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A First Course in the Numerical Analysis of Differential Equations

Book Description

Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The exposition maintains a balance between theoretical, algorithmic and applied aspects. This new edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; and a variety of algorithms to solve large, sparse algebraic systems.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface to the second edition
  7. Preface to the first edition
  8. Flowchart of contents
  9. I: Ordinary differential equations
    1. 1. Euler’s method and beyond
      1. 1.1 Ordinary differential equations and the Lipschitz condition
      2. 1.2 Euler’s method
      3. 1.3 The trapezoidal rule
      4. 1.4 The theta method
      5. Comments and bibliography
      6. Exercises
    2. 2. Multistep methods
      1. 2.1 The Adams method
      2. 2.2 Order and convergence of multistep methods
      3. 2.3 Backward differentiation formulae
      4. Comments and bibliography
      5. Exercises
    3. 3. Runge–Kutta methods
      1. 3.1 Gaussian quadrature
      2. 3.2 Explicit Runge–Kutta schemes
      3. 3.3 Implicit Runge–Kutta schemes
      4. 3.4 Collocation and IRK methods
      5. Comments and bibliography
      6. Exercises
    4. 4. Stiff equations
      1. 4.1 What are stiff ODEs?
      2. 4.2 The linear stability domain and A-stability
      3. 4.3 A-stability of Runge–Kutta methods
      4. 4.4 A-stability of multistep methods
      5. Comments and bibliography
      6. Exercises
    5. 5. Geometric numerical integration
      1. 5.1 Between quality and quantity
      2. 5.2 Monotone equations and algebraic stability
      3. 5.3 From quadratic invariants to orthogonal flows
      4. 5.4 Hamiltonian systems
      5. Comments and bibliography
      6. Exercises
    6. 6. Error control
      1. 6.1 Numerical software vs. numerical mathematics
      2. 6.2 The Milne device
      3. 6.3 Embedded Runge–Kutta methods
      4. Comments and bibliography
      5. Exercises
    7. 7. Nonlinear algebraic systems
      1. 7.1 Functional iteration
      2. 7.2 The Newton–Raphson algorithm and its modification
      3. 7.3 Starting and stopping the iteration
      4. Comments and bibliography
      5. Exercises
  10. II: The Poisson equation
    1. 8. Finite difference schemes
      1. 8.1 Finite differences
      2. 8.2 The five-point formula for ∇2u × f
      3. 8.3 Higher-order methods for ∇2u × f
      4. Comments and bibliography
      5. Exercises
    2. 9. The finite element method
      1. 9.1 Two-point boundary value problems
      2. 9.2 A synopsis of FEM theory
      3. 9.3 The Poisson equation
      4. Comments and bibliography
      5. Exercises
    3. 10. Spectral methods
      1. 10.1 Sparse matrices vs. small matrices
      2. 10.2 The algebra of Fourier expansions
      3. 10.3 The fast Fourier transform
      4. 10.4 Second-order elliptic PDEs
      5. 10.5 Chebyshev methods
      6. Comments and bibliography
      7. Exercises
    4. 11. Gaussian elimination for sparse linear equations
      1. 11.1 Banded systems
      2. 11.2 Graphs of matrices and perfect Cholesky factorization
      3. Comments and bibliography
      4. Exercises
    5. 12. Classical iterative methods for sparse linear equations
      1. 12.1 Linear one-step stationary schemes
      2. 12.2 Classical iterative methods
      3. 12.3 Convergence of successive over-relaxation
      4. 12.4 The Poisson equation
      5. Comments and bibliography
      6. Exercises
    6. 13. Multigrid techniques
      1. 13.1 In lieu of a justification
      2. 13.2 The basic multigrid technique
      3. 13.3 The full multigrid technique
      4. 13.4 Poisson by multigrid
      5. Comments and bibliography
      6. Exercises
    7. 14. Conjugate gradients
      1. 14.1 Steepest, but slow, descent
      2. 14.2 The method of conjugate gradients
      3. 14.3 Krylov subspaces and preconditioners
      4. 14.4 Poisson by conjugate gradients
      5. Comments and bibliography
      6. Exercises
    8. 15. Fast Poisson solvers
      1. 15.1 TST matrices and the Hockney method
      2. 15.2 Fast Poisson solver in a disc
      3. Comments and bibliography
      4. Exercises
  11. III: Partial differential equations of evolution
    1. 16. The diffusion equation
      1. 16.1 A simple numerical method
      2. 16.2 Order, stability and convergence
      3. 16.3 Numerical schemes for the diffusion equation
      4. 16.4 Stability analysis I: Eigenvalue techniques
      5. 16.5 Stability analysis II: Fourier techniques
      6. 16.6 Splitting
      7. Comments and bibliography
      8. Exercises
    2. 17. Hyperbolic equations
      1. 17.1 Why the advection equation?
      2. 17.2 Finite differences for the advection equation
      3. 17.3 The energy method
      4. 17.4 The wave equation
      5. 17.5 The Burgers equation
      6. Comments and bibliography
      7. Exercises
  12. Appendix Bluffer’s guide to useful mathematics
    1. A.1 Linear algebra
      1. A.1.1 Vector spaces
      2. A.1.2 Matrices
      3. A.1.3 Inner products and norms
      4. A.1.4 Linear systems
      5. A.1.5 Eigenvalues and eigenvectors
      6. Bibliography
    2. A.2 Analysis
      1. A.2.1 Introduction to functional analysis
      2. A.2.2 Approximation theory
      3. A.2.3 Ordinary differential equations
      4. Bibliography
  13. Index