trans-portnı metodou, vcetne poslednıch vylepsenı paralelne vyvıjeneho difuznıhoresice – konformnıho zobrazenı a homogenizace vstupnıch dat, pochazejıcıchz termohydraulickych vypoctu provedenych spolecnostı Skoda JS, a. s.

Klıcova slova:

reaktorova fyzika, rovnice transportu neutronu, uloha s pevnym zdrojem ne-utronu, sfericke harmonicke funkce, Legendreovy polynomy, 1D geometrie,metoda sferickych harmonickych funkcı, PN , SPN , rovnice difuze neutronu,mnohagrupova aproximace, kriticke cıslo reaktoru, uloha na vlastnı cısla,mocninna metoda, metoda konecnych objemu, CMFD, FMFD, neutronovytok, nodalnı metoda, metoda prıcne integrace, sestihranna kazeta, homoge-nizace, albedo.

Acknowledgements

I would like to thank my supervisor Ing. Marek Brandner, Ph.D., for hissupport and kind guidance throughout the course of thesis preparation. Manythanks also belong to the remaining members of our work-team:Ing. Tomas Berka, Ing. Ales Matas, Ph.D. and Ing. Roman Kuzel, Ph.D.,for their insight into the huge amount of problems that arose during the de-velopment of the code, for the stimulating discussions that we had in orderto solve them and for the joyful moments when the solution finally poppedup.

A special thanks goes to Roman Kuzel, for his invaluable advices onimplementation of the methods in MATLAB and his help not only duringthe tedious debugging phases.

Finally, I am deeply grateful to all who had patience with me in timeswhen progress became slow and time-consuming. Especially to my girlfriendJana Bohumila.

This work was supported by project 1M0545.

i

Contents

List of Symbols iii

List of Figures xi

1 Introduction 11.1 Past work and motivation for improvements . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Improving the mathematico-physical model . . . . . . . . . . . . . . . . 21.1.2 Improving the numerical method . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Present work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.1 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Neutron transport theory 82.1 Neutron transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Phase space of neutrons and related notation . . . . . . . . . . . . . . . 82.1.2 Boltzmann’s transport equation for neutrons . . . . . . . . . . . . . . . . 102.1.3 Steady state transport equation . . . . . . . . . . . . . . . . . . . . . . . 142.1.4 Fundamental physical quantities . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Mathematical setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.1 Physical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.2 Boundary and interface conditions . . . . . . . . . . . . . . . . . . . . . 172.2.3 Operator form of the Boltzmann’s equation . . . . . . . . . . . . . . . . 18

2.3 Two problems in steady state transport theory . . . . . . . . . . . . . . . . . . . 202.3.1 Eigenvalue calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Fixed source calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Representation of the angular variable . . . . . . . . . . . . . . . . . . . . . . . 252.4.1 Scattering kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.2 Fission and external sources . . . . . . . . . . . . . . . . . . . . . . . . 312.4.3 The final equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5 The slab-geometry transport equation . . . . . . . . . . . . . . . . . . . . . . . 322.5.1 Streaming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5.2 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5.3 Fission and external sources . . . . . . . . . . . . . . . . . . . . . . . . 342.5.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

CONTENTS ii

2.5.5 The final equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Solution methodology 363.1 General principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Energy discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 Practical generation of multigroup constants . . . . . . . . . . . . . . . . 393.2.2 Operator form of the multigroup equations . . . . . . . . . . . . . . . . 403.2.3 Source iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.4 Eigenvalue problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Angular approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.3.1 The method of discrete ordinates (SN) . . . . . . . . . . . . . . . . . . . 473.3.2 Methods of Galerkin type . . . . . . . . . . . . . . . . . . . . . . . . . 493.3.3 The method of spherical harmonics (PN) . . . . . . . . . . . . . . . . . . 503.3.4 Passage to diffusion approximation . . . . . . . . . . . . . . . . . . . . 563.3.5 Simplified spherical harmonics method (SPN) . . . . . . . . . . . . . . . 563.3.6 Three-dimensional SPN approximation . . . . . . . . . . . . . . . . . . 583.3.7 Theoretical analysis of the 3D SPN approximation . . . . . . . . . . . . 61

3.4 Multigroup 3D SP3 approximation . . . . . . . . . . . . . . . . . . . . . . . . . 643.4.1 Multigroup total collision cross-section . . . . . . . . . . . . . . . . . . 643.4.2 Multigroup boundary conditions . . . . . . . . . . . . . . . . . . . . . . 653.4.3 Multigroup SP3 diffusion coefficients . . . . . . . . . . . . . . . . . . . 653.4.4 Practical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 The multigroup SP1 and SP3 methods in the hex-z geometry 704.1 Spatial discretization by finite volumes . . . . . . . . . . . . . . . . . . . . . . . 71

4.1.1 The finite volume method – basic principle . . . . . . . . . . . . . . . . 714.1.2 The space-angle-group scheme . . . . . . . . . . . . . . . . . . . . . . . 754.1.3 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2 Transverse integrated nodal methodology . . . . . . . . . . . . . . . . . . . . . 824.2.1 Improvements of the NODWAG code . . . . . . . . . . . . . . . . . . . 82

4.3 Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.3.2 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 Conclusion 93

Bibliography 96

A The KNK-II model problem 101

iii

List of Symbols

Reader’s guideline

The list of letters used throughout the text is divided into two categories: those written in Roman(or calligraphic) script and those written in Greek script. Letters not used beyond first few linesafter their definition are normally not listed. If the letter denotes some quantity for which there isa numbered definition relation in the text, the equation number is referenced after the descriptionof the quantity. Dependency of functions is written in terms of variables used in the place of theirfirst appearance.

Some letters occur in the text in both normal and bold form, the latter usually denotinga matrix or vector representation of the former. Since the same bold symbols are understooddifferently in different sections, including them here would only be confusing. Instead, blocksentitled N are placed at places where the notational convention changes anddefine the precise meaning of the symbols for the ensuing part of the text.

In addition to explanations of indexed letters in the just mentioned lists, commonly usedindices are also enumerated separately in another two tables (for subscripts, resp. superscripts).Listing of acronyms concludes this List of Symbols. Abbreviations of named methods are listedseparately.

L S iv

R

C SP3 matrix used in the definition of inhomogeneous boundary conditions(eq. 3.61)

E energetic range

G SP3 matrix used in the definition of albedo boundary conditions (eq. 3.61)

Hd down-scattering part of the scattering matrix H (eq. 3.15)

Hs self-scattering part of the scattering matrix H (eq. 3.15)

Hu up-scattering part of the scattering matrix H (eq. 3.15)

I identity operator/matrix

keff effective multiplication factor

LpΣ(X) space weighted with the total cross-section Σt. (eq. 2.30)

L∞ space of bounded measurable functions

Lp(X) space of functions integrable in the Lebesgue sense with their p-th powerover X. (eq. 2.30)

N matrix of normal vectors (eq. 3.60)

Pk(µ) Legendre polynomial of order k

Pml (µ) associated Legendre polynomial of degree l and order m (eq. 2.44)

Rn an n-dimensional Euclidean vector space

P(V) thermal power production rate (eq. 2.25)

X The slab phase space (eq. 2.54)

W pk (X) Sobolev space of functions whose (generalized) partial derivatives up to or-

der k are in Lp. (eq. 2.30)

Yml spherical harmonic function of degree l and order m (eq. 2.43)

D(z) Diffusion coefficient (eq. 3.46)

D1(z), D3(z) SP3 diffusion coefficients (eq. 3.51)

E Hilbert space in which we look for the solution in a Galerkin methods

E energy of neutrons

EN N-dimensional subspace of the space of solutions E in which we look for theapproximation of the true solution in Galerkin methods

F fission operator (eq. 2.19)

L S v

H scattering operator (eq. 2.19)

I number of nodes in a radial plane

IA number of assemblies in the core (= number of nodes in the CMFD nodal-ization)

J number of axial cuts through the core

j scalar neutron current

j±(r, E) partial neutron currents (eq. 2.10)

J1(z), J3(z) SP3 currents (eq. 3.52)

jn PN currents – the odd Legendre moments of angular flux: φn for n = 1, 3, . . .(eq. 3.36)

j+n n-th partial PN current (Legendre moment) in the positive z-direction(eq. 3.37)

j−n n-th partial PN current (Legendre moment) in the negative z-direction(eq. 3.37)

K degree (or order) of scattering anisotropy (index of the last kept Legendrepolynomial in the scattering kernel expansion)

L streaming operator (eq. 2.19)

N order of the spherical harmonics method (index of the last kept Legendrepolynomial in the expansion of angular flux)

P core power level

Q(r, E,Ω, t) external neutron sources (independent of neutron density)

S n n-th Legendre expansion coefficient of the fixed sources term (eq. 3.35)

Vz a section through the core domain V along the z-axis

wn quadrature weight in the method of discrete ordinates

r position vector

S unit sphere (x ∈ R3 : ‖x‖ = 1B transport operator (eq. 2.23)

c mean number of secondary neutrons following a collision, excluding fission

G number of energy groups

J(r, E) net neutron current density (eq. 2.9)

j(r, E,Ω, t) angular neutron current density (eq. 2.3)

n unit normal vector

L S vi

Qg group external neutron sources (eq. 3.7)

R∗(r, E) reaction rate (concrete type of reaction substituted by asterisk) (eq. 2.11)

R∗ total volumetric reaction rate (eq. 2.12)

G

φi integral average of flux over nodeVi (eq. 4.2)

α albedo coefficient (eq. 2.15)

αg′g multigroup albedo coefficient, representing the fraction of neutrons with en-ergies in group g appearing in the core as a result of reflection of neutronsleaving with energies in group g′ into the reflector (eq. 3.10)

ψ(r, E,Ω, t) angular neutron flux density

ψg group angular flux (eq. 3.4)

φgk k-th Legendre expansion coefficient of the group angular flux (eq. 3.5)

φkm coefficient of the Laplace expansion of angular flux (eq. 2.47)

φk Coefficient of the Legendre expansion (azimuthally symmetric equivalent toLaplace expansion) of angular flux (eq. 2.60)

ϕ azimuthal angle

β integral albedo operator (eq. 2.16)

Ω unit vector of neutron’s direction of motion

ΩR specularly (mirror-like) reflected direction

Ωin inward direction

χg group fission spectrum (eq. 3.6)

χ∗(E′ → E,Ω′Ω, t)energy and direction transfer kernel(∗ = f , s, β for, respectively, fission, scattering or albedo-type reflection)

δi j Kronecker’s delta symbol

δx Dirac’s delta function

εf energy per fission

γ albedo coefficient (eq. 3.42)

λ multiplication eigenvalue

µ cosine of the polar component of the direction vector Ω, i.e. µ = cosϑ

µ0 scattering cosine (µ0 = cosϑ0)

L S vii

µn discrete ordinate

µ0 mean scattering cosine

νΣ f fission yield

ϑ polar angle

ρ(A) spectral radius of operator A (eq. 2.28)

σ(A) spectrum of operator A (eq. 2.28)

Σ∗ macroscopic cross-section for reaction substituted for the asterisk(see the listing of subscripts for possible reaction abbreviations)

Σs(r, E′E,Ω′Ω, t)differential scattering cross section (eq. 2.4)

Σan Legendre moments of the absorption cross-section (eq. 3.36)

Σgan SPN group removal moments (eq. 3.68)

Σsk the k-th coefficient of the scattering kernel expansion (eq. 2.38)

Σtr mono-kinetic transport cross section (eq. 3.47)

Σgtr multigroup transport cross-section (eq. 3.71)

τ optical path length (or optical distance)

Σ f minimum value attained by Σ f

ζ replacement for several independent variables to shorten the notation

dΩ elementary solid angle around direction vector Ω

Φ0(z), Φ2(z) SP3 angular flux moments (eq. 3.50)

φ(r) total neutron flux (eq. 2.7)

φ(r, E) scalar neutron flux (eq. 2.6)

φ(V) total volumetric neutron flux (eq. 2.8)∣∣∣Γi,ξ

∣∣∣ area of the surface Γi,ξ

Σgt group-discretized total cross section (eq. 3.9)

νΣg′f group-discretized fission cross section (eq. 3.9)

Σg′gs cross section for neutron scattering from energy group g into group g′ (eq. 3.9)

S

(a, b] a left-open interval on the real line between a and b (notation of the otherpossible types of intervals is obvious)

L S viii

[·]g g-th component of a vector

d· Lebesgue integration sign∫g

dE integral over the energy group Eg =(Eg, Eg−1]

Γi,ξ surface of nodeVi oriented by a unit outward normal nξa · b inner (dot) product of two vectors a, b

∇ gradient operator (eq. 2.3)

O big ’O’ symbol used to express asymptotic order

f complex conjugate of f , or integral average of f (according to context)

∂X+ outgoing (exiting) boundary of the slab phase space (eq. 2.63)

∂X− incoming (entering) boundary of the slab phase space (eq. 2.63)

∂V0 part of boundary ∂V with prescribed inhomogeneous conditions

∂Vh part of boundary ∂V with prescribed homogeneous conditions

∂V0z part of boundary ∂Vz (in a slab geometry) with prescribed inhomogeneous

conditions

∂Vhz part of boundary ∂Vz (in a slab geometry) with prescribed homogeneous con-

ditions

∂X± exiting/entering boundary of the phase space

f some modification of f , described in the text

f ≈ g f is approximated by g

f ≡ g f is by definition equivalent to g, or, in some contexts, function f is identi-cally equal the value of g everywhere in its definition domain

∂V boundary of domain V

S

∗ substitution symbol standing for some other subscripts

β associated with the albedo operator

← limit from left along the specified direction

in incoming (direction, current)

→ limit from right along the specified direction

ξ belonging to the direction of vector nξa absorption

L S ix

B bottom (point on the z-axis)

d (at a matrix) down-scattering part of the matrix

f fission

h homogeneous

i belonging to nodeVi

i + ξ belonging to nodeVi+ξ adjacent to nodeVi in the direction of nξ (i.e. sharingthe withVi face Γi,ξ)

k index of Legendre expansion of scattering kernel

l degree of a spherical harmonic function

n index of Legendre expansion of angular flux

R reflection

r removal (cross section)

s scattering

s (at a matrix) self-scattering part of the matrix

T top (point on the z-axis)

t total (cross section)

u (at a matrix) up-scattering part of the matrix

x,y,z components of vectors in Cartesian coordinate system

S

∗ dual space or adjoint operator

g energy group index

m order of a spherical harmonic function

+ in the direction of the outward normal, or, in a 1D case, in the positive direc-tion of the coordinate axis

- in the opposite direction of the outward normal, or, in a 1D case, in thenegative direction of the coordinate axis

A

a homogenized version of variable a

ADF Assembly discontinuity factor

L S x

ADS Accelerator Driven Systems

ET Equivalence theory

GET Generalized equivalence theory

LWR Light Water Reactor

MATLAB MATrix LABoratory

MOX Mixed Oxide Fuel

ORNL Oak Ridge National Laboratory

PUREX Plutonium - URanium EXtraction

VVER Voda-Vodyanoi Energetichesky Reaktor (Russian term for the pressurizedwater reactor)

SJS Skoda Jaderne strojırenstvı (Skoda Nuclear Machinery)

M

PN Spherical harmonics method of order N

SN Method of discrete ordinates of order N

SPN Simplified spherical harmonics method of order N

CMFD Coarse-mesh, finite-difference method

FLIP Gauss-Seidel type (with respect to angular moments) iterative solution of theSPN equations ([BL00])

FMFD Fine-mesh, finite-difference method

FV(M) Finite volume method

IRAM Implicit Restarted Arnoldi Method

MFP mean free path of neutrons

MOC Method of characteristics

NODCONF Hexagonal nodal method applying the conformal mapping of the hexagon toa rectangle

NODWAG Hexagonal nodal method with transverse leakage approximated according toM. R. Wagner ([Wag89])

xi

List of Figures

2.1 Neutrons’ phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Cartesian coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Solid angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Evolution of neutron flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.6 Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.7 Slab geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1 Hexagonal node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2 Hexagon subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3 Triangular node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.4 Structure of the L matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.5 Detail of the L matrix structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.6 VVER-1000 core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.7 Diffusion FMFD solution compared to DIF3D . . . . . . . . . . . . . . . . . . . 794.8 The SP3 results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.9 Comparison of NODWAG and NODCONF . . . . . . . . . . . . . . . . . . . . 834.10 NODCONF results with flat transverse leakage . . . . . . . . . . . . . . . . . . 844.11 NODCONF results with linear transverse leakage at the boundary . . . . . . . . 854.12 Distribution of νΣ1

f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.13 S-JS test problem – volume averaging . . . . . . . . . . . . . . . . . . . . . . . 904.14 S-JS test problem – symmetry BC, no ADF . . . . . . . . . . . . . . . . . . . . 904.15 S-JS test problem – true BC, no ADF . . . . . . . . . . . . . . . . . . . . . . . . 914.16 S-JS test problem – symmetry BC, ADF . . . . . . . . . . . . . . . . . . . . . . 914.17 S-JS test problem – true BC, ADF inside . . . . . . . . . . . . . . . . . . . . . . 92

1

Chapter 1

Introduction

It is a common agreement nowadays that nuclear energy is the only available source able to fulfillcurrent and future energetic demands of mankind without polluting the Earth any further. Pre-ceding its useful utilization, such as generation of electricity, materials irradiation or for medicalpurposes, its extraction within a nuclear reactor must occur. Nuclear reactors are sophisticateddevices composed of diverse materials like fuel, coolant, regulation and structural materials. Allthese constituents are arranged in a highly heterogeneous manner due to various safety, tech-nological and economical considerations. Depending on the time spent within the reactor, boththeir physical and mechanical properties change and some of them (like fuel elements of controlrods) are also subject to repeated rearrangement.

Design of such reactors and analysis of their various operational modes is therefore a com-plicated task that encompasses several areas of science and engineering. At its start, however,determination of neutronic conditions within the reactor core plays a crucial role and receiveda substantial attention in the field of reactor physics in past decades. Main objective of suchneutronic analyses is to describe and predict the states of the reactor under various circumstancesand to find its optimal configuration, in which it is capable of long-term self-sustained operationwith only a minimal human intervention.

The optimal operation is achieved by automating the regulation devices to balance neutronproduction from fission chain reaction and their loss due to capture and out of core leakage. Suchautomation foremost requires knowledge of a long-term behaviour of any tested core configura-tion, which may be obtained by performing a steady-state analysis of the reactor core. Methodsused for these analyses should accurately calculate the neutron multiplication factor (character-izing the departure of the core from the desired equilibrium state) and spatial distribution ofneutron flux (from which subsequently the important reaction rates, power and heat distributioncan be obtained for further steps of reactor design or assessment). Since there is usually a mul-titude of possible core configurations, all calculations should proceed swiftly to be practicallyapplicable.

1.1 Past work and motivation for improvements 2

1.1 Past work and motivation for improvements

Formulation and implementation of a method suitable for the just mentioned calculation tasks hasbeen the main subject of my previous work [Han07]. It thoroughly describes the developmentof a two-dimensional solver capable of efficient calculation of the fission multiplication factorand corresponding neutron flux distribution for an arbitrary core composition. The method hasbeen implemented in the MATLAB development environment and successfully tested on a fewbenchmark problems.

At the end of the thesis, several directions for future investigation and enhancements havebeen outlined. Also, many new requirements have come from the company Skoda-JS, a.s., theassigner of the project whose part the work has constituted, or simply from the ongoing scientificprogress. To identify the major areas for modernization, I will now step through the thesis andreflect on its main parts from the current point of view.

1.1.1 Improving the mathematico-physical model

Expressing the balance of the processes occurring in the core mathematically leads in full gener-ality to an integro-differential equation with seven independent variables – the Boltzmann equa-tion of neutron transport. This has been thoroughly described in Section 3.1 of the thesis. Thefollowing section takes the reader through various simplifications of the Boltzmann equation, re-sulting in a system of partial differential equations with only three independent variables (spatialcoordinates in three dimensions) for the steady-state calculations. The two major approximationsperformed on the course are the assumption of diffusion-like behaviour of neutrons within thecore and their categorization into several groups of constant energy.

Limitation of diffusion approximation

The diffusion theory has long served a standard tool for whole-core neutronic calculations. Itallows an efficient numerical formulation and yet produces plausible results for many realisticreactor configurations. However, the assumptions made in its derivation severely restrict thecircumstances under which the approximation is reasonable. Neutrons can be safely expected tomove from places of their highest concentration (corresponding to highest neutron flux) to thoseless occupied (according to the well-established Fick’s law of diffusion) only in highly scatteringand weakly absorbing media. The scattering is the only process contributing to neutron fluxduring the derivation (i.e. no sources and sinks have been assumed) and should be isotropicor only ”slightly” anisotropic (what is meant by that will become clear in Sec. 3.3.4). Highabsorption would lead to rapid spatial variation of neutron flux similarly to the proximity tointerfaces separating different materials and invalidate another assumption made in the originalderivation of the theory ([Sta01, Chap. 3]).


Consequently, at the peripheral regions of the core where structural material of the pres-sure vessel replaces the inner-core composition, near the control rods and localized sources incurrently operating production and research reactors, in cores loaded with high-burnup fuel (anobvious and logical trend in todays fuel management) or in cores containing the MOX fuel ad-jacent to standard UO2 fuel (an effective way to dispose of plutonium originating from weaponsor PUREX-type spent fuel reprocessing), the theory warns us against using the diffusion approx-imation and the practical results of some recent difficult problems corroborate the warning.

There are also cases where the diffusion becomes inadequate at all due to its isotropic nature,i.e. the inability to capture the phenomena associated with neutrons’ direction of motion, likemodeling the ADS systems or shielding applications. In order to develop a solution methodcapable of treating these cases, a full transport theory must be used as a basis instead of thediffusion approximation.

Limitation of the two-group approximation

Discretization of the continuous energetic dependence of neutrons into several intervals (groups)relies on a careful precalculation of group-wise constant parameters for the resulting few-groupsystem of governing equations. This is usually done by a multigroup calculation over single orseveral neighboring fuel assemblies (using a high-fidelity physical model), providing detailedspatial and energetic distribution of neutron flux (i.e. its spectrum), which is used in turn tocollapse the multigroup data into the desired number of groups (the so called group condensa-tion, or collapsing). The resulting set of few-group constants is expected to capture most of thespectral characteristics of the given material. Due to a very complicated energetic dependenceof core constituents and their non-trivial spectral interaction (especially in future cores, as ex-emplified e.g. by the combination of MOX/UO2 fuel), however, this becomes hardly achievablewhen collapsing into only a small final number of energy groups. Newly developed methodsshould therefore be applicable in finer energy group structures or even use a better approxima-tion of energy spectrum and effects associated therewith (e.g. the resonance self-shielding, asdemonstrated in [ZDX+06]).

1.1.2 Improving the numerical method

Perhaps the simplest numerical method for solving boundary value problems like that of neutrondiffusion is based on the finite difference approximation of the involved differential operatorsand has become a workhorse of computer neutronic calculations since their beginning around1960s. More computationally and physically pleasing methods have then been developed basedon the variational or conservation principles, but the stringent demands on the fineness of thecomputational grid remained major obstacles to their efficient application. To overcome thisdrawback, the so called nodal methods were devised in mid seventies and have matured untiltoday. They are now used to speed-up majority of both diffusion and transport codes.


Nodal method

Modern nodal methods are built upon a conservative discretization of the governing equationsby the classical finite volume method (FVM). Somewhat against the theory of the method, how-ever, the finite volumes are intentionally kept quite large so as to facilitate efficient solution ofthe equations for the whole core. Usually, the computational volumes (or nodes) are chosen ascoarse as the whole fuel assembly. Discretization errors thus introduced are accounted for by aclever correction scheme, which (similarly to a classical multigrid method) employs an additionallevel of calculation at several stages of the coarse mesh calculation. The more accurate approx-imation (commonly termed ”higher order approximation”) of desired quantities obtained fromthis second level is compared with that of the finite volume method on the coarse mesh to deter-mine the correction. The solution of this iterative process converges to that of the more accuratemethod but with substantially less computational demands. This correction procedure is com-monly known as a coarse-mesh, finite-difference method (since the primary part of its iterationmatrix represents the finite-difference equations obtained from the finite volume discretizationover coarse volumes), or CMFD.

There are several ways of constructing the higher order solution and many actually do notrequire finer mesh. These methods either use knowledge of the analytic solution of the govern-ing equations within each node (i.e. fuel assembly) or approximate the nodal flux shape by itsprojection into a space spanned by a finite set of suitable basis functions defined over the node.The former are hence termed analytic nodal methods (ANM) and some successful examples aredescribed e.g. in [GRMK05] or [CL06]. Methods in the other class are called nodal expansionmethods (NEM). They differ among each other primarily in the choice of basis functions usedfor approximating the unknown flux function, leading to methods like PNM (polynomial nodalmethod) or SANM (semi-analytic nodal method). Nodal expansion methods are easier to formu-late and implement, particularly when finer energetic discretization is required, than the analyticmethods. They are also more easily used in conjunction with the transport model. Since resultsof many numerical experiments with methods from both categories indicate comparative solutionaccuracy, nodal expansion method has become the method of choice for our previous work andremains the same for the current work.

Homogenization and reconstruction

In the derivation of the nodal method equations, one assumes that material properties of eachnode are spatially constant, i.e. the node is physically homogeneous. Since one node typicallyencompasses the whole fuel assembly, however, this condition almost certainly does not holdeven for the fresh assembly at the start of a new reactor campaign (which contains complicatedarrangement of fuel rods, or pins, with varying fuel material composition, their cladding and themoderator). Physically heterogeneous nodes are therefore first converted into homogeneous onesby a (purely mathematical) homogenization procedure, which ensures that the solution consid-ering the simpler homogenized configuration will be equivalent to one that would be obtainedwithout it, at least in the sense of preserving important quantities like reaction rates and multipli-


cation factor. This is often done in a similar fashion as the energy group condensation, perform-ing a single-assembly calculation with fictitious boundary conditions (usually corresponding toan infinite lattice of nodes with the same properties as the one being homogenized).

Although there are applications, such as fuel optimization, where global reactor quantities(multiplication factor, assembly average power distribution) are the only required results, thereare also others (like assembly power peaking estimation or thermal-hydraulic calculations) thatrequire knowledge of detailed, pin-by-pin neutron flux distribution. Because nodal methodsyield only the integral averages of neutron flux over the nodes, determination of node-wise fluxdistribution from these averaged values is certainly not self-evident. A common way to resolvethis issue of nodal methods is to utilize for the pinwise flux reconstruction the information aboutits distribution, previously obtained during homogenization.

Calculation procedure

The solution of a steady state core problem translates either into a fixed source calculation, inwhich the neutron flux produced by specified external neutron source is determined, or into aneigenvalue problem, in which the dominant eigenpair is sought as it is the only one correspond-ing to real physical quantities, namely the reactor multiplication factor and neutron flux ([Han07,Sec. 3.3.1.1]). My previous thesis dealt exclusively with the latter as it is the most frequent typeof calculation performed in reactor analyses. As has been demonstrated there, traditional nu-merical methods like the power or Rayleigh-Ritz method can be used to determine the dominanteigenpair.

However, in many real-world reactor problems, these simple iterative methods convergevery slowly. Many techniques have been devised in need for an effective acceleration, like theWielandt’s shift method, Chebyshev or asymptotic source extrapolation. Although these meth-ods can significantly reduce the number of iterations in some cases, there are problems in whicheven their convergence rate may become too slow. Furthermore, they depend rather sensitivelyon one or more parameters that must be guessed a priori. Recently, however, Krylov subspacemethods began to be used in numerical core studies as they are believed to bring a solution atleast to the last problem. In a summary, acceleration schemes for the eigenvalue calculationsare certainly a valuable addition to a core calculation code, although their usage should not beautomatic and requires an expert input.

There are many areas of nuclear reactor studies where the calculation performance is highlyimportant. In the fuel loading optimization process, for example, the steady state equation has tobe evaluated many times in order to find the optimal condition. Another area, where the need fora fast core neutronic solver is even aggravated because a quasi-static approach cannot be used, isthe analysis of the transient states of the reactor, such as the startup, shutdown or abrupt controlrod movement during an accident. Although not being the subject of this thesis, a good methodfor steady-state reactor calculations should be prepared for a dynamical extension. Selection ofmathematical and numerical models should therefore be guided with computational efficiency inmind.

1.2 Present work 6

1.2 Present work

The previous section demonstrated what a multi-faceted task the development of a core neutroniccalculation code truly presents. Therefore, in order to solve as much of the undisclosed or newlyapparent problems, some of which were briefly overviewed in that section, a team-work hasproven to be absolutely necessary. The talk about the code would not be complete without atleast mentioning the members of the team1.

The work on the code has been coordinated by Ing. Roman Kuzel, PhD., who has also beenthe main programmer and whose primary aim has been on the nodal method. Ing. Tomas Berkahas begun the works on the homogenization module and, at the moment of writing this text,prepares the reconstruction module. Many valuable insight has been provided by the supervisorof this thesis and many fruitful discussions were started and contributed to also by Ing. AlesMatas, PhD.

I have been involved in several areas of the development, including the work on enhancingthe existing CMFD and nodal methods. During the process, however, a new requirement hasarisen, to develop a fine-mesh, finite-difference (FMFD) solver. Although this may seem as astep backwards, the fine mesh heterogeneous solver is actually a key ingredient in the homoge-nization procedure, as will be described in Sec. 4.3. Moreover, after its calibration on availableacknowledged results from literature, it has proven to be an invaluable aide in validating thedeveloped nodal method against the input data from S-JS.

Most notably, however, the FMFD discretization approach provides a simple way to deal withspatial dependence of the Boltzmann equation and allows thus to concentrate on the remainingindependent variables in the equation. The main proposal of the thesis is to explore the transporttheoretical methods available for improving the accuracy of the diffusion based solvers, that is,without the need for rewriting the whole existing code from scratch. I tried to provide as muchcomplete (although not at all exhaustive) overview of the process of transforming the neutrontransport equation into a readily programmable numerical method as the time and also the scopeof a Master’s thesis allow.

1.2.1 Structure of the thesis

Chapter 2 is devoted to the mathematical theory of neutron transport, which I found in the liter-ature to be often either silently omitted (in the physicists’ works) or tremendously involved (themathematicians’ works). Section 2.1 presents the fundamental equation of the theory togetherwith a physical explanation of its terms and some basic simplifications.

Section 2.2 then describes a functional-analytic setting of the whole theory, which provesvery useful to describe the methods generally applicable to any transport approximation (includ-ing diffusion) in the subsequents parts of the thesis.

1All affiliated at the Department of Mathematics, Faculty of Applied Sciences, University of West Bohemia inPilsen, CZ

1.2 Present work 7

The two problems of primary interest for applications are presented next, in Sec. 2.3 – namelythe eigenvalue (core criticality) calculation and the calculation with specified non-changing neu-tron sources. Overview of the mathematical conditions necessary to prove the existence of theirsolution and a connection to the corresponding physical conditions is also given.

The chapter is concluded with sections 2.4 and 2.5, which concentrate on the complicateddirectional dependence of the processes governed by the transport equation and develops theirconvenient mathematical representation. This is often described very briefly (if at all) in theliterature, without explanation of various factors that arise during the derivation and also with-out mentioning the link to the well-known mathematical theories, originally used in differentcontexts. I hope that the two sections at least partially fill this gap.

Chapter 3 focuses on the actual discretization of the energetic and directional dependenceof Boltzmann’s equation. Energetic dependence is attended first in Sec. 3.2, which describesthe multigroup approximation and the natural solution technique of the resulting quasi-discreteequations (the source iteration).

The angular variable is addressed next in Sec. 3.3. Two widely used methods – that of discreteordinates and that of spherical harmonics) – are described in the section, again with a link to theclassical methods of numerical mathematics. The latter is then chosen for closer study, ultimatelyleading to the so called simplified spherical harmonics approximation.

In the final Sec. 3.4, the results of the energetic discretization are bound with those of thedirectional one to formulate the multigroup simplified spherical harmonics approximation.

Chapter 4 addresses the remaining continuous variable in the transport equation by describ-ing the spatial discretization using a finite volume scheme. As the code’s main application do-main is going to be the neutronic analysis of thermal light water reactors (LWR) with hexagonalfuel assemblies, the spatial discretization is performed accordingly – by the coarse-mesh, finite-difference method accompanied by a nodal method to reduce discretization errors, and by thefine-mesh, finite-difference method. The first approach is described for the diffusion approxi-mation, highlighting the biggest improvements over our former nodal method on some modelexamples. The latter is shown to give consistent results with the worldwide acknowledged fine-mesh codes in the three-dimensional diffusion calculations first and then some preliminary resultsof its application to transport theory problems is also presented.

The final Chapter 5 provides a summary of the thesis and outlines some possible directionsof future research.

8

Chapter 2

Neutron transport theory

The primary factor that determines the behaviour of a nuclear reactor is the distribution of neu-trons within its core. Assuming a continuum description of the core where neutrons are repre-sented by point particles with position and velocity, its best mathematical model is provided bytransport theory, originally developed by Ludwig Boltzmann to describe kinetics of gases.

2.1 Neutron transport equation

2.1.1 Phase space of neutrons and related notation

The fundamental equation of the theory is a non-linear integro-differential equation describingdensity changes of arbitrary particles under an arbitrary force field. Its application to neutrons,whose density is so smaller than that of the scattering atoms that interactions among themselvescan be neglected, simplifies the equation by making it linear. However, it is still quite involved asthe neutrons possess three spatial, two directional and one energetic degrees of freedom, definingtheir phase space:

X = V × E × S ≡ x = (r, E,Ω) : r ∈ V, E ∈ E,Ω ∈ S. (2.1)

Here r = (x, y, z)1 specifies the position of the particle in the bounded domain (e.g. the reactorcore) V ⊂ R3 with a piecewise smooth boundary ∂V , Ω its direction of motion (a vector frompoint r to the surface of the unit sphere S centered at r, measured by its polar angle ϑ andazimuthal angle ϕ in the sense shown in Fig. 2.1) and E = 1

2mv2 its energy corresponding to speedv (m = 1.675 × 10−27 kg is the mass of neutron). Energy can attain any value in E = [0, Emax),where Emax is chosen sufficiently large, so that it covers any energy that a neutron in an actualapplication can have. For instance, neutrons in nuclear reactors attain maximum kinetic energy of

1To reduce the number of symbols, x is used to denote both the general phase space variable and the firstcomponent of the position vector; the precise meaning of the letter will be obvious from context.

2.1 Neutron transport equation 9

at most a few MeV, immediately after their liberation from fuel nuclei by fission (most probablyabout 1 MeV, practically always less than 10 MeV – see e.g. [Sta01, p. 12]).

All sets V , E and S are measurable and their respective Lebesgue measures define the mea-sure of the whole phase space

dx = dµ(X) = dµ(V × E × S) = dµ(V) dµ(E) dµ(S) = dr dE dΩ

with the usual notational convention. The measure will be written in integrals first, followed bythe integrand. In order not to clutter the expressions with parentheses, all terms appearing afterthe integration sign (including sums) will be assumed to belong to the integrand until the next +

or − sign. The same convention applies in this thesis to the summation sign. Further, I will writeintegrals over the angular part of the phase space (the unit sphere S) as

∫

SdΩ f (r, E,Ω) ≡

∫

4πdΩ f (r, E,Ω)

to make the integration range more clear and, for the sake of brevity, the integrals over the wholephase space as ∫

SdΩ

∫

EdE

∫

Vdr f (r, E,Ω) ≡

∫

Xdx f (x)

(and likewise for their respective subsets).

x

y

z

ex

ey

ez

r

v(E)Ω

|dΩ|

ϕ

ϑ

Figure 2.1: Neutrons’ phase space

When working with the direction vector, it will be sometimes convenient to switch betweenthe spherical coordinate system and the basic Cartesian coordinate system2. The transformation

2Other systems, like cylindrical, will not be dealt with in this thesis.


is given by (see Fig. 2.2):

Ω =

Ωx

Ωy

Ωz

=

sinϑ cosϕsinϑ sinϕ

cosϑ

.

Note that since we will always consider unit direction vectors, the radius variable r appearing inthe ordinary definition of the spherical coordinate system is set to unity.

x

y

z

ex

ey

ez

r

Ω

dΩ

Ωx

Ωy

Ωz

ϕ

ϑ

Figure 2.2: Cartesian coordinate system

The corresponding component of the phase space measure – the cone of directions dΩ , de-fined as a solid angle subtended at the center of S and measured thus in steradians – is given by(see Fig. 2.3):

dΩ =|dΩ |

r2 =r2 sinϑdϑ dϕ

r2 = sinϑdϑ dϕ .

2.1.2 Boltzmann’s transport equation for neutrons

The equation can be derived from the fundamental principle of conservation of neutrons withinan arbitrary element of the phase space as shown in [Han07, Sec. 3.1.1]. That is, the time-rateof change of neutron density N within the element ought to be equal to the difference betweenthe production rate and the loss rate. The actual reactions that contribute to production or lossesdepend on the intended application of the equation. The primary application considered in thisthesis is that to neutron-physical characterization of nuclear reactor core. In this device, neutronscan be introduced into the elementary phase space volume by their liberation either from thefissile nuclei during a fission event or from an external source, or by scattering into the elemen-tary energy range dE and elementary cone of directions dΩ . On the other hand, neutrons can


x

y

z

R=

rsin

ϑ

Rdϕ

r

ϑdϑ

Ω

|dΩ|

Figure 2.3: Schematic of the solid angle of directions (scaled up for clarity)

leave the elementary volume by leakage through its boundaries or by collisions with the nucleiinside, that either scatter them out of the volume or lead to their complete loss by absorption.Representing the just mentioned reactions as in [Han07, Sec. 3.1.1.2] and putting them in theright place in the conservation statement results in the integro-differential Boltzmann’s equationof neutron transport, [Han07, Eq. 3.18]:

[1

v(E)∂

∂t+ Ω · ∇ + Σt(r, E, t)

]ψ(r, E,Ω, t)

=

∫

4πdΩ′

∫

EdE′ Σs(r, E′E,Ω′Ω, t)ψ(r, E′,Ω′, t)

+

∫

4πdΩ′

∫

EdE′ ν(r, E′)Σ f (r, E′E,Ω′Ω, t)ψ(r, E′,Ω′, t)

+ Q(r, E,Ω, t).

(2.2)

The terms appearing in the equation are described below, together with some first simplifyingassumptions. For more detailed description and physical background, see [Han07] and the ref-erences therein. As a basic assumption leading to eq. (2.2), the core composition is consideredisotropic, i.e. with same properties in any direction, which is a classic, physically acceptableassumption (based on spherical symmetry of nuclei comprising the core materials). Note thatthis does not mean that the direction of motion of neutrons can be neglected.

• ψ(r, E,Ω, t) = v(E)N(r, E,Ω, t)

Angular neutron flux, expressing the track length that neutrons positioned within a unitvolume, moving in any direction from a unit solid angle, with speeds corresponding to en-


ergies within a unit range, travel per unit time. To characterize neutron streaming, angularneutron current is defined as

j(r, E,Ω, t) = Ωψ(r, E,Ω, t), (2.3)

so that the scalar j = j · n quantifies the number of neutrons (per unit time) crossingin direction Ω a surface of unit area, oriented by the unit normal vector n which pointsoutwards the volume enclosed by the surface. These quantities should be understood asdensities with respect to the phase-space measure dr dE dΩ .

•1

v(E)∂ψ(r, E,Ω, t)

∂tTime rate of change of neutron density.

• Ω · ∇ψ(r, E,Ω, t)

Neutron streaming in direction Ω that accounts for leakage out of the elementary volume.Specifically in Cartesian coordinates:

Ω · ∇ψ = Ωx∂ψ

∂x+ Ωy

∂ψ

∂y+ Ωz

∂ψ

∂z,

where Ωi are Cartesian components of the unit vector Ω (see Fig. 2.2).

• Σt(r, E, t)ψ(r, E,Ω, t)

Characterizes the rate at which neutrons disappear from the elementary phase-space vol-ume, due to scattering and absorption. Capital Greek Sigma conventionally denotes themacroscopic cross-section that allows to quantify neutron-nuclear interactions at macro-scopic scale. It is a material property that represents the probability that a neutron withspecific energy and motion direction will trigger the particular reaction upon hitting thenucleus. The dependence on neutron energy is highly non-trivial, while the dependenceon angle of hitting is unimportant due to the assumption of isotropic core. Values of cross-sections are obtained either experimentally or theoretically with the help of quantum me-chanics and are available for computer processing in public databases (e.g. [OEC07]). Inthis particular case, Σt is called total macroscopic cross-section,since it includes all reac-tions that may happen when a neutron of given energy E and motion direction Ω strikesthe particular nucleus (i.e. both non-fissioning and fissioning absorption, scattering intothe other angles and energies (Ω′, E′), etc.).

•∫

4πdΩ′

∫

EdE′ Σs(r, E′E,Ω′Ω, t)ψ(r, E′,Ω′, t)

Rate at which neutrons moving in an arbitrary direction with an arbitrary speed scatter intothe elementary phase-space volume, given in terms of the differential scattering cross sec-


tion. It may be written in the following factorized form

Σs(r, E′E,Ω′Ω, t) = Σs(r, E′, t)χs(E′E,Ω′Ω, t), (2.4)

in which Σs is the ordinary cross section for scattering induced by neutrons with energyE′ hitting the nucleus from direction Ω′ (with no dependency on the latter again due toisotropy of the medium) and χs (scattering kernel) is the density of probability that suchneutrons will be scattered into the cone of directions dΩ around Ω and energy range dEaround E. Isotropy of material properties by no means leads to isotropy of scattering, al-though it does simplify this reaction by making it dependent only on the scattering angleinstead of both the incoming and outgoing direction. I shall postpone further discussionabout this simplification until Sec. 2.4 and keep for now the general form of directionaldependency of scattering.

•∫

4πdΩ′

∫

EdE′ ν(r, E′)Σ f (r, E′E,Ω′Ω, t)ψ(r, E′,Ω′, t)

Rate at which new neutrons appear in the balance domain as a result of fission induced byneutrons hitting the fissile nucleus from an arbitrary direction with arbitrary speed. Simi-larly to differential scattering cross section, the factorized form is

Σ f (r, E′E,Ω′Ω, t) = Σ f (r, E′, t)χ f (E′E,Ω′Ω, t),

and the probability density function χ f (fission kernel) describes the directional and en-ergetic distribution of newly released neutrons, depending on the momentum of the inci-dent neutrons. The mean number of neutrons emitted from a fissioned nucleus (∼ 2.5 for23592U) is denoted ν. In cross-section libraries, ν is typically included in the cross-section

itself – values of νΣ f (the fission yield) are usually provided for each isotope, dependingon the energy of the inducing neutron. Therefore, I will accordingly drop the explicit de-pendence of ν from now on.

An important simplification made at this point is the neglect of delayed neutrons (re-leased during decay of the fission products), i.e. only prompt neutrons emitted immediatelyafter the fission event are considered. This assumption is allowable for steady state mod-eling (which is the subject of this thesis) but rules out the application of the method fordynamic or reactor control studies.

• Q(r, E,Ω, t)

This term represents neutrons originating from sources other than fission (independent ofneutron density), hence the name external sources (although they are actually located in-side the core). The spontaneously fissioning isotope 252

98Cf or the Am-Be system generatingneutrons by conversion from spontaneously emitted α particles are typical examples. Be-sides these, external sources may also include beams of neutrons injected into the corefrom an external device, e.g. from the spallation target in the accelerator driven systems,where they model their propagation into the core.


2.1.3 Steady state transport equation

The steady state of a reactor is characterized by a non-changing neutron population, which trans-lates into a mathematical condition

1v(E)

∂ψ(r, E,Ω, t)∂t

= 0.

Also, time dependence of all quantities appearing in the time-dependent Boltzmann’s equation(2.2) is dropped. The equation then takes the following form:

[Ω · ∇ + Σt(r, E)

]ψ(r, E,Ω) =

∫

4πdΩ′

∫

EdE′ Σs(r, E′E,Ω′Ω)ψ(r, E′,Ω′)

+

∫

4πdΩ′

∫

EdE′ νΣ f (r, E′E,Ω′Ω)ψ(r, E′,Ω′)

+ Q(r, E,Ω).

(2.5)

It describes the equilibrium distribution of neutrons at any point in the core, travelling in anydirection and with any speed, in the presence of a steady state source. It is the starting point forfurther analysis.

2.1.4 Fundamental physical quantities

Instead of the angular flux, reactor physicists rather work with the integrated quantities, that arealso experimentally measurable by various detector mechanisms. In particular, the followingglobal quantities are important for reactor studies and will be used in following sections:

• Scalar neutron flux density

φ(r, E) =

∫

4πdΩψ(r, E,Ω). (2.6)

Total number of neutrons with energy within a unit range about E, passing per unit timethrough a unit area at r, regardless of their flight direction. To calculate reaction rates,spatial distribution of neutron flux is required rather than its precise spectral characteristics.Then

φ(r) =

∫

4πdΩ

∫

EdE ψ(r, E,Ω) (2.7)

will be the relevant quantity, called total neutron flux or simply neutron flux (e.g. [Her81]).Total volumetric neutron flux is finally obtained by integrating the last quantity over aselected volumeV ⊂ V:

φ(V) =

∫

Vdr

∫

4πdΩ

∫

EdE ψ(r, E,Ω). (2.8)


• Net neutron current density

J(r, E) =

∫

4πdΩ j(r, E,Ω) =

∫

4πdΩ Ωψ(r, E,Ω). (2.9)

The corresponding scalar quantity j = J · n quantifies net number of neutrons that crossthe unit surface at point r in the direction of its unit outward normal n. As before, totalneutron current density is obtained by integrating (2.9) over the range of energies, totalvolumetric neutron current density by further integrating over the spatial domain.

• Partial neutron current densitySome numerical methods are more conveniently formulated in terms of partial currentsinstead of the net currents. Partial neutron currents are defined by the expression

j±(r, E) =

∫

Ω·n≷0dΩ j(r, E,Ω) · n (2.10)

as the number of neutrons that pass through a unit surface in the direction or in the oppositedirection of its outward normal n. Note that by definition, J · n ≡ j = j+ − j−, and eq.(2.10) is sometimes rewritten using the definition of angular current (2.3) with absolutevalue to emphasize this fact:

j±(r, E) =

∫

Ω·n≷0dΩ |Ω · n|ψ(r, E,Ω).

• Reaction rate densityReaction rate density is defined as the product of scalar flux and the corresponding macro-scopic cross section,

R∗(r, E,Ω) = Σ∗(r, E,Ω)φ(r, E). (2.11)

Rather than the pointwise values, total volumetric reaction rates produced by flux in a givenregion are used in practical reactor analyses. They are usually represented in a functionalform:

R∗(ψ;V) =

∫

Vdr

∫

4πdΩ

∫

EdE Σ∗(r, E)ψ(r, E,Ω) =

∫

Vdr

∫

EdE Σ∗(r, E)φ(r, E).

(2.12)If the reaction is described by a differential cross-section, the correct definition of thecorresponding reaction rate would be (e.g. the total volumetric fission rate):

R f (ψ;V) =

∫

Vdr

∫

4π×4πdΩ′ dΩ

∫

E×EdE′ dE Σ f (r, E′)χ f (E′E,Ω′Ω)ψ(r, E′,Ω′).

(2.13)In the following section 2.2.1, I will show that it is actually equivalent to definition (2.12).

2.2 Mathematical setting 16

2.2 Mathematical setting

2.2.1 Physical considerations

In order to correspond with physical reality, the cross sections, angular neutron fluxes and exter-nal sources will be expressed by measurable bounded functions:

0 ≤ Σ∗(x) < ∞,0 ≤ ψ(x) < ∞,0 ≤ Q(x) < ∞

a.e. in X and also Σ∗, ψ . 0.

We allow for merely piecewise-continuous cross-section data (which is the case of real hetero-geneous reactor cores), so we assume Σ∗ ∈ L∞(X). Infinite fluxes may be allowed at singularpoints of sources distribution ([Sta01]), while trivial flux is usually discarded since it providesno helpful information. This setting ensures existence of integrals over the phase space or itssubsections and their interchangeability via Fubini’s theorem. The physical quantities from theprevious section Sec. 2.1.4 are therefore well-defined and suggest a natural setting for angularflux: ψ ∈ L1(X), and similarly for the external neutron source: Q ∈ L1(X).

A general, non-fissioning collision may end up by non-productive capturing of the neutron,elastic scattering, in which the neutron only changes direction of motion, or inelastic scattering,in which the neutron also transfers some of its energy to the target nucleus. This last reactionmay result in subsequent emission of neutrons from the nucleus (such reaction is then denoted(n,2n), (n,3n), etc.) and hence fission is not the only reaction that introduces new neutrons intothe system. Denoting the mean number of neutrons, liberated by an arbitrary neutron-nucleuscollision barring fission, by c (the so called secondary scattering neutrons), the differential scat-tering cross-section of eq. (2.4) can be (in analogy to fission) also written as

Σs(r, E′E,Ω′Ω) = c(r, E′)Σt(r, E′)χs(E′E,Ω′Ω).

Inelastic scattering may make c exceed unity. However, it requires high neutron energies and isnot as frequent as the other collision types in thermal reactors. Thus, we can assume 0 < c < 1 3.

The distribution kernels χ obviously obey the following normalization condition∫

4πdΩ

∫

EdE χs(E′E,Ω′Ω) =

∫

4πdΩ

∫

EdE χ f (E′E,Ω′Ω) = 1, (2.14)

which represents the certainty that there will always be some direction and energy that the neu-tron take if it triggers the appropriate reaction. Using this in eq. (2.13), we can see that

R f (ψ;V) =

∫

Vdr

∫

4πdΩ′

∫

EdE′ Σ f (r, E′)ψ(r, E′,Ω′)

∫

4πdΩ

∫

EdE χ f (E′E,Ω′Ω)

=

∫

Vdr

∫

EdE′ Σ f (r, E′)φ(r, E′),

which is precisely the definition (2.12) of a fission rate functional (analogically for scattering).3This is also a necessary condition for subcriticality (defined later), which is the desired state of reactor operation.


2.2.2 Boundary and interface conditions

Boundary conditions

The first-order spatial differential operator appearing in the steady state integro-differential for-mulation theoretically admits an infinite set of solutions. The one that corresponds to the givenphysical system (i.e. the core geometry and its composition) is obtained by imposing an ad-ditional condition at the boundary. This condition specifies the angular flux coming from theoutside of the core, i.e. in the direction Ωin such that Ωin · n < 0, where n denotes, as before,the unit outward normal to the boundary. The condition can be decomposed into a homogeneousand an inhomogeneous part

ψ(r, E,Ωin)∣∣∣r∈∂V

= ψin + ψh,

where ψin is any function defined on the segment ∂V0 ⊂ ∂V and satisfying requirements for theangular flux. For the homogeneous part, one of the following conditions is usually prescribed onthe segment ∂Vh (∂V = ∂V0 ∩ ∂Vh), or their combination on subsets of ∂Vh:

• Vacuumψh(r, E,Ωin)

∣∣∣r∈∂Vh

= 0.

Vacuum or purely absorbing material around the core allows no neutron that escapes fromthe core to return.

• Specular reflectionψh(r, E,Ωin)

∣∣∣r∈∂Vh

= ψ(r, E,ΩR)∣∣∣r∈∂Vh

,

where ΩR is the direction in which the neutron leaves the domain and which reflects intoΩin according to the law of mirror-like reflection: ΩR = Ωin − 2n(Ωin · n), in particular|Ωin · n| = |ΩR · n| (the angle of incidence equals the angle of reflection) and (ΩR×n)·n = 0(escaped neutrons are reflected back to the core in the same plane in which they left). Thiscondition is typically used to model planes of symmetry within the system.

• Albedo conditionψh(r, E,Ωin)

∣∣∣r∈∂Vh

= αψ(r, E,ΩR)∣∣∣r∈∂Vh

, (2.15)

generalizes the specular reflection condition so that only a fraction α of escaped neutronsis reflected back. For α = 1, specular reflection is retained, while vacuum condition canbe represented by setting α = 0. Values of α between 0 and 1 are generally used to modela reflector surrounding the core.

The albedo coefficient α may in general vary with the reflector properties and should alsocapture angular and energetic redistribution of the reflected neutrons due to their diffusionthrough the reflector. For a correct description, a general integral albedo operator must beused:

ψh(x)∣∣∣∂X− = (βψ)(x) =

∫

∂X+

dbx′ β(x′ x)ψ(x′), (2.16)

where ∂X± = x = (r, E,Ω) : r ∈ ∂Vh, E ∈ E,Ω ∈ S : Ω · n ≷ 0 is the exiting (resp.entering) boundary of the phase space and dbx = |Ω · n| dS dE dΩ the boundary measure,


in which dS is an elementary spatial boundary segment oriented by outward normal n.Like scattering and fission, the kernel of the albedo operator can be factorized into:

β(x′ x) = α(r′, E′)χβ(r′r, E′E,Ω′Ω)

(the first argument of χβ represents spatial redistribution and may for example account forreentrant boundaries or periodic conditions), whence the scalar albedo coefficient of eq.(2.15) is recovered by localization to point (r, E,ΩR):

χβ = δ(r′ − r)δ(Ω′ −ΩR)δ(E′ − E).

Without delving into details and referring to literature (e.g. [Reu08, App. C.1]), Dirac’s”delta functions”are introduced here to perform the localization using their properties(vaguely speaking)

∫D

du′ δ(u − u′) f (u′) = f (u) and δ(u1, u2, . . . , un) = u1 · u2 · · · un, whereD is the definition domain of f . As the solution method developed in Chap. 3 is not tra-jectory based, the albedo operator will be expected to capture only energy redistribution,

χβ = δ(r′ − r)δ(Ω′ −ΩR), (2.17)

although the results presented in the following section generally do not depend on thisassumption. For a more thorough discussion of transport boundary conditions applicableto trajectory based methods, see the works of Sanchez et al., in particular [San02].

Interface conditions

LetV andW be two regions within the solution domain V that share the same interface, ∂V. Thenumber of neutrons crossing that interface in an arbitrary direction from one region to anothermust be conserved, hence the angular flux must be continuous:

ψ(r, E,Ω)∣∣∣∂V→ = ψ(r, E,Ω)

∣∣∣∂V←, ∀E, Ω, (2.18)

where ∂V→ stands for the limit in which r approaches ∂V from regionV while ∂V← for the limitin which ∂V is approached from the other region,W.

2.2.3 Operator form of the Boltzmann’s equation

The steady state transport equation (2.5) can be cast into the following operator form:

Lψ = Hψ + Fψ + Q in X,ψ = βψ + ψin on ∂X−,

(2.19)

where the differential streaming operator is defined by

L = Ω · ∇ + Σt, (2.20)


the integral operators of scattering and fission by

(Hψ)(x) =

∫

4πdΩ′

∫

EdE′ Σs(r, E′E,Ω′Ω)ψ(r, E′,Ω′), (2.21)

(Fψ)(x) =

∫

4πdΩ′

∫

EdE′ νΣ f (r, E′E,Ω′Ω)ψ(r, E′,Ω′) (2.22)

and the general albedo boundary reflection by

(βψ)(x) =

∫

∂X+

dbx′ α(r′, E′)χβ(r′r, E′E,Ω′Ω)ψ(r′, E′,Ω′).

Functional space setting of the operators must take into account the physically desirableproperties of the neutron flux function they are acting on. Its precise specification howeverdepends also on the concrete application and the properties of the space that the applicationrequires for its accomplishment (e.g. existence of an inner product). A physically natural settingwill be discussed in the following sections.

The inhomogeneous boundary conditions must be generally understood in the sense of tracesof ψ on ∂X , whose existence is ensured for typically considered spaces by well-known tracetheorems ([DL00], [DM07]). However, the construction of a correct functional setting for thealbedo operator is non-trivial and requires invocation of some special trace theorems – for details,see [DL00], in particular the Appendix of §2.

One may notice that since by definition

Ω · ∇ψ = Ωx∂ψ

∂x+ Ωy

∂ψ

∂y+ Ωz

∂ψ

∂z=

dxds∂ψ

∂x+

dyds∂ψ

∂y+

dzds∂ψ

∂z=

dψds,

where s is the path traveled by the neutron along the direction Ω (the characteristic), the differ-ential operator L may be inverted by integration along these characteristics to obtain an integralformulation of the neutron transport equation. This transformation is of both theoretical (proofsof many results mentioned in Sec. 2.3 use it) and practical (it is the basis of the method of colli-sion probabilities or the method of characteristics) value. Basically, numerical methods foundedon the global integral formulation of Boltzmann’s transport equation reduce to problems withfull matrices as opposed to sparse matrix problems arising from the original integro-differentialequation. They are hence less suited for efficient solution of large (e.g. whole core) regionsand will not be discussed in any more detail. A great deal of literature about both the abovementioned integral based methods is available – a good overview is provided e.g. by Sanchez in[SM81] or in [Sta01, Chap. 9.2].

2.3 Two problems in steady state transport theory 20

2.3 Two problems in steady state transport theory

There are two fundamental configurations of the core that are studied with the steady state trans-port theory – with or without the presence of external neutron sources.

2.3.1 Eigenvalue calculation

First, consider a source-free steady state problem, characterized by a homogeneous operatorequation with homogeneous boundary conditions:

Bψ = Fψ in X,ψ = βψ on ∂X−,

(2.23)

where the transport operator B is given as B = L − H. This equation admits either infinitelymany solutions (real multiples of one selected solution) or no solution at all.

The first case calls for normalization, which is done either to one fission neutron in the core:

1 = N (ψ; V) ≡∫

Vdr

∫

EdE′ νΣ f (r, E′)φ(r, E′) (2.24)

or to the given core thermal power level P (which is important for safety, economical and tech-nical reasons)

P = P(ψ; V) ≡∫

Vdr

∫

EdE′ εfΣ f (r, E′)φ(r, E′). (2.25)

where εf is the energy obtained from fission (e.g. for 23592U, it is about 200 MeV, [Her81, Tab. 3]).

In order to assure existence of the solution, one introduces a variable parameter into the equa-tion that may be adjusted until the steady state solution is found. Mathematically, this transformsthe problem into an eigenvalue problem. There are several places at which the tunable parame-ter can be placed within the homogeneous steady state equation. Here, I will use the so calledmultiplication eigenvalue λ, by which the fission intensity is varied:

Bψ =

1λ

Fψ in X,

ψ = βψ on ∂X−.(2.26)

As an example of another possible eigenvalue formulation, assuming time dependence of fluxin the form of an exponential yields the time eigenvalue equation, which characterizes the rateat which the associated eigenfunction will decay in the absence of external source. For detailsabout this formulation, more appropriate for modeling subcritical accelerator-driven systems, seee.g. [Lat03].


Solvability

A great deal of work has been done on characterizing the conditions under which the neutrontransport eq. (2.26) has a physically plausible solution. The main result of these works is thestatement that the greatest eigenvalue of the (generalized) eigenvalue problem (2.26) is simple,positive and associated with a positive eigenvector, and any other eigenvector changes sign andthus cannot represent realistic neutron flux. It can be shown ([Han07, Sec. 3.1.4]) that thedominant eigenvalue actually corresponds to the effective multiplication factor keff, which has thephysical interpretation as a ratio between the number of neutrons in two successive generations,or, symbolically,

keff ≡ neutron productionneutron loss

. (2.27)

Provided that the transport operator is invertible, this correspondence can be written as

ρ(B−1F) ≡ maxλ∈σ(B−1F)

λ = keff, (2.28)

(ρ and σ denote, respectively, the spectral radius and the spectrum of the operator). With keff < 1,keff > 1 and keff = 1, respectively, the system is said to be subcritical, critical and supercritical,respectively, meaning that without an additional neutron source, the neutron population will,respectively, continuously diminish, increase or be maintained only through the balance betweenleakage, absorption and fission. In this way, keff , 1 measures departure of the system fromthe steady (or critical) state characterized by keff = 1. The associated eigenfunction representsthe shape of the asymptotic neutron flux in the steady reactor core. Note that only the dominanteigenpair possesses the just described physical significance.

Most of the proofs of the main statement basically proceed in the following sequence

1. Prove that the transport operator B is invertible. This permits the traditional transcriptionof the eigenvalue equation (2.26):

B−1Fψ = λψ

and also yields regularity results for the solution.

2. Prove that operator B−1F is (strongly) positive and compact. As such, it has countablymany eigenfunctions. Positivity can be deduced from physical properties of the involvedoperators, compactness is much harder.

3. Invoke the Krein-Rutmann theorem for positive linear compact operators ([DM07, Thm.5.4.33]) to prove that the spectral radius of B−1F is a simple eigenvalue associated with theunique positive eigenfunction.

Depending on the chosen functional setting, various additional assumptions need to be madein order to prove the above steps. These mathematical assumptions restrict either the boundaryconditions, geometry or material composition of the solution domain, or energetic dependence(or all) and may not always coincide with physical reality. For instance, strong positivity would


require Σ f ≥ Σ f > 0 ([San06]), implying that fission occurs everywhere, which it certainly doesnot (consider for instance the area between fuel rods). For only a non-strongly positive compactoperator (the realistic case of B−1F), one can still use the weak form of the Krein-Rutmanntheorem ([DM07, Prop. 5.4.32]). That theorem however does not guarantee uniqueness of theeigensolution and a separate demonstration is required. Also, early mathematical analyses of thetransport operator (see the references in [San06]) were carried out in the Hilbert space L2 withonly homogeneous (vacuum) boundary conditions. Later, the results were generalized to otherboundary conditions and general Lp spaces, but in simplified geometric4 or energetic5 setting.Condition of everywhere positive total cross-section Σt has been classically assumed for generalLp spaces, which rules out void regions (that may appear e.g. in the VVER-440 reactors).

Sanchez ([San06]) overviews the previous analytical works more thoroughly. He suggeststo work in the space of Lebesgue-integrable functions L1(X), since only the ”total” quantitiesdefined according to Sec. 2.1.4 as integrals over the phase space X (or its section) are of physicalsignificance (see also [DL00]). To lift some of the aforementioned restrictions, he proposes tofurther equip the space with measure

dτ = Σt(x)dx , (2.29)

defining physically an elementary optical path length, i.e. the probability of interaction withinthe elementary dx , and work instead of L1(X) in the weighted space L1

Σ(X) (physically the ”space

of reaction rates”). In this setting he proves the main existence result for arbitrary geometries(with arbitrarily large void regions), energetic structure described by the whole continuum E orunion of its subintervals (the multigroup treatment as described in Sec. 3.2) and general boundaryconditions (2.16) (incorporated into the transport operator B):

Theorem 1: Let B, F : L1Σ(X) → L1(X), β : L1(∂X+) → L1(∂X−), maxx′∈∂X+ α(x′) ≤ 1 and

c < 1. Furthermore, let the scattering, fission and albedo kernels χs, χ f and χβ be limits ofequicontinuous kernels and either Σ f ≥ Σ f > 0 a.e. in X, or at least in a nonempty subset XF ⊂ Xthat is trajectory-connected with whole X. Then the problem (2.26) has a countable numberof eigenvalues and associated eigenfunctions which belong to W1

1,Σ(X). There exists a largesteigenvalue ρ(B−1F) ∈ σ(B−1F), which is algebraically simple and its associated eigenfunction isthe only one that does not change sign in X. ♦

The functional spaces are defined as follows:

L1(X) =

u :

∫

Xdx |u(x)| < ∞

, L1

Σ(X) =

u :

∫

Xdx Σt(x) |u(x)| < ∞

,

L1(∂X±) =

u :

∫

∂X±dbx |u(x)| < ∞

,

W11,Σ(X) =

u : u ∈ L1

Σ(X) ∧Ω · ∇u(x) ∈ L1Σ(X)

.

(2.30)

4Assumption of convexity of the domain, which is not true e.g. in the case of the VVER-1000 core investigatedin Chap. 4, see Fig. 4.6.

5One speed, energy independent transport theory.


The first two conditions are satisfied by virtue of the earlier discussions in sections 2.2.1 and2.2.2. The physically non-restrictive condition on the kernels is required to prove the compact-ness of the transport operator, while the last condition is required for uniqueness. The notionof trajectory connectivity is so far rather heuristic and basically means that particles producedin XF may reach any other point by direct streaming or through collisions. Essentially similarconditions are often used to circumvent the unphysical restriction of almost everywhere strictlypositive fission cross-sections ([MK98, AB99]).

2.3.2 Fixed source calculation

Suppose now that an additional neutron source exists within the given multiplying medium. Thesteady state source equation has the following operator form

Bψ = Fψ + Q in X,ψ = βψ + ψin on ∂X−.

(2.31)

This equation is often encountered in ADS studies. Nuclear reactors utilize fixed neutron sourcesduring start-up phases or to ensure high enough count rates for detector devices. Note that whenthe medium is non-multiplying, only the external source term Q remains on the right hand sideof eq. (2.31), which then describes a fixed-source transport problem with scattering:

Bψ ≡ (L − H)ψ = Q in X,ψ = βψ + ψin on ∂X−.

(2.32)

This equation finds its use e.g. in radiation shielding studies, but surprisingly also in reactorcriticality (i.e. eigenvalue) calculations at every stage of an iteration process to determine themultiplication source (a more concrete description will be provided in Sec. 3.2.3).

Solvability

If the multiplying properties of the medium were exactly critical (steady) without the source, itis physically intuitive that adding one is going to disturb the equilibrium and lead to a continuingincrease of neutron population (further accelerated by chain reaction). The same will be true fora supercritical source-free composition, of course. Thus one may achieve the time independentneutron distribution with an external neutron source only with subcritical composition, as maybe demonstrated by plotting (Fig. 2.4) the evolution of neutron flux obtained by solving a simplepoint kinetics equation with external source (see e.g. [Sta01, Chap. 5]).

Rigorous derivation of existence conditions for the fixed source problem is based on theFredholm’s alternative theorem. The corresponding equation (2.31) may be considered as theeigenvalue equation (2.26) with fixed value of parameter λ, i.e. λ = λ′, and an added externalsource term Q. Utilizing invertibility of B in L1

Σ(X), the equation is more familiarly written as

(λ′I − B−1F)ψ′ = Q, (2.33)


0.2 0.4 0.6 0.8 1.0t

20

40

60

80

100

Φ

0.2 0.4 0.6 0.8 1.0t

200

400

600

800

Φ

0.2 0.4 0.6 0.8 1.0t

50 000

100 000

150 000

200 000

250 000

Φ

Figure 2.4: Time evolution of neutron flux for the three states of the core

where Q = λ′B−1Q ∈ L1Σ(X) and I is the identity operator on L1

Σ(X). Note that the homoge-

neous albedo boundary conditions may be incorporated into the transport operator B and theinhomogeneous conditions into the source term as a surface contribution. If we stay in the func-tional setting of previous sections, then eq. (2.33) represents an operator equation with a nonself-adjoint, non-compact (although power compact) operator B−1F, mapping the Banach spaceL1

Σ(X) onto itself. A special version of Fredholm’s alternative applicable for such operators exists

(see e.g. [DM07, Thm. 2.2.9 and Rem. 2.2.11]) and can be used to obtain the following theorem:

Theorem 2: Let B, F be as in theorem 1 and its conditions are satisfied. Let Q ∈ L1(X). Thenone of the following alternatives holds:

(i) If λ′ is an eigenvalue of B−1F, i.e. λ′ ∈ σ(B−1F), then eq. (2.33) has a solution (uniqueup to addition of an arbitrary eigenfunction of B−1F) if and only if 〈Q, ψ∗〉 = 0 (bracketsrepresent duality pairing between L1

Σand L∞

Σ) for every solution ψ∗ of the adjoint problem6

B∗−1F∗ψ∗ = λ′ψ∗ in X,ψ∗ = β∗ψ∗ + ψ← on ∂X+,

(2.34)

where B∗ = −Ω · ∇+ Σt −H∗ and H∗, F∗ and β∗ are formed by switching in their respectivekernels the incoming phase-space variable (x′) for the outgoing (x) and vice versa (see e.g.[San01, Sec. 2.1]).

(ii) If λ′ < σ(B−1F), then eq. (2.31) admits a unique solution ψ′ ∈ W1Σ(X).

(a) If λ′ < keff (the supercritical case), then the solution changes sign.

(b) If λ′ > keff (the subcritical case), then the solution is positive a.e. in X (and satisfiesboundary conditions on ∂X−). ♦

More information about points (ii/a) and (ii/b) may be found in part 4 of the article [San01].

6For a physical interpretation of the adjoint equation, you can consult e.g. [Rav00] or almost any literature aboutreactor physics.

2.4 Representation of the angular variable 25

2.4 Representation of the angular variable

In this section I will be concerned with the impact of the physically well-founded assumptionof isotropy (either of the medium or, in the case of fission, the reaction itself) on the terms ofthe transport equation (2.19)-(2.22). Since the focus is here on the angular variable Ω, I willsomewhat loosely group the remaining independent variables under one variable ζ to simplifythe notation, i.e. either ζ = (E′E), ζ = (r, E) or ζ = (r, E′) (according to context).

2.4.1 Scattering kernel

Material isotropy of the scattering medium, already discussed in connection with the differentialscattering, implies that the scattering kernel χs is invariant under rotation in R3 and depends onlyon the scattering angle or, equivalently, its cosine µ0 = cosϑ0 = Ω′ ·Ω:

χs(ζ,Ω′Ω) = χs(ζ,ΩΩ′) = χs(ζ,Ω′ ·Ω) = χs(ζ, µ0).

As it is customary in literature, I will describe angular dependency of the whole unseparated dif-ferential scattering cross-section, which with the above expression for its kernel readsΣs(r, E′E, µ0), and, for the sake of brevity, refer to it also as to the scattering kernel.

x

y

z

ϑ

ϕ

Ω

ϑ′

Ω′

ϑ0

ϕ′

Figure 2.5: Geometry of scattering

The assumption of material isotropy reduces the domain of angular dependence of the scat-tering process from a Cartesian product of two unit spheres to a single interval [−1, 1]. Assuming


a Hilbert space setting, the scattering kernel may be equivalently written in terms of its Fourierseries expansion with respect to a suitable complete orthonormal basis. In studying angular de-pendence of spherically symmetric operators, appropriately normalized Legendre polynomialsPk(µ0) are the appropriate and most often used choice for reasons that will become apparent laterin subsection 2.4.1. Hence

Σs(ζ, µ0) =

∞∑

k=0

2k + 14π

Σsk(ζ)Pk(µ0), (2.35)

where first few Legendre polynomials are

P0(µ) = 1, P1(µ) = µ, P2(µ) =12

(3µ2 − 1), P3(µ) =12

(5µ3 − 3µ),

higher order Legendre polynomials can be generated from the three-term recurrence

(2k + 1)µPk(µ) = (k + 1)Pk+1(µ) + kPk−1(µ) (2.36)

and they satisfy the following orthogonality relation∫ 1

−1dµ P j(µ)Pk(µ) =

2δ jk

2k + 1(2.37)

([Sta01, p. 331], additional properties of Legendre polynomials can be found in almost any liter-ature dealing with orthogonal polynomials, see e.g. [RVCK95, Sec. 16.5]). The k-th coefficientof the expansion is obtained by operating with

∫ 1

−1dµ0 Pk(µ0) on both sides of (2.35) and using

relation (2.37):

Σsk(ζ) = 2π∫ 1

−1dµ0 Pk(µ0)Σs(ζ, µ0). (2.38)

Physical interpretation of the expansion coefficients

From the computational point of view, the full Fourier expansion of scattering kernel is unreal-izable, so in practice it is truncated after the term with finite index k = K. The truncation indexK represents the degree (or order) of scattering anisotropy. In reference to Fig. 2.5,

Σs0(ζ) = 2π∫ 1

−1dµ0 P0(µ0)Σs(ζ, µ0) =

∫ 2π

0dϕ

∫ π

0sinϑ0dϑ0 Σs(ζ, cosϑ0)

= Σs(ζ)∫

4πdΩ χs(ζ,Ω′ ·Ω) = Σs(ζ)

(2.39)

(the physical condition (2.14) has been used in the final equality), so the zeroth expansion coef-ficient is the isotropic component of scattering and hence truncating the expansion after K = 0implies the assumption of isotropic scattering. Similarly, the first coefficient accounts for linearvariation of scattering in µ0:

Σs1(ζ) = 2π∫ 1

−1dµ0 µ0Σs(ζ, µ0)


and can be written in terms of the isotropic scattering cross-section as

Σs1(ζ) = µ0(ζ)Σs(ζ), (2.40)

where

µ0(ζ) =2π

∫ 1

−1dµ0 µ0Σs(ζ, µ0)

2π∫ 1

−1dµ0 Σs(ζ, µ0)

(2.41)

is the mean scattering cosine, explicitly obtainable in the case of elastic scattering off a given

nucleus with mass number A as µ0 =2

3A.

Spherical harmonics

Motion of neutrons is described in terms of the unit vector Ω (or its polar and azimuthal com-ponents ϑ, ϕ) and it is therefore desirable to express the scattering kernel in the same variablesinstead of the scattering angle cosine. Despite the complicated relation:

µ0 = Ω′ ·Ω =[sinϑ′ cosϕ′, sinϑ′ sinϕ′, cosϑ′

]T · [sinϑ cosϕ, sinϑ sinϕ, cosϑ]

= µ′µ +√

(1 − µ′2)(1 − µ2) cos(ϕ′ − ϕ),

where µ = cosϑ is the cosine of the polar component of the direction vector Ω (see Fig. 2.5),a Legendre polynomial of this argument can be conveniently transcribed using the well knownaddition theorem:

Pk(µ0) = Pk(µ)Pk(µ′) + 2k∑

m=1

(k − m)!(k + m)!

cos(m(ϕ − ϕ′))Pm

k (µ)Pmk (µ′)

=4π

2k + 1

k∑

m=−k

Ymk (ϑ, ϕ)Ym

k (ϑ′, ϕ′) (2.42)

further rewritten on the second line in terms of the spherical harmonic functions (or shortlyspherical harmonics). Spherical harmonics Ym

l are formally defined ([AH01, Def. 6.5.9]) asrestrictions of homogeneous harmonic polynomials to a unit sphere (in our case S ⊂ R3). Severalpossible explicit expressions of spherical harmonics appear in literature, depending on chosennormalization. I use here one in which the functions are orthonormalized on L2(S):

∫

4πdΩ Ym

l (Ω)Yml (Ω) =

∫ 2π

0dϕ

∫ π

0sinϑdϑYm

l (ϑ, ϕ)Ym′l′ (ϑ, ϕ)

=

∫ 2π

0dϕ

∫ 1

−1dµ Ym

l (µ, ϕ)Ym′l′ (µ, ϕ) = δmm′δll′

where the overline denotes complex conjugate and δi j is the standard Kronecker delta symbol.The (complex) spherical harmonic function of degree l and order m (l ∈ N0, m ∈ Z, 0 ≤ m ≤ l)


then assumes the form

Yml (Ω) = Ym

l (ϑ, ϕ) =

√2l + 1

4π(l − m)!(l + m)!

Pml (cosϑ)eimϕ, (2.43)

or (since the dependence on polar angle is only through its cosine)

Yml (µ, ϕ) =

√2l + 1

4π(l − m)!(l + m)!

Pml (µ)eimϕ,

where

Pml (µ) =

(−1)m√

(1 − µ2)m dmPl(µ)dµm if 0 ≤ m ≤ l,

(−1)m (l − m)!(l + m)!

Pml (µ) if − l ≤ m < 0.

(2.44)

are the associated Legendre functions. Note that ordinary Legendre polynomials are recoveredfor m = 0. Also, real spherical harmonics can be defined in terms of trigonometric functionsinstead of complex exponentials ([RVCK95, Sec. 16.5]).

Important properties of spherical harmonics. The fact that makes spherical harmonics ap-pealing for problems exhibiting spherical symmetry is that the spherical harmonic functions ofgiven degree l generate a rotationally invariant subspace of L2(S):

Λl = SpanYm

l ;−l ≤ m ≤ l, (2.45)

in the sense that R(S l) ⊂ S l for the rotation transformation R. In fact, Λl is the eigenspaceassociated with the l-th eigenvalue of the Laplace-Beltrami operator on the surface S ([Reu08]).Being eigenfunctions of the (spherical) Laplacian operator, spherical harmonics can be viewedas fundamental modes of vibration of a unit sphere. Several of them are plotted in Fig. 2.6, fromwhich one may also see the importance of keeping the higher degree terms in the expansion ofscattering kernel (the following equation (2.46)) if stronger scattering anisotropy or, with m = 0,its preferential directionality has to be captured.

It can be shown ([Reu08]) that a general rotationally invariant linear operator has theeigenspaces Λl and spherical harmonics are thus the eigenfunctions of such operators. Theyform a complete orthonormal system over the unit sphere and generalize the trigonometric poly-nomials that are eigenfunctions of rotationally invariant operators in a plane. As such, they can beused to generalize the Fourier series for expanding functions defined on S (the so called Laplaceexpansion). Further theoretical properties of spherical harmonics and the Laplace expansion (forinstance its pointwise convergence for bounded integrable functions like those for which themethod is used in the neutron transport field) are extensively studied e.g. in [San04, Chap. 3].


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Spherical harmonic representation of scattering

Let us now return to the primary task of this subsection, i.e. expressing the scattering kernelin terms of the direction vectors Ω′ and Ω. The Legendre expansion of the scattering kernel(2.35) allows us to use the addition theorem (2.42) to express the rotationally invariant scatteringoperator H of eq. (2.21) in terms of its eigenfunctions, the spherical harmonics:

(Hψ)(ζ,Ω) =

∫

EdE′

∫

4πdΩ′ Σs(ζ,Ω′ ·Ω)ψ(ζ,Ω′)

=

∫

EdE′

∫

4πdΩ′

∞∑

k=0

Σsk(ζ)k∑

m=−k

Ymk (Ω)Ym

k (Ω′)ψ(ζ,Ω′)

=

∫

EdE′

∞∑

k=0

Σsk(ζ)k∑

m=−k

Ymk (Ω)

∫

4πdΩ′ Ym

k (Ω′)ψ(ζ,Ω′)

=

∫

EdE′

∞∑

k=0

Σsk(ζ)k∑

m=−k

Ymk (Ω)φkm(ζ),

(2.46)

whereφkm(ζ) =

∫

4πdΩ Ym

k (Ω)ψ(ζ,Ω) (2.47)

are the coefficients of the Laplace expansion in angular variable of the angular neutron flux,

ψ(ζ,Ω) =

∞∑

k=0

k∑

m=−k

φkm(ζ)Ymk (Ω), (2.48)

i.e. the angular moments of flux. If only a finite expansion of the scattering kernel is used,cut after k = K, it implies that only the angular expansion of flux into functions from the firstK + 1 eigenspaces is used to describe the scattering of neutrons. Even so, thanks to the rotationalinvariance of the eigenspaces, all neutron directions lying on the unit sphere are treated equallyand the fundamental physical property of the process is preserved7.

Description of neutron scattering via the angular moments of flux motivates the developmentof a method having these variables as primary unknowns. This is precisely the method of spher-ical harmonics discussed in Sec. 3.3.3. In practice, there may be other possibilities of angularapproximation of scattering, specifically suited for a particular solution method (e.g. the direc-tion sampling for the SN methods, reviewed in Sec. 3.3.1). However, the amount of expansioncoefficients that is required for a sufficiently accurate approximation is usually lower than storagerequirements of the other approaches ([OA87]). Therefore, the spherical harmonic representa-tion of the scattering operator is often used also in codes which employ a different method toapproximate angular dependence of the remaining operators and of the angular flux.

7I remark here again that it is generally not the isotropy of the collision itself (which would mean that neutronsmay leave the reaction in any direction with the same probability), but rather the invariance of the cross-sectionagainst rotation of coordinates.


2.4.2 Fission and external sources

Fission neutrons are generally assumed to be emitted isotropically and independently on thedirection of the neutron that induced the reaction (see [Lep07, p. 35]). Therefore, we put

νΣ f (ζ,Ω′Ω) = νΣ f (ζ,Ω) =νΣ f (ζ)

4π, (2.49)

so that, in accord with the second normalization condition in (2.14),∫

4πdΩ νΣ f (ζ,Ω′Ω) =

∫

4πdΩ νΣ f (ζ,Ω) = νΣ f (ζ). (2.50)

When substituted into the definition relation (2.22) of the fission operator, we can see that itsaction is fully specified by the scalar flux (2.6):

(Fψ)(ζ,Ω) =

∫

EdE′

νΣ f (ζ)4π

∫

4πdΩ′ ψ(ζ,Ω′) =

∫

EdE′

νΣ f (ζ)4π

φ(ζ). (2.51)

External sources of neutrons, if present, are usually assumed isotropic in reactor studies:

Q(ζ,Ω) =Q(ζ)4π

. (2.52)

Means of capturing anisotropy of neutron emission from external sources (important for instancefor modeling accelerator driven systems) vary according to the solution method and its approachto representation of angular dependence of flux and may include expansion of the term intospherical harmonics or sampling at chosen directions.

2.4.3 The final equation

All the previous steps of representing the angular dependence of the initial transport equation(2.19)-(2.22) lead to its following explicit form:

[Ω · ∇ + Σt(r, E)]ψ(r, E,Ω) =

∫

EdE′

∞∑

k=0

k∑

m=−k

Ymk (Ω)Σsk(r, E′E)φkm(r, E′)

+1

4π

∫

EdE′ νΣ f (r, E′E)φ(r, E′) +

14π

Q(r, E).

(2.53)

However is the representation of angular properties via spherical harmonics pleasing from amathematical point of view, it is not computationally convenient because the spherical harmonicsand corresponding angular moments are complex quantities. To resolve this issue, one may usethe following property of spherical harmonic functions:

Y−ml (Ω) = (−1)mYm

l (Ω),

2.5 The slab-geometry transport equation 32

or directly utilize the real spherical harmonics. In any way, the number of required sphericalharmonic moments that have to be stored in order to describe the scattering process grows like(K + 1)2 for the order of anisotropy K. An effort to reduce this number has led to derivation ofthe so called simplified spherical harmonics method, which has been chosen for implementationinto our existing diffusion solver and is described in more detail in section 3.3.5 and those thatfollow. Its principle lies in local approximation of neutron transport as an azimuthally symmetricprocess in a slab-like geometry, thus converting it to a one-dimensional process. Correspondingform of the transport equation is examined in the following section.

2.5 The slab-geometry transport equation

The slab-geometry neutron transport is characterized by a special case of neutron distribution,namely one that is spatially varying only along one coordinate direction and, moreover, that issymmetric with respect to rotations about that direction. Without loss of generality, we maychoose the z-axis to define the principal direction of variation. This situation may arise forexample when the system is composed of slabs, each with homogeneous properties and extentsin the x and y directions much larger than in the principal direction, so that dependence on x andy may be neglected. Angular flux (and also the other involved quantities) will be the same at anyplane perpendicular to the principal direction, i.e. ψ(r, E,Ω) ≡ ψ(z, E,Ω), and, because of theazimuthal rotation symmetry, also in any direction lying on a cone with the same apex angle ϑ,see Fig. 2.7. To describe distinct neutronic states, it is therefore sufficient to consider the reducedphase space

X =(z, E, µ) : z ∈ Vz, E ∈ E, µ ∈ [−1, 1]

, (2.54)

where Vz is an arbitrary section parallel to the z-axis through the whole core domain V . Similarlyto the isotropic (i.e. invariant with respect to any direction from the full solid angle) fission ineqns. (2.49) and (2.50), azimuthally invariant flux is related to the directionally dependent fluxby the identity

ψ(r, E,Ω) =ψ(z, E, µ)

2π, (2.55)

so that

ψ(z, E, µ) =

∫ 2π

0dϕψ(z, E,Ω) ≡

∫ 2π

0dϕψ(r, E,Ω). (2.56)

2.5.1 Streaming

As a consequence of invariability of angular flux in transverse directions, ∂ψ

∂x =∂ψ

∂y = 0, andbecause Ωz = cosϑ = µ, eq. (2.20) yields the following form of the streaming operator:

L = µ∂

∂z+ Σt.


x

y

z

ϕ

ϑ

Figure 2.7: Direction vector in azimuthally independent slab geometry

2.5.2 Scattering

In order to transform the scattering operator into the new coordinate system, let us investigatehow the azimuthal independence of angular flux affects its expansion into spherical harmonics.Inserting eq. (2.56) into eq. (2.47), written in terms of the polar and azimuthal components,gives

φkm(z, E) =

∫ 1

−1dµ

∫ 2π

0dϕ Ym

k (µ, ϕ)ψ(z, E, µ, ϕ) =

∫ 1

−1dµ

ψ(z, E, µ)2π

∫ 2π

0dϕ Ym

k (µ, ϕ).

For m , 0, the last integral vanishes8, while for m = 0 we have

Y0k (µ, ϕ) =

√2k + 1

4πPk(µ), 9 (2.57)

and

φk0(z, E) =

∫ 1

−1dµ

√2k + 1

4πPk(µ)ψ(z, E, µ). (2.58)

Hence, the Laplace expansion of angular flux into spherical harmonics, eq. (2.48), becomes inthe azimuthally independent case its ordinary Fourier expansion into a series of Legendre poly-nomials (which are in fact the ϕ-independent eigenfunctions of the spherical Laplace-Beltrami

8Spherical harmonics of nonzero order m depend on the azimuthal angle only through the complex exponential,which can be written as a combination of cos mϕ and sin mϕ, whose integral over 2π is zero.

9Note that, except for a factor, Legendre polynomials are therefore the ϕ-independent spherical harmonics,known also as the zonal spherical harmonics.


operator):

ψ(z, E, µ) =

∞∑

k=0

2k + 12

φk(z, E)Pk(µ), (2.59)

where

φk(z, E) =

∫ 1

−1dµ Pk(µ)ψ(z, E, µ) (2.60)

is the k-th Legendre moment of angular flux. This will be used later to derive the PN equations inslab geometry. At this moment, I will integrate the last equality in (2.46) over the full azimuthalangle and use relation (2.56) to define the action of the scattering operator in the simplifiedgeometrical setting, using identities (2.58) and (2.57) in the process10:

(Hψ)(z, E, µ) =

∫ 2π

0dϕ (Hψ)(z, E, µ, ϕ)

=

∫ 2π

0dϕ

∫

EdE′

∞∑

k=0

Σsk(z, E′E)k∑

m=−k

Ymk (µ, ϕ)φkm(z, E′)

=

∫

EdE′

∞∑

k=0

Σsk(z, E′E)2π

√2k + 1

4πPk(µ)

∫ 1

−1dµ′

√2k + 1

4πPk(µ′)ψ(z, E′, µ′)

=

∫

EdE′

∞∑

k=0

2k + 12

Σsk(z, E′E)Pk(µ)∫ 1

−1dµ′ Pk(µ′)ψ(z, E′, µ′)

=

∞∑

k=0

2k + 12

Pk(µ)∫

EdE′ Σsk(z, E′E)φk(z, E

′).

(2.61)

2.5.3 Fission and external sources

If we assume isotropic sources (as discussed in Sec. 2.4.2), their slab equivalents obtained analo-gously to slab angular flux (2.56) by azimuthally integrating (2.51) and (2.52), respectively, havethe form:

(Fψ)(z, E, µ) =

∫ 2π

0dϕ

∫

EdE′

νΣ f (z, E′E)4π

φ(z, E′) =12

∫

EdE′ νΣ f (z, E′E)φ(z, E′),

Q(z, E, µ) =

∫ 2π

0dϕ

Q(z, E)4π

=12

Q(z, E).

If anisotropic emission from external neutron source has to be taken into account, we put

Q(r, E,Ω) =Q(z, E, µ)

2π(2.62)

10One could alternatively use the azimuthally-invariant addition theorem, Pk(µ0) = Pk(µ)Pk(µ′), directly in thescattering expansion (2.35) to arrive at the same final expression.


and address the angular dependence by Legendre expansion or sampling from interval [−1, 1], asappropriate for the given method.

2.5.4 Boundary conditions

The exiting and entering boundary of the slab phase space is defined, respectively, by

∂X+ =(zB, µ) : zB ∈ Vz ∩ ∂V, µ ∈ [−1, 0)

∪

(zT , µ) : zT ∈ Vz ∩ ∂V, µ ∈ (0, 1]

× E

∂X− =(zB, µ) : zB ∈ Vz ∩ ∂V, µ ∈ (0, 1]

∪

(zT , µ) : zT ∈ Vz ∩ ∂V, µ ∈ [−1, 0)

× E

(2.63)

The two distinct points zB and zT are the positions where the boundary slabs cross the z-axis andeither one may be subject to either the inhomogeneous or homogeneous boundary condition. Forinstance, let there be one side with the inhomogeneous condition at point zB ∈ Vz ∩ ∂V0 ≡ ∂V0

zand one with the homogeneous condition at point zT ∈ Vz ∩ ∂Vh ≡ ∂Vh

z . Then

ψ(zB, E, µ) = ψin(µ), for 0 < µ ≤ 1,

ψ(zT , E, µ) =

∫

EdE′ α(zT , E′E)ψ(zT , E′,−µ), for −1 ≤ µ < 0.

(2.64)

For the homogeneous part, the general albedo condition has been used, localized in space andexiting directions by applying eq. (2.17) and using the relation between the polar angle cosinesbelonging to the incoming direction and its reflection: µR = −µ. Conditions of eq. (2.18),representing angular flux continuity at slab interfaces, are modified likewise.

2.5.5 The final equation

Collecting the results of the previous subsections leads to the final transcription of the steady stateneutron transport equation (2.5) in the one-dimensional slab geometry with azimuthal symmetry:

[µ∂

∂z+ Σt(z, E)

]ψ(z, E, µ) =

∞∑

k=0

2k + 12

Pk(µ)∫

EdE′ Σsk(z, E′E)φk(z, E

′)

+12

∫

EdE′ νΣ f (z, E′E)φ(z, E′) +

12

Q(z, E, µ)

(2.65)

with appropriate boundary conditions given above. This equation will be the subject of study inthe following parts of this thesis, until Sec. 3.3.6.

36

Chapter 3

Solution methodology

In this chapter, I will transform the exact (within the margins of the physical and geometricalmodel) problem of neutron transport developed so far into a form suitable for numerical so-lution on a computer. To make the key ideas more clear, I will place the exposition into theone-dimensional slab geometry studied in Sec. 2.5 of preceding chapter and only mention thepossible generalizations of intermediate results into higher dimension, referring to literature fordetailed derivations. Only at the end of the chapter, from Sec. 3.3.6 onward, I will lift off to threedimensions to formulate the final method.

The starting point of this chapter will be the one-dimensional neutron transport equation(2.65), written in the operator form as:

Lψ = Hψ + Fψ + Q in X,ψ = βψ + ψin on ∂X−,

(3.1)

where the interior and boundary domains are given, respectively, by defs. (2.54) and (2.63), and

(Lψ)(z, E, µ) = µ∂

∂zψ(z, E, µ) + Σt(z, E)ψ(z, E, µ),

(Hψ)(z, E, µ) =

∫

EdE′

∞∑

k=0

2k + 12

ΣskPk(µ)φk(z, E′)

(Fψ)(z, E, µ) =12

∫

EdE′ νΣ f (z, E′E)φ(z, E′),

(βψ)(z, E, µ)∣∣∣z∈∂Vh

z= α(z, E′E)ψ(z, E′,−µ)

∣∣∣z∈∂Vh

z,

(3.2)

The Legendre moments of angular flux in the scattering operator are given by eq. (2.60).

Although the phase space integral of squared angular flux does not have a direct physicalinterpretation, the operators will be considered as acting in the L2(X) space (definition domainof the streaming operator being a subset of W2

1 (X)) instead of the physically more meaningful L1

space. This proviso allows, in particular, to use the ordinary inner product of the Hilbert spaceL2 when constructing the classical numerical methods.

3.1 General principles 37

3.1 General principles

There are two main directions that can be taken when one wants to solve the transport equationon a computer – the deterministic and the stochastic. In the former, continuous dependenceof equation (3.1) on the phase space parameters is first discretized and numerical methods arethen employed to solve the finite problem. On the other hand, the latter (commonly called aMonte Carlo approach) is based on a direct simulation of interactions between neutrons andthe surrounding matter by probabilistic laws and employs statistical tools to obtain the expectedvalues of the quantities of interest.

Both approaches, of course, introduce additional distortions of physical reality – determin-istic methods due to loss of information about the parts of phase space neglected during dis-cretization, Monte Carlo methods due to statistical uncertainty arising from insufficient numberof simulated particles. The pursuit of reducing these errors leads in both cases to increase ofcomputing time. However, broadly speaking, deterministic methods are often capable of produc-ing reasonable results in a fraction of time required by the stochastic method, although the latteris considered more reliable when sufficiently low statistical uncertainty is achieved ([Kot07]).Therefore, routine reactor calculations are usually performed by deterministic methods and arevalidated using the reference results produced by a Monte Carlo type method.

Because the supposed application sphere of the method described in this thesis is a fast whole-core calculation, a deterministic approach has been selected for its development. In the followingsections, the discretization of each of the continuous independent variables appearing in equation(3.1) will be examined, starting with energy.

3.2 Energy discretization

Virtually all deterministic solvers use the energy-grouping method to treat the continuous de-pendence of transport equation on neutron energy. In principle, the range of all energies attain-able by neutrons in the considered physical system is divided into G discrete intervals calledgroups. Indexing of these intervals is chosen so that group g contains neutrons with energiesEg < E ≤ Eg−1. The structure of the multigroup discretization can then be written as

0 = EG < EG−1 < . . . < Eg+1 < Eg < Eg−1 < . . . < E1 < E0 = Emax.

To obtain the multigroup approximation of the transport problem, we proceed as in [Han07,Sec.3.3], starting from the explicit form of the transport equation (2.65). First, it is physicallyjustifiable to assume that in thermal reactors, energetic distribution of fission neutrons does notdepend on energy of the neutron that induces the reaction ([Wil71, p. 16]). Using this fact tosimplify the fission cross-section:

νΣ f (z, E′)χ f (E′E) = νΣ f (z, E′)χ f (E),

3.2 Energy discretization 38

then integrating equation (2.65) over any energy group Eg =(Eg, Eg−1], g = 1, . . . ,G, and using

linearity of integral, we obtain the steady state, group-g neutron transport equation1:

µ∂

∂z

∫

gdE ψ(z, E, µ) +

∫

gdE Σt(z, E)ψ(z, E, µ)

=

∞∑

k=0

2k + 12

Pk(µ)G∑

g′=1

∫

gdE

∫

gdE′ Σsk(z, E′E)φk(z, E

′)

+12

∫

gdE χ f (E)

G∑

g′=1

∫

gdE′ νΣ f (z, E′)φ(z, E′) +

12

∫

gdE Q(z, E, µ). (3.3)

This equation can now be equivalently rewritten in terms of the group angular flux

ψg(z, µ) =

∫

gdE ψ(z, E, µ), (3.4)

its Legendre expansion coefficients

φgk(z) =

∫

gdE φk(z, E) =

∫ 1

−1dµ Pk(µ)

∫

gdE ψ(z, E, µ), (3.5)

group fission spectrum

χgf =

∫

gdE χ f (E) (3.6)

and group external source

Qg(z, µ) =

∫

gdE Q(z, E, µ) (3.7)

as

µ∂

∂zψg(z, µ) + Σ

gt (z)ψg(z, µ) =

∞∑

k=0

2k + 12

Pk(µ)G∑

g′=1

Σgg′sk (z)φg′

k (z)

+12χ

gf

G∑

g′=1

νΣg′f (z)φg′(z) +

12

Qg(z, µ),

(3.8)

provided that the group constants are defined as the following weighted averages:

Σgt (z, µ) =

∫g

dE Σt(z, E)ψ(z, E, µ)

ψg(z, µ), νΣ

g′f (z) =

∫g′ dE′ νΣ f (z, E′)φ(z, E′)

φg′(z),

Σgg′s (z) =

∫g

dE∫

g′ dE′ Σsk(z, E′E)φk(z, E′)

φg′k (z)

.

(3.9)

1To simplify the notation, I denote∫

EgdE ≡

∫ Eg−1

EgdE by

∫g dE .


Multigroup albedo coefficient may be defined like the multigroup scattering cross-section:

αgg′(z, µ) =

∫g

dE∫

g′ dE′ α(z, e′E)ψ(z, e′,−µ)

ψg′(z,−µ)(3.10)

Note that the total cross-section and originally ”non-directional” albedo coefficient become inthe multigroup approximation dependent on the direction of neutron flight. Solution of thisissue depends on the actual method used to approximate angular dependence of flux and will beaddressed in Sec. 2.4.

Although the above definitions of multigroup constants have the nice property of preservingreaction rates, they are only formal since they involve the unknown solution ψ. They can be useddirectly only if the cross-sections are truly constant with respect to energy in the group range, orthe flux is separable into a spatio-angular part and a known energy part (the flux spectrum):

ψ(z, E, µ) = ψ(z, µ) f (E). (3.11)

In that case the multigroup constants are obtained simply as weighted spectrum averages, forinstance

Σgt (z) =

∫g

dE Σt(z, E) f (E)∫

gdE f (E)

(3.12)

Although the separability assumption (3.11) is usually not realistic in nuclear reactors, it mayserve as a starting point to obtain the full spatial-energetic-angular approximation of type

ψ(z, E, µ) ≈ Ψ(z, E, µ) (3.13)

which, if performed well, is the best thing one may use in eqns. (3.9) to resolve the problematiccycled definition of the multigroup constants.

3.2.1 Practical generation of multigroup constants

The most common method of generating the multigroup constants has already been outlined inthe introductory Chapter 1 and is called energy group condensation ([Trk00, DH76]). It startswith calculating the neutron energy spectrum, considering a very fine energetic dependence andeffects associated with cross-section resonances (e.g. Doppler broadening, [Sta01, Sec. 4.3]). Inthis phase, one uses various theories of neutron thermalization (often analytically) and detailed(e.g. 1000 groups, or energy-pointwise) cross-section data for each finest-scale representativeregion in the core (typically a single fuel pin cell, assumed to form a uniform infinite lattice so thatspatial dependence may be neglected) and various operating conditions (taking into account fuelburn-up, thermodynamical state of moderator, boron concentration, etc.). These cross-sectionsare then collapsed (or condensed) over the generated local spectrum to prepare ”many-group”data (eqs. (3.12), where the integration range corresponds to one of the ”many-groups”).


The number of ”many-groups” is chosen to be computationally acceptable within subsequenttransport-theoretical, spatially and directionally dependent calculations, which provide a furtherreduced set of few-group (a standard terminology) data in each region. The procedure is thesame, i.e. collapsing the ”many-group” data with energetic distribution of the resulting fluxby using it in equations like (3.9), where the integration interval now spans one of the few-groups. This time, however, spatial effects such as spatially dependent energy self-shielding([Sta01, Sec. 11.5]) or those caused by spectral interactions between different fuel types (like theincreasingly common combination of MOX and UO2 fuel) are also considered. Therefore, thecalculation is usually performed over each representative fuel assembly with interacting pin-cells(single-assembly calculation) or even taking into account the effects of adjacent assemblies (theso called colorset calculation).

For the purposes of nodal methods, the resulting flux (which is actually the approximation Ψ

in eq. (3.13)) is also used in a similar fashion to homogenize the data in the region (see Sec. 4.3),so that the final result are both group-wise and region-wise constant cross-sections (plus anyadditional homogenization data). Although a homogenization module has been incorporatedinto the final solver, the multigroup data are yet assumed to be given and the complex task ofenergy-group collapsing will not be discussed in any more detail in this thesis. Those interestedmay consult, for instance, the papers [Trk00], or [Kul00], describing in details the multigroupdata preparation procedure, or the thesis of Palmtag [Pal97], in which he provides a descriptionof and a solution to some issues involved with the infinite lattice calculations.

3.2.2 Operator form of the multigroup equations

N

Only in this section 3.2 will the bold letters denote the G×G operator matrices or G×1 vectors asdescribed below. Their elements correspond to energetic discretization and are continuous withrespect to the angular and scalar spatial variable.

In order to improve readability and come closer to conventional mathematical notation, I willfurther change for these matrices and vectors the convention of superscripted group indexing andput the indices into subscript instead, for instance in the following eq. (3.14)

H =[Hgg′

]g,g′=1,...,G, ψ =

[ψg

]g=1,...,G.

The multigroup approximation of the slab transport equation, eq. (3.8), may be convenientlyrewritten in terms of the operators from (2.19) by broadening their domains into the Cartesianproduct L2(X)G and reconsidering them as matrix operators acting on vector functions:

Lψ = Hψ + Fψ + Q in X,ψ = βψ + ψin on ∂X−.

(3.14)


The individual terms in equation (3.14) are specified as follows:

ψ is a G × 1 vector whose components are the group fluxes ψg(z, µ),

L is a G ×G diagonal matrix with diagonal filled with

Lgg = µ∂

∂z+ Σ

gt (z),

H is a G ×G full matrix with elements

Hgg′ =

∞∑

k=0

2k + 12

Σgg′sk (z)Pk(µ)

∫ 1

−1dµ′ Pk(µ′),

F is a G ×G full matrix with elements

Fgg′ =12χ

gf νΣ

g′f (z)

∫ 1

−1dµ′ Pk(µ′),

Q is a G × 1 vector of external sources Qg(z, µ) that do not depend on neutron density

β is a G × G lower triangular matrix (gaining energy in the reflector is impossible) withelements αgg′ ,

ψin is a G × 1 vector containing the prescribed incident angular fluxes with energies withinthe individual group ranges.

Note that the matrix H represents all forms of scattering into each group, while out-scatteringis contained in the L matrix inside Σ

gt . The structure of the multigroup equations depend on the

selection of bounds of the individual energy groups and a careful one may much simplify theequations without significant distortions of the true physical processes.

For example, if the incident neutron is so fast that its energy is substantially greater than thatof the target nuclei (which is typically less than 0.1 eV), it will not gain energy by the collision(we say that it will not up-scatter). Denoting by gF the index of the last group whose neutrons stillsatisfy this condition2, we may safely assume that Hgg′ = 0 for g′ > g and g ≤ gF . This resultsin an almost lower triangular matrix H with a full block at rows and columns correspondingto the up-scattering range. In many practical LWR calculations, the last energy group collectsall thermal neutrons with energies less than about 1 eV, so the probability of up scatter will benegligible and H becomes a truly lower triangular matrix. The sub-diagonal part of the scatteringmatrix may be also simplified by choosing the group spacing so that neutrons may only scatterto the next lower group. Together with the previous assumption, H becomes a simple lowerbidiagonal matrix, defining the very often used directly-coupled multigroup equations.

2Recall that higher group index means lower energy.


3.2.3 Source iteration

The multigroup discretization of the energetic dependence leads to a natural solution strategy thatreflects the physical process of neutron slowing-down. Its overall structure is independent on thesubsequently used transport approximation (including diffusion theory) and I will therefore de-scribe it in this subsection in the just presented general operator matrix form. Upon the particularspatio-angular discretization, the operator matrices eventually become ordinary numeric matricesand the abstract scheme becomes a readily programmable numerical method.

Solution of the matrix operator equation (3.14) can be found by simply inverting the ma-trix operator L − H − F. In numerical practice, however, direct solution of the (in some waydiscretized) equation (3.14), to which the inversion actually translates, is computationally infea-sible. Therefore, the solution is rather looked for iteratively. To mimic the neutron behaviour dur-ing moderation, we split the matrix H into the down-scattering, self-scattering and up-scatteringparts:

H = Hd + Hs + Hu, (3.15)

where

Hd is the lower triangular part that accounts for neutron transfer from lower groups intohigher groups, i.e. from higher energies into lower energies (down-scattering),

Hs is the diagonal part that accounts for scattering collisions by which the energy of the col-liding neutron is changed so little that it still remains within the same group(self-scattering),

Hu is the upper triangular part that accounts for neutron scattering from higher groups intolower groups, i.e. from lower energies into higher energies (up-scattering),

and write the iteration procedure with iteration index p as

(L −Hd −Hs)ψ(p) = (Hu + F)ψ(p−1) + Q, p = 1, 2, . . . (3.16)

In every iteration, eq. (3.16) is solved by stepping in the order of decreasing energy (i.e. fromthe first group to the last), solving the individual group problems of form (3.8). At each suchstep, flux from a previous iteration is used to calculate the contribution from lower energies (up-scattering) and fission (which couples all energies together) and the current flux from previousenergy steps is used to calculate the down-scattered source. In analogy to ordinary systemsof linear equations, down-scattering represents the forward substitution in solution of a lower-triangular system (which here however comprises abstract operator equations) and the wholeiteration procedure can be viewed as analogy to the Gauss-Seidel method.

This procedure is implemented in almost every neutronics code (diffusion and transport the-ory based alike) and is known as source iteration. Note that by ”source”, it is meant (besides thepossible external sources) the fission and, if present, the up-scatter source terms, whose energycoupling called for the iterative approach. Note that these terms are specified and fixed for everyiteration and so the whole process comprises a series of fixed-source transport problems withscattering, as introduced in eq. (2.32).


Originally in the SN methods, another iteration level has been introduced to account for theangular coupling of all the angular fluxes, caused by the self-scattering term. In such a two-leveliteration (or more commonly outer-inner iteration, where ”outer” stands for the so-far discussedsource iteration), each in-group equation is solved iteratively as

Lggψ

(p,q)g = Hs,ggψ

(p,q−1)g + S (p)

g ,

S (p)g =

[Hdψ

(p) + (Hu + F)ψ(p−1) + Q]g

, q = 1, 2, . . . , (3.17)

where the outer iteration index p is fixed, q denotes the inner iteration index and [·]g the g-thvector component. In the SN methods, Lgg can now be efficiently inverted by sweeping onedirection after the other (see e.g. [OA87] or [Sta01, Sec. 9.8]). It has been found that theother methods, including those based on diffusion approximation, may also benefit from iterativesolution of the in-group problems and the outer-inner strategy has found a firm place in thesetoo (it is used in our current diffusion solver, as well as in many codes from the references, e.g.[Hil75, ABB+95, Tur94]).

Initiation and stopping. The first iterate ψ(0) in the outer iteration as well as ψ(p,0) in everyinner iteration can be chosen arbitrarily. In numerical applications, both the outer and inneriterations are alternately iterated until a predefined number of iterations has been taken or whentwo successive flux iterates are close enough in the sense of low difference of successive iterates:

∥∥∥ψ(p) − ψ(p−1)∥∥∥ ≤ ε,

although the closeness criterion ε is usually different for both iteration levels. The norms arechosen such that they represent the fundamental physical quantities, usually the total flux in thecore

‖ψ‖1 =

G∑

g=1

∫

Vz

dz∫ 1

−1dµψg(z, µ) =

G∑

g=1

∫

Vz

dz φg(z) = φ(Vz) (3.18)

or in the outer iteration sometimes the total core fission source3

‖ψ‖ f =

G∑

g′=1

∫

Vz

dz νΣg′f (z)

∫ 1

−1dµ′ Pk(µ′)ψ(z, µ′) =

G∑

g′=1

∫

Vz

dz νΣg′f (z)φg′(z) = N (ψ; Vz), (3.19)

since this is actually the quantity updated at that level by successive flux iterates. It is alsocommon to test the uniform convergence of scalar fluxes to decide whether to stop the iteration,by using the (L∞)G norm:

‖ψ‖∞ = maxg∈1,...,G

[ess sup

z∈Vz

φ(z)], (3.20)

which in the numerical implementation amounts to taking maximum of all the discrete values ofscalar flux over the spatial-energy grid. Note that also the integrals in the previous norms are ina practical numerical scheme converted into sums over spatial points of the discretization.

3Note that this norm is equivalent to other L1(X)G norms by the boundedness of fission cross section.


Summarizing this paragraph, the outer-inner iteration procedure is used to converge to thesolution of the multigroup transport equation (3.14), where, at every outer iteration, fission andup-scattering (if not absent thanks to a carefully chosen group spacing) is determined first andused as a source in a sequence of within-group problems. These are solved one after the other bythe inner iteration (driven by self-scattering), using the most recently calculated fluxes to specifythe down-scattering component of the group source. After all groups are traversed, next outeriteration begins, using the current results to update the fission and up-scattering sources.

Convergence properties. The source iteration may be viewed as a means of finding the fixedpoint of the mapping

ψ 7→ B−1Fψ + Q,

where B = (L − H) and Q = B−1Q. Existence of the fixed point is ensured by the Banach fixedpoint theorem in the subcritical case in which the spectral radius ρ(T) (or more simply ρ) of thematrix operator T = B−1F is less than one (as a consequence of extension of eq. (2.28) to themultigroup setting4). By successive application of the operator T to iteration errors5, one mayfind out that for any suitable angular flux norm,

‖ψ − ψ(p)‖ ≤ ‖Tp‖‖ψ − ψ(0)‖ = ρp‖ψ − ψ(0)‖ 6,

so for any reasonable choice of initial approximation ψ(0), the iteration converges to the fixedpoint ψ. The convergence may however become arbitrarily slow if the spectral radius is close tounity. This is the case of a nearly critical system, which is unfortunately one of the most frequentproblems in nuclear reactor analyses. Also, such case corresponds to optically thick systems withhigh optical path lengths7 τ, combined with scattering ratios only slightly less than unity. Thescattering ratio for neutrons in group g is given by

cg =Σ

gs

Σgt

=Σ

gs

Σgs + Σ

ga

=

∑Gg′=1 Σ

gg′s∑G

g′=1 Σgg′s + Σ

ga

,

and the properties of the medium discussed above essentially imply that the neutrons undergomany scattering collisions before they get finally absorbed or leak out of the system. Similarsituation occurs also with the inner iteration, which also suffers from slow convergence in real-istic reactor problems (see e.g. [AL02]). Various acceleration techniques have been devised toimprove the convergence rate of both the inner and outer iterations and are overviewed in [AL02]or [OA87].

4It is also possible to extend the other results of Sec. 2.3, as described for example in [San06], [The04] or morethoroughly in [DL00].

5This is a standard procedure of finite-dimensional analysis of numerical schemes for iterative solution of linearsystems, see e.g. [MPB06].

6For the last equality, it is necessary to realize that the transport operator with symmetric scattering kernel is anunbounded anti-self adjoint operator and use an appropriate version of the spectral theorem; for details, see [DL00].

7Meaning high collision probability or equivalently low escape probability, see eq. (2.29).


3.2.4 Eigenvalue problem

The second, arguably an even more important problem of steady state neutronic analyses – de-termination of the critical eigenvalue – has the following multigroup operator representation:

(L −H)ψ =

1λ

Fψ in X,

ψ = βψ on ∂X−

with the operator matrices given in the previous subsection. We know from section 2.3.1 that thisgeneralized eigenvalue problem can be rewritten for B = L −H as

λψ = B−1Fψ in X,ψ = βψ on ∂X−

The spectral properties of operator B−1F admit using the simple power method to calculate itsdominant eigenpair. As in the previous case of fixed sources calculation, it is imperative toavoid direct inversion of matrix arising from discretization of B to obtain an efficient numericalscheme. The power method is therefore executed in the following sequence, for an arbitraryinitial estimates of λ(0) and ψ(0):

For p = 1, 2, . . . (outer iterations)

obtain B−1Fψ by solving

Bψ(p) =1

λ(p−1) Fψ(p−1), (3.21)

update the eigenvalue

λ(p) = λ(p−1) ‖ψ(p)‖‖ψ(p−1)‖ , (3.22)

repeat until convergence (to be discussed later).

When used with the ”fission yield” norm of (3.19), a modification of the standard power methodthat is very frequently used in reactor criticality calculations is obtained. The reason why it ispreferred over the standard numerical method ([RVCK95, §30.11]) may be that it is compatiblewith the physical interpretation of the eigenvalue. Indeed, in (3.22) we actually compute

λ(p) =N (p)

1λ(p−1) N

(p−1)=

N (p)

∑

g

∫

Vz

dz1

λ(p−1) Fψ(p−1)≈ N (p)

∑

g

∫

Vz

dz Bψ(p−1),

where the nominator represents neutron production and the denominator net losses of neutronsthrough streaming and absorption and hence the formal definition (2.27) is recovered.

Equation (3.21) makes it clear that at every iteration of the power method, a fixed-sourcetransport problem with scattering and a given fission source is solved. The splitting (3.16) or


(3.17) with the previous eigenvalue estimate in the fission term at the right-hand side is actuallyused to solve the multigroup equations in the group-after-group manner.

Stopping condition. In view of the source iteration for fixed-source problems, the eigenvaluecalculations alter their outer iteration phase so as to take into account the multiplication factorwhen updating the fission sources. Accordingly, a different stopping criterion for the outer itera-tion has to be used instead (or additionally to) monitoring just the difference of successive fissionsources, for instance: ∣∣∣∣∣∣

λ(p) − λ(p−1)

λ(p−1)

∣∣∣∣∣∣ < ε.

Convergence properties. The power iteration is known to converge to the dominant eigenpair

λ(p) → keff , ψ(p) → ψkeff, as p→ ∞,

provided that the problem has appropriate spectral structure (a complete set of eigenfunctions,sequence of decreasing eigenvalues with the first one being strictly larger in magnitude than theothers) and that the initial estimate ψ(0) is chosen properly8 ([RVCK95]). It is also well-knownthat the rate of convergence is governed by the ratio of the two largest eigenvalues, known as thedominance ratio:

κ =|λ2|λ1

= maxn≥2

|λn|λ1,

where the eigenvalues are assumed to be ordered as keff = λ1 > |λ2| ≥ |λ3| ≥ · · · . The convergenceis as κk and, unfortunately again, κ / 1 in nearly critical nuclear reactors with optically thick fuelregions. Some typically used acceleration techniques thus aim to somehow reduce the dominanceratio, like the shifted power iteration method. This method is however very sensitive on problem-dependent parameters which have to be supplied to it so as to make it work ([AL02]).

There are some other possibilities to improve the convergence of eigenvalue calculations.The one that has received large popularity recently is the implicit restarted Arnoldi method,which utilizes as much information from the iteration process as possible at current iteration.The power method (represented by eq. (3.21) and (3.22)) generates successively better approx-imations of the final eigenfunction by actually computing the powers (B−1F)pψ(0). However, atevery step, it throws away the previously generated approximations. The IRAM method, on theother hand, uses the whole Krylov subspace generated by approximations (B−1F)pψ(0) up to thecurrent iteration p in a clever way to construct the best possible approximation from all the avail-able information at the moment. For a detailed theory about the method, see e.g. [Ste01], foran implementation in a transport solver e.g. [WWM+01]. I used this method as a ”black-box”via the MATLAB’s function eigs to replace the custom written source iteration module. Theonly tunable parameter is the size of the maximum Krylov subspace used in the method. Resultsreported in Sec. 4.1.3 indeed corroborate its capability to accelerate the eigenvalue calculation.

8In our numerical experiments, we did not encounter any convergence problems with the simple choice ψ(0) ≡ 1.

3.3 Angular approximation 47

3.3 Angular approximation

In this section, I will describe two mostly used methods that transform the continuous depen-dence of angular flux on neutrons direction of motion Ω, or µ in the azimuthally symmetric case,to a computer-representable form. I will assume that all neutrons share the same energy (theone-speed approximation) to more clearly present the ideas concerning only the angular vari-able. This, however, does not restrict the range of applicability of the described methods, as themultigroup solution technique eventually also reduces into solution of within-group equations,which effectively represent the one-speed situation. Therefore, equation of type (3.17), writtenwithout the unneeded indices as

Lψ(z, µ) = Hψ(z, µ) + S (z, µ), (3.23)

or in an explicit form as

µ∂ψ(z, µ)∂z

+ Σt(z)ψ(z, µ) =

K∑

k=0

2k + 12

ΣskPk(µ)∫ 1

−1dµ′ Pk(µ′)ψ(z, µ′) + S (z, µ) (3.24)

(with approximate scattering anisotropy of order K) will be considered in this section. Thesource term S is assumed as fixed; if eq. (3.23) is solved as part of a fission source iteration, thenit contains the in-scattering contribution as well as external sources and fission sources (possiblymultiplied by the eigenvalue if external sources are missing) contribution. Its precise form insuch case will be given later in Sec. 3.4.3.

To describe both types of boundary conditions, the example setting given by eq. (2.64) willbe used. That is, without the explicit energy dependence,

ψ(zB, µ) = ψin(µ), for 0 < µ ≤ 1,ψ(zT , µ) = α(zT )ψ(zT ,−µ), for −1 ≤ µ < 0.

(3.25)

Although the first method, described in the following section, is not directly used in thedeveloped code, it leads to a computationally very efficient scheme that had for a long timedominated the field of production-oriented neutron transport codes. In my opinion, it thereforedeserves an overview here as well, even more as its implementation and comparison with thecurrent method could be a fruitful direction of future research.

3.3.1 The method of discrete ordinates (SN)

This method is derived upon a basic discretization approach of sampling the continuous indepen-dent variable and evaluating the solution at the generated points. We use this approach to resolvedirectional dependence of angular flux in the (slab) neutron transport equation in which scatter-ing has been represented by spherical harmonic expansion (as suggested at the end of Sec. 2.4.1).


It amounts to picking N discrete directions9 (or ordinates) µnn=1,...,N and assigning to each ofthem a weight wn. The pair µn,wn defines a quadrature set for numerical approximation of theangular integrals appearing in operators H and F:

φk(z) ≡∫ 1

−1dµ′ Pk(µ′)ψ(z, µ′) ≈

N∑

n=1

wnPk(µn)ψ(z, µn).

By using this approximation on the right hand side of the transport equation and then specializingit into the selected directions, we obtain N first-order ordinary differential equations (partialdifferential in more than one dimension)

µndψn(z)

dz+ Σt(z)ψn(z) =

K∑

k=1

2k + 12

Pk(µn)Σsk(z)N∑

m=1

wmPk(µm)ψm(z) + S n(z)

for the N unknown angular fluxes ψn(z) = ψ(z, µn), n = 1, 2, . . . ,N in the selected directions. Theboundary conditions are also evaluated in the selected ordinate directions.

One of the crucial moments in implementing the discrete ordinates method is constructionof the quadrature set. The directions should be chosen to preserve any a priori known propertiesof the flux (like symmetries or directions of preferential streaming); if those are unknown, theangular set should at least be symmetric with respect to µ = 0 and excluding this ordinate. Thisfacilitates specification of boundary conditions and also reduces several difficulties connectedwith the directions perpendicular to principal coordinate axis (i.e. µ = 0), in which the neutronsare assumed to mostly flow ([Sta01, Reu08]). The weights should be positive and fully parti-tion the integration domain10 for numerical reasons, and chosen so that the quadrature exactlyintegrates Legendre polynomials up to N-th order. This last condition is important for not intro-ducing any additional error into the scattering term (and perhaps the anisotropic external sourcesterm) besides its approximation by a truncated Legendre polynomial expansion.

Even if those conditions are satisfied by a chosen quadrature set (often used is the classicalGauss-Legendre set known from other areas of numerical mathematics), inevitable omission ofsome directions leads to inadequate determination of neutron flux received by some regions thatmay be important for the problem at hand (in two or more dimensions). This results in the socalled ray-effects, characterized by non-physical oscillations of solution11. Also in more dimen-sions, satisfaction of all the above conditions is further complicated and requires compromisesto be made – see [OA87, Sec. IV.A] for more details.

After selecting a suitable ordinates set and weights, spatial discretization follows to obtainfinal numerical scheme. This is usually done by dividing the domain into a set of finite intervalsand using the finite volume method to write the equation for each interval in terms of angularfluxes at the ends of the interval (in a classical finite volume jargon numerical fluxes) and anintegral average of angular flux over the interval. Using then a suitable relationship between the

9More exactly direction cosines.10That is,

∑wn =

∣∣∣[−1, 1]∣∣∣ = 2 for the azimuthally symmetric case.

11Physically, this problem is caused by the loss of rotational invariance of the discretized scattering operator.


side and average quantities, the discrete equations are formulated in a way that allows to performthe inner iteration phase (i.e. inversion of the transport operator) by simply sweeping acrossthe domain, using the left boundary condition to start, calculating in-domain angular fluxes inpositive µ > 0 directions (which at their common interfaces link one interval to the other), andrepeating the whole process in the opposite direction using the right boundary condition to start.

Although this solution scheme is quite simply and efficiently programmable, it requires acarefully performed discretization and a choice of numerical flux approximation in order to workproperly. The simple central difference scheme (known among the discrete ordinates communityas the diamond difference scheme) is sufficiently accurate for most problems in reactor physics,but is not unconditionally positive – it may produce negative angular fluxes if the interval spacingis larger than some number depending on the currently computed direction and material prop-erties inside the interval. If one does not want to use an ad-hoc fix-up that sets the negativefluxes to zero, an alternative in the form of an ”upwind” scheme is available, which is howevernot as accurate and does not possess some other desirable properties (like the diffusion limit).Of course, many advanced numerical flux approximations have been devised, most notably thestep characteristic scheme that is based on analytical solution along the characteristics of thetransport equation, ultimately leading to the very promising method of characteristics (MOC)([CKL+07, Cho05]).

The method of discrete ordinates was presented above in the simplest possible slab-geometryform. Although its extension into more complicated (but also more applicable) geometries isconceptually straightforward, it brings many complications as delineated by the discussion aboutthe discrete ordinate sets. In the remaining part of this section, I will describe the method ofspherical harmonics, which, in its original formulation, is even more difficult to use in three di-mensions. Fortunately, since the appearance of Gelbard’s seminal works in early 1960’s (see thereferences in [BL00] or in [LMM95]), we know of a way of how to take advantage of its merits(mainly its correct physical behaviour and high accuracy even with low-order approximation)in multi-dimensional calculations without the overly complicated mathematical formulations. Iwill describe this simplified spherical harmonics method later in section Sec. 3.3.5.

3.3.2 Methods of Galerkin type

The original method of spherical harmonics originates from a broader class of methods that maybe employed for discretization of general boundary value problems – the methods of Galerkintype ([MPB06, Sec. 4.1.1]). In these methods, one looks for the solution of a given boundaryvalue problem in a form of a (for numerical purposes finite) linear combination of convenientlyselected linearly independent functions:

ψ ≈ ψ =

N∑

n=1

ψnun (3.26)

and attempts to determine the unknown coefficients so that the expansion is the best possibleapproximation of the exact solution. In order to arrive at a usable discrete formulation, the


expansion functions unn=1,...,N , by which we try to capture the shape of the exact solution in(3.26), are expected to form a complete orthogonal basis of some finite-dimensional subspaceEN of the (separable Hilbert) space E, in which we look for the solution.

The best possible quality of the approximation is ensured by minimizing the residual

r = (L − H)ψ − S .

There are several approaches that perform this task and allow for convenient realization, e.g. thecollocation method, the least-squares method or the method of weighted residuals. A particularcase of the last one is the classic Galerkin method, in which we minimize the residual by re-quiring that its projection onto the approximation subspace EN vanishes. This may be generallywritten as

Pr = P(L − H)ψ − S

= 0 ⇔ P(L − H)ψ = PS , (3.27)

where P : E → EN is the orthogonal projection operator given by

P f =

N∑

n=1

(un, f )un (3.28)

with the standard inner product on E:

( f , g) =

∫

Ddx f g. (3.29)

Because of the linearity of the transport operator L − H and linear independence of functions un,projection (3.27) leads to a uniquely solvable set of linear algebraic equations for the unknownexpansion coefficients (or moments) ψn:

N∑

n=1

Amnψn = S m, Amn = ((L − H)un, um) , S m = (S , um), m = 1, . . . ,N. (3.30)

3.3.3 The method of spherical harmonics (PN)

When we apply the classic Galerkin method to approximate the variation of angular flux withrespect to Ω, we have E = L2(S) and a suitable choice for EN is the collection of its rotation-ally invariant subspaces Λl for l ≤ N as introduced by (2.45), i.e. a functional space spannedby spherical harmonics up to degree N. By this choice, we get the appealing physical proper-ties and useful mathematical relations of spherical harmonics presented in Sec. 2.4, notably theconvergence ψ→ ψ as N → ∞. Even with the rich mathematical apparatus at our disposal, how-ever, the system of equations (3.30) arising from the projection is quite involved (note that theequations will be complex in nature), particularly at higher dimensions (recall that the number ofLaplace expansion moments and hence the equations grows like (N + 1)2 in three dimensions).


You may consult e.g. [Sta01, Sec. 9.2] for a general form or the recent thesis [Smi09] for explicitexamples of multidimensional spherical harmonics equations. Therefore, the spherical harmon-ics method has not gained a widespread popularity as a production-oriented, three-dimensionalcore calculation method and has been used primarily as a benchmark and analytical tool or as acomplement to other methods (e.g. to mitigate the ray-effects in SN methods). However, as earlyas in 1970’s, Fletcher introduced a spherical harmonics program that found its use in the ORNLcode package MARC-PN ([RSI81]). For a more recent implementation, see e.g. [CTGV08].

Let us now return to our simplified one dimensional setting with azimuthal symmetry and seewhat the method of spherical harmonics looks like there. Now, E = L2([−1, 1]) and the chosensubspace EN becomes the space PN of Legendre polynomials up to order N 12, as discussed inSec. 2.5. Expansion (3.26) becomes (I omit the tilde for simpler notation)

ψ(z, µ) =

N∑

n=0

2n + 12

φn(z)Pn(µ), with φn(z) =

∫ 1

−1dµ Pn(µ)ψ(z, µ). (3.31)

By inserting the expansion into eq. (3.24):

N∑

n=0

2n + 12

µPn(µ)dφn(z)

dz+Σt(z)

N∑

n=0

2n + 12

φn(z)Pn(µ) =

K∑

k=0

2k + 12

ΣskPk(µ)φk(z)+S (z, µ) (3.32)

and using the recurrence relation (2.36) of Legendre polynomials to transform the first term13

N∑

n=0

2n + 12

µPn(µ)dφn(z)

dz=

N∑

n=0

n + 12

Pn+1(µ)dφn(z)

dz+

N∑

n=0

n2

Pn−1(µ)dφn(z)

dz, (3.33)

we obtain the transport equation in a convenient form for projection onto the space PN as givenby eqns. (3.27) through (3.29). It leads to a system of N + 1 equations of the form (3.30), that weobtain by multiplying eq. (3.32) (with (3.33)) by Pm(µ), m = 0, 1, 2, . . . ,N, and integrating over−1 ≤ µ ≤ 1. Using the orthogonality relation of Legendre polynomials (eq. 2.37), we finallyarrive at the system of N + 1 first-order ordinary differential equations

n + 12n + 1

dφn+1(z)dz

+n

2n + 1dφn−1(z)

dz+

[Σt(z) − Σsn(z)

]φn(z) = S n(z),

n = 0, 1, . . . ,N,(3.34)

which are called spherical harmonics equations (like the multidimensional equations), shortlyPN equations. The scattering and source expansion coefficients (moments) are defined by14

S n(z) =

∫ 1

−1dµ Pn(µ)S (z, µ), Σsn(z) = 2π

∫ 1

−1dµ0 Pn(µ0)Σs(z, µ0) (3.35)

12More precisely, it is the space EN+1, counting the first Legendre polynomial P0.13The P−1(µ) term in the last sum is to be ignored.14The factor 2π by the scattering moment originates from the form of its expansion (eq. 2.35).


The terms with out-of-range indices are set to 0, which is only a notational convenience in thecase of φ−1, but represents an approximation in the case of φN+1

15, that is nevertheless required toclose the system. Note that in the spherical harmonics method, the angular representation of thestreaming operator L is consistent with that of the scattering operator H and allows to simplifythe third term of the PN equations by defining the absorption moments (corresponding to theabsorption cross-section Σa = Σt − Σs):

Σan(z) = Σt(z) − Σsn(z). (3.36)

It also seems to effectively reduce the order K of scattering anisotropy to that of the expansionby taking into account only the first N Legendre moments of the scattering cross-section – notehowever, that the Legendre fitting of the cross-section from which the first N moments are takenhas been performed with the full desired order K.

Properties of the PN representation

The above equations may be solved even analytically, providing a useful insight about the struc-ture of the approximation ([SM81, Sec. II.B]) – for example, the reason for preferring the odd-order PN expansion (see also [Smi09, Sec. 3.4]).

The angular dependence is in the PN equations contained in the flux expansion coefficientsand becomes thus in a sense a property of the functional space, whose elements are guaranteedto possess the desired physical characteristics. The problems of choosing a right discretizationscheme for the angular variable, typical to SN methods, are therefore elegantly avoided. More-over, thanks to the convergent behaviour of spherical harmonics, a better angular approximationis obtained by simply enlarging the projection subspace.

However, in contrast to the experimentally observed preferential motion of neutrons in thedirection of principal coordinate axes, the expansion contains components in perpendicular di-rections and the rate of its convergence may deteriorate. This problem is even aggravated atmaterial interfaces and boundaries, since there (except for a perfect reflection) the angular flux isa discontinuous function of µ at µ = 0 and the approximation ψ, continuous over the whole range−1 ≤ µ ≤ 1, is unsatisfactory. This is often solved simply by increasing the expansion order or,more elegantly, by performing the expansion separately for the forward (µ > 0) and backward(µ < 0) directions, using the half-angle Legendre polynomials Pn(2µ∓ 1) for the basis (the resultis sometimes called method of hemispherical harmonics or double PN method, see [Sta01, Sec.9.6] for more details).

As a final note, in one dimension, equivalency of the PN−1 and SN16 equations may be proved

if the Gauss-Legendre quadrature is used in the latter ([Sta01, Sec. 9.7]) – this motivated theextension of this idea to more dimensions and utilizing of the spherical harmonics method to

15One can imagine that we are assuming some behaviour of the solution ”beyond” the target space of its projection– see [Cul01] for some non-trivial conclusions that may be drawn from this assumption.

16Note that for odd-order PN equations, there is an even number of equivalent ordinates, convenient for formulat-ing the discrete ordinates method as discussed in Sec. 3.3.1.


produce ”physically correct” quadrature sets or other modifications for the computationally moreefficient discrete ordinates methods ([OA87]).

Interpretation of flux expansion coefficients

Like the scattering cross-sections in Sec. 2.4.1, the first two Legendre moments allow a directphysical interpretation as the scalar neutron flux and the z-directed net neutron current17:

φ0(z, E) =

∫ 1

−1dµ P0(µ)ψ(z, E, µ) =

∫ 1

−1dµψ(z, E, µ) · 1

2π

∫ 2π

0dϕ

=

∫ 2π

0dϕ

∫ π

0sinϑdϑ

ψ(z, E, cosϑ)2π

=

∫

4πdΩψ(r, E,Ω) = φ(r, E)

and

φ1(z) =

∫ 1

−1dµ P1(µ)ψ(z, µ) =

∫ 1

−1dµ µψ(z, µ) · 1

2π

∫ 2π

0dϕ

=

∫ 2π

0dϕ

∫ π

0sinϑdϑ cosϑ

ψ(z, cosϑ)2π

=

∫

4πdΩ (Ω · ez)ψ(r,Ω) = jz(r).

This motivates the extension of the notion of current to higher-order angular moments, leadingto the definition of net PN currents:

jn(z) ≡ φn(z) for n = 1, 3, 5, . . .

The PN moments of the upward and downward oriented partial currents (or shortly PN partialcurrents), respectively, are obtained by a hemispherical projection of angular flux for odd n:

j+n (z) =

∫ 1

0dµ Pn(|µ|)ψ(z, µ), resp. j−n (z) =

∫ 0

−1dµ Pn(|µ|)ψ(z, µ); (3.37)

the first partial current moment corresponds to the real partial current from eq. (2.10), and alsothe relationship

jn = j+n − j−n

holds for all odd n. Note that this common definition of slab partial currents changes the originalmeaning of the three dimensional partial currents (2.10), since they are defined with respect tothe positive z-axis instead of the outward normal. Hence, at the bottom, j+n will be the incomingcurrent while at the top, it will represent the exiting neutron current.

17The same holds also for the first two Laplace coefficients in the spherical harmonics method in a general two-angle coordinate system. in accord with their respective definition relations (2.7) and (2.9).


Boundary and interface conditions

As the true boundary conditions (3.25) entail discontinuity of angular flux at µ = 0, they cannotbe satisfied exactly by the continuous finite expansion of angular flux. The two most often usedapproximations are due to Mark and Marshak.

In the so-called Mark boundary conditions, we require that the approximate angular fluxexpansion (3.31) satisfies the bottom boundary condition at the (N +1)/2 discrete values (in viewof Sec. 3.3.1: discrete ordinates)

µn : µn > 0 ∧ PN+1(µn) = 0

and the top boundary condition

forµn : µn < 0 ∧ PN+1(µn) = 0

. These conditions were developed from an analytical solution

of a model problem of source-free, purely absorbing medium and have been shown ([SM81]) toperform well in similar situations.

Except for a few places, the real reactor is usually a weakly absorbing environment with a lotof scattering. Therefore, the other Marshak conditions are used more frequently in core calcu-lations and have been shown to be generally more accurate than the Mark conditions ([Sta01]).They are naturally obtained in the framework of Galerkin methodology as weak conditions (sat-isfied in an integral sense) by projecting the exact boundary conditions onto the same space asthe angular flux (in the case of interface conditions) or onto the ”half-space” to obtain a correctnumber of boundary conditions. This latter operation amounts to multiplication by odd Legendrepolynomials (based on the fact that only the odd Legendre polynomials represent ”directionality”as they attain different values for µ and −µ) and integrating in the range of entering directions(i.e. over ∂X−). Specifically, Marshak approximation of conditions (3.25) becomes

∫ 1

0dµ Pm(µ)

N∑

n=0

2n + 12

Pn(µ)φn(zB) =

∫ 1

0dµ Pm(|µ|)ψin(zB, µ),

∫ 0

−1dµ Pm(|µ|)

N∑

n=0

2n + 12

Pn(µ)φn(zT ) = α(zT )∫ 0

−1dµ Pm(|µ|)

N∑

n=0

2n + 12

Pn(−µ)φn(zT ),

(3.38)

where m = 1, 3, . . . ,N. In terms of the PN partial currents (3.37), these conditions read:

j+m(zB) = j+in,m(zB), j−m(zT ) = α(zT ) j+m(zT ), m = 1, 3, . . . ,N. (3.39)

(the latter was obtained by substituting µ → −µ in the last integral in (3.38). In the first, j+in,mrepresents the PN moments of the incoming current through the bottom boundary, completelyspecified by the prescribed incoming flux ψin. Boundary adjacent to a vacuum is characterizedby ψin = 0. This condition is also equivalent to the albedo boundary condition with α = 0. Forα = 1, i.e. specular symmetry conditions, we conclude from the second equality in (3.38), byusing parity properties of Legendre polynomials and their completeness, that φn must vanish forall odd n. This is the obvious condition for specular symmetry ψ(zT , µ) = −ψ(zT ,−µ) .

Marshak interface conditions are obtained in the same way, leading because of the complete-ness of the Legendre system to the requirement of spatial continuity of all flux moments acrossthe interfaces (which includes the continuity of real scalar fluxes and net currents).


Special cases of PN equations

The two following sets of PN equations will be studied more closely:

• P1 equationsIf N = 1, i.e. we assume that angular flux is at most linearly anisotropic, then

ψ(z, µ) ≈ ψ(z, µ) =

1∑

n=0

2n + 12

Pn(µ)φn(z) =φ0(z)

2+

32µφ1(z), (3.40)

and the spherical harmonic equations (3.34) become the P1 equations:

dφ1(z)dz

+ Σa0(z)φ0(z) = S 0(z),

13

dφ0(z)dz

+ Σa1(z)φ1(z) = S 1(z),(3.41)

with boundary conditions following from Marshak conditions (3.38), using φ1 ≡ j1:

φ0(zB)4

+j1(zB)

2= j+in(zB), j1(zT ) = γφ0(zT ), γ =

1 − α2(1 + α)

, (3.42)

where γ is the albedo coefficient defined in [Han07, eq. (3.40)].

• P3 equationsFor N = 3, we have

ψ(z, µ) ≈ φ0(z)2

+32µφ1(z) +

54

(−1 + 3µ2

)φ2(z) +

74

(−3µ + 5µ3

)φ3(z)

and the PN equations become

dφ1(z)dz

+ Σa0(z)φ0(z) = S 0(z),

13

dφ0(z)dz

+23

dφ2(z)dz

+ Σa1(z)φ1(z) = S 1(z),

25

dφ1(z)dz

+35

dφ3(z)dz

+ Σa2(z)φ2(z) = S 2(z),

37

dφ2(z)dz

+ Σa3(z)φ3(z) = S 3(z),

(3.43)

subject to boundary conditions (with γ defined as in the P1 case)

φ0(zB)4

+j1(zB)

2+

5φ2(zB)16

= j+in,1(zB), j1(zT ) = γφ0(zT ) +5γ4φ2(zT ),

−φ0(zB)16

+j3(zB)

2+

5φ2(zB)16

= j+in,3(zB), j3(zT ) = −γ4φ0(zT ) +

5γ4φ2(zT ).

(3.44)


3.3.4 Passage to diffusion approximation

Expanding the absorption moments according to (3.36), interpreting the second scattering mo-ments as in eq. (2.40): Σs1 = µ0Σs, and assuming that the source is isotropic18, the second of theP1 equations (3.41) yields

φ1(z) = − 13[Σt(z) − µ0Σs(z)

] dφ0(z)dz

, (3.45)

which is the Fick’s law of diffusion with the diffusion coefficient

D(z) =1

3[Σt(z) − µ0Σs(z)

] ≡ 13Σtr(z)

, (3.46)

whereΣtr = Σt(z) − µ0Σs(z) (3.47)

is the commonly introduced mono-kinetic transport cross-section. If scattering is isotropic, thenµ0 = 0 and

D(z) =1

3Σt(z).

Under the additional assumption that Σt ≡ Σa + Σs ≈ Σs (a weak absorption dominated byscattering), this definition of diffusion coefficient becomes equivalent to relation (3.34) in myformer thesis [Han07], where it had been derived from physical principles and eventually gaverise to the primary subject of the thesis – the diffusion equation [Han07, eq. 3.46]. To obtain thisequation in the current framework, it is only necessary to use the relation (3.45) in the first of theP1 equations (assuming sufficient differentiability of the first moment):

− ddz

[D(z)

dφ0(z)dz

]+ Σa(z)φ0(z) = S 0(z).

The main assumptions leading towards diffusion theory are therefore that the angular flux is only”slightly” anisotropic (so that its directional variation can be described by a linear function in µas in eq. (3.40)) and that the sources have no linearly anisotropic component.

3.3.5 Simplified spherical harmonics method (SPN)

The procedure by which the P1 equations were converted into the diffusion equation in the previ-ous subsection, i.e. solving the odd-order equations for the odd-order flux moments in terms of agradient of the even-order flux moments and using the result to eliminate the odd-order flux mo-ments from the even-order equations, can be successfully applied to a PN approximation of anyodd order. This is called a simplified spherical harmonics approximation with the abbreviation

18That means the contribution from external sources is isotropic as fission is mostly an isotropic process asdiscussed in Sec. 2.4.2.


SPN . The SP1 equation is thus the familiar diffusion equation and generally, the SPN equationsare a reformulation of the N + 1 equations of the PN approximation into a system of (N + 1)/2diffusion-like equations (containing second-order spatial derivatives of unknowns).

In particular, the SP3 equations are derived as follows. First, as in the case of diffusionapproximation, we assume that sources are isotropic, i.e.

S n(z) ≡ 0 for n ≥ 1. (3.48)

Straightforward derivation including higher degrees of sources anisotropy introduces first-orderderivatives into the resulting equations and hence complicates their numerical solution, particu-larly in the multigroup case. Workarounds to this issue exist, however, and may be used whensuch behaviour of neutron sources is important (see [Kot07]).

From the second and fourth equations of (3.43), the odd-order expansion moments can beexpressed in terms of the even-order moments as

φ1(z) = − 13Σa1(z)

ddz

[φ0(z) + 2φ2(z)

],

φ3(z) = − 37Σa3(z)

ddzφ2(z).

(3.49)

Defining the SP3 flux moments

Φ0(z) = φ0(z) + 2φ2(z), Φ2(z) = φ2(z) (3.50)

and SP3 ”diffusion” coefficients

D1(z) =1

3Σa1(z)=

13[Σt(z) − Σs1(z)

] ,

D3(z) =3

7Σa3(z)=

37[Σt(z) − Σs3(z)

] ,(3.51)

equations (3.49) become the SP3 generalization of Fick’s law, defining the SP3 currents (i.e., oddSP3 moments) J1 and J3 (note their equivalency with the P3 currents):

j1(z) ≡ φ1(z) = −D1(z)dΦ0(z)

dz≡ J1(z),

j3(z) ≡ φ3(z) = −D3(z)dΦ2(z)

dz≡ J3(z).

(3.52)

When used in the first and third equations of (3.43) and the zeroth angular flux moment (i.e.the scalar neutron flux) is expressed from (3.50) in terms of the SP3 moments:

φ(z) ≡ φ0(z) = Φ0(z) − 2Φ2(z), (3.53)


the SP3 equations are obtained19:

− ddz

[D1(z)

dΦ0(z)dz

]+ Σa0(z)Φ0(z) − 2Σa0(z)Φ2(z) = S 0(z),

− ddz

[D3(z)

dΦ2(z)dz

]+

[43

Σa0(z) +53

Σa2(z)]Φ2(z) − 2

3Σa0(z)Φ0(z) = −2

3S 0(z),

(3.54)

where S 0 contains the same elements as the original source S 0, only expressed in terms of theSP3 moments. Note that this system of two second-order ordinary differential equations is cou-pled only through the SP3 unknowns (and not their derivatives) which facilitates its numericalsolution.

Appropriate conditions for the SP3 moments are the Marshak P3 boundary conditions (3.44),written in terms of the SP3 moments as:

Φ0(zB)4

+J1(zB)

2− 3Φ2(zB)

16= j+in,1(zB), J1(zT ) =

γ

4

[4Φ0(zT ) − 3Φ2(zT )

],

−Φ0(zB)16

+J3(zB)

2+

7Φ2(zB)16

= j+in,3(zB), J3(zT ) = −γ4

[Φ0(zT ) − 7Φ2(zT )

],

(3.55)

and continuity of the SP3 moments and currents at the interfaces.

3.3.6 Three-dimensional SPN approximation

N

In this section 3.3.6, bold letters will denote matrices and vectors arising from the SP3 approxima-tion of continuous angular dependence and should by no means be interpreted as in the previoussection 3.2, i.e. as matrices and vectors of group-discretized operators or variables. In order todistinguish the traditional Euclidean vectors and matrices over R, these will be decorated by adouble (for matrices) or single (for vectors) underline.

When compared to the PN approximation, the SPN equations, apart from their lower num-ber, are also amenable to more efficient numerical solution by diffusion based solvers, withoutabandoning the higher-order transport physics. As nice as the last sentence sounds, however, itis problematic to realize it in practical calculations which are almost always multidimensional.When the general, two-angle dependent, spherical harmonics equations are reformulated intoa system of second-order partial differential equations, they become even more complicated([BL00]). Their number also grows quadratically with the order of approximation N as op-posed to linear growth in planar geometry. A natural question thus arose: ”Is there any wayto extend the slab-geometry SPN formulation into more dimensions without losing the structure

19 Note that some authors define the SP3 diffusion coefficients differently – e.g. to obtain Beckert’s, [BG08], or

Brantley’s, [BL00], definition, the following transformation has to be made: D1 → D0, D3 → 35

D2.


of the equations?” It was answered for the first time by Gelbard in 1960 (see the references in[BL00, LMM95]).

To transform the planar equations into more dimensions (I will consider three for concrete-ness), Gelbard attempted to utilize the formal correspondence betweenone-dimensional differentiation and its three-dimensional analogue, realized by the gradientoperator ∇. Consider first the effect of formally replacing d

dz → ∇ on the P1 approximation.Equation (3.45) becomes the standard three-dimensional Fick’s law, j = −D∇φ. Note that thistransformation changed the first angular moment φ1 into a vector function with values in R3.Therefore, in order to use this vector in the first P1 equation of (3.41) to formally obtain the SP1

approximation, we need to replace the ddz operator in that equation by the divergence operator ∇·

that operates on vectors. In this way, we eventually arrive at the three-dimensional SP1 equation,equivalent to the classical three-dimensional neutron diffusion equation. Boundary conditionsare derived by replacing φ1

∣∣∣∂Vz

with n · j∣∣∣∂V

.

Motivated by the success of this formal approach in obtaining the three-dimensional SP1

equation, Gelbard proposed to use the same simple idea for the higher order PN equations toobtain corresponding second-order SPN formulations. That is, replace

• z→ r in all problem’s parameters (cross-sections, fission spectra, etc.)

• ddz → ∇· in those 1D equations that appear at even positions in the PN system,

• ddz → ∇ in those 1D equations that appear at odd positions in the PN system,

• scalar function φn : Vz ⊂ R1 → R1 for even n with a scalar field φn : V ⊂ R3 → R1,

• scalar function φn : Vz → R1 for odd n with a vector field φn ≡ jn

: V → R3,

• φn(z)∣∣∣z∈∂Vz→ n · φn(r)

∣∣∣r∈∂V

for odd n (n . . . unit outward normal to ∂V at r)

(all vectors are implicitly treated as column vectors, except for n, which is assumed to be 1 × 3).

Note that the last item implies that at the interfaces, transformation from the local one-dimensional reference frame to the global three-dimensional one is done in direction of the out-ward normal. This means that the outgoing SP3 partial currents in the global three-dimensionalframe will always correspond to the upward (positive z) directed SP3 partial currents J+

n in thelocal one-dimensional frame and likewise the incoming three-dimensional SP3 partial currentswill always correspond to the downward (negative z) directed one-dimensional SP3 partial cur-rents J−n . Consequently, both 1D SP3 boundary conditions that are to be extended to the global3D system will have the form of top boundary conditions, which amounts to changing in (3.55)the plus sign at the SP3 currents and in the superscript of the prescribed incoming SP3 currentsto the minus sign as these conditions are derived from j−m(zT ) = j−in,m(zT ) instead of the first eq. in(3.39). Also note that the prescribed currents are azimuthally extended (cf. eq. (2.56)) as:

j−in,n(r) =

∫ 0

−1dµ Pn(|µ|)

∫ 2π

0dϕψin(r, µ, ϕ).

Using now a more succinct matrix notation and dropping spatial dependence (every following


variable depends on r), the generalized SP3 Fick’s law becomes

J ≡[

J1

J3

]= −

[D1 00 D3

]·[ ∇Φ0

∇Φ2

]≡ −D∇Φ, (3.56)

where Φ =[Φ0,Φ2

]T is the vector of even SP3 angular moments and the gradient (as wellas divergence) of a vector (as well as matrix) is conventionally taken element-wise. The SP3

equations (3.54) then become:[ −∇ · D1∇ + Σa0 −2Σa0

− 23Σa0 −∇ · D3∇ + Σa2

]·[

Φ0

Φ2

]=

[1−2

3

]S 0, (3.57)

with Σa2 =43

Σa0 +53

Σa2, or equivalently

−∇ · D∇Φ + ΣaΦ = S3, (3.58)

where

D =

[D1 00 D3

], Σa =

Σa0 −2Σa0

− 23Σa0

43Σa0 + 5

3Σa2

, S3 =

[1− 2

3

]S 0 (3.59)

(index at the source term will be explained in a moment).

They are supplemented with the conditions of Marshak form (3.55) (with the minus sign atthe SP3 currents as explained above):

[n oo n

]·[

J1

J3

]=

[1/2 −3/8−1/8 7/8

] [Φ0

Φ2

]− 2

[j−in,1j−in,3

]

at the inhomogeneous boundary (r ∈ ∂V0)

[n oo n

]·[

J1

J3

]=γ

4

[4 −3−1 7

] [Φ0

Φ2

]

(o is the 1×3 vector of zeros) at the homogeneous boundary (r ∈ ∂Vh). Defining the 2×6 matrix

N =

[n oo n

](3.60)

and vector j−in =[2 j−in,1, 2 j−in,3

]Tof prescribed incoming SP3 currents, the generalized Fick’s law

allow us to rewrite the three-dimensional SP3 boundary conditions in the final compact form:

N · D∇Φ + CΦ = j−in, C =

[1/2 −3/8−1/8 7/8

], at ∂V0, (3.61a)

N · D∇Φ + γGΦ = o, G =

[1 −3/4

−1/4 7/4

], at ∂Vh, (3.61b)


which are, respectively, the inhomogeneous and homogeneous Robin boundary conditions forthe SP3 moments. Analogically, the conditions at the inner interfaces of dissimilar materials are

Φ∣∣∣∂V→ = Φ

∣∣∣∂V←, and

(N · J)

∣∣∣∂V→ =

(N · J)

∣∣∣∂V←. (3.61c)

All the steps performed on previous lines are summarized by the operator matrix formulation:

B3Φ = S3, where B3 = −∇ · D∇ + Σa, (3.62)

matrices D, Σa and vector S3 are defined by (3.59) and boundary and interface conditions aregiven by (3.61). The subscript index 3 corresponds to the SP3 approximation. Note also, that thescattering part H3 of the operator B3 ≡ L3 −H3 is contained in Σa.

In the same way, the SP1, or diffusion model, could be represented as

B1Φ0 ≡ B1φ = S , where B1 = −∇ · D1∇ + Σa, (3.63)

where the diffusion coefficient is defined in (3.51) (or (3.46)).

As a final remark, the procedure leading to three-dimensional SP3 equations (3.62) couldbe readily extended to higher order SPN approximations. Kotiluoto [Kot07] or Montagnini[CCMR02], for example, present the coefficients of the SPN equations for general N. How-ever, numerical evidence shows that the SP3 equations are usually sufficiently accurate for widerange of practical problems (more on that in the discussion at the end of this subsection) and theyallow an illustrative derivation suitable for this thesis.

3.3.7 Theoretical analysis of the 3D SPN approximation

Over the first thirty years that followed Gelbard’s initial idea, numerous practical calculationresults have proven it right. Only at the beginning of 1990’s, however, first mathematical analysesof the completely formal approach have been performed in order to settle the (rightful) disputesabout theoretical basis of the method. The two most influential approaches that strengthened thetheoretical foundations of the SPN approximation are the asymptotic analysis and the variationalanalysis. Both allow mathematically precise derivation of the SPN equations, but differ in whatadditional information do they provide about the approximation.

Overview of asymptotic analysis

The asymptotic analysis, pioneered by Edward Larsen ([Ada04]), is based on the observation ofthe behaviour of a general problem as it approaches a particular physical limit that is of greaterinterest. To this end, a small fudge factor, ε 1, is introduced into the parameters of theproblem (e.g. the cross-sections) so that in the limit ε → 0 the desired special case emerges.In his early work, Larsen demonstrated that in an interior of optically thick (large optical path


length (2.29), or equivalently small mean free path MFP = 1/Σ = O(ε)20 and highly scattering(Σt ≈ Σs) medium, the diffusion equation (or P1 approximation) asymptotically approaches thetransport equation (for a sketchy but very illustrative explanation, see [Ada04]). Later, Larsentogether with Morel and McGhee ([LMM95]) generalized the result by proving that, under theassumptions

1. Σt = O(ε−1) (⇒ MFP = O(ε)) – an optically thick system,

2. that probability of absorption is small (of order ε2)

3. scattering is not highly forward-peaked21,

leading to a collection of fixed source transport problems symbolically written as22

(Bψ

)(·; ε) ≡ (

(L − H)ψ)(·; ε) = Q(·; ε),

where dependence of the operators (and consequently the solution) signifies the scaling of the ac-tual problem parameters by ε, the following holds (using the notation for approximate operatorsfrom previous subsection):

•(Bψ

)(·; ε) =

(B1φ

)(·; ε) + O(ε3)

•(Bψ

)(·; ε) =

[(B3Φ

)(·; ε)

]1

+ O(ε7) and(Bψ

)(·; ε) =

[(B3Φ

)(·; ε)

]2

+ O(ε7), provided that

4. the system is homogeneous or the transport solution shows a nearlyone-dimensional behaviour in the vicinity of interfaces of different materials by hav-ing there sufficiently weak tangential directional derivatives.

By the analogy with traditional numerical analysis of approximations of a continuous problemby difference schemes, we might say that the SP1 (i.e., diffusion theory) is a physically consistentapproximation, of order O(ε3), of the multigroup neutron transport problem in a physical regimedefined by the conditions 1 – 3 above. Under the additional assumption 4, the SP3 equationsare the higher-order (O(ε7)) approximations of the transport problem and may be viewed asasymptotic corrections of the diffusion model.

Overview of variational analysis

The largest merit of the variational analysis lies in revealing the right interface and outer bound-ary conditions during the derivation. The seminal paper in this area has been written by Brantleyand Larsen, [BL00], who derived the multigroup SP3 equations as approximate Euler-Lagrangeequations in the space of trial functions spanned by the same angular basis as used in the fullP3 approximation. The functional that is minimized by their solution characterizes deviations

20In this context, I understand the symbolism f (x; ε) = O(ε) as that there exists a bounded function C(x) suchthat | f (x)| ≤ |C(x)| ε; for a classical definition of the O symbol, see [RVCK95, Def. 11.4.6].

21This can be equivalently expressed by the requirement that the mean scattering cosine is not close to unity.22The authors actually proved the results in a multigroup setting.


from an arbitrary reaction rate (eq. (2.12) in multigroup approximation). Marshak boundary andinterface conditions of type (3.38) are derived along with the equations and they together definea practical method that is in the above variational sense optimal for calculating the reaction rates.The derivation relies on an assumption of one-dimensional behaviour at internal interfaces andboundaries, which is consistent with the asymptotic derivation.

Comments

Both the asymptotic and variational analyses provided valuable information about the theoreticalproperties of the SPN approximation and thus increased confidence in its practical usage. Con-clusions drawn from recent numerical experimenting have been very helpful in deciding aboutwhich transport method to implement in the current diffusion solver to improve its accuracy.

The ”diffusive” physical regime characterized by the conditions 1 through 4 above is rea-sonably well attained in real nuclear reactors. Then, argumenting by the asymptotic analysis,the diffusion equation is expected to adequately model the neutron transport processes and thissecured its firm position in practical core-calculation codes. However, by the same argument,more accuracy can be gained by using the SPN equations and indeed, this is what the researchersexperimenting numerically with SPN approximations generally observed. However, as the as-sumptions under which the asymptotic analysis is valid indicate, the SPN approximation is prob-ably not a right choice for problems with strong multidimensional transport effects and complexspatial heterogeneities, although some numerical results show that it is not always the case (seethe discussion in [BL00]). Anyway, improvement over diffusion theory is naturally expected alsofor such problems.

Downar [Dow05] compared the S16, SP3 and diffusion approximations over several modelproblems, with results that the SP3 method well agrees with the high-order transport solution ofthe S16 method and provides more than 80% improvement in reactor critical number and 50% to30% improvement in pin powers over the diffusion approximation. Somewhat smaller but stillwell-noticeable improvement has been obtained by Brantley and Larsen in [BL00]. Similarlyto Downar and others, however, they conclude that SP3 captures most of the transport effects indiffusive regimes of nuclear reactors (and that higher orders than 3 are not usually necessary, asalso shown e.g. by Cho et al. in [CKL+07]).

The authors also mention the generally observed computational efficiency of the method,which is for low orders much greater than that of solving the corresponding SN or PN equationsand only twice or three-times worse than that of solving the diffusion problem. They also warn,however, that more careful spatial discretization than in the diffusion methods is required in orderto capture the sharper boundary layer behaviour of the more transport-like SP3 approximation.Employing the SP3 approximation within the higher-accuracy nodal methods would thereforebe highly desirable. Beckert and Grundmann ([BG08]) or Downar developed and tested thenodal SP3 methods for rectangular geometries, Kim et al. ([KYHK09]) recently implemented ahexagonal SP3 solver and. Again, they conjointly reported good improvements over the diffusiontheory.

3.4 Multigroup 3D SP3 approximation 64

3.4 Multigroup 3D SP3 approximation

In this last section of Chap. 3, the energetic approximation of the neutron transport equationvia the multigroup theory and its angular approximation via the simplified spherical harmonicsmethod of third Legendre order, developed in previous sections, will be collected together.

It has been already shown that the mono-kinetic SPN equations can be straightforwardlyderived from the mono-kinetic PN equations (3.32) under the assumption of zero higher-orderLegendre moments of sources. In order to use the multigroup scheme to include energy transfer,specification of the multigroup physical parameters such as cross-sections or fission spectra isrequired. This has been discussed in Sec. 3.2, although the question of the angular dependenceintroduced into multigroup total cross-section and non-directional albedo coefficient has been leftopen. A natural answer is now provided in the framework of spherical harmonics methods andwill be presented below. In addition, multigroup SPN methods require specification of multigroupgeneralized diffusion coefficients, which is a highly non-trivial task (see e.g. [Sta01, Chap. 10]). Iwill only mention one difficulty connected with anisotropic group-to-group scattering and, basedon its resolution, define the SP3 diffusion coefficients in one of the many possible ways.

A practical procedure for solving the resulting equations will be given at the end. The lastobstacle before its transformation into a computer program – the spatial discretization – will beaddressed in the following Chapter 4.

3.4.1 Multigroup total collision cross-section

The one-dimensional PN equations can be written in the multigroup form analogically to eq.(3.8). Using the group variables (3.5), (3.6), (3.7) and the last two of (3.9), it reads

n + 12n + 1

dφgn+1

dz+

n2n + 1

dφgn−1

dz+ Σ

gt φ

gn = Σgg

snφgn + S g

n, n ≥ 0, (3.64)

where the group source moments are defined (assuming isotropic fission and external sources,using (3.35) and orthogonality of Pn) as

S g0 =

G∑

g′=1,g′,g

Σgg′s0 φ

g′0 + χ

gf

G∑

g′=1

νΣg′f φ

g′0 + Qg, or S g

0 =

G∑

g′=1,g′,g

Σgg′s0 φ

g′0 +

χgf

λ

G∑

g′=1

νΣg′f φ

g′0 ,

S gn =

G∑

g′=1,g′,g

Σgg′sn φ

g′n , n ≥ 1.

(3.65)

Note that I have separated from the group sources the self-scattering part, which will prove usefullater. At this moment, notice that in order to exactly satisfy the group-integrated continuousenergy PN equation, group moments of the total cross-section would have to be defined as

Σgtn(z) =

∫g

dE Σt(z, E)φn(z, E)

φgn(z)

=

∫g

dE Σt(z, E)φn(z, E)∫

gdE φn(z, E)

, n ≥ 0. (3.66)


In three dimensions, z would be replaced by r as before. These energy-wise constant total cross-sections could be obtained for each grup by the same approach as the other group constants,described in Sec. 3.2.1.

Instead of introducing new total cross-section in every moment equation, however, a simpleflux-weighted parameter is used in all the relevant publications available to me at the moment ofwriting the thesis ([BG08, BL00, Dow05, KYHK09]):

Σgtn = Σ

gt0 ≡ Σ

gt , n ≥ 0.

This is not only convenient for implementation, but also avoids possible problems arising fromthe fact that angular flux moments are not necessarily positive and hence could make the integralin the denominator of (3.66) close to zero ([San09]). This, of course, applies also to the inter-group scattering cross-sections.

3.4.2 Multigroup boundary conditions

Similar considerations apply to the multigroup albedo coefficient, which should be correctlyweighted by the partial current moments. Description of the theoretical aspects of obtainingthe multigroup albedo coefficients for transport theory goes, however, beyond the scope of thisthesis. Therefore, I will assume they are given, via the multigroup albedo matrix β of eq. (3.14).

This matrix shows that the albedo condition produces a group-to-group coupling, although– being lower triangular – only from lower groups to the higher. Nevertheless, this coupling isinconvenient for a simple implementation (yielding additional terms in the SPN prescription foralbedo conditions, eq. (3.61b), in higher groups) and will not be taken into account, as it is alsonot in boundary input data of any of the available benchmark and validation problems.

The complete multigroup boundary conditions are then specified for g = 1, . . . ,G as:

N · Dg∇Φg + CΦg = j−,gin , at ∂V0, (3.67a)

N · Dg∇Φg + γgGΦg = o, at ∂Vh, (3.67b)

where

Dg =

[Dg

1 00 Dg

3

], Φg =

[Φ

g0

Φg2

], j−,gin =

[2 j−,gin,12 j−,gin,3

], γg =

1 − αgg

2(1 + αgg)

and the matrices C and G are given in (3.61). Interface conditions are defined analogically.

3.4.3 Multigroup SP3 diffusion coefficients

In the framework of the multigroup approximation, the mono-kinetic SP3 (or SP1) equations(3.62) (or (3.63)) constitute the within-group equations, in which the inter-group coupling iscontained in the fission and scattering components of the SP3 group source terms S n. As can


be seen by comparing the multigroup PN equations (3.64) with the mono-kinetic eq. (3.34), thescattering component of the mono-kinetic ”absorption moments” (3.36) is actually represented inthe multigroup scheme by the within-group self-scattering, yielding the group removal moments:

Σgan(r) = Σ

gt (r) − Σgg

sn(r), n ≥ 0, (3.68)

corresponding to the usual group removal cross-section ([Han07, eq. 3.47]): Σgr = Σ

ga0.

In the mono-kinetic case, the absorption moments have been used to define the SPN diffusioncoefficients by eq. (3.51), allowing a straightforward derivation of the final equations by usingthe generalized Fick’s law. However, the necessary condition for doing so required S n(r) = 0for n ≥ 1. Returning into the multigroup world and looking back at equation (3.65) reveals,that by neglecting the higher order group source moments we would actually neglect the higherorder Legendre moments of group-to-group scattering and possibly also of fission and externalsources. This is usually acceptable for the latter two but almost never for the first.

Nevertheless, the appealing mathematical form of the original SPN equations made someresearchers ([BL00]) actually use the approximation of no anisotropic group-to-group scatteringin their multigroup formulation23:

Σgg′sn (r) = 0, g′ , g, n ≥ 1.

The SP3 (or analogically diffusion) equations are then expressed as in Sec. 3.3.6, using the SP3

group diffusion coefficients

Dg1(r) =

13Σ

ga1(r)

, Dg3(r) =

37Σ

ga3(r)

, (3.69)

and the SP3 group source terms (derived from eq. (3.65) using eq. (3.53)):

S g0(r) =

G∑

g′=1,g′,g

Σgg′s0 (r)

[Φ

g′0 (r) − 2Φ

g′2 (r)

]+ χ

gf

G∑

g′=1

νΣg′f (r)

[Φ

g′0 (r) − 2Φ

g′2 (r)

]+ Qg(r),

S gn(r) = 0, n ≥ 1.

(3.70)

Definition of group currents Jg1, Jg

2 by the multigroup SP3 Fick’s law is now obvious, as is themodification of eq. (3.70) for eigenvalue calculations.

On the other hand, Beckert and Grundmann [BG08] report large errors in pin-by-pin calcu-lations when anisotropic energy transfer is wholly neglected. They propose (and have successwith) including the first order anisotropic group-to-group scattering by using in the first multi-group SP3 diffusion coefficient Dg

1 the multigroup transport cross-section24:

Σgtr ≡ Σ

gt −

G∑

g′=1

Σg′gs1 = Σ

gt − Σ

gs1 = Σ

gt − µg

0Σgs0 (3.71)

23Recall that some modifications may be applied in order to include the anisotropic source contributions, which,however, generally break the structure of the SPN equations.

24Condition (2.14) has been used to obtain the second identity.


instead of the ordinary group removal cross-section. This is analogical to the traditional transportcorrection of the diffusion theory and is based on the same assumption

G∑

g′=1

Σgg′s1 (r)φg′

1 (r) =

G∑

g′=1

Σg′gs1 (r)φg

1(r).

This approximation has been shown to be sufficiently accurate in diffusive environments withweak absorption for which the SPN model ought to be applied (see the references in [BG08]).

3.4.4 Practical implementation

Having now a set of multigroup constants at our disposal, we may obtain the multigroup SP3

equations from the mono-kinetic equations (3.62) in essentially two ways.

Angle-group scheme

In this first approach, we define the multigroup SP3 operators

Bg3 = −∇ · Dg∇ + Σg

a, g = 1, 2, . . . ,G, (3.72)

where

Dg =

[Dg

1 00 Dg

3

], Σg

a =

Σ

ga0 −2Σ

ga0

− 23Σ

ga0

43Σ

ga0 + 5

3Σga2

, Sg3 =

[1−2

3

]S g

0 (3.73)

and solve, at every outer iteration, a sequence of complete within-group SP3 problems

Bg3Φ

g = Sg, g = 1, 2, . . . ,G, (3.74)

with boundary conditions (3.67).

The solution proceeds from lower energy groups to the higher while progressively updatingthe down-scattering source in Sg in a Gauss-Seidel manner. This way are the SP3 equationssolved by Brantley and Larsen [BL00]. Both SP3 moments could be solved at the same time bydirect inversion of the within-group matrix operator Bg

3 or by an inner iteration25. A Gauss-Seideltype iteration over the SP3 moments, solving for Φ0 first and then for Φ2, has been used by thecited authors who called it a FLIP iterative scheme.

The original formulation of this scheme is obtained by transferring the off-diagonal terms inmatrix Σa to the right-hand side and appropriately adding the inner iteration index q. The result

25This is the inner iteration of eq. (3.17) applied to the SPN method. Note that the constituent equations of eachinner iteration (produced by spatial discretization) could be solved by still another iteration, leading to a total ofthree iteration levels.


is explicitly written as follows (group and outer iteration index omitted):

−∇ · D1∇Φ(q)0 + Σa0Φ

(q)0 = 2Σa0Φ

(q−1)2 + S 0,

−∇ · D3∇Φ(q)2 +

(43

Σa0 +53

Σa2

)Φ

(q)2 =

23

(Σa0Φ

(q)0 − S 0

), q = 1, 2, . . .

(3.75)

Boundary conditions (3.67) for the given group are modified accordingly by manipulating matri-ces C and G like Σa:

n · D1∇Φ(q)0 +

12

Φ(q)0 =

38

Φ(q−1)2 + 2 j−in,1

n · D3∇Φ(q)2 +

78

Φ(q)2 =

18

Φ(q)0 + 2 j−in,3

at ∂V0, (3.76a)

n · D1∇Φ(q)0 + γΦ

(q)0 =

3γ4

Φ(q−1)2

n · D3∇Φ(q)2 +

7γ4

Φ(q)2 =

γ

4Φ

(q)0

at ∂Vh. (3.76b)

The converged SP3 moments are inserted into eq. (3.53) to obtain the scalar flux and updatedown-scattering source for the FLIP iteration in the next lower group. After all groups are swept,fission sources (and possibly the eigenvalue) are updated and the next outer iteration begins.

Group-angle scheme

Rather than defining the multigroup SP3 operators via the matrices of multigroup physical pa-rameters (3.73), we may consider the physical parameters themselves to be G ×G matrices anduse them in the mono-kinetic SP3 equations (3.62). Since this leads to ordering of the vectorof unknown SP3 moments such that the group index is placed innermost and the moment indexoutermost, the angle-by-angle FLIP (Gauss-Seidel) iterative scheme is natural in this case. It isnow written as

−∇ · D1∇Φ0(q) + Σa0Φ0

(q) = 2Σa0Φ2(q−1) + S0,

−∇ · D3∇Φ2(q) +

(43

Σa0 +53

Σa2

)Φ2

(q) =23

(Σa0Φ0

(q) − S0

), q = 1, 2, . . . ,

(3.77)

where

Dn =

D1n

. . .

DGn

, Σan =

Σ1an

. . .

ΣGan

, Φn =

Φ1n...

ΦGn

, S0 =

S 10...

S G0

.

Boundary conditions are specified for the two moment equations (3.77) by eq. (3.76), writtenwith the multigroup vector Φn, analogically defined multigroup vector of incident current mo-ments and the diagonal multigroup matrix Γ = diag

(γ1, . . . , γG)

.


At every outer iteration, the 0th moment equations are solved in all groups first, using Φ2,fission and scattering sources computed in the previous iteration. Since the inter-group couplingrepresented by the latter two is via the scalar flux that depends on both SP3 moments, only aJacobi-type inner iteration over the groups may be used to solve the system of multigroup 0th

moment equations. Once the moments Φg0 are obtained for all groups, they enter the ensuing

solution of the 2nd moment equations as sources on their right-hand side. In the group-by-groupsolution of the 2nd moment equations, down-scattering could be already treated in a Gauss-Seidelmanner by using the lower-groups portion of the actual 0th order SP3 moments and the the alreadycalculated 2nd order SP3 moments to determine the scalar flux by eq. (3.53). After the momentsΦ

g2 are obtained for all groups, fission sources (and possibly the eigenvalue) are updated and the

next outer iteration begins.

Conclusion

Spatial discretization performed in the following chapter still increases the ordering possibilitiesfor the final system of linear algebraic equations for values of the SP3 moments on a spatialgrid. Their results remain the same in all cases, of course, although the properties of the systemmatrix change and may influence the performance. Downar analyzes various ordering schemes in[Dow05], concluding that the space-group-angle choice is superior over the angle-space-group.Since the primary aim of this thesis is to show that the SP3 approximation actually works andmay be used in future in conjunction of the nodal method, I have chosen the space-angle-groupapproach as it best fits into the current diffusion code (which will be shown in the followingchapter). More thorough analysis of the ordering impacts could be one of the possible ways forimproving the code in the future.

70

Chapter 4

The multigroup SP1 and SP3 methods in thehex-z geometry

The final step leading towards a computer-executable solution scheme is the discretization ofthe multigroup equations (3.74) with respect to spatial variable. In this step, we can take fulladvantage of the SP3 formulation since the resulting equations have a form of coupled ellipticpartial differential equations, as can be seen from eq. (3.72) and (3.73). The same approach thathas been used to discretize the diffusion equation (i.e. the SP1 equation) can therefore be takento discretize the higher order SPN equations.

In my Bachelor’s thesis [Han07], I derived the finite volume scheme (FV) as a discrete rep-resentation of the multigroup diffusion equations in a two-dimensional cross-section of a corewith hexagonally shaped fuel assemblies. The finite volumes, called nodes, have been deliber-ately chosen as large as the assemblies to facilitate an efficient numerical solution. Of course,large spatial discretization errors would arguably make the results nonsensical if so-constructedFV method had been used alone. Therefore, a non-linear procedure, called coarse-mesh, finite-difference method (CMFD) has been employed, in which the CMFD equations are graduallycorrected by a more accurate nodal method (as already outlined in the Introduction, Sec. 1.1.2).

Accuracy of the nodal method crucially influences accuracy of the results and we devoteda great deal of effort to its improvement over the past two years. It is beyond the scope of thisthesis to give a detailed account of even any of them. The nodal method is also still a workin progress, which is from the largest part performed by my colleague Roman Kuzel, who istruly the person that deserves main credit for the reported achievements. Therefore I only brieflydescribe the principle of the method and summarize the biggest enhancements to the previousversion in Sec. 4.2.

The focus of the next part of this chapter will rather be on a fine-mesh, finite-difference FMFDsolver, which improves the solution accuracy by traditional mesh refinement. The decision todevelop the fine-mesh solver along with the production-oriented CMFD solver has been madepartly because the need for some easily customizable reference solver for verification purposes,

4.1 Spatial discretization by finite volumes 71

partly because the need for heterogeneous solution for homogenization purposes. The latterprocedure will be briefly outlined in Sec. 4.3, again with numerical results from the diffusionsolver applied to a real heterogeneous problem provided by S-JS.

4.1 Spatial discretization by finite volumes

Thanks to the conservative form1 of the SPN equations, formally equivalent with that of the diffu-sion equation, derivation of the corresponding finite volume scheme closely copies that presentedin a two-dimensional geometry in [Han07, Sec. 4.3] and later extended into three dimensions in[HBB+08]. I shall therefore only outline its basic principle on the diffusion equation (3.63).

4.1.1 The finite volume method – basic principle

By integrating the equation (3.63) over each of the homogeneous bounded volumes (nodes) Vi,such that

⋃iVi = V,

⋂iVi = ∅ (zones with the same material composition comprising the whole

core), and using the divergence theorem of Gauss and Ostrogradskij, we obtain the followingdiscrete expression of nodal balance in group g:

1|Vi|

∑

ξ

∣∣∣Γi,ξ

∣∣∣ jgi,ξ + Σ

gi,rφ

gi =

G∑

g′=1,g′,g

Σgg′i,s φ

g′i + χg

G∑

g′=1

νΣg′i, fφ

g′i + Qg

i . (4.1)

where Σgir is the removal cross-section as defined by (3.68), the index i denotes association of a

quantity to the volume Vi, and ξ denote the surfaces Γi,ξ ⊂ ∂Vi,⋃

ξ Γi,ξ = ∂Vi, oriented by theunit outward normal nξ, and

∣∣∣Γi,ξ

∣∣∣ is the measure (area) of face Γi,ξ. The discrete unknowns ofthis equation are the volume-averaged scalar flux and the face-averaged neutron currents:

φgi ≡

1|Vi|

∫

Vi

dr φg(r), jgi,ξ ≡

1∣∣∣Γi,ξ

∣∣∣∫

Γi,ξ

dA jg(r) · nξ, (4.2)

which are also used to express the boundary conditions. The average external sources are definedanalogously and the cross-sections are spatially constant by the assumption of homogeneity.

To obtain a set of well-defined linear algebraic equations of only one type of unknown (mostusually the average scalar flux), a relation between the surface currents and nodal fluxes has to beprovided. In this case, it may be obtained from the Fick’s law j = −D∇φ·n. By approximating thedirectional derivative at the interface Γi,ξ between nodesVi and and its neighbour in the directionof the interface normal nξ (further denoted Vi+ξ) with the finite-difference quotients from bothsides of the interface and combining the two expressions so that the resulting approximation of

1Here I mean by conservative form that if the approximate streaming operator, i.e. first term in eq. (3.62), isintegrated over a spatial region, the result can be directly interpreted as leakage through the region’s surface


interface current be conservative ([Han07, Sec. 4.3.1])2, we obtain

Jgi,ξ = −Dg

i,ξ(φgi+ξ − φg

i), Dg

i,ξ ≡Dg

i Dgi+ξ

h+i Dg

i+ξ + h−i+ξDgi

, (4.3)

where h+i and h−i+ξ, respectively, denote the distance from the centerpoint of the interface Γi,ξ to

the center of nodes Vi and Vi+ξ, respectively. If any face of node Vi coincides with the coreboundary ∂V , the specified boundary conditions must be used to obtain the relationship. For theexample of albedo boundary conditions of the form j · n − γφ = 0, this equation is discretizedat the boundary by the same finite-difference principle and the additionally appearing unknownboundary flux is eliminated, leading in the diffusion case to the relation ([Han07])

Jgi,ξ = Dg

i,ξφgi , Dg

i,ξ ≡2Dg

i γg

h+i γ

g + 2Dgi

. (4.4)

Coarse-mesh discretization

In the CMFD method, a core with hexagonal assemblies (typical, but not exclusive, to the RussianVVER reactors) is modelled in three dimensions by cutting it along the z axis into a set of radialplanes and overlaying each plane by nodes coinciding with the real assemblies loaded into thereactor. Denoting by IA the number of assemblies and byVi the specific nodes, the core domainis thus discretized as V =

⋃IA Ji=1Vi, where J is the number of axial cuts. The node has eight faces:

ξ ∈ x±, u±, v±, z±, by pairs perpendicular to the four principal directions defined by the fourpositive-direction normals nx, nu, nv and nz

3 – see the illustration in Fig. 4.1.

Discretization in the axial direction need not be uniform:

h+i , h−i+z,

allowing a finer subdivision if more accuracy is desired. Of course, the heightshi,z = h+

i + h−i of nodes in each column along the z axis sum to the total height of the core,denoted H. Discretization is however fixed in each radial plane to

h+i = h−i+ξ ≡ hi,ξ ≡ h,

where h is the assembly pitch. Correcting the discretization errors for large assemblies4 is thetask left to the nodal method. Nodal dimensions are finally given by

` =h√3,

∣∣∣Γi,ξ

∣∣∣ =

`hi,z for ξ ∈ x, u, v±,h2√

32 for ξ = z± , |Vi| =

∣∣∣Γi,z

∣∣∣ hi,z.

2Besides conservativity, the approximation of interface currents has to be also consistent in order to prove con-vergence of the FV scheme, i.e. it must approach the exact surface-averaged current from eq. (4.1) in the limit ofvanishing nodal spacing (for more details see [Han07, Sec. 4.3.1] and the references therein).

3i.e. the ± faces share the same normal direction, but differ in their outward orientation4h = 23.6 cm in a typical VVER-1000 reactor, which is much more than the neutron’s MFP in the core, signal-

izing large spatial discretization errors ([Sta01])


nx+

nu+nv+

Γi,z−

Γi,u+

Γi,x+

Γi,v−

Γi,u−

Γi,v+

Γi,x−

nz+

x

yz

`

hhz

Γi,z+

Coordinates of radial cross section Hex-Z prismatic node

Figure 4.1: A node in the hex-z geometry

The set of linear algebraic equations that results from writing the balance eq. (4.1) for eachnode of the just described core nodalization and expressing the internodal leakage by the approx-imate Fick’s law (4.3) is explicitly stated in [Han07] (2D) or in the form of group-block matrixin [HBB+08] for the 3D case, including the boundary conditions. This latter concise form ofthe multigroup equations remains the same also in the fine mesh discretization and formally alsowhen the spatially discretized equations arise from the angle-group SPN discretization scheme,as presented in the previous chapter. This will be shown shortly by eq. (4.6).

Fine-mesh discretization

In the FMFD method, the finite volumes (I shall call them for simplicity also ”nodes” as in theCMFD method) are formed by radially subdividing the hexagonal assemblies into finer zones,mostly the equilateral triangles5 as shown in Fig. 4.2 (for simplicity, I consider at this momentonly a uniform subdivision of all radial hexagons).

Two levels of such subdivision are shown in the figure – the number of triangles per hexagonin a general subdivision level T is equal to

U = 6T 2.

The axial discretization is performed in the same way as in the coarse-mesh method and thereare consequently I = UIA nodes per radial plane and a total of IJ nodes comprising the whole

5the other possibility being subdivision into lozenges


Figure 4.2: Hexagon subdivision for T = 1 and T = 2, respectively

core. Dimensions of the nodes are given with respect to the hexagonal assembly dimensions as:

h =hhex

3T, ` =

`hex

T,

∣∣∣Γi,ξ

∣∣∣ =

`hi,z for ξ ∈ x, u, v±,h2

hex

√3

2U for ξ = z±, |Vi| =

∣∣∣Γi,z

∣∣∣ hi,z. (4.5)

h

hz

Figure 4.3: Two neighbouring nodes in the fine-meshed hex-z geometry


4.1.2 The space-angle-group scheme

N

In this section 4.1.2, the bold letters will denote the matrices whose elements represent the nodalquantities, arising from the FVM discretization (their order is given by the total number of nodesof the discrete finite volume mesh). The SP3 moment matrices for which the bold notation wasused in the previous chapter will be now denoted by square brackets around the letter.

If the angle-group scheme, given by eqns. (3.74), (3.72), (3.73) and (3.76), is discretized by theFMFD procedure described above, a set of linear algebraic equations of the space-angle-groupscheme is obtained. It may be written in a ”group-block” matrix form (for simplicity for onlytwo groups with an obvious means of more-groups extension):

( [L]1

+[Σa

]1 −[Σs]12

−[Σs]21 [

L]2

+[Σa

]2

)

︸︷︷︸[[B3]]

·([

Φ]1

[Φ

]2

)

︸︷︷︸[[Φ]]

=

(χ1[νΣ f

]1χ1[νΣ f

]2

χ2[νΣ f]1

χ2[νΣ f]2

)

︸︷︷︸[[F]]

·([

Φ]1

[Φ

]2

)

︸︷︷︸[[Φ]]

+

([Q

]1

[Q

]2

)

︸︷︷︸[[Q]]

(4.6)

where the bold letters indicate spatial discretization, the square brackets around them the SP3

angular discretization and the final superscripts the group indices. A system of an analogic formwould be obtained for SPN approximations of different orders, including diffusion, where thebrackets around the spatially discretized blocks of the multigroup matrix would disappear (see[Han07, eq. 4.59]). Equations for the eigenvalue problem are also simply obtained from eq. (4.6)

[[B3]] · [[Φ]] =1λ

[[F]] · [[Φ]]. (4.7)

Anyway, taking note of that [[B]] = [[L]] − [[H]], one may recognize equation (3.16), definingthe source iteration scheme for solving the equations.

The ”bracketed matrices” are all rectangular sparse matrices of order 2IJ (IJ equations foreach angular moment), so there is a total of 2GIJ equations to be solved in the multigroupSP3 approximation with G energy groups. The Σs and Σ f matrices are diagonal, composed ofthe scattering and fission cross-section constants for each homogeneous node. The L matricesrepresent the streaming and are composed of the ”discrete diffusion coefficients” Dg

i,ξ from eq.(4.3), written, of course, in terms of the particular SP3 diffusion coefficient. Boundary conditionsare incorporated to the diagonal of the matrix at rows corresponding to the boundary nodes.They are obtained analogically to the diffusion case by writing their equations (3.76) in termsof the auxiliary boundary surface flux moments, approximating the gradient terms by the finite-difference principle and finally eliminating the auxiliary surface variables. For instance, thediscrete albedo conditions for a boundary nodeVi are given by

Jg1,i,ξ =

2γgDg1

[(25hγg + 32Dg

3

)Φ

g0 − 24Dg

3Φg2

]

8Dg1

(7hγg + 8Dg

3

)+ hγg

(25hγg + 32Dg

3

)

Jg3,i,ξ =

2γgDg3

[(25hγg + 56Dg

1

)Φ

g2 − 8Dg

1Φg0

]

8Dg1

(7hγg + 8Dg

3

)+ hγg

(25hγg + 32Dg

3

) .(4.8)


Note that this form of boundary conditions introduces coupling between the two angularmoment equations. Another such coupling is produced by the Σa matrices. This consequentlyleads to the structure of the Lg matrix in each group shown in Fig. 4.4. It corresponds to theFMFD discretization of the one sixth of the KNK-II core from the Appendix with IA = 29, T = 2,J = 5. The detail on the part corresponding to the 0th moment, 1st group equations and thenfurther on the 0th moment, 1st group, 1st radial plane equations is given in figures 4.5a and 4.5bfor illustration. Except for the small block at the beginning6, the matrices have a symmetricalstructure, but are non-symmetric due to the form of the Σa matrices (see eq. (3.73)).

Figure 4.4: Structure of the L matrix

6arising, due to the rotational symmetry, from the coupling of the central assembly to the adjacent assembly No.2 through all of its six sides, but only through one side of the assembly No. 2 of the computed symmetrical segment


(a) 0th moment, 1st group

(b) 0th moment, 1st group, 1st radial plane

Figure 4.5: Selected parts of the L matrix


4.1.3 Numerical tests

Diffusion theory (SP1)

The FMFD method has been tested on several benchmarks from the ref. [CS95b]. As an illus-tration, Fig. 4.7 compares the diffusion variant of the FMFD code with the code DIF3D ([Der],used by Chao and Shatilla to generate their reference results) on the Benchmark No. 6 from thecited reference. It is an eigenvalue problem set in a VVER-1000 type core with 163 assemblieswith a pitch of 23.6 cm and a total height of 200 cm. The core loading was homogenized intofive different materials, whose two-group physical parameters are available either in [CS95b] orin my Bachelor’s thesis [Han07]. The core reflector was not modelled explicitly – instead, thealbedo boundary conditions of eq. (3.42) were specified. At the periphery of each radial plane,the albedo coefficient γ was set to value 0.125, while γ = 0.15 at the top and bottom boundariesof the core (for both groups).

Figure 4.6: The model composition of the VVER-1000 core from [CS95b, Bench. No. 6]

The radial section through the upper half of the core (with the numbers representing the ho-mogenized materials), shown in Fig. 4.6, reveals the one sixth symmetry of the core, so only onesymmetric segment consisting of 28 assemblies was used used in the calculation. Compositionof the lower and the upper part of the core differ only in assembly No. 4, which has control rodsinserted into its upper half (material No. 4 in the upper half, No. 3 in the lower one).

Figure 4.7 shows for each assembly its number (the top number) and the relative differencesof the mean assembly powers (discrete equivalent of eq. (2.25)), normalized to the total corepower of unity, in the upper core and in the lower core, respectively (the two numbers below).The core critical number keff and its relative difference to that reported by Chao and Shatilla isalso shown, in the units of pcm – percent-milli, or 10−5.


Figure 4.7: Diffusion FMFD solution compared to DIF3D. The numbers in the hexagons are –top: assembly No., middle and bottom: relative difference of mean assembly powers in the upperand bottom halves of the core from the reference [CS95b]. The same homoegeneous materialsare coloured by the same shade.

The results demonstrate a near perfect match. The small differences could be attributed toa different calculation setup and procedure (which the authors do not describe in much detail)and possibly also to different numerical methods employed to solve the system of algebraicequations. Our results have been obtained with the radial subdivision level T = 9, i.e. 486triangles per hexagon, and a uniform axial division by 5 cm, i.e. into 40 radial planes (the samenumber used by Chao and Shatilla). As for the numerical method, the stabilized bi-conjugategradients method provided by MATLAB (function bicgstab) has been used, preconditioned byblock incomplete LU factorization (function ilu) with drop tolerance7 of 10−2. This method isclassical for the solution of large sparse non-symmetric systems, and I refer the reader to [Saa96,Chap. 10] and [vdV92] for more details. The eigenvalue calculation continues while both

∣∣∣∣∣∣λ(p) − λ(p−1)

λ(p−1)

∣∣∣∣∣∣ < 10−6 and maxg,Vi

∣∣∣φg,(p) − φg,(p−1)∣∣∣ < 10−5.

It is worth noting that the unaccelerated source iteration took 2 458 s, while the IRAM methodwith the maximum subspace dimension 7 (recall Sec. 3.2.4) took only 375 s to converge to thesame result. These numbers are only illustrative, of course, but point at the latter method as at apromising candidate for the final solver8.

7Numbers greater than this threshold that appear during the factorization at originally zero positions will bedropped to reduce fill-in.

8Even more as it is publicly available in the library ARPACK ([Ric])


Transport theory (SP3)

The SP3 variant of the code has been tested on a Model problem No. 4 from [TI91]. It isa small fast breeder core similar to a KNK-II core proposed by the researchers at the formerKernforschungszentrum Karlsruhe in Germany. It consists of 169 hexagonal assemblies in a30-degrees symmetric structure. The core is depicted in the Appendix, together with the fullspecification of the problem and the four-group physical data.

Time permitted to test only the first type of problem with rods withdrawn and the last casewith the rods fully inserted. For preliminary testing of the transport method, only the calculatedkeff has been compared with the results in the cited reference. Thanks to the relatively smallassembly pitch, only a radial subdivision with level T = 3 has been chosen; axial subdivisionwas uniform by 10 cm, with a finer division by 5 cm where necessary to fit the specified geometry(see the Appendix). The results are summarized in Tab. 4.1, together with running times ofthe diffusion and SP3 methods (both used the same convergence criteria as the example in theprevious subsection and the same calculation method). The Monte Carlo (MC) reference resultsand the full PN (i.e., not SPN), for N = 1, 3, 7, reference results (those reported by Fletcher) arecopied from [TI91] to facilitate the comparison.

Method case 1 case 3 time for case 1SP1 1.07969 0.89155 21 sSP3 1.08494 0.90401 69 sP1 1.07860 0.87217P3 1.09558 0.89166P7 1.09570 0.89203

MC 1.09510 0.87990

Table 4.1: Results of the KNK-II model problem (case 1: rods withdrawn, case 2: rods in)

It is clear, that the SP3 method provides slightly better results than the diffusion method,although the difference on the selected model problem is not too convincing to justify the three-times longer computing time (which is however in agreement with the other researchers observa-tion, recall the discussion at the end of Sec. 3.3.7). However, it is in par with the results reportedby the participants in the cited report. Also, for the first case, the developed SPN methods behaveexpectedly in comparison with the full PN methods, that is the diffusion (SP1) results are in parwith the P1 results and the SP3 results are in between the full P1 and P3 results.

In the second case, it seems that the difference from the Monte Carlo result is higher for theSP3 method than for the diffusion method, but this agrees with Fletcher’s results, whose P1 resultsare also closer to the MC results than the higher-order ones. A more thorough testing would beneeded to explain this a little surprising result (differences of the average fluxes will arguably tella different story), but this I have to leave at this moment for the future.


(a) Case 1 (rods out)

(b) Case 3 (rods in)

Figure 4.8: Distribution of the first group scalar flux in the the central plane of the model KNK-IIcore, as computed by the SP3 solver

4.2 Transverse integrated nodal methodology 82

4.2 Transverse integrated nodal methodology

Nodal equations in most modern nodal methods are derived by performing the spatial transverseintegration procedure on the initial three-dimensional neutron diffusion or transport equationover each coarse mesh (node), reducing it into a set of simpler lower-dimensional equations ofthe same type and hence facilitating some better, more accurate approximation than that used toobtain the fully three-dimensional CMFD (or FV) scheme. The lower-dimensional equations arecoupled through the transverse leakage terms, representing neutron migration in the remainingdirections. These terms require approximation and in fact, it is the principal source of error (theonly one in the analytic nodal methods). However, guessing the lower-dimensional transverseleakage shape is generally easier than approximating the three-dimensional flux shape, whichsomeway justifies the whole procedure.

First level of the transverse integration procedure splits the three-dimensional equation intoa two-dimensional equation in radial directions and a one-dimensional axial equation. Nodalmethods that are based on such splitting are represented e.g. by the AFEN or DYN3D codes.Following the same argument and continuing the procedure one step further, a set of one-dimensional equations for each coordinate direction is obtained. For the relative simplicity ofone-dimensional equations and their solution methods, this approach has been chosen for thenodal method developed as part of my previous thesis. However, when applied to non-rectangularnodes (in particular those with hexagonal shape, required for nodal modelling of the RussianVVER reactors), the two-level transverse integration suffers from both mathematical and physi-cal difficulties connected with the definition of transverse leakage.

The local coordinate system of a hexagonal node shown in Fig. 4.1 defines in the radial planethree principal directions and hence the two-fold transverse integration results in four coupledone-dimensional equations, as opposed to three in a Cartesian coordinate system for rectangularnodes. In the radial plane, boundaries of the hexagon transverse to any of the three principaldirections are not described by a smooth function, which introduces a non-physical singularity tothe transverse leakage function ([CS95b]). Moreover, the transverse boundary surfaces are notaligned with the principal solution directions. This further complicates the transverse leakageterm by involving a contribution of neutron migration through the transverse surfaces to thesolution in the chosen direction, in the form of the (fundamentally unknown) transverse boundaryfluxes. Several approximate resolutions have been tried by various researchers in the past andone successful due to Wagner ([Wag89]) has been chosen and verified in our previous work (it isdescribed in my Bachelor’s thesis). I will refer to this nodal code shortly as to ”NODWAG”.

4.2.1 Improvements of the NODWAG code

In this subsection, I will compare subsequent versions of the NODWAG code as they have comeinto being throughout the past two years’ development process. I choose for illustration purposesthe Benchmark No. 1 from ref. [CS95b], which is a two-dimensional variant of the previouslymentioned benchmark problem No. 6, with vacuum outer boundary conditions. Figure 4.9


shows the best NODWAG results next to the best results obtained by the present code (PD standfor the normalized mean assembly power, the subscript re f for the reference solution by Chaoand Shatilla and finally the subscript err for the relative difference from the reference).

(a) NODWAG

(b) NODCONF

Figure 4.9: Comparison of the NODWAG and the NODCONF code.


Evolution of the NODCONF code

To put the transverse integrated hexagonal nodal methods on a firm mathematical ground, Chaoand Shatilla ([CS95a, CS95b]) developed a method in which the hexagon is transformed to arectangle via the conformal mapping. By performing the transverse integration in the mappedrectangular domain (three times, each time aligning the rectangle’s base with one of the prin-cipal directions), the above mentioned problems with transverse leakage are elegantly avoided.An important feature of the transformation that makes the whole procedure possible is that theLaplacian operator is invariant under the mapping9. Hence, the Laplacian neutron diffusion equa-tion (or some higher-order SPN moment equation) with constant coefficients (due to the assumedhomogeneity of each node) remains after the mapping still elliptic in nature, only its coeffi-cients become scaled by the mapping area scale function10. The coefficients (cross-sections) inthe equations thus become spatially varying, but an explicit mathematical expression may beobtained for the variation ([CS95b]).

Having implemented into NODWAG the conformal mapping procedure (hence the nameNODCONF of the final code) with the simplest flat transverse leakage ([Han07, HBB+08])yielded first notable accuracy improvement, shown for the model problem in Fig. 4.10. Al-though in some assemblies, the errors have risen, overalls is their distribution more even and inparticular, they do not exceed 5% anywhere in the core.

Figure 4.10: Results of the early version of the NODCONF code with flat transverse leakage

9This has been well recognized in many branches of physics where the processes are governed by elliptic differ-ential equations, in particular fluid dynamics.

10in classical multivariable calculus represented by the Jacobian determinant


Noting that the largest errors occur at the boundaries of the core, we aimed at these partsnext. Rather than by a constant shape, we approximated the one-dimensional transverse leakageby a linear function constructed using the information about the boundary. This idea has beenoriginally proposed by V. Zimin [ZB02] and led to further improvement of both the mean as-sembly powers and critical number estimate (see Fig. 4.11). These results are in par with theacknowledged ones of V. Zimin (also for different problems).

Figure 4.11: Results of the NODCONF code with flat transverse leakage inside the core andlinear approximation at the core outer boundary

The results shown in Fig. 4.9 were obtained by further improvements relating to the twoproblematic nodes clearly identifiable from Fig. 4.11 ([Kuz09]).

4.3 Homogenization 86

4.3 Homogenization

In deriving the CMFD as well as the nodal equations, the material properties of each node haveto be spatially constant. Since the fuel assemblies inside a real reactor are highly heterogeneous,either due to their construction or due to various effects associated with their long-term stay in thecore (such as fuel burn-up), the procedure of assembly homogenization has to be performed onthe initial data first to provide the required piecewise-constant description of the core propertiesfor the numerical methods. Although a recent trend is to avoid this procedure and perform thecore calculations with all the fine-scale heterogeneities taken into account ([Cho05, CKL+07]),it is prohibitive in terms of computation time and homogenization still plays an important role incore calculation methods. Although it is beyond the scope of this thesis to thoroughly describethe homogenization procedure (see e.g. [Sta01, Chap. 14] for a detailed account), I will mentionat least its fundamental principle.

4.3.1 Theory

Thirty years after Koebke started the massive use of homogenization in nodal methods by for-mulating the so called equivalence theory (ET , later generalized by Smith into GET), it is still amethod of choice for generating equivalent homogeneous input data for majority of nodal codes.By the word equivalent in the previous sentence, an equivalency between the real problem andthe homogenized one is meant in the following sense. Consider a heterogeneous multigroupeigenvalue problem, written in a global conservation form (obtained by integrating eq. (2.53)over all directions and using the definition of net current (2.9)) as ([Smi86])

∇ · Jg(r) + Σgt (r)φg(r) =

G∑

g′=1

Hgg′φg′(r) +

1λ

G∑

g′=1

Fgg′φg′(r), g = 1, . . . ,G. (4.9)

Solution of this equation may be considered as a reference solution, that we wish to attain insome sense by solving a (desirably simpler) homogenized equation (with terms decorated by thehat)

∇ · Jg(r) + Σgt (r)φg(r) =

G∑

g′=1

Hgg′ φg′(r) +

1λ

G∑

g′=1

Fgg′ φg′(r), g = 1, . . . ,G. (4.10)

with coefficients constant over the homogenization region Vi. Specifically, if the homogenizedsolution reproduces the average reference reaction rates (compare with def. (2.12)) over thehomogenization region as well as the average reference leakage through its boundaries,

∫

Vi

dr Σg∗(r)φg(r) =

∫

Vi

dr Σg∗(r)φg(r) (4.11a)

∫

Γi,ξ

dA Jg(r) · n =

∫

Γi,ξ

dA Jg(r) · n (4.11b)


then also the eigenvalue will be reproduced and we attain with the homogeneous solutions thereference solution in a weak integral sense. The spatially constant homogenized coefficients inthe given region that allow this are formally given as

Σg∗,i ≡

∫Vi

dr Σg∗(r)φg(r)

∫Vi

dr φg(r).

Satisfaction of eq. (4.11b) depends on the transport model used for the homogenized prob-lem. If it is diffusion (or SPN in general), evaluation of the net currents according to Fick’slaw would require to appropriately define the homogenized diffusion coefficient. Smith [Smi86]recognized the problem of satisfying both (4.11a) and (4.11b) with a single constant diffusion co-efficient, ultimately leading to a solution in which the diffusion coefficient is defined arbitrarily,but the homogenized flux is allowed to be discontinuous at the interface between two adjacenthomogeneous regions (group index omitted):

fiφi

∣∣∣Γi,ξ = fi+ξφi+ξ

∣∣∣Γi,ξ

with the assembly discontinuity factors (ADF) determined exactly so that the real heterogeneousflux is continuous:

fi

∣∣∣Γi,ξ =

φi

φi

∣∣∣∣∣∣Γi,ξ

, fi+ξ

∣∣∣Γi,ξ =

φi+ξ

φi+ξ

∣∣∣∣∣∣Γi,ξ

. (4.12)

As before,→and← indicate the limits as r approaches the interface from left and right, respec-tively, along the direction ξ.

It is evident that spatial homogenization is analogical to energy group condensation (Sec. 3.2),where the multigroup constants were formally defined by relations (3.9) so that solution of thecondensed (group) equation (3.8) with these constants agrees in terms of integral average reac-tion rates with the continuous energy equation integrated over the ”energy condensation region”(i.e. group) Eg, eq. (3.3). It thus faces the same conceptual problems, namely that the solution ofthe heterogeneous reference problem is not known, since otherwise we would not have to botherwith homogenization, and further the homogeneous solution in the formal definition of the ho-mogenized coefficients is not known either as it requires these very coefficients as an input). Thesame approach may be used to resolve these issues, i.e. using approximation of type (3.13):

φ ≈ ΦSA (4.13)

of the unknown heterogeneous flux by the result of an isolated single-assembly (or colorset)calculation. Symmetry (specular reflection) boundary conditions are prescribed (i.e., zero netcurrent on the outer boundaries of the assembly or the colorset) to model the infinite lattice. Thenormalization condition ∫

Vi

dr φ(r) =

∫

Vi

dr φ(r)

is used to resolve the second issue. If the zero net current boundary conditions are used, thehomogeneous solution in the node is constant and this normalization condition allow us to use


the results of the single-assembly heterogeneous calculation in denominator of (4.12). If weknew some more information about the boundary of the node (for instance when it is placedat the core boundary), we could arguably use it to obtain a better approximation (4.13) thanwith the zero net current boundary conditions. In this case, the surface homogeneous flux ineq. (4.12) is evaluated using the results of the single-assembly calculation by the nodal method,for which the homogenization is performed, with input data11 from the heterogeneous single-assembly calculation with the more precise boundary conditions.

In practice, the energy-group condensation and spatial homogenization are performed at thesame time, using the same single-assembly flux to weight the heterogeneous, continuous energycross-sections both spatially and energetically (i.e. there will be double integrals in eqns. (3.9)).The homogenized multigroup parameters for each node are thus obtained and may be used in theCMFD and nodal methods to perform the whole-core calculation.

The whole procedure is grossly dependent on the assumption of infinite, periodical structureof assemblies in the core, characterized by zero net leakage between them. Significant leakageis however usual in real cores, e.g. near the strong absorbing regions or at boundaries. Colorsetcalculations may help in this case or the homogenization and nodal solutions may be intertwinedin a non-linear iteration, using the intermediate nodal solution to define more accurate bound-ary conditions for the single-assembly calculation, which in turn provides refined homogenizedparameters for the nodal solver. This is an area of active research and a good overview of re-cent progress can be found in [San09]. The author also generalizes the ADF’s to a general SPN

method by allowing discontinuity of even angular flux moments.

4.3.2 Practice

Assembly homogenization was tested on a two-dimensional section of a VVER-1000 core withmaterial composition corresponding to the beginning of fourth fuel campaign. The two-groupheterogeneous data were provided by S-JS for 54 triangles in each of the 28 assemblies of thesymmetrical segment as a result of previous thermal-hydraulical calculations. The core wassupposed to be surrounded by vacuum. Heterogeneity at both core and assembly level is apparentfrom Fig. 4.12, where values of νΣ1

f are plotted for each assembly (the interassembly distributioncorresponds with the numbering shown in Fig. 4.2).

The calculation procedure consisted of two main parts – preparation of the homogeneousdata, spatially constant within each assembly, and using them afterwards in the NODCONFsolver to obtain the core critical eigenvalue and the mean assembly powers. The results from thefine mesh heterogeneous solution with T = 12 (i.e. 864 triangles per hexagon) were used as areference (note that subdivision of the basic 54 triangles with specified material properties hadto be performed, so only a multiple of three could have been chosen for T ).

Relatively large errors occurred when only a simple volume-averaging was used to obtain

11In the case of NODCONF, these are the node average flux, estimate of the eigenvalue and net currents forevaluating boundary conditions and transverse leakage.


Figure 4.12: Distribution of νΣ1f in the S-JS core example

homogeneous data for each node – see Fig. 4.13. When we used the homogenization procedurewith zero net current boundary conditions for each single-assembly calculation and without con-sidering the ADF’s (i.e. setting them all to unity), the results even a bit worsened (Fig. 4.14).This was an indication of the importance of either the boundary information or the discontinu-ity factors (or both). Therefore, as a first step, we incorporated the information about the coreboundary by using at the outer boundary faces of the peripheral nodes the appropriate bound-ary conditions instead of the zero net current conditions. The results hugely improved, in termsof both the eigenvalue and mean assembly powers, as it is clear from Fig. 4.15. When, on theother hand, we implemented the discontinuity factors and kept the zero net current boundaryconditions, we got an improvement over the simple volume averaging as well – the eigenvalueestimate was better by almost one half and the power estimates for assemblies inside the coretoo (Fig. 4.16). However, large errors occurred at the boundary, suggesting to combine the twoapproaches, i.e. using the true boundary conditions where possible and the ADF’s in the nodalcalculations. However, this proved to be non-trivial since some theoretical aspects of the ho-mogenization are violated by considering the large leakage characteristic for the specified albedoconditions. Therefore, as a first approximation, we simply used the ADF’s for the inner nodes,while setting them to unity for the boundary nodes. The results are in Fig. 4.17.


Figure 4.13: S-JS test problem – volume averaging

Figure 4.14: S-JS test problem – homogenization with pure reflective symmetry boundary con-ditions and no ADF’s


Figure 4.15: S-JS test problem – homogenization with true boundary conditions where possibleand no ADF’s

Figure 4.16: S-JS test problem – homogenization with pure reflective symmetry boundary con-ditions and the ADF’s


Figure 4.17: S-JS test problem – homogenization with true boundary conditions where possibleand ADF’s considered only inside the core

93

Chapter 5

Conclusion

The subject of this thesis has been the study of Boltzmann’s transport equation, as applied tomodelling of neutrons in nuclear reactors with assemblies of hexagonal cross-section. A methodfor solving this equation has been developed as a part of my previous thesis [Han07], based onits diffusion approximation. The primary aim of the current thesis was to assess the possibil-ities of improving the diffusion solver via the transport theoretical model, without extensivelyaffecting the existing code that is prepared for a production-oriented implementation. During thework, many other improvements have also been made to the diffusion solver that, in my opinion,deserved at least a brief mentioning in the thesis too.

The transport theory was overviewed after the introductory first chapter in the followingChapter 2. This not only served for introducing the notation, but also provided a connection be-tween the firm functional analytic ground of the theory with the underlying physical reality. Afterreviewing some work related to mathematical analysis of two fundamental steady-state neutrontransport problems – the core criticality calculations (an eigenvalue boundary value problem)and the fixed source calculations (an inhomogeneous boundary value problem) – ended by twotheorems clearly stating their solvability conditions, a mathematically convenient representationof directionality of the transport processes was developed. The chapter was finished by the slabneutron transport equation, which is a simple one-dimensional, azimuthally symmetric approx-imation, which, nevertheless, has a large number of applications and leads to an efficient, fullythree-dimensional approximation of the Boltzmann’s equation.

Before coming to this approximation later in Chapter 3, discretization of the energetic de-pendence of the neutron transport equation was performed at its beginning in Sec. 3.2. The samemultigroup approach that has been used in the diffusion solver was used, although for a generalnumber of G energy groups (as opposed to only a two-group approximation discussed in myformer thesis). The abstract matrix operator notation allowed to formulate the source iterationmethod for solving the multigroup equations with an as yet non-specified arbitrary directionaland spatial approximation. Some well-known convergence properties of the source iteration werealso reviewed for both the fixed source and eigenvalue problems, proposing the way of improvingthe convergence rate.

94

Neglecting the energetic dependence, directional dependence was approximated next inSec. 3.3 by the spherical harmonics method (PN). Its relationship to the classical Galerkin meth-ods was revealed first, followed by derivation of its constituent equations and their properties.Explicit expressions for the Marshak boundary conditions were also provided, including thealbedo boundary conditions, which I could not find in any available literature but which may beuseful for a future application of the method to problems provided by the S-JS company. Thesecond-order rewriting of the PN equations, called simplified spherical harmonics method (SPN)was presented afterwards, with a notice that the slab geometry SP1 equation is actually the slabgeometry diffusion equation as we know it, and this also holds in three dimensions. The exten-sion from slab geometry to three dimensions was presented in the following Sec. 3.3.6, basedon the formal Gelbard’s approach. The recently developed mathematical basis for this approachwas overviewed at the end of the section.

The third chapter was finished by combining the energetic and directional approximations ofthe transport equation into the multigroup, 3D, SPN transport method.

The final chapter before this one dealt with the spatial discretization. The method of finitevolumes was described, briefly for the coarse volumes discretization (CMFD) and then for thefine mesh discretization (FMFD). Results of numerical calculations performed with the FMFDmethod were presented for three dimensional problems and compared with both diffusion andtransport theory based codes, being encouraging enough to draw us in future experimenting.

Finally, the improvements of the nodal component of the CMFD solver were described insections 4.2 and 4.3. Although these two sections are not directly related to the transport theorymethod, the results presented in them utilized the fine-mesh solver developed for testing thetransport method and both the nodal method and the homogenization module are going to beimbued with the SP3 approximation in the near future. The two sections demonstrate that evenwithout it, these two modules are capable of producing high-fidelity results, comparable with oreven better than the cited literature on benchmark problems and similarly accurate also for theheterogeneous real world data provided by S-JS.

Future directions

As I mentioned several times in the text, the presented method is a work in progress and thereforemany new ideas are currently being discussed, implemented and experimented with. They rangefrom further improvements of the conformally mapped nodal method (mainly the ubiquitousproblem of transverse leakage approximation), to the incorporation of heterogeneous data, tothe improvement of the CMFD iteration procedure, to the implementation of some lower-levelprogramming language like C++, etc.

From the transport theoretical point of view, the most important work that lies ahead will bea thorough testing against the available benchmark problems. As soon as we are fully confidentof the FMFD implementation, the implementation of the derived SP3 expressions (which are

95

already on the paper) into the nodal method will be performed. Also, the SP3 option of theFMFD solver will be tested for the homogenization purposes, hoping for improved results on theheterogeneous real world problems.

The numerical method used for solving either the FMFD or the CMFD equations for large,three dimensional whole core problems should also be more carefully selected and tested, as therehave been developed a huge number of them in the past. This holds for both the fixed-source cal-culations (currently performed by the Bi-CG stabilized method) and the eigenvalue calculations(currently performed by either the source iteration or the IRAM method). The question of theirefficient preconditioning and acceleration has been left open in this thesis.

As a final point in this, certainly not exhaustive list of possible research directions, it wouldbe also beneficial to explore the other formulations of the SP3 methods (like the space-group-angle scheme or the others) or even some different transport theoretical methods (like the SN

method of discrete ordinates) and compare them to the current method.

96

Bibliography

[AB99] Gregoire Allaire and Guillaume Bal. Homogenization of the criticality spectralequation in neutron transport. RAIRO - Modelisation mathematique et analysenumerique, 33(4):721–746, 1999.

[ABB+95] Raymond E. Alcouffe, Randal S. Baker, Forrest W. Brinkley, Duane R. Man,R. Douglas ODell, and Wallace F. Walters. DANTSYS: A Diffusion AcceleratedNeutral Particle Transport Code System. Los Alamos National Laboratory, cic-14edition, June 1995.

[Ada04] Marvin L. Adams. I have an idea! An appreciation of Edward W. Larsen’s contri-butions to particle transport. Annals of Nuclear Energy, 31:19631986, 2004.

[AH01] Kendall Atkinson and Weimin Han. Theoretical Numerical Analysis: A FunctionalAnalysis Framework, volume 9 of Texts in Applied Mathematics. Springer, 2001.

[AL02] Marvin L. Adams and Edward W. Larsen. Fast iterative methods for discrete-ordinates particle transport calculations. Progress in Nuclear Energy, 40:3–159,2002.

[BG08] C. Beckert and U. Grundmann. Development and verification of a nodal approachfor solving the multigroup SP3 equations. Annals of Nuclear Energy, 35:75–86,2008.

[BL00] Patrick S. Brantley and Edward W. Larsen. The simplified P3 approximation. Nu-clear Science and Engineering, 134:1–21, 2000.

[CCMR02] R. Ciolini, G. G. M. Coppa, B. Montagnini, and P. Ravetto. Simplified PN and ANMethods in Neutron Transport. Progress in Nuclear Energy, 40(2):237–264, 2002.

[Cho05] Nam Zin Cho. Fundamentals and recent developments of reactor physics methods.Nuclear Engineering and Technology, 37:25–78, 2005.

[CKL+07] Jin-Young Cho, Kang-Seog Kim, Chung-Chan Lee, Sung-Quun Zee, and Han-GyuJoo. Axial SPN and radial MOC coupled whole core transport calculation. Journalof Nuclear Science and Technology, 44:1156–1171, 2007.

[CL06] Nam Zin Cho and Jaejun Lee. Analytic Function Expansion Nodal (AFEN) Methodin Hexagonal-Z Three-Dimensional Geometry for Neutron Diffusion Calculation.

BIBLIOGRAPHY 97

Journal of Nuclear Science and Technology, 43(11):1320–1326, 2006.

[CS95a] Y. A. Chao and Y. A. Shatilla. Conformal Mapping and Hexagonal Nodal Methods– I: Mathematical foundation. Nucl. Sci. Eng., 121:202–209, 1995.

[CS95b] Y. A. Chao and Y. A. Shatilla. Conformal Mapping and Hexagonal Nodal Methods– II: Implementation in the ANC-H Code. Nucl. Sci. Eng., 121:210–225, 1995.

[CTGV08] M. Capilla, C. F. Talavera, D. Ginestar, and G. Verdu. A nodal collocation approxi-mation for the multi-dimensional pl equations - 2d applications. Annals of NuclearEnergy, 35:1820–1830, 2008.

[Cul01] D. E. Cullen. Why are the PN and SN Methods Equivalent? Technical report,Lawrence Livermore National Laboratory, 2001.

[Der] K. L. Derstine. DIF3D, Standard Code Description. Available from: http://www.ne.anl.gov/codes/dif3d/. Last checked: 09-16-2007.

[DH76] James J. Duderstadt and Louis J. Hamilton. Nuclear Reactor Analysis. John Wiley& Sons, Inc., 1976.

[DL00] Robert Dautray and Jacques-Louis Lions. Mathematical Analysis and NumericalMethods for Science and Technology, volume 6 – Evolution Problems II. Springer,2000.

[DM07] Pavel Drabek and Jaroslav Milota. Methods of Nonlinear Analysis. Birkhauser,2007.

[Dow05] Thomas J. Downar. Adaptive nodal transport methods for reactor transient analysis.Technical report, Purdue University, 2005.

[GRMK05] U. Grundmann, U. Rohde, S. Mittag, and S. Kliem. DYN3D Version 3.2, De-scription of Models and Methods. Wissenschaftlich-Technische Berichte FZR-434,Forschungszentrum Rossendorf, 2005.

[Han07] Milan Hanus. Numerical modelling of neutron flux in nuclear reactors. Faculty ofApplied Sciences, University of West Bohemia in Pilsen, June 2007. Bachelor’sThesis. Supervisor: Marek Brandner.

[HBB+08] Milan Hanus, Tomas Berka, Marek Brandner, Roman Kuzel, and Ales Matas.Three-dimensional numerical model of neutron flux in hex-z geometry. In Pro-gramy a algoritmy numericke matematiky, 2008.

[Her81] Bedrich Hermansky. Jaderne reaktory. SNTL, Praha, 1981.

[Hil75] T. R. Hill. ONETRAN: A Discrete Ordinates Finite Element Code for the Solutionof the One-Dimensional Multigroup Transport Equation. Technical report, LosAlamos Scientific Laboratory, June 1975.

[Kot07] Petri Kotiluoto. Adaptive tree multigrids and simplified spherical harmonics ap-proximation in deterministic neutral and charged particle transport. PhD thesis,

BIBLIOGRAPHY 98

University of Helsinki, 2007.

[Kul00] Teresa Kulikowska. Reactor lattice codes, 2000. Lecture given at the Workshop onNuclear Data and Nuclear Reactors: Physics, Design and Safety, Trieste, 13 March- 14 April 2000.

[Kuz09] Roman Kuzel. Private communication. [email protected], 2009.

[KYHK09] Yeong-il Kim, Jaewoon Yoo, Dohee Hahn, and Chang Hyo Kim. A conformalmapped nodal SP3 method for hexagonal core analysis. Annals of Nuclear Energy,36:498–504, 2009.

[Lat03] D. Lathouwers. Iterative computation of time-eigenvalues of the neutron transportequation. Annals of Nuclear Energy, 30:1793–1806, 2003.

[Lep07] Jaakko Leppanen. Development of a New Monte Carlo Reactor Physics Code. PhDthesis, Helsinki University of Technology, 2007.

[LMM95] Edward W. Larsen, Jim E. Morel, and John M. McGhee. Asymptotic derivation ofthe multigroup p1 and simplified pn equations with anisotropic scattering. Techni-cal report, Los Alamos National Laboratory, 1995.

[MK98] M. Mokhtar-Kharroubi. Mathematical topics in neutron transport theory. WorldScientific Pub. Co. Inc., 1998.

[MPB06] Stanislav Mıka, Petr Prikryl, and Marek Brandner. Specialnı numericke metody.Vydavatelsky servis, Pilsen, 2006.

[OA87] R. Douglas O’Dell and Raymond E. Alcouffe. Transport calculationsfor nuclearanalyses – theory and guidelines for effective use of transport codes. Technicalreport, Los Alamos National Laboratory, 1987.

[OEC07] OECD / Nuclear Energy Agency. Janis 3, 2007. Available from: http://www.

nea.fr/janis/.

[Pal97] Scott P. Palmtag. Advanced Nodal Methods for MOX Fuel Analysis. PhD thesis,Massachusetts Institute of Technology, August 1997.

[Rav00] P. Ravetto. Problems in the neutron dynamics of source-driven systems, 2000.Lecture given at the Workshop on Nuclear Data and Nuclear Reactors: Physics,Design and Safety, Trieste, 13 March - 14 April 2000.

[Reu08] Paul Reuss. Neutron Physics. EDP Sciences, 2008.

[Ric] Rice University. Arpack. Available from: http://www.caam.rice.edu/

software/ARPACK/.

[RSI81] Tennessee RSICC, Oak Ridge. MARC-PN: a neutron diffusion code system withspherical harmonics option, 1981. Available from: http://www-rsicc.ornl.

gov/codes/ccc/ccc3/ccc-311.html.

[RVCK95] Karel Rektorys, Vaclav Vilhelm, Tomas Cipra, and Frantisek Kejla. Prehled uzite

BIBLIOGRAPHY 99

matematiky I, II. Prometheus, 1995.

[Saa96] Yousef Saad. Iterative Methods for Sparse Linear Systems. PWS Publishing Com-pany, 1996.

[San01] Richard Sanchez. Some bounds for the effective multiplication factor. Annals ofNuclear Energy, 31:1207–1218, 2001.

[San02] Richard Sanchez. Treatment of boundary conditions in trajectory-based determin-istic transport methods. Nuclear Science and Engineering, 140:23–50, 2002.

[San04] G. Sansone. Orthogonal Functions. Dover Publications, 2004.

[San06] Richard Sanchez. The Criticality Eigenvalue Problem for the Transport Opera-tor with General Boundary Conditions. Transport Theory and Statistical Physics,35(5):159–185, 2006.

[San09] Richard Sanchez. Assembly homogenization techniques for core calculations.Progress in Nuclear Energy, 51:14–31, 2009.

[SM81] Richard Sanchez and Norman J. McCormick. A Review of Neutron TransportApproximations. In Critical Reviews, number 11. American Nuclear Society, 1981.

[Smi86] Kord S. Smith. Assembly homogenization techniques for Light Water Reactoranalysis. Progress in Nuclear Energy, 17:303–335, 1986.

[Smi09] Martina Smitkova. Numericke modelovanı transportu neutrono u. Master’s thesis,University of West Bohemia in Pilsen, 2009.

[Sta01] Weston M. Stacey. Nuclear Reactor Physics. John Wiley & Sons, Inc., New York,2001.

[Ste01] G. W. Stewart. Matrix Algorithms, Volume II: Eigensystems. SIAM Philadelphia,2001.

[The04] Laurent Thevenot. On the optimization of the fuel distribution in a nuclear reactor.SIAM J. Math. Anal., 35:1133–1159, 2004.

[TI91] Toshikazu Takeda and Hideaki Ikeda. 3-d neutron transport benchmarks. Technicalreport, Department Of Nuclear Engineering, Osaka University, Japan, 1991.

[Trk00] Andrej Trkov. From basic nuclear data to applications, 2000. Lecture given atthe Workshop on Nuclear Data and Nuclear Reactors: Physics, Design and Safety,Trieste, 13 March - 14 April 2000.

[Tur94] Paul J. et al. Turinsky. NESTLE few-group neutron diffusion equation solver utiliz-ing the nodal expansion method for eigenvalue, adjoint, fixed-source steady-stateand transient problems. Technical report, Electric Power Research Center, NorthCarolina State University, Raleigh, NC 27695-7909, June 1994.

[vdV92] Henk A. van der Vorst. BI-CGSTAB: A fast and smoothly converging variant ofBI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput.,

BIBLIOGRAPHY 100

13:631–644, 1992.

[Wag89] M. R. Wagner. Three-Dimensional Nodal Diffusion and Transport Theory Methodsfor Hexagonal-z Geometry. Nuclear Science and Engineering, 103:377–391, May1989.

[Wil71] M. M. R. Williams. Mathematical Methods in Particle Transport Theory. Butter-worths, London, 1971.

[WWM+01] James S. Warsa, Todd A. Wareing, Jim E. Morel, John M. McGhee, and Richard B.Lehoucq. Krylov subspace iterations for deterministic k-eigenvalue calculations.Technical report, Los Alamos National Laboratory, 2001.

[ZB02] Vyacheslav G. Zimin and Denis M. Baturin. Polynomial nodal method for solvingneutron diffusion equations in hexagonal-z geometry. Annals of Nuclear Energy,29:1105–1117, 2002.

[ZDX+06] Zhaopeng Zhong, Thomas J. Downar, Yunlin Xu, Mark L. Williams, andMark D. DeHart. Continuous-Energy Multidimensional SN Transport for Problem-Dependent Resonance Self-Shielding Calculations. Nuclear Science and Engineer-ing, 154(2):190–201, 2006.

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Appendix A

The KNK-II model problem

Numerical Modeling of Neutron Transport

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