Nuclear Computational Science: A Century in Review 9048134102, 9789048134106

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Nuclear Computational Science

Yousry Azmy

•

Enrico Sartori

Nuclear Computational Science A Century in Review

123

Prof. Yousry Azmy North Carolina State University Department of Nuclear Engineering Raleigh, NC 27695 1110 Burlington Engineering Labs USA [email protected]

Enrico Sartori Organisation for Economic Co-operation and Development (OECD) 12 bd. des Iles 92130 Issy-les-Moulineaux France [email protected]

ISBN 978-90-481-3410-6 e-ISBN 978-90-481-3411-3 DOI 10.1007/978-90-481-3411-3 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009944067 Mathematics Subject Classification (2010): 82D75, 65C05 c Springer Science+Business Media B.V. 2010  No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Ely Gelbard November 6, 1924–April 18, 2002

Preface

Scheduled on the heels of the atomic century, the American Nuclear Society’s international topical meeting on Mathematics and Computation seemed like an opportune moment in time to capture accomplishments in this area during the first half-century of nuclear engineering. Held in a semi-secluded part of the city of Gatlinburg, Tennessee, April 6–10, 2003, this gathering of prominent experts in the field and young professionals embarking on exciting careers in what promises to develop into a nuclear renaissance turned out to be the perfect venue for such a review. The conference was co-sponsored by three divisions of the American Nuclear Society, namely the Mathematics and Computation Division, the Reactor Physics Division, and the Radiation Protection and Shielding Division. The Technical Program of the conference revolved around the theme of its title, Nuclear Mathematical & Computational Sciences: A Century in Review, A Century Anew. The Anew component comprised contributed papers organized in 25 regular and special sessions on a broad variety of topics, plus a poster session and a panel session. The Review component of the conference comprised the lecture series that grew into this book. As Technical Program Chair (YYA) and Assistant General Chair (ES) of the conference, we decided to break with the traditional format of plenary sessions standard in technical meetings and organize a lecture series that takes stock of the state of the art in nuclear computational science at the turn of a new century. Thus the concept of the lecture series that led to the chapters of this book was born. One of the first experts we solicited to present a lecture in the series was the late Dr. Ely Gelbard of Argonne National Laboratory at the time. In his gentle, but firm and persuasive manner, he declined preferring instead to participate as co-organizer of the lecture series. We jumped on the opportunity recognizing his long-standing, distinguished, and generous contributions to many subareas in nuclear computational science, and his many years of service in the field positioned him well to know the major areas to cover in the lectures and to nominate world-renowned lecturers. In short order the three of us came up with a slate of topics and a corresponding list of lecturers. The response of the nominated lecturers was supportive and enthusiastic, and by mid Fall 2001 what has later become known as the Gelbard Lecture Series was fully conceived, and a tentative idea of ultimately documenting the lecture contents in book chapters was initiated. Our charge to the invited lecturers was to provide an overview of the assigned topic aiming primarily at breadth of coverage,

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Preface

with a sharp focus on its mathematical and computational aspects. Specifically we requested that each author provide a historical perspective of the conception of their topic as a major area of research in nuclear computational science, and to identify landmarks for the evolution of the topic through the end of the twentieth century. We further requested that the lecturers delineate the current state of the art in their assigned topic and to project into the future by exposing perceived challenges and opportunities for advancing the frontier of knowledge. Our renowned lecturers did not disappoint and the lecture series was a smashing success, thanks to their dedicated effort and professionalism. The lectures, scheduled to open each half-day of the conference, were well attended, with conference participants packing the lecture hall on a consistent basis. Perhaps the only sour note that tainted the lecture series was the passing on April 18, 2002, of Dr. Ely Gelbard whose contributions to the success of the lecture series, and ultimately to the publication of this book, cannot be overstated. This great loss to the field of nuclear computational science overshadowed the conference leading to various observances of this sad event. The conference banquet included a memorial celebrating Dr. Gelbard’s life and his significant contributions to nuclear computational science, and the lecture series was named after him in recognition of his involvement that propelled the series to success. Later, the contributing authors to this book agreed to dedicate it to the memory of Dr. Ely Gelbard. Unfortunately death struck again with the passing of Dr. Richard Hwang on December 20, 2007, shortly after he completed the final revisions to his chapter appearing in this book. We are grateful for Richard’s contribution to the success of the lecture series, for the chapter he composed in this book, and for his dedication to his research over the past 5 decades. While the original list of topics envisioned in our early planning of the lecture series has not changed, the reader will notice a few differences between the lectures lineup and the chapters herein. First, Dr. Dan Cacuci who, for unforeseen circumstances, was unable to deliver his lecture on Sensitivity and Uncertainty Analysis at the conference has graciously composed the corresponding chapter for this book. Second, Dr. Kord Smith who presented the lecture on Reactor Physics at the conference apologized from composing the corresponding book chapter due to increased job-related responsibilities. We are grateful to Dr. Robert Roy for accepting to undertake such burden and for the excellent job he did in composing his chapter on Reactor Core Methods. Lastly, in composing Chapter 7, Elliott Whitesides recruited Mike Westfall and Calvin Hopper to help with the composition. This book would not have been possible without the support and active involvement of many people over the span of 6 years. Most of all we wish to thank the authors who willingly and cheerfully accepted this additional burden to their normally hectic schedules. We are confident that the benefit to the field of nuclear computational science and the gratitude of its practitioners, especially the young scientists who will carry the torch into the future, will reward the authors’ perseverance and patience during this long an arduous journey. We are grateful to Argonne National Laboratory’s Dr. Roger Blomquist for composing the memorials to Ely Gelbard and Richard Hwang, and for reviewing the final version of Richard’s

Preface

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Chapter 5. The support and encouragement of Bernadette Kirk, Director of Oak Ridge National Laboratory’s Radiation Safety Information Computational Center (RSICC) and General Chair of the Gatlinburg conference, was invaluable to the completion of this project. The technical help by Alice Rice of RSICC with bringing together the pieces of this book into a single volume is greatly appreciated. In addition, we wish to acknowledge the tacit approval and support of our respective institutions, The Pennsylvania State University and North Carolina State University (YYA), and the Nuclear Energy Agency of the Organisation for Economic Cooperation and Development (ES). June 2009

Yousry Y. Azmy Enrico Sartori

Obituary Composed by Dr. Roger Blomquist for Dr. Ely Meyer Gelbard

Ely Gelbard was born in New York City on November 6, 1924. He was the son of immigrants. His undergraduate work was at the City Colleges of New York and after World War II he earned his Ph.D. in physics from the University of Chicago. During the war, he served in the US Army Air Corps as a radar technician. He was a Senior Scientist at Argonne National Laboratory and a Fellow of the American Nuclear Society. Ely started his postgraduate career when the use of digital computers to solve the neutron balance equations for fission reactor core design and analysis was just starting to receive wide application. At Bettis (1954–1972), he participated in the efforts that put the numerical methods for the solution of the finite difference form of the neutron transport equation on a firm mathematical basis, and he devised several approximation schemes that were suitable for numerical methods and also developed efficient algorithms for their solution. While at Bettis, he earned international stature in the field, authoring important papers in many variants of the solution procedures (spherical harmonics, Sn , synthetic methods, and Monte Carlo), including the book, Monte Carlo Principles and Neutron Transport Problems, with J. Spanier. He was the first physicist at Bettis to attain the rank of Consulting Scientist, and earned the Atomic Energy Commission’s prestigious E. O. Lawrence Award. Since 1972, when Dr. Gelbard joined Argonne National Laboratory, fast reactors have been the focus of ANL’s reactor program, with its emphasis on more accurate computation of the neutron spectrum. His work in this area produced fundamental advances in the analysis of neutron streaming, collision probabilities, improvements in Monte Carlo methods, and neutron diffusion and transport within the nodal approximation. He also brought improved iterative solution strategies to bear on the equations of single-phase computational thermal-hydraulics analysis of passively safe metal-cooled reactor systems. He was consulted by many at ANL, at other labs, and at universities on a wide variety of technical issues, and invariably provided important insights. Ely’s sustained record of high productivity of the highest-quality technical work attracted a series of bright and vigorous visiting scholars and students whose participation magnified his work. He excelled at distilling complex technical issues to their essence, then performing the relevant mathematical analysis and, finally, computationally confirming the analysis. He was always careful, honest, and thoroughly

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Obituary Composed by Dr. Roger Blomquist for Dr. Ely Meyer Gelbard

scrupulous in his work. He earned the ANS Special Award for Computer Methods for the Solution of Problems in Reactor Technology, the ANS Mathematics and Computations Division Distinguished Service Award, the ANS Reactor Physics Division Eugene Wigner Award, and the University of Chicago Distinguished Performance Award. In spite of his great stature and many accomplishments, Ely was a mild and modest gentleman who always gave full credit to others’ work, and was very approachable and an excellent listener. His technical questions at meetings were insightful, probing, and gentle. He also pursued the understanding of others’ points of view in personal and political matters with both intellect and sensitivity. His restaurant adventures at meetings and other venues have provided a rich array of gastronomic experiences and many fond memories to his many friends in our profession.

The Gelbard Review Lecture Series Conducted during the American Nuclear Society’s Conference Nuclear Mathematical and Computational Sciences: A Century in Review, A Century Anew Gatlinburg, Tennessee, April 6–10, 2003

Back row: Richard Hwang, Elmer Lewis, Kord Smith, Enrico Sartori Front row: Yousry Azmy, Elliott Whitesides, Jerry Spanier, Jack Dorning, Ed Larsen

Contents

Preface .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . vii Obituary Composed by Dr. Roger Blomquist for Dr. Ely Meyer Gelbard .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . xi 1

Advances in Discrete-Ordinates Methodology .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . Edward W. Larsen and Jim E. Morel

1

2

Second-Order Neutron Transport Methods . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 85 E.E. Lewis

3

Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .117 Jerome Spanier

4

Reactor Core Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .167 Robert Roy

5

Resonance Theory in Reactor Applications . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .217 R.N. Hwang

6

Sensitivity and Uncertainty Analysis of Models and Data.. . . . .. . . . . . . . . . .291 Dan Gabriel Cacuci

7

Criticality Safety Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .355 G.E. Whitesides, R.M. Westfall, and C.M. Hopper

8

Nuclear Reactor Kinetics: 1934–1999 and Beyond . . . . . . . . . . . . . .. . . . . . . . . . .375 Jack Dorning

Index . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .459

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Chapter 1

Advances in Discrete-Ordinates Methodology Edward W. Larsen and Jim E. Morel

1.1 Introduction In 1968, Bengt Carlson and Kaye Lathrop published a comprehensive review on the state of the art in discrete-ordinates .SN / calculations [10]. At that time, SN methodology existed primarily for reactor physics simulations. By today’s standards, those capabilities were limited, due to the less-developed theoretical state of SN methods and the slower and smaller computers that were then available. In this chapter, we review some of the major advances in SN methodology that have occurred since 1968. These advances, combined with the faster speeds and larger memories of today’s computers, enable today’s SN codes to simulate problems of much greater complexity, realism, and physical variety. Since 1968, several books and reviews on general numerical methods for SN simulations have been published [32, 46, 71], but none of these covers the advanced work done during the past 20 years. The specific purpose of this chapter is to describe how the field of SN calculations has matured through the lens of three important physical problems that can be simulated today but could not be realistically simulated in 1968. By discussing these problems and the methods developed to overcome their calculational difficulties, we hope to (i) show how dramatically the field of SN simulations of the transport equation has advanced and (ii) provide an introduction to the new algorithmic techniques that have enabled these advances. An outline of the remainder of this review follows. In Section 1.2, we briefly introduce the transport equation and discuss its basic temporal (implicit), energy (multigroup), directional .SN /, and spatial (finite-difference) discretizations, together with iterative solution procedures – as of 1968. The purpose of this section is to establish notation and set the stage for the later sections, which describe more recent developments. E.W. Larsen () Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, 48109-2104 Michigan, USA e-mail: [email protected] J.E. Morel Department of Nuclear Engineering, Texas A&M University, College Station, Texas, USA e-mail: [email protected] Y. Azmy and E. Sartori, Nuclear Computational Science: A Century in Review, c Springer Science+Business Media B.V. 2010 DOI 10.1007/978-90-481-3411-3 1, 

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E.W. Larsen and J.E. Morel

Section 1.3 discusses three important physical problems that could not be simulated in 1968 but can be realistically simulated today: thermal radiation transport, charged-particle transport, and oil-well logging tool design. In Section 1.4, we discuss advanced spatial discretizations (characteristic methods, discontinuous finite-element methods [DFEMs], and nodal methods) and the asymptotic thick diffusion limit (a technique to predict the validity of SN spatial discretizations for diffusive systems with optically thick spatial cells). Section 1.5 describes advances in discretizations of the angular derivatives associated with curvilinear geometries and treatments of anisotropic scattering. Section 1.6 covers advances in angular and energy discretizations for charged particles; Section 1.7 describes advances in time discretizations. In Section 1.8, we discuss major advances in iteration acceleration: diffusionsynthetic acceleration (DSA), linear multifrequency-grey acceleration for thermal radiation transport, fission source acceleration for time-dependent calculations, and upscatter acceleration. Section 1.9 outlines the recent application of preconditioned Krylov methods. Section 1.10 concludes with a brief discussion of challenges for the future: robust finite-element methods on nonorthogonal grids, positive and monotone methods, efficient parallel sweep algorithms for unstructured grids, further development of Krylov methods for solving the SN equations, methods for charged-particle calculations with pencil-beam sources, Galerkin quadrature with positive generalized weights, and ray-effect mitigation.

1.2 Basic Concepts The physical process discussed in this chapter is the interaction of radiation with matter (radiation transport, or particle transport). The archetypical equation that describes these interactions is the linear Boltzmann equation (LBE) [2, 3, 7, 13]: 1 @ .r; ; E; t / C   r .r; ; E; t / C †t .r ; E/ .r ; ; E; t / v @t Z 1Z   D †s r; 0  ; E 0 ! E .r; 0 ; E 0 ; t / d0 dE 0 0 4 Z Z      .r; E/ 1 Q .r; E; t/ C †f r; E 0 : (1.1) r; 0 ; E 0 ; t d0 dE 0 C 4  4  0 4 

In full generality, this equation has seven independent variables: three spatial variables .r/, two direction-of-flight (or angular) variables , energy .E/, and time .t/. Particle transport problems are difficult and costly to simulate because, in part, of the high dimensionality of phase space. In this section, we discuss the basic numerical methods used to solve Eq. (1.1) in the principal large computer codes of the 1960s [4, 5, 10, 14]. We assume that the reader understands the physical meaning and basic mathematical properties of each of the terms in Eq. (1.1), and we have used notation that is broadly standard. The discussion in this section is terse; we refer the reader to standard texts [13, 71] for details.

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Advances in Discrete-Ordinates Methodology

3

The LBE given in Eq. (1.1) describes neutron transport with scattering and fission interactions. Variations of this equation primarily involve the types of interactions that are included. For instance, a gamma-ray transport equation would not have a fission term. Systems of coupled transport equations, each similar to Eq. (1.1), are required to describe the coupled transport of multiple types of particles, e.g., coupled neutron gamma-ray transport in which neutrons interact with nuclei to create gamma-rays and gamma-rays interact with nuclei to create neutrons. The principal computational difficulties associated with Eq. (1.1) are common to essentially all variations of this equation that are associated with different physical applications. In this chapter, we describe numerical methods in terms of Eq. (1.1), or simpler versions of that equation whenever possible. We consider variations of Eq. (1.1) that correspond to different physical applications only when necessary. To begin, we mention a few technical details. First, the differential scattering cross section is commonly written as a Legendre polynomial expansion: 1 X     2m C 1 Pm 0   †s;m r; E 0 ! E : †s r;   ; E ! E D 4  mD0



0



0

(1.2)

The Legendre moments †s;m are typically calculated and stored for each material region. Also, initial and boundary conditions must be specified for Eq. (1.1). If V denotes the physical system and t D 0 is the initial time, then Eq. (1.1) holds for all r 2 V;  2 4; 0 < E < 1, and t > 0. At t D 0, must be fully specified in V : .r ; ; E; 0/ D

i

.r; ; E/; r 2 V;  2 4 ; 0 < E < 1:

(1.3)

Also, must be specified on the boundary @V for directions of flight pointing into V : .r ; ; E; t/ D

b

.r; ; E; t/; r 2 @V;   n < 0; 0 < E < 1; 0 < t: (1.4)

Here, n is the unit outer normal vector at the boundary point r 2 @V . Many important algorithmic concepts can be explained most easily for problems with planar-geometry symmetry, in which the geometry and solution depend on only one spatial variable x and one angular variable  D   i . (The unit vector i points in the positive x-direction.) For a planar-geometry system 0  x  X; Eq. (1.1) simplifies to 1 @ .x; ; E; t/ @ .x; ; E; t/ C C †t .x; E/ .x; ; E; t/ v @t @x Z 1Z 1     †s x; 0 ; ; E 0 ! E x; 0 ; E 0 ; t d0 dE 0 D 0

1

.x; E/ C 2

Z 0

1Z 1 1

  †f x; E 0

  Q.x; E; t/ x; 0 ; E 0 ; t d0 dE 0 C ; (1.5) 2

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E.W. Larsen and J.E. Morel

where 1 X       2m C 1 Pm ./Pm 0 †s;m x; E 0 ! E : (1.6) †s x; 0 ; ; E 0 ! E D 2 mD0

The initial condition for Eq. (1.5) is .x; ; E; 0/ D

i

.x; ; E/; 0 < x < X; 1    1; 0 < E < 1;

(1.7)

and the boundary conditions are .0; ; E; t/ D

l

.X; ; E; t/ D

.; E; t/; 0 <   1; 0 < E < 1; 0 < t; r

(1.8a)

.; E; t/; 1   < 0; 0 < E < 1; 0  t: (1.8b)

Because D vN , where v is the particle speed and N is the particle density, physically must be non-negative. If the cross sections, inhomogeneous source, initial conditions, and boundary conditions in Eqs. (1.5) through (1.8) are all non-negative (as they must be physically), then it can be shown that the solution of these equations is non-negative. However, the positivity of does not necessarily hold when approximations (discretizations) of the LBE are imposed. A desirable feature of a discretization for the LBE is that the resulting approximate solution should be positive – or nearly so. We now sketch the basic discretization and solution methods for Eqs. (1.5) through (1.8), which existed in computer codes in the late 1960s. We begin with the discretization of time. The most widely used time-discretization technique for transport problems, even today, is implicit time differencing. For a time interval tk1=2 0 (flow from left to right), Eq. (1.52) can be analytically solved for   †t .xxj 1=2 /=n .x/ D e n

(1.52) n .x/:

 †  s †t .xxj 1=2 /=n C 1  e

j : n;j 1=2 2†t (1.53)

Using this expression, the flux exiting the j th cell is n;j C1=2

  D e †t xj =n

n;j 1=2

 †  s C 1  e †t xj =n

j ; 2†t

(1.54)

and the cell-average flux is n;j

D

 n  1  e †t xj =n n;j 1=2 †t xj   † n  s †t xj =n 1e C 1

j : †t xj 2†t

(1.55)

The exiting flux (Eq. (1.54)) is used as the incident flux for the next, i.e., .j C 1/, cell, and the cell-average flux (Eq. (1.55)) is folded into an array that, on completion of the transport sweep in all discrete directions, yields a new estimate for j .

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Advances in Discrete-Ordinates Methodology

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The solution is considered to be converged when the new value of j differs from the old value by less than a prespecified convergence criterion for all j . This algorithm, with .x/ represented as a constant in each cell, is the StepCharacteristic (SC) method [12, 25]. For 1-D problems, the SC method can be formulated as a weighted-diamond scheme, i.e., of the form described by Eqs. (1.28) and (1.30). The SC method readily generalizes to multidimensional problems on irregular grids, but in these circumstances it cannot be formulated as a weighteddiamond scheme. The SC method is currently used in several multidimensional production neutron transport codes. In applications of these codes, the spatial grid is optically thin, and accurate solutions are obtained. However, the SC method is not accurate for problems demanding optically thick meshes, e.g., of the type described in Section 1.3. Thus, more complicated characteristic methods have been proposed and implemented, in which the scattering source is represented within each spatial cell as a linear, or even quadratic function of the spatial variables [25]. As can be expected, with each increase in the polynomial order of the representation of .x/:  The accuracy of the resulting solution increases.  The computational effort required to process the extra algebraic complexity

increases.  The computer memory needed to store the extra problem unknowns increases.

1.4.2 Linear Discontinuous Method Among the most flexible and successful of the noncharacteristic methods are the discontinuous finite-element (DFE) methods [18, 70, 98, 105, 109, 116]. The linear discontinuous (LD) method is perhaps the archetypical method in this class. This method is based on representations of n .x/ and .x/ that are linear within each cell, but discontinuous at cell edges. In the LD method, Eq. (1.52) is approximated by n

d

n .x/

dx

C†t

n .x/ D

 †s 2 .0/ .1/

j C .x xj / j ; xj 1=2 < x < xj C1=2 ; 2 xj (1.56)

where xj is the center of the j th cell; now the representation of .x/ requires two unknowns per cell, j.0/ and j.1/ . If the operator on the left side of Eq. (1.56) were inverted exactly, as described previously, we would obtain a (linear) characteristic method. However, discontinuous finite-element methods are based on an approximate, rather than an exact, inversion of this operator.

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E.W. Larsen and J.E. Morel

The LD method employs the following linear-discontinuous representation of n .x/:

n .x/

D

8 ˆ ˆ ˆ
0; xj 1=2 0 and from the right for n < 0, but is discontinuous otherwise. The unknowns nj and n;j ˙1=2 in Eq. (1.57) and .0/ .1/

j and j in Eq. (1.56) are related by

j.0/ D

N X

nj wn ;

(1.58a)

nD1 .1/

j

D

X 





n;j C1=2

nj

wn C

n >0

To obtain equations for

X  nj



n;j 1=2



wn : (1.58b)

n 0 obtain n  xj

n;j C1=2

C

n;j 1=2

2

 nj

C

†t  3

n;j C1=2



 nj

D

†s .1/

: (1.61) 6 j

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Advances in Discrete-Ordinates Methodology

27

Equations ( 1.59) and ( 1.61) are two linear algebraic equations for nj and n;j ˙1=2 . [For n < 0, the term . n;j C1=2  nj / in Eq. (1.61) is replaced by . n;j  n;j 1=2 /.] The resulting LD method can be generalized to multidimensional problems on structured (rectangular or orthogonal) and unstructured (nonorthogonal) spatial grids [97]. Variants of the LD method have also been developed, such as the lumped LD method, which is more robust for optically thick spatial cells but less accurate for optically thin cells, and various corner balance methods [76, 83, 90, 96]. In general, LD-like methods are much more accurate and robust for the difficult physical problems described in Section 1.3 than finite-difference methods. LD methods require greater computer arithmetic and storage than finite-difference methods, but their increased accuracy and robustness usually more than compensates for these disadvantages.

1.4.3 Nodal Methods Another class of spatial differencing techniques that have been developed and widely used in the nuclear reactor community are nodal methods [28–30, 43–45]. In essence, a nodal method approximates a multidimensional transport equation by a coupled system of 1-D transport equations. Discretization techniques that are highly accurate for 1-D can then be utilized. To illustrate, let us consider an .x; y/-geometry version of Eq. (1.51a) on a spatial cell (assuming for simplicity Q D 0): n

@

n .x; y/

C n

@

n .x; y/

C †t

n .x; y/

D

@x @y xi 1=2 < x < xi C1=2 ; yj 1=2 < y < yj C1=2 :

†s

.x; y/; 4  (1.62)

Transversely integrating this equation by the operator <  >y;j D

1 yj

Z

yj C1=2

./ dy;

(1.63a)

>y;j ;

(1.63b)

yj 1=2

and defining n;y;j .x/

D
x;i D

1 xi

Z

xi C1=2

./dx;

(1.65a)

>x;i ;

(1.65b)

xi 1=2

and defining n;x;i .y/

D
0 and  > 0 : n;y;j .x/

D

  †t .x  xi 1=2 /  n    †t .x  xi 1=2 / 1  exp  ; n

n;y;j .xi 1=2 / exp

C

qn;y;j;i †t

(1.70)

where qn;y;j;i D

†s 4 xi

Z

xi C1=2

y;j .x/dx xi 1=2

n  yj

n;x;i .yj C1=2 /



n;x;i .yj 1=2 /



:

(1.71)

Following the same procedure for Eq. (1.69), we obtain   †t .y  yj 1=2 / n;x;i .y/ D n;x;i .yj 1=2 / exp  n    †t .y  yj 1=2 / qn;x;i;j C 1  exp  ; †t n

(1.72)

where qn;x;i;j D

†s 4 yj

Z

yj C1=2

x;i .y/ d y yj 1=2

n  xi

n;y;j .xi C1=2 /



n;y;j .xi 1=2 /



:

(1.73)

Equation (1.70) is now evaluated at x D xi C1=2 , Eq. (1.72) is evaluated at y D yj C1=2 , and the resulting two equations [together with Eqs. (1.71) and (1.73)] are solved for the outgoing edge fluxes n;x;i .yj C1=2 / and n;y;j .xi C1=2 /. This procedure constitutes the 2-D Constant-Constant Nodal (CCN) method. This method is so-named because the transverse derivative term and the scattering source are approximated as constants in each transverse-integrated equation. The more accurate (but more expensive) 2-D Linear-Linear Nodal (LLN) method has four transverse-integrated equations with linear transverse derivatives and linear scattering sources in each equation. Extending these methods to 3-D is straightforward. Nodal transport (and diffusion) methods have played an extremely important role in the nuclear engineering community during the past 20 years. As discussed in Section 1.3.3 nodal methods (on rectangular cells) were applied to oil-well logging problems in the 1980s but were abandoned for that application because of their lack of suitability to nonorthogonal grids. The applicability of nodal methods to nonorthogonal grids remains a research topic.

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E.W. Larsen and J.E. Morel

1.4.4 Solution Accuracy in the Thick Diffusion Limit The last topic in this section is not a discretization scheme, but rather a theoretical technique, which, in the past 20 years, has become essential in predicting the accuracy of discretization schemes for diffusive problems with optically thick spatial cells .†t x 1/. Early (finite difference) spatial differencing schemes for the transport equation were experimentally and theoretically understood to be accurate only when spatial cells were optically thin .†t x  1/, and the accuracy of these schemes was generally measured by the order of their truncation error [25]. For example, an nth-order scheme would satisfy k

exact



x k

D O. n /;  1;

where jj  jj is a suitable error norm and D †t x. In slab geometry, the DD and SC schemes are second-order, while the LD method is third-order. Analyses to mathematically prove the order of convergence were always carried out in slab geometry; in multidimensional geometries the SN solutions have singular characteristics, across which the solution is not smooth [20], so the truncation error analyses that can be carried out in slab geometry are not applicable. In fact, computer experiments have shown [33] that because of the singular characteristics, the order of convergence of the DD scheme in x,y-geometry depends on the definition of the error norm. Worse yet, the difficult thermal radiation and charged-particle transport problems described in Section 1.3 are so optically thick that it is impossible, because of limits in computer memory, to assign spatial grids for them that are optically thin. However, such calculations are associated with the diffusion and Fokker–Planck limits, respectively, and the spatial scale lengths for the solution associated with these limits are much larger than a mean free path. It is not unreasonable to expect a transport spatial discretization scheme to yield accurate results with optically thick cells if the scale length of the solution is well-resolved by the mesh. Indeed, one would intuitively expect to get accurate results with such mesh resolution. The difficulty is that a truncation error analysis does not provide useful information for these types of problems. Such an analysis tells us only that accurate results will be obtained by using optically thin cells. To determine if accurate results can be obtained with a mesh that is optically thick but resolves the spatial scale length of the solution, it is necessary to perform a discrete asymptotic analysis [49, 113]. Although the system is optically thick in both the diffusion and Fokker–Planck limits, the requirements associated with each limit for spatial differencing schemes are quite different. Accurate and robust spatial discretization schemes are generally required for charged-particle transport, but the highly anisotropic scattering treatment is of primary importance in the Fokker–Planck limit rather than the spatial differencing scheme [113]. In contrast, the spatial discretization is of primary importance in the diffusion limit. We focus on the diffusion limit here.

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A diffusive problem is optically thick with weak absorption; it is a problem for which the transport solution is well-approximated by the diffusion solution. A spatial discretization of the SN equations is of practical use for diffusive problems if it possesses the optically thick diffusion limit [49, 52, 66, 86, 88, 105]. Such a discretization scheme will yield accurate results for diffusive problems if the spatial mesh cells are thin with respect to a diffusion length (the spatial scale length for the diffusion solution), even if these cells are thick with respect to a mean free path. To describe the diffusion limit, let us consider the monoenergetic planargeometry SN equations n

d

n .x/

dx

C †t

n .x/

D

N †s X 2 0

n0 .x/wn0 C

n D1

Q.x/ ; 1  n  N; 2

(1.74)

and their diffusion approximation 

d 1 d

.x/ C †a .x/ D Q.x/: dx 3†t dx

(1.75)

Here, we have used the standard notation †a D †t  †s D absorption cross section; and

.x/ D

N X

n .x/wn

D scalar flux:

(1.76a)

(1.76b)

nD1

To motivate the subsequent analysis, we multiply the diffusion Eq. (1.75) by a positive constant ": d " d

.x/ C "†a .x/ D "Q.x/: (1.77)  dx 3†t dx Clearly, the solution of the diffusion equation is unchanged. This shows that if we define the following scaled cross sections and source, †t ; " †a ! "†a ;

(1.78b)

Q.x/ ! "Q.x/;

(1.78c)

†t !

(1.78a)

which implies †s D †t  †a !

†t  "†a ; "

(1.78d)

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then the diffusion equation is invariant under this scaling, for any choice of ". However, the SN equations are not invariant; they become n

d

n .x/

dx

 C

†t "





1 2

n .x/ D

†t  "†a "

Q.x/ ; 2

C"

X N

n0 .x/wn0

n0 D1

1  n  N:

(1.79)

Now, one can show that for "  1, the solution of Eqs. (1.79) satisfies n .x/

.x/ C O."/; 2

D

(1.80)

where .x/ satisfies the diffusion Eq. (1.75). To derive this result, we solve Eqs. (1.79) by assuming a solution that, for "  1, depends on " by a simple asymptotic expansion: 1 X .x/ D "i n.i / .x/: (1.81) n i 0

Introducing Eq. (1.81) into Eq. (1.79) and equating the coefficients of different powers of ", we obtain for i  0 the following system of equations: †t

.i / n .x/

!

N 1 X  2 0

.i / 0 n0 .x/wn

n D1

D n .1/

d dx

.i 1/ .x/ n



N †a X 2 0

.i 2/ .x/ wn0 n0

n D1

C ıi;2

Q.x/ ; (1.82) 2

.2/

where n .x/ D n .x/ D 0. We solve this system recursively, by solving the first .i D 0/ equation, then the second .i D 1/ equation, etc. The first .i D 0/ equation is ! N 1 X .0/ .0/ 0 D 0: (1.83) †t n .x/  n0 .x/wn 2 0 n D1

Assuming that the quadrature set satisfies N X

Z wn D

nD1

1

d D 2;

(1.84)

1

Eq. (1.83) has the general isotropic solution .0/ n .x/

D

.0/ .x/ ; 2

where .0/ .x/ is – for now – undetermined.

(1.85)

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The second .i D 1/ equation is, using Eq. (1.85), .1/ n .x/

†t

N 1 X  2 0

! .1/ 0 n0 .x/wn

D

n D1

n d .0/

.x/: 2 dx

(1.86)

Assuming that the quadrature set satisfies N X

Z n wn D

1

d D 0;

(1.87)

1

nD1

the general solution of Eq. (1.86) is .1/ n .x/

D

n d .0/

.1/ .x/  .x/; 2 2†t dx

(1.88)

where .1/ .x/ is undetermined. The third .i D 2/ equation, using Eqs. (1.85) and (1.86), is X t

.2/ n .x/

N 1 X  2 0

! .2/ 0 n0 .x/ wn

n D1

d D n dx

n d .0/ .x/

.1/ .x/  2 2†t dx

! 

Q.x/ †a .0/

.x/ C : 2 2

(1.89)

Unlike Eqs. (1.83) and (1.86), this third equation does not automatically possess a solution. To see this, we multiply Eq. (1.89) by wn and sum over 1  n  N ; assuming that the quadrature set satisfies N X

Z 2n wn D

nD1

1

2 d D 1

2 ; 3

(1.90)

we obtain the solvability condition 0D

d 1 d .0/ .x/  †a .0/ .x/ C Q.x/: dx 3†t dx

(1.91)

If this equation is satisfied, then it can easily be shown that solutions to Eq. (1.89) exist. Equations (1.91), (1.85), and (1.81) confirm the result (1.80). The asymptotic analysis outlined above provides a direct mathematical link between the SN equations (1.74) and the diffusion equation (1.75). This analysis shows that if the cross sections and source in the SN equations are scaled by Eqs. (1.78) with "  1, the diffusion equation (1.75) is obtained. The condition "  1 is

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consistent with the physical understanding of neutron diffusion: the mean free path [ D 1=†t D O."/] is small, the absorption rate [†a D O."/] is small, and the source [Q D O."/] is small – all of these “smallnesses” being balanced so that the resulting angular flux is O(1) and satisfies the diffusion equation. In the thick diffusion limit, two limits occur that have significance for spatial discretizations:   1 , 1. †t ! O "

.x/ . 2. n (x) ! 2 Thus, the total cross section becomes unbounded, yet the SN solution limits to an O.1/ diffusion solution. This result applies to the spatially continuous SN equations (no spatial discretization). We now ask: What happens if this same asymptotic analysis is applied to the spatially discretized SN equations? More precisely, let us consider a spatially discrete SN problem posed on a fixed spatial grid. We scale the cross sections and source in this problem exactly as in Eqs. (1.78). For "  1, we seek a solution of this discrete system in the form of Eqs. (1.81), i.e., we expand all unknowns (cell-average fluxes, cell-edge fluxes, etc.) as power series in ", and we solve the resulting hierarchy of equations as described above for the continuous SN equations. What happens to the spatially discrete SN solution in this limit? There are two possible answers to this question. First, because as " ! 0 the SN solution smoothly limits to the diffusion solution, it is plausible to hope that the spatially discrete SN solution will smoothly limit to the solution of a spatially discrete diffusion solution. (Then, if the chosen spatial grid is adequate to resolve the solution of this discrete diffusion problem, the resulting discrete solution will be accurate.) However, because †t D O."1 /, the optical thickness of spatial cells †t x ! 1 as " ! 0. This and the fact that SN solutions generally become inaccurate as †t x increases suggest that spatially discrete SN solutions may not limit to an accurate result as " ! 0. Which of these two possibilities is correct? The answer to this question depends on the chosen spatial discretization scheme. Some schemes are accurate in the thick diffusion limit; others are not. For example, the Step-Characteristic (SC) scheme fails as " ! 0 (the SC solution ! 0). The Diamond-Difference (DD) scheme fails unless all the diffusive regions of the problem have isotropic incident boundary fluxes (in the presence of nonisotropic boundary fluxes, DD solutions become corrupted by unphysical spatial oscillations). LD-like schemes perform successfully in the thick diffusion limit in 1-D geometries. LD methods also perform well in multi-D geometries with triangular (2-D) or tetrahedral (3-D) spatial grids, but they fail in quadrilateral (2-D) or hexahedral (3-D) grids. (However, bilinear-discontinuous methods work well for quadrilateral grids and trilinear-discontinuous methods work well for hexahedral grids.) The thick diffusion limit analysis, which has been applied to these discretization schemes and many others, accurately predicts the performance of approximation schemes in realistic calculations. This analysis has enabled the successful development of spatial discretization methods for problems with optically thick, diffusive

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systems – in particular, for the thermal radiation transport and charged-particle transport problems discussed in Section 1.3. A reader may ask that if a transport problem is diffusive, then why not solve a simpler diffusion problem instead? The answer is that in many applications, only a part of the physical system is diffusive, and it may not be obvious where this diffusive part is. Also, some energy groups may be diffusive, while others are not. Finally, for time-dependent problems, some regions of space-energy phase space may be diffusive for certain times but not for others. For these reasons, it is generally infeasible to calculate accurate transport solutions by using the diffusion approximation in subregions of phase space where it is accurate. This leads to an important issue that can be discussed only briefly here: the behavior of SN spatial discretization schemes in the presence of unresolved boundary layers. (These are thin volumes, typically only a few mean free paths in width, containing the material boundaries that separate diffusive and nondiffusive subregions of a problem.) Across boundary layers, the flux usually has a rapid spatial variation; if the spatial grid is not sufficiently fine to resolve this fast variation, the boundary layer is said to be unresolved.) Many problems exist in which, due to computer memory limitations, it is not practical to prescribe a spatial grid that adequately resolves all boundary layers. Thus, one is led to the question of whether a given discretization scheme is accurate across an unresolved boundary layer. In particular, if an optically thick, diffusive region is adjacent to a nondiffusive region, can anything be said about the ability of a given discretization scheme to predict the changes in the flux across an unresolved boundary layer between two such regions? The asymptotic thick diffusion limit analysis does make it possible to study unresolved boundary layers; the conclusions so far are that no known differencing scheme is completely adequate to model unresolved boundary layers accurately. For example, LD methods are generally inaccurate in the first cell (containing the boundary layer) within the thick diffusive region, and they incorrectly predict that the flux exiting the diffusive region is isotropic. Generally, to be certain that a discrete solution is accurate, all spatial boundary layers must be adequately resolved by the spatial grid. For charged-particle transport problems, which are optically thick and have highly forward-peaked scattering, a more complicated asymptotic limit exists in which the total cross section †t D O."1 / and the mean scattering cosine 0 D 1 O."/. As " ! 0, the solution of the continuous transport equation limits to the solution of a Fokker–Planck equation (see the discussion in Section 1.3.2). Spaceangle discretization schemes have also been successfully analyzed in this asymptotic limit [113]. Ensuring that the discretized SN equations limit to a valid discretization of the Fokker–Planck equation is primarily related to the treatment of anisotropic scattering rather than the spatial differencing scheme. Nonetheless, the presence of very large and very small eigenvalues in the spectrum of the angular Fokker–Planck operator necessitates the use of accurate and robust spatial differencing schemes in Fokker–Planck calculations. In the thick diffusion and Fokker–Planck problems discussed above, it is generally impossible, given computer memory limitations, to use optically thin spatial

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grids for the entire problem. To successfully simulate these problems, discretization schemes must produce accurate solutions for optically thick spatial grids away from boundary layers; and a theory is needed to justify the use of these schemes for these problems. The discontinuous finite-element schemes (such as LD and its variants) and the asymptotic (thick diffusion and Fokker–Planck) theories were developed to deal with just these practical difficulties.

1.5 Advances in Angular Discretizations Next we discuss (i) advances in SN discretizations for the angular derivative terms that appear in curvilinear coordinates systems and (ii) improvements to the standard SN treatment for highly anisotropic scattering. Perhaps surprisingly, very little has been accomplished during the past 40 years to successfully reduce the classic ray effects in SN simulations [21, 24].

1.5.1 Angular Derivatives The transport equation in curvilinear geometries contains one or more angular derivatives, in addition to spatial derivative terms. The traditional technique for treating the angular derivative term in the 1-D spherical geometry equation, which is representative of the traditional treatment used in essentially all curvilinear geometries, is described in Section 1.2. This technique is characterized by:  The use of special ˇ-coefficients to represent the quantity .12 / at each angular

cell edge (see Eq. (1.23a))  The use of the diamond-in-angle relationship to express each cell-average angu-

lar flux in terms of the adjacent cell-edge angular fluxes (see Eq. (1.24))  The use of a starting-direction flux equation to obtain initial values for the angular

flux at  D 1 (see Eq. (1.25))

This treatment has a deficiency, known as the discrete-ordinates flux dip, which consists of an erroneous suppression in the flux at the center of a sphere. Although the existence of the flux dip was recognized in the early 1960s, it was not eliminated until the early 1980s. Three features of the original method contributed to the existence of the flux dip:  The starting-direction flux equation is a slab-geometry equation, but this was

originally put in the following curvilinear-like form before being spatially discretized [22]: 

d 2 r dr D

1=2 N X n0 D1

C 2r

1=2

C †t .r/r 2

r 2 †s .r; 1; n0 /

1=2

n0 .r/wn0

C

r 2Q 2

(1.92)

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This was thought to make the discretization for the starting-direction flux equation consistent with that of the other directions. However, it actually contributed to truncation errors that enhanced the flux dip.  A boundary condition corresponding to specular reflection was used at the center

of the sphere, even though the angular flux at the center of a sphere is rigorously isotropic and equal to the starting-direction flux. This incorrectly allowed the angular flux at r D 0 to be anisotropic.  The diamond-in-angle equation is inconsistent with the location of the quadrature cosines within each angular bin. As a result, the diamond-in-angle scheme does not preserve solutions that are linear in . In the late 1970s, it was proposed that the slab-geometry form of the startingdirection flux equation be discretized rather than the curvilinear-like form, and that all of the angular fluxes at the center of the sphere be set equal to the startingdirection flux value [26]. These two steps significantly reduced the severity of the flux dip. In the early 1980s, an angular weighted-diamond equation was proposed that related the angular edge and average fluxes in a manner consistent with the location of the cosine in each angular bin [38]:  n .r/

D

n  n1=2 wn



 nC1=2 .r/

C

nC1=2  n wn

 n1=2 .r/:

(1.93)

When all three of these measures were combined, the resulting angular discretization scheme eliminated the flux dip [38]. This scheme has been generalized to 2-D cylindrical geometry [38]. Very few practical improvements beyond the elimination of the flux dip have been made in SN angular derivative treatments. Discontinuous finite-element discretizations might have been expected to have had an impact, but this has not happened, partly because it is difficult to develop a discontinuous angular finiteelement method that is compatible with the standard SN method in multidimensional geometries. It is interesting that discontinuous angular derivative treatments do not require a starting-direction flux. This would appear to be an advantage, but one of the few linear-discontinuous SN angular derivative treatments ever developed for the 1-D spherical geometry equation was found to be less accurate than the weighteddiamond scheme (Eq. (1.93)) for a series of test problems [60]. A reason for this is that the starting-direction flux is computed (by Eq. (1.25)) with greater accuracy than the other directions; hence, significant accuracy is actually lost if the starting-direction flux plays no role in the angular derivative treatment. However, superior accuracy relative to the weighted-diamond scheme was obtained by using a quadratic-continuous approximation in the first angular cell and using a lineardiscontinuous approximation in the remaining angular cells [60]. All of these factors make it challenging to develop advanced SN angular derivative treatments [100].

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1.5.2 Anisotropic Scattering The standard SN treatment for the scattering source, which is based on a Legendre polynomial expansion for the scattering cross section in conjunction with quadrature-generated spherical-harmonic moments of the angular flux, is still the workhorse for modern discrete-ordinates calculations, even though it is not always satisfactory. There are several reasons why it remains in widespread use:  Fundamentally different approaches usually require significant processing of raw

cross-section data.  Such techniques often have memory requirements that are significantly larger

than those of the standard treatment.  The standard technique is often much more accurate than one would expect, even

when highly truncated cross-section expansions are used in a calculation. Next, we describe the standard method, together with an improvement that has had a notable impact on charged-particle calculations [54]. For simplicity, we consider the monoenergetic 1-D slab-geometry scattering source denoted by S : Z S .x; / D

C1 1

1 X 2m C 1 Pm ./Pm .0 /†s;m 2 mD0

!



 x; 0 d0 :

(1.94)

We assume that the angular flux is a Legendre series of degree L, .x; / D where

L X 2m C 1 Pm ./ m .x/; 2 mD0

Z

m .x/ D

(1.95)

C1

1

Pm ./ .x; /d:

(1.96)

Substituting Eq. (1.95) into Eq. (1.94) and using the orthogonality of the Legendre polynomials, we find that the scattering source is S .x; / D

L X 2m C 1 Pm ./†s;m m .x/: 2 mD0

(1.97)

Thus, the scattering source generated by an angular flux that is a Legendre series of degree L is itself a Legendre series of degree no higher than L. Furthermore, the only cross-section information appearing in the scattering source is the first L C 1 moments of the scattering cross section. This same result is obtained if a cross-section expansion of degree L is used, rather than an exact expansion of infinite degree. In this case, the convergence of the cross-section expansion is irrelevant. This powerful result is not widely appreciated. An analogous result holds for multidimensional calculations when the angular flux takes the form of a finite

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spherical-harmonic expansion. These results follow from the fact that the sphericalharmonic functions (which include the Legendre polynomials) are eigenfunctions of the Boltzmann scattering operator. We now discuss how this property impacts SN calculations. For simplicity, we consider the 1-D slab-geometry scattering source. Assuming that an N -point angular quadrature set is used in conjunction with a cross-section expansion of degree N  1, the SN scattering source takes the following form: S

n .x/ D

N 1 X

2m C 1 Pm .n /†s;m m .x/; 2 mD0

where

m .x/ D

N X

Pm .n /

n .x/wn :

(1.98)

(1.99)

nD1

We assume a Gauss–Legendre quadrature set. With N quadrature points, one can uniquely interpolate those points with a polynomial of degree N  1. Furthermore, since an N -point Gauss–Legendre set exactly integrates polynomials of degree 2N  1 [19], the Legendre moments in Eq. (1.99) are exactly the moments of the interpolatory polynomial. Considering our previous results regarding the scattering source for a polynomial angular flux representation, we see that the discrete scattering source values given in Eq. (1.98) are exactly those of the scattering source generated with the polynomial interpolation for the angular flux. Thus, if the true angular flux is well-represented by the polynomial interpolation of the discrete angular flux values, the true scattering source will similarly be well-represented by the polynomial interpolation of the discrete scattering source values. We again stress that this is true regardless of the convergence of the cross-section expansion. This property does not guarantee positive discrete scattering source values, given positive discrete angular flux values, because the polynomial interpolation of the discrete angular fluxes can be negative at some points, even though the discrete values themselves are positive. However, since polynomial interpolation at the Gauss points is known to be stable, any negativities in the angular flux interpolation will be small relative to the maximum discrete angular flux value. Therefore, any negativities in the discrete scattering source values will also be small relative to the maximum discrete scattering source value. Hence, accurate SN solutions for angle-integrated quantities can be obtained in a wide variety of problems in 1-D slab geometry with highly anisotropic scattering using Gauss–Legendre quadrature, even if the scattering cross-section expansion is highly truncated. If a Gauss–Legendre quadrature set is not used, some of the scattering source moments of the interpolatory polynomial will be properly computed, but others will not, depending on the accuracy of the quadrature set. It can be seen from Eq. (1.98) that the mth moment of the scattering source is just the product of the mth moment of the scattering cross section and the mth moment of the angular flux. Thus, any flux moment that is erroneous yields a corresponding scattering source moment that

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is erroneous. This deficiency can be treated by generating a separate set of quadrature weights for each moment. In particular, for each 0  m  N  1, one can generate N weights, fwm;n gN nD1 that are defined by the N linear equations N X

Pm .n /Pj .n /wm;n D

nD1

2 ımj ; 1  j  N : 2m C 1

(1.100)

Pm .n /

(1.101)

Then Eq. (1.99) is replaced by

m .x/ D

N X

n .x/wn :

nD1

This method gives the desirable properties of Gauss quadrature to non-Gauss quadrature for the purpose of calculating the scattering source. (However, there is no guarantee that the weights generated in this way will be positive.) This is one variant of a more general technique known as Galerkin quadrature [54]. To present the more general method, we reexpress the standard SN technique for calculating the scattering source in terms of matrix algebra. In particular, we write Eqs. (1.98) and (1.99) as follows: SE D M†D E ;

(1.102)

where E is the vector of discrete angular flux values: E .

1;

2; : : : ;

N/

T

;

(1.103)

D is the N N matrix, Dm;n  Pm .n / wn ;

(1.104)

† is the N N diagonal matrix: †  diag.†0 ; †1 ; †2 ; : : :/;

(1.105)

and M is the N N matrix: Mn;m 

2m C 1 Pm .n /: 2

(1.106)

The discrete-to-moment matrix D maps a vector of discrete angular flux values to a corresponding vector of Legendre flux moments. We note from Eq. (1.104) that the first row of this matrix consists of the standard quadrature weights, because P0 ./ D 1. The matrix † is the scattering matrix in the Legendre basis, or equivalently, the scattering matrix for the PN 1 approximation. It maps a vector of Legendre flux moments to a corresponding vector of Legendre scattering source

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moments. The moment-to-discrete matrix M maps a vector of Legendre scattering source moments to a corresponding vector of discrete scattering source values. Using the orthogonal property of the Legendre polynomials, one can show that with Gauss quadrature, M D D1 , DM D

N X kD1

Di;k Mk;j D

N X kD1

Pi .k /

2j C 1 Pj .k /wk D ıi;j : 2

(1.107)

Thus, using Eq. (1.107), we can reexpress Eq. (1.102) as follows: SE D D1 †D E :

(1.108)

Equation (1.108) shows that the SN scattering matrix represents a similarity transformation of the Legendre scattering matrix, †. This means that the standard SN scattering source with Gauss quadrature (and a Legendre cross-section expansion of degree N  1) is equivalent to the scattering source of the PN 1 approximation. This is to be expected, considering the well-known equivalence between the SN and PN approximations in 1-D slab geometry [71]. If Gauss quadrature is not used, then M ¤ D1 , which is an undesirable result. The matrix D maps a vector of N discrete function values to N Legendre moments, and the matrix M maps a vector of N Legendre moments to N discrete function values. One can uniquely define a polynomial of degree N  1 either in terms of N Legendre moments or in terms of N discrete function values at N distinct points. Therefore, D and M should be inverses of one another. The moment-dependent weights defined in Eq. (1.100) ensure that this will be the case. We note that it is not necessary to actually generate the moment-dependent weights; one can directly obtain the correct matrix D simply by calculating the inverse of M. This Galerkin quadrature method is useful for 1-D calculations when quadratures with special directions are desired. For example, Lobatto and double Radau quadrature sets, which have quadrature points at  D ˙1, are particularly useful for simulating a normally incident plane-wave of radiation [54]. In 2-D and 3-D, the Galerkin quadrature method is based on spherical-harmonic interpolation of the discrete angular fluxes rather than polynomial interpolation. Choosing the correct spherical harmonics for interpolation is more complicated in multidimensions because the number of spherical-harmonic functions of order N  1 does not equal the number of discrete directions in a multidimensional SN quadrature set. Nonetheless, suitable interpolation functions have been defined for triangular quadrature sets [54]. The Galerkin quadrature method in 2-D and 3-D can be much more accurate than the standard quadrature method with highly anisotropic scattering because there is no analog of Gauss quadrature in 2-D and 3-D, i.e., there is no 2-D or 3-D quadrature set that will exactly calculate all of the sphericalharmonic moments of the interpolated angular flux. In fact, fewer than half of the moments are exactly calculated with typical sets, e.g., even-moment symmetric sets [14], etc.

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The Galerkin quadrature method can also accommodate nonpolynomial or nonspherical-harmonic interpolation functions [54]. To demonstrate this in 1-D, we consider a general interpolatory basis set for a given set of N discrete directions: N X

.x; / D

n .x/Bn ./;

(1.109)

nD1

where Bi .j / D ıij :

(1.110)

Multiplying Eq. (1.109) by Pm ./ and integrating over all directions, we obtain

m .x/ D

N X

Z n .x/

C1 1

nD1

 Pm ./Bn ./d :

(1.111)

It follows from the definition of the discrete-to-moment matrix M and Eq. (1.110) that the components of D are Z Dm;n D

C1

1

Pm ./Bn ./d:

(1.112)

Equation (1.112) is valid for all types of interpolation functions, including polynomials. We note that the first row of the discrete-to-moment matrix consists of standard quadrature weights that are exact for integrating the interpolated angular flux: Z C1 N X .x; /d D (1.113)

.x/ D n wn ; 1

where

nD1

Z wn D

C1 1

Bn ./d:

(1.114)

These are called the companion quadrature weights. Nonpolynomial interpolation requires much more computational effort to generate the discrete-to-moment matrix, because the interpolatory basis functions must be explicitly formed and their products with the Legendre polynomials must be integrated. Also, one must invert the discrete-to-moment matrix to obtain the moment-to-discrete matrix, because the standard SN expression for the moment-to-discrete matrix, Eq. (1.106), is only correct for polynomial interpolation. We refer to the scattering source obtained by operating on the interpolated angular flux with the exact scattering kernel as the exact interpolation-generated scattering source. When the interpolation functions are nonpolynomial, the exact interpolation-generated scattering source is generally not expressible in terms of the interpolation functions. Thus, if the discrete scattering source values obtained from the Galerkin quadrature method are interpolated, one generally does not obtain

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the exact interpolation-generated scattering source. Rather, one obtains a scattering source that has the same Legendre moments of degree 0 through N  1 as the exact interpolation-generated scattering source [54]. As an example of a useful nonpolynomial interpolation scheme, we consider a linear-discontinuous angular trial space in 1-D spherical geometry. Such a trial space is fully compatible with a linear-discontinuous treatment for the angular derivative term [60]. An “SN ” trial space of this type is defined to consist of N=2 equalwidth piecewise-linear segments in , where N is even and N > 2. There are two discrete angular flux unknowns per segment located p at the local Gauss S2 quadrature points, i.e., the points corresponding to ˙1= 3 obtained by linearly mapping Œ1; C1 onto each segment. A Galerkin quadrature set is generated for this trial space by exactly evaluating the Legendre angular flux moments of degree 0 through N  1 associated with the linear-discontinuous interpolation of the N discrete flux values [60]. The companion quadrature set corresponding to the Galerkin set, i.e., the standard quadrature set having the same quadrature points as the Galerkin set with quadrature weights that exactly integrate the interpolated angular flux representation, corresponds to a local Gauss S2 set on each linear segment. Since each local Gauss set exactly integrates cubic polynomials, it follows that the companion set will exactly evaluate the zeroth, first, and second Legendre moments of the interpolated angular flux. However, all higher flux moments will be inexactly evaluated, regardless of the quadrature order N . This is in contrast to the Galerkin quadrature set of order N , which always exactly evaluates the Legendre angular flux moments of degree 0 through N  1. Furthermore, because the companion quadrature set never exactly integrates polynomials of degree greater than 3, one cannot use a Legendre cross-section expansion of degree greater than 3 with the companion quadrature set (otherwise particle conservation will be lost). Thus, the accuracy of the scattering source with highly anisotropic scattering can be greatly improved for linear-discontinuous angular trial spaces in 1-D by using Galerkin quadrature. This enables one to use a linear-discontinuous approximation for the angular derivative term in 1-D spherical geometry in conjunction with an accurate treatment for highly anisotropic scattering. Perhaps the most important property of the Galerkin quadrature method, independent of the type of functions used to interpolate the discrete angular flux values, is that straight-ahead delta-function scattering is exactly treated. This has a very strong impact on charged-particle calculations, because it enables the total scattering cross section to be dramatically reduced (with an attendant decrease in the scattering ratio) while leaving the SN solution invariant. To demonstrate how straight-ahead scattering is exactly treated, let us consider the following differential scattering cross section (1.115) †s .0 / D ˛ı.0  1/; where ˛ is an arbitrary constant. The Boltzmann scattering operator associated with this cross section is ˛ times the identity operator: Z S

D

C1 1

˛ı.0  1/

 0 0  d D ˛ ./:

(1.116)

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Furthermore, the Legendre moments of this cross section are all equal to alpha: Z †m D ˛

C1 1

  ı.0  1/Pm 0 d0 D ˛Pm .1/ D ˛; 0  m  1:

(1.117)

Thus, the diagonal matrix of cross-section moments used to construct the vector of discrete scattering source values is ˛ times the identity matrix: † D ˛I:

(1.118)

Substituting from Eq. (1.118) into Eq. (1.108), and recognizing that M D D1 , we obtain SE E D M˛ID E D ˛MD E D ˛ E ; (1.119) which agrees with Eq. (1.116). We have explicitly considered only the 1-D case, but this result also applies in multidimensions. For charged particles, the scattering ratio for each group is generally very close to unity, and the mean free path is very small; nonetheless, the transport process is not diffusive. This is because the “transport-corrected” scattering ratio, .†0  †1 /=†t , is not close to unity. Within-group straight-ahead scattering is equivalent to no scattering at all, since the particle scatters into the same group and direction it had before the scattering. Thus, one can add or subtract a straight-ahead differential scattering cross section from any physically correct within-group cross section without changing the analytic transport solution. Since all Galerkin quadratures treat straight-ahead scattering exactly, one can subtract the truncated expansion for a within-group straight-ahead scattering cross section from the physically correct cross-section expansion without changing the SN solution. For instance, let us consider the total Boltzmann scattering operator (outscatter minus inscatter) associated with a 1-D SN Galerkin quadrature: †0 E  SE D ŒM†0 D  M†D E D MŒ†s  †D E D MŒdiag.†0  †0 ; †0  †1 ; : : : †0  †N 1 /D E : (1.120) Subtracting the delta-function cross section given in Eq. (1.115) from the physically correct cross section, we obtain the following modified outscatter matrix: †0 D .†0  ˛I/;

(1.121)

and the following modified inscatter matrix

where

.M†D/  D M† D;

(1.122)

† D diag.†0  ˛; †1  ˛; : : : ; †N 1  ˛/:

(1.123)

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We note that while the outscatter and inscatter matrices are modified by subtraction of the straight-ahead scattering cross section, the total Boltzmann matrix, Eq. (1.120), does not change, i.e., †0  M† D D †0  M†D:

(1.124)

Thus, the SN solution does not change. However, the convergence properties of the source iteration process can be dramatically changed. A discussion of the optimal choice of ˛ is beyond the scope of this review, but the traditional choice (which is nearly optimal) is to set ˛ D †N . This extended transport correction can greatly reduce both the total scattering cross section and the scattering ratio in relativistic charged-particle transport calculations [9, 27]. We note that the significance of this cross-section modification depends on the convergence of the cross-section expansion. If the expansion is essentially converged, †N 1 will be very small relative to †0 , resulting in a negligible reduction in †0 ; and if the cross-section expansion is highly truncated, †N 1 will be comparable to †0 , resulting in a significant reduction in †0 . Acceptable computational efficiency often cannot be achieved without the use of the extended transport correction in charged-particle calculations. Thus, with Galerkin quadrature, the extended transport correction (correctly) leaves the SN solution invariant. This is a powerful motivation for using the Galerkin method in charged-particle calculations.

1.6 Advances in Fokker–Planck Discretizations Next, we discuss advances in Fokker–Planck angle and energy discretizations for charged-particle transport. We first consider the continuous-scattering operator, and then the continuous-slowing-down operator.

1.6.1 The Continuous-Scattering Operator The continuous-scattering operator in 1-D slab geometry is   @ †r;t r @  2 .x; ; E/:  .x; ; E/ D  1 2 @ @

(1.125)

Taking the zeroth and first angular moments of Eq. (1.125), we obtain Z

C1 1

 .x; / d D 0

(1.126)

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and

Z

C1 1

 .x; /d D †r;t r J.x/;

(1.127)

respectively, where J.x/ is the current Z J.x/ D

C1

 .x; / d:

(1.128)

1

It is highly desirable for a numerical approximation to the continuous-scattering operator to preserve both the zeroth and first angular moments of that operator. Also, since the continuous-scattering operator is a diffusion operator on the unit sphere [see Section 1.3.2], it is highly desirable that the discretization for this operator yield a coefficient matrix that is symmetric and monotone. These two properties ensure that the matrix (like the analytic operator) will have only positive real eigenvalues and will yield positive solutions given positive sources. A straightforward discretization of Eq. (1.125) is     †r;t r 1  nC1  n n  n1 2 2 1  nC1=2 ;  1  n1=2 . /n D  2 wn nC1  n n  n1 (1.129) where nC1=2 D n1=2 C wn ; 1  n  N; 1=2 D 1:

(1.130)

This discretization results in a symmetric and monotone coefficient matrix and preserves Eq. (1.126) under numerical integration, but it preserves Eq. (1.127) only if each quadrature point lies at the center of its associated angular interval. (As previously noted, this never occurs with standard quadrature sets.) A discretization that does preserve Eq. (1.127) with standard quadrature sets is [40] †r;t r 1 . /n D  2 wn

 ˇnC1=2

 n n  n1  ˇn1=2 nC1  n n  n1 nC1

 ;

(1.131)

where the ˇ-coefficients are defined by Eq. (1.23b) and all else remains as previously defined. Thus the ˇ-coefficients used to admit the constant solution in the discretization of the angular derivative term in the spherical geometry transport equation are also used in the discretization of the continuous-scattering operator to preserve Eq. (1.127). This moment-preserving approach has been extended to multidimensions for product quadratures [120]. For instance, in three dimensions, the continuousscattering operator is 

†r;t r D 2



  @ 1 @2 2 @ .1   / C : @ @ 1  2 @! 2

(1.132)

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Taking the zeroth and first angular moments of Eq. (1.132), we obtain Z

2

Z

C1

dd! D 0;

 and

Z

2 

Z

C1

dd! D †r;t r J ;

 1

0

(1.133)

1

0

(1.134)

respectively, where J is the current Z

2

Z

C1

J D

 0

dd!:

(1.135)

1

Standard triangular SN quadrature sets do not represent a rectangular angular mesh on the unit sphere, but product sets do. Hence, it is reasonably straightforward to derive a discretization for the continuous-scattering operator assuming a product quadrature set. An SN product quadrature set has 2N 2 directions and is formed by the tensor product of an N -point quadrature defined over the polar cosine and a 2N point quadrature defined over the azimuthal angle. Each direction can be uniquely referenced in terms of a polar index n and an azimuthal index j . A momentpreserving discretization for the 3-D continuous-scattering operator is 2 . /n;j D 

†r;t r 6 4 2

1 p wn

 ˇnC1=2

n 1 C 1 2 wa n j

nC1;j 

 ˇn1=2

n;j

nC1 n



n;j C1 

n;j

!j C1 !j



n;j 

n1;j

n n1

n;j 

n;j 1



3 7 5:

!j !j 1

(1.136) p

Here, wn and waj are weights associated with the polar and azimuthal quadratures that sum to 2 and 2 , respectively, the ˇ-coefficients are identical to those defined for the 1-D case, and  2 K n     ; 1  n  2N; 2N 1  cos N p Kn D 2.1  2n / C cn 1  2n; n D

ˇnC1=2 dnC1=2  ˇn1=2 dn1=2 ; p wn p p 1  2nC1  1  2n D : nC1  n

cn D dnC1=2



(1.137) (1.138) (1.139) (1.140)

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The above discretization is defined only for product quadrature sets constructed with azimuthal quadrature sets of the Chebychev type: 2j  1 ; 1  j  2N; N   ; 1  j  2N: waj D N !j D  

(1.141) (1.142)

This discretization has a 5-point stencil, and is symmetric positive-definite (SPD) and monotone. It preserves Eqs. (1.133) and (1.134). The restriction to Chebychev azimuthal quadrature arises from the fact that three first-moment equations must be met, but the ˇ-coefficients and the  coefficients can only be defined to meet two of them. The ˇ-coefficients are defined to preserve the moment equation associated with the polar cosine, i.e., cos , and for a general quadrature set, the  -coefficients can preserve one of the two moment equations associated with the cosines that depend on the azimuthal angle, i.e., sin cos! or sin sin!. However, when a Chebychev azimuthal quadrature set is used, the  -coefficients can be defined to preserve both azimuthal cosines. An alternative to a finite-difference representation for the continuous-scattering operator is a Legendre moment representation. Because the total Boltzmann scattering operator (outscatter minus inscatter) and the continuous-scattering operator have the same eigenfunctions, one can define effective cross-section moments to represent the continuous-scattering operator [31]. In particular, the mth eigenvalue of the continuous-scattering operator is c;m D †r;t r

m.m C 1/ ; 2

(1.143)

while the mth eigenvalue of the total Boltzmann scattering operator is b;m D †0  †m :

(1.144)

Without loss of generality, we assume that a 1-D SN calculation is performed with Galerkin quadrature. In this case, an effective cross-section expansion of degree N  1 is required. The first step in defining the effective cross-section moments is to equate the eigenvalues defined by Eqs. (1.143) and (1.144): †e0  †em D †r;t r

m.m C 1/ ; 0  m  N  1: 2

(1.145)

This step does not uniquely define the effective cross-section moments, but rather leaves †0 a free parameter. The only consideration for choosing †0 is to minimize the effective scattering ratio. In analogy with the choice of ˛ in the extended transport correction,†0 is defined so that the last moment in the expansion is zero: †e0 D †r;t r

.N  1/N : 2

(1.146)

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Substituting from Eq. (1.146) into Eq. (1.145), we obtain an expression for the remaining effective moments: †em D †r;t r

.N  1/N  m.m C 1/ ; 1  m  N  1: 2

(1.147)

When used in conjunction with Galerkin quadrature, the number of eigenvalues preserved is always equal to the number of discrete directions. This is to be contrasted with finite-difference approximations that only preserve the zeroth and first moments. Thus, the moment representation can be considered to be more accurate than the finite-difference representations, but the coefficient matrix for the moment representation is not monotone. Thus the moment representation is less robust than the finite-difference representation.

1.6.2 The Continuous-Slowing-Down Operator We next consider the continuous-slowing down operator: C .x; ; E/ D

@ˇ .x; ; E/ : @E

(1.148)

Discretizations of this operator have advanced from a multigroup or step-like treatment [31] through a diamond treatment [41] to a linear-discontinuous finite-element treatment [47]. Because standard SN codes were not originally intended to solve the charged-particle transport problems, early efforts in treating the continuousslowing-down operator were spent defining effective cross sections to implement various discretization schemes via the standard SN scattering source representation [31, 40, 47]. Modern codes that were designed to solve the charged-particle transport equation use a linear-discontinuous finite-element discretization in space, but treat the energy derivative similar to the spatial derivatives. Thus, this operator is inverted via a space-energy sweep [101]. Self-adjoint codes must still treat the term as a scattering source, and invert it via source iteration [112]. Because the spectral radius associated with iterations on the continuous-slowing down term can be very close to unity, these iterations must be accelerated. A synthetic acceleration technique, based on the diamond-difference approximation as the low-order operator, has been used for this purpose [47]. Although the pure Fokker–Planck equation can be solved, most charged-particle calculations are carried out with the Boltzmann–Fokker–Planck equation [37]. In this case, the Boltzmann scattering operator is treated with the standard multigroup approximation. The resulting hybrid multigroup/linear-discontinuous operator is formally treated as a linear-discontinuous operator. One simply takes the scattering kernel to be piecewise-constant in energy; equivalently, one takes all energy slopes associated with the scattering kernel to be zero.

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For instance, let us assume that the angular flux within group g has the following linear-discontinuous dependence: .E/ D where

a;g

a;g

C

e;g

 2  E  Eg ; EgC1=2  E < Eg1=2 ; Eg

is the group average flux:

a;g

e;g

(1.149)

D

1 Eg

Z

Eg1=2

.E/ dE;

(1.150)

EgC1=2

is the group energy slope:

e;g

D

6 Eg2

Z

Eg1=2

  E  Eg .E/ dE;

(1.151)

EgC1=2

and Eg D Eg1=2  EgC1=2 is the group width for group g. We note from Eq. (1.149) that the angular flux at the interface energy between two groups is defined by the solution in the higher energy group. Since the continuous-slowing-down operator causes particles to lose energy, this choice is consistent with the direction of particle flow in energy. Under the assumptions of a Legendre expansion of degree L and 0 energy slopes for the multigroup scattering kernel, and a constant dependence of the restricted stopping power within each group, the discretized 1-D transport equation takes the following form for group g: 

@

a;g

@x

C †t;g

a;g

D

G X L X 2m C 1 m Eg 0 †g 0 !g

a;m;g 0 Pm ./ C Qa;g 2 Eg 0 mD1

g D1

1  ˇr;g1 . a;g1  e;g1 /  ˇr;g . a;g  e;g / ; C Eg   3  @ e;g C †t;g e;g D Qe;g C ˇr;g1 a;g1  e;g1  @x Eg  2ˇr;g a;g C ˇr;g a;g 

(1.152a)

 e;g

:

(1.152b) Here, 'a;m;g denotes the group average Legendre flux moment of degree m for group g, Qa;g and Qe;g respectively denote the group average source and group source energy slope for group g, and †m g 0 !g is the standard multigroup-Legendre coefficient of degree m for a transfer from group g0 to group g: Z Eg1=2 Z Eg0 1=2 Z C1   1 D †s E 0 ! E; 0 Pm .0 /d0 dE 0 dE: †m g 0 !g Eg 0 EgC1=2 Eg0 C1=2 1 (1.153)

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Equations (1.152a) and (1.152b) are solved via source iteration, but the continuousslowing-down operator is inverted during the sweep. In particular, the source iteration process takes the form 

@

.`C1/ a;g

.`C1/ C †t;g a;g @x  1 h .`C1/ ˇr;g1 a;g1   Eg

D Qa;g C C @

.`C1/ e;g

@x



 ˇr;g



.`C1/ a;g



.`C1/ e;g

i

L X 2m C 1 m .`/ †g!g a;m;g Pm ./ 2 mD1

G X g 0 DgC1



.`C1/ e;g1

C †t;g 2ˇr;g

L X 2m C 1 m Eg 0 .`C1/ †g 0 !g

0 Pm ./; 2 Eg a;m;g mD1

 3 h ˇr;g1 Eg  .`C1/ .`C1/ Cˇ  r;g a;g a;g

.`C1/ e;g



.`C1/ a;g1

 i

.`C1/ e;g

.`C1/ e;g1

(1.154a)



D Qe;g ; (1.154b)

where ` is the iteration index. If the full linear-discontinuous treatment for the scattering source were used, one would have nonzero scattering source energy slopes. In this case, one would iterate on both the scattering source averages and energy slopes. If the scattering ratio for a given group is sufficiently large to require convergence acceleration, the scattering source energy slopes could require acceleration in addition to the scattering source averages. Accelerating the source energy slopes significantly complicates the acceleration process. (We will address this point again, regarding the convergence acceleration of the temporal scattering source slopes associated with a linear-discontinuous discretization of the time derivative.) When combining the linear-discontinuous energy approximation with discontinuous finite-element approximations in space, one generally assumes a single energy slope per spatial cell rather than a separate energy slope for each spatial unknown within a cell. Although the latter assumption is more accurate, it can be excessively expensive. For example, if a trilinear-discontinuous spatial approximation is used for a 3-D rectangular cell, one gets eight spatial unknowns per cell per angle per group. If a separate energy slope is used for each spatial unknown, the number of unknowns per cell per angle per group increases to 16; but if only a single energy slope is used for the entire spatial cell, the number of unknowns only increases to nine. The linear-discontinuous discretization of the continuous-slowing-down operator represents a major improvement relative to step and diamond-difference discretizations [47]. In particular, the linear-discontinuous method is much less numerically diffusive than the step method and much less oscillatory than the diamond method. Nodal methods can also be applied to the continuous-slowing down operator. One would expect such methods to be comparable to discontinuous finite-element methods. However, because they have rarely been used in practice, we will not explicitly discuss nodal methods here.

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1.7 Advances in Time Discretizations Next, we discuss advanced discretization techniques for the time derivative. As is the case for most of the derivatives terms in the Boltzmann equation, the time derivative has been treated with the discontinuous finite-element method [91,98] and the nodal method [59]. The linear-discontinuous method assumes an angular flux dependence of the following form over the kth time step: .t/ D where

k a

C

k a

k t

 2  k t  t ; t k1=2 < t  t kC1=2 ; t k

is the average flux: k a

k t

(1.155)

D

1 t k

Z

t kC1=2

.t/dt ;

(1.156)

  t  tk .t/dt ;

(1.157)

t k1=2

is the temporal slope: k t

6 D .t k /2

Z

t kC1=2 t k1=2

t k D .t k1= 2 C t k1= 2 / = 2 is the midpoint of the time step, and t k D .t kC1= 2  t k1= 2 / is the width of the time step. We note from Eq. (1.155) that the angular flux at the interface between two time steps is defined by the solution from the previous time step. For simplicity, let us consider the 1-D time-dependent slab-geometry monoenergetic transport equation with isotropic scattering and an isotropic inhomogeneous source: @ Q †s 1@ C C †t D

C : (1.158) v @t @x 2 2 We obtain the following equations after applying the linear-discontinuous finiteelement approximation in time to Eq. (1.158): 1 h vt k

k a

C

k t

3 h vt k

k a

C

k t

D





†s k

C Qtk ; 2 t





2

k1 a

k a

C

C 

k1 t

k1 a

i

C

C

k1 t

@ k C †t @x i

C

k

D

†s k

C Qak ; 2 (1.159a)

@ tk C †t;g @x

k t

(1.159b)

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where Qak and Qtk respectively denote the source temporal average and source temporal slope for time step k. Equations (1.159a) and (1.159b) can be simultaneously solved via source iteration:   i 1 h k;.`C1/ k;.`C1/ k1 k1  C C t a a t vt k k;.`C1/ @ a † s k;.`/ C †t ak;.`C1/ D

C C Qak ; (1.160a) @x 2 a  i  3 h k;.`C1/ k;.`C1/ k;.`C1/ k1 k1  2 C C C t a a a t vt k k;.`C1/ @ †s k;.`/ C †t;g tk;.`C1/ D

C t C Qtk ; (1.160b) @x 2 t where ` is the iteration index. We note that one must iterate on both the temporal averages and the temporal slopes of the scattering source. If the scattering ratio is close to unity, the source iterations for both the averages and slopes must be accelerated. Deriving fully consistent diffusion acceleration equations from Eqs. (1.160a) and (1.160b) yields a complicated and difficult-to-solve system of coupled diffusion equations. If one uses step or diamond differencing in time, the diffusion-synthetic acceleration algorithm requires the solution of only one diffusion equation and is essentially identical to the algorithm for steady-state calculations. An approximate method has been developed in which the fully coupled system of diffusion acceleration equations associated with the linear-discontinuous temporal discretization scheme is replaced by two independent diffusion equations [98]. This approximate method appears to work quite well, resulting in a cost increase for performing DSA of about a factor of 2 relative to that associated with traditional temporal differencing schemes. While the linear-discontinuous finite-element approximation in time is more accurate than the step scheme and more robust than the diamond scheme, it is also more expensive. As with the continuous-slowing down operator, when one combines the linear-discontinuous temporal approximation with discontinuous finite-element approximations in space, one generally assumes a single temporal slope per spatial cell rather than a separate temporal slope for each spatial unknown within a cell. Although the latter assumption is more accurate, it can be excessively expensive. As we noted before, if a trilinear-discontinuous spatial approximation is used for a 3-D rectangular cell, one gets eight spatial unknowns per cell per angle per group. If a separate temporal slope is used for each spatial unknown, the number of unknowns per cell per angle per group increases to 16; but if only a single temporal slope for the entire spatial cell is used, the number of unknowns only increases to nine. In analogy with the derivation of Eqs. (1.70) and (1.72), we apply the constantconstant nodal method to Eq. (1.158) to obtain the following equations for  > 0 (assuming for simplicity Q D 0):   h i  k1=2 k1=2 t exp † .t/ D v t  t n;x;i n;x;i t h io  qn;x;i;k n 1  exp †t v .t  t k1=2 ; (1.161) C †t

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where qn;x;i;k D

†s 4 t k

Z

t kC1=2 t k1=2

and n;t;k .x/ D

where †s qn;t;k;i D 4 xi

Z

n  xi

n;t;k

  xi C1=2 



n;t;k .xi 1=2 /

t;k .x/ dx 

;

(1.162)

  †t .x  xi 1=2 / .x / exp  n;t;k x1=2 n    †t .x  xi 1=2 / qn;t;k;i C 1  exp  ; †t n

xi C1=2 xi 1=2

x;i .t/ dt

1 h vt k

n;x;i

  t kC1=2 

n;x;i

(1.163)

 i t k1=2 : (1.164)

The considerations for applying nodal methods in time are analogous to those for applying discontinuous finite-element methods. For instance, with a linear nodal method, one must be concerned with accelerating the temporal slopes of the scattering source. With a linear nodal method in both time and space [59], there would be only one temporal slope per space cell, but if one were to apply a nodal method in time in conjunction with another type of spatial discretization, multiple temporal slopes per space cell could arise. The practical need for advanced temporal discretization schemes relative to traditional discretization schemes is not as strong for the time variable as for the space and energy variables. This is due to the relative ease with which adaptive techniques can be applied to time integration, making it feasible to avoid the regimes in which simple discretization schemes perform poorly. In any event, the transport community has little experience with advanced temporal discretization schemes, and little research has been performed in this area.

1.8 Advances in Iteration Acceleration Next, we discuss major advances in iteration acceleration. A plethora of SN iterative acceleration techniques have been developed over the years (we refer the reader to the recent comprehensive review by Adams and Larsen [110]), but there is little doubt that the practical application of diffusion-synthetic acceleration (DSA) to source iteration has been the most significant advance in iteration acceleration techniques in the history of discrete-ordinates methods. Early attempts to utilize DSA were marred by successful performance only for problems with optically thin spatial grids [16]. Later, the subtleties concerning how these methods should be discretized became understood. To motivate the DSA and DSA-like methods, we first discuss the Fourier analysis technique, which has become an invaluable theoretical tool for predicting the convergence rate of iterative solutions of continuous and discrete problems.

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1.8.1 Fourier Analysis The Source Iteration (SI) method is described in Section 1.2; see Eqs. (1.31) and (1.32). For a model infinite, homogeneous-medium transport problem with no discretization, 1 @ .x; / C †t .x; / D Œ†s .x/ C Q.x/; @x 2 Z 1  

.x/ D x; 0 d0 ;



(1.165a) (1.165b)

1

the SI process begins with an initial guess .0/ .x/ of the scalar flux, and then for `  1, the `th source iteration is defined by 

@

.`/

.`1=2/

.x; /

C †t

@x .x/ D

.`1=2/

Z .`1=2/

.x; / D

1

.x/ 

.`1=2/



i 1h †s .`1/ .x/ C Q.x/ ; (1.166a) 2

 x; 0 d0 :

(1.166b)

1

We now write an exact transport equation for the scalar and angular flux errors: ı .`1/ .x/ D .x/  .`1/ .x; /; ı

.`1=2/

.x; / D

.x; / 

(1.167a)

.`1=2/

.x; /:

(1.167b)

To do this, we subtract Eqs. (1.166) from Eqs. (1.165), obtaining 

@ı

ı

.`1=2/

.x; /

@x .`/

.x/ D ı

C †t ı

.`1=2/

Z .`1=2/

.x; / D

1

.x/ 

ı

.`1=2/



1 †s ı .`1/ .x/; (1.168a) 2

 x; 0 d0 ;

(1.168b)

1

which define ı .`/ .x/ in terms of ı .`1/ .x/. Clearly, the rate of convergence of Eq. (1.166) is equal to the rate at which ı .`/ .x/ ! 0. To calculate this rate, we introduce the Fourier transforms Z 1 .`1/ ı

.x/ D a.`1/ ./e i †t x d; (1.169a) 1

ı

.`1=2/

.x; / D

Z

1 1

b .`1=2/ .; /e i †t x d;

(1.169b)

56

E.W. Larsen and J.E. Morel

into Eqs. (1.168) to obtain c .i  C 1/b .`1=2/ .; / D a.`1/ ./; 2 Z 1   a.`/ ./ D b .`1=2/ ; 0 d0 ;

(1.170a) (1.170b)

1

where c D †s =†t D scattering ratio. Equation (1.170a) gives b .`1=2/ .; / D

1 c a.`1/ ./ ; 2 1 C i 

(1.171)

and then Eq. (1.170b) gives a.`/ ./ D

  Z 1 d0 c a.`1/ ./ 2 1 1 C i 0

D ! ./ a.`1/ ./ D    D Œ!./` a.0/ ./; where c !./ D 2

Z1 1

d0 c D tan1 ./ 1 C i 0 

(1.172a)

(1.172b)

is the iteration eigenvalue. Equations (1.172) and (1.169a) yield Z1 ı

.`/

.x/ D

! ` ./a.0/ ./e i †t x d:

(1.173)

1

Thus, the rate at which the Fourier mode corresponding to wave number  limits to zero is determined by !./. If j!./j  1, the corresponding mode converges rapidly. If j!./j < 1 and j!./j  1, the mode converges slowly. If j!./j  1, the mode does not converge. The overall rate of convergence is determined by the most slowly converging error mode, i.e., the largest value of j!./j over the Fourier variable . For large `, Eq. (1.173) implies



.`/

ı .x/  ` A;

(1.174)

where A is a constant, and

.`/

ı .x/



D sup j!./j D lim `!1 ı .`1/ .x/

1 0; n > 0; n > 0/, and assume that the cell-centered flux is constant inside the volume and that the surface (or cell-edged) fluxes are also constant over each of the cell’s six faces. The balance Eq. 4.21 can be approximated by n n n x ˆn C y ˆn C z ˆn D Qijk  †ijk ˆn;ijk xi yj zk

(4.22)

where constant differences are taken for each coordinate: 8 C  ˆ ˆ