A Student’s Guide to Numerical Methods

✍ Scribed by Ian H. Hutchinson

• Tongue: English
• Leaves: 260
• Category: Library

✦ Table of Contents

Halftitle page
Title page
Dedication
Copyright page
Contents
Preface
1 Fitting functions to data
1.1 Exact fitting
1.2 Approximate fitting
1.3 Tomographic image reconstruction
1.4 Efficiency and non-linearity
2 Ordinary differential equations
2.1 Reduction to first order
2.2 Numerical integration of initial-value problems
2.3 Multidimensional stiff equations: implicit schemes
2.4 Leap-Frog schemes
3 Two-point boundary conditions
3.1 Examples of two-point problems
3.2 Shooting
3.3 Direct solution
3.4 Conservative differences, finite volumes
4 Partial differential equations
4.1 Examples of partial differential equations
4.2 Classification of partial differential equations
4.3 Finite-difference partial derivatives
5 Diffusion. Parabolic partial differential equations
5.1 Diffusion equations
5.2 Time-advance choices and stability
5.3 Implicit advancing matrix method
5.4 Multiple space dimensions
5.5 Estimating computational cost
6 Elliptic problems and iterative matrix solution
6.1 Elliptic equations and matrix inversion
6.2 Convergence rate
6.3 Successive over-relaxation
6.4 Iteration and non-linear equations
7 Fluid dynamics and hyperbolic equations
7.1 The fluid momentum equation
7.2 Hyperbolic equations
7.3 Finite differences and stability
8 Boltzmann’s equation and its solution
8.1 The distribution function
8.2 Conservation of particles in phase-space
8.3 Solving the hyperbolic Boltzmann equation
8.4 Collision term
9 Energy-resolved diffusive transport
9.1 Collisions of neutrons
9.2 Reduction to multigroup diffusion equations
9.3 Numerical representation of multigroup equations
10 Atomistic and particle-in-cell simulation
10.1 Atomistic simulation
10.2 Particle-in-cell codes
11 Monte Carlo techniques
11.1 Probability and statistics
11.2 Computational random selection
11.3 Flux integration and injection choice
12 Monte Carlo radiation transport
12.1 Transport and collisions
12.2 Tracking, tallying, and statistical uncertainty
13 Next steps
13.1 Finite-element methods
13.2 Discrete Fourier analysis and spectral methods
13.3 Sparse-matrix iterative Krylov solution
13.4 Fluid evolution schemes
Appendix A Summary of matrix algebra
A.1 Vector and matrix multiplication
A.2 Determinants
A.3 Inverses
A.4 Eigenanalysis
References
Index