Numerical Methods of Reactor Analysis

Nuclear Science and Technology, Volume 3: Numerical Methods of Reactor Analysis presents the numerical analysis frequently used in the nuclear reactor field. This book discusses the numerical approximation for the multigroup diffusion method, which results in simple algebraic equations. Organized into six chapters, this volume starts with an overview of the simplified formulation of linear algebra by defining the matrices and operations with matrices. This text then discusses the properties of special matrices and reviews the elementary properties of finite difference equations. Other chapters consider a variety of methods of obtaining numerical solutions to the approximating equations. The final chapter deals with Monte Carlo method, which is a statistical method for solving statistical or deterministic problems. This book is a valuable resource for nuclear engineers. Students at the graduate level who had an introductory course in reactor physics and a basic course in differential equations will also find this book useful.

PrefaceChapter I Linear Equations and Matrix Algebra 1.1 Linear Equations and Matrix Notation 1.2 Matrix Operations 1.3 Determinants 1.4 Solution of Simultaneous Equations 1.5 Special Matrices and Their Properties 1.6 Vector Interpretation 1.7 Matrix Functions and Similarity Transformations 1.8 Linear Independence of Vectors and Orthogonalization of Vectors 1.9 Eigenvalues and Eigenvectors 1.10 Nonsymmetric Matrices 1.11 Geometric Interpretation 1.12 Biorthogonal Vectors 1.13 Nonnegative Matrices 1.14 Special Forms and Matrix Factorization References ProblemsChapter II Difference Equations 2.1 A Simple Example 2.2 Difference and Summation Operators 2.3 Formation of Difference Equations and Truncation Error 2.4 Analytic Solution of Difference Equations 2.5 Partial Difference Equations 2.6 Convergence of Difference Solutions 2.7 Matrix Form of Difference Equations References ProblemsChapter III Numerical Solutions of Equations 3.1 Numerical Integration 3.2 Ordinary Differential Equations 3.3 Partial Differential Equations 3.4 Hyperbolic Equations 3.5 Parabolic Equations 3.6 Elliptic Equations and Iterative Methods References ProblemsChapter IV Multigroup Diffusion Methods 4.1 Age-Diffusion Approximation 4.2 Adjoint Equations 4.3 Formation of Multigroup Equations 4.4 Adjoint Multigroup Equations 4.5 Multigroup Difference Equations 4.6 Matrix Form of Multigroup Equations 4.7 Numerical Solution of the Multigroup Equations References ProblemsChapter V Transport Methods 5.1 The PN Approximation 5.2 Double PN Approximation 5.3 Multigroup Transport Methods 5.4 Discrete Ordinate Methods 5.5 The SN Method 5.6 Time Dependent Transport Methods 5.7 Moments Method References ProblemsChapter VI The Monte Carlo Method 6.1 Introduction 6.2 Random Numbers 6.3 Distribution Functions 6.4 Statistical Estimation 6.5 Analogs of Two Simple Problems 6.6 Monte Carlo Calculation of the Fast Fission Factor 6.7 Variance Reduction Methods 6.8 Concluding Remarks References ProblemsAppendix A The Boltzmann Transport Equation Text References ProblemsAppendix - Velocity Relations for Nuclear Events B.1 Kinematical Relations B.2 Conservation of Momentum B.3 Conservation of Energy B.4 Relation between the Initial and Final Speeds and the Angle of Scattering B.5 Relation between the Scattering Angles in the Laboratory and the Center-of-Mass System B.6 Relations among the Scattering Angles and the Initial and Final Energies B.7 Relations between the Direction Cosines of the Velocity of a Scattered Neutron in the Laboratory System and in a Center-of-Mass System B.8 Relations between the Direction Cosines of a Scattered Neutron in Two Center-of-Mass Systems B.9 Transfer Probabilities for Elastic, Isotropic Scattering and Fission References ProblemsAppendix C Moments Method for Neutrons Text ReferencesAppendix D Special Functions Text ReferencesIndex