First Course in the Numerical Analysis of Differential Equations

ISBN-10: 0521734908
ISBN-13: 9780521734905
Edition: 2nd 2008 (Revised)
List price: $77.00 Buy it from $29.82
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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  More...

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Book details

List price: $77.00
Edition: 2nd
Copyright year: 2008
Publisher: Cambridge University Press
Publication date: 11/27/2008
Binding: Paperback
Pages: 480
Size: 7.00" wide x 9.50" long x 1.25" tall
Weight: 2.068
Language: English

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.

Preface to the second edition
Preface to the first edition
Flowchart of contents
Ordinary differential equations
Euler's method and beyond
Ordinary differential equations and the Lipschitz condition
Euler's method
The trapezoidal rule
The theta method
Comments and bibliography
Exercises
Multistep methods
The Adams method
Order and convergence of multistep methods
Backward differentiation formulae
Comments and bibliography
Exercises
Runge-Kutta methods
Gaussian quadrature
Explicit Runge-Kutta schemes
Implicit Runge-Kutta schemes
Collocation and IRK methods
Comments and bibliography
Exercises
Stiff equations
What are stiff ODEs?
The linear stability domain and A-stability
A-stability of Runge-Kutta methods
A-stability of multistep methods
Comments and bibliography
Exercises
Geometric numerical integration
Between quality and quantity
Monotone equations and algebraic stability
From quadratic invariants to orthogonal flows
Hamiltonian systems
Comments and bibliography
Exercises
Error control
Numerical software vs. numerical mathematics
The Milne device
Embedded Runge-Kutta methods
Comments and bibliography
Exercises
Nonlinear algebraic systems
Functional iteration
The Newton-Raphson algorithm and its modification
Starting and stopping the iteration
Comments and bibliography
Exercises
The Poisson equation
Finite difference schemes
Finite differences
The five-point formula for &#8711;<sup>2</sup>u = f
Higher-order methods for &#8711;<sup>2</sup>u = f
Comments and bibliography
Exercises
The finite element method
Two-point boundary value problems
A synopsis of FEM theory
The Poisson equation
Comments and bibliography
Exercises
Spectral methods
Sparse matrices vs. small matrices
The algebra of Fourier expansions
The fast Fourier transform
Second-order elliptic PDEs
Chebyshev methods
Comments and bibliography
Exercises
Gaussian elimination for sparse linear equations
Banded systems
Graphs of matrices and perfect Cholesky factorization
Comments and bibliography
Exercises
Classical iterative methods for sparse linear equations
Linear one-step stationary schemes
Classical iterative methods
Convergence of successive over-relaxation
The Poisson equation
Comments and bibliography
Exercises
Multigrid techniques
In lieu of a justification
The basic multigrid technique
The full multigrid technique
Poisson by multigrid
Comments and bibliography
Exercises
Conjugate gradients
Steepest, but slow, descent
The method of conjugate gradients
Krylov subspaces and preconditioners
Poisson by conjugate gradients
Comments and bibliography
Exercises
Fast Poisson solvers
TST matrices and the Hockney method
Fast Poisson solver in a disc
Comments and bibliography
Exercises
Partial differential equations of evolution
The diffusion equation
A simple numerical method
Order, stability and convergence
Numerical schemes for the diffusion equation
Stability analysis I: Eigenvalue techniques
Stability analysis II: Fourier techniques
Splitting
Comments and bibliography
Exercises
Hyperbolic equations
Why the advection equation?
Finite differences for the advection equation
The energy method
The wave equation
The Burgers equation
Comments and bibliography
Exercises
Appendix Bluffer's guide to useful mathematics
Linear algebra
Vector spaces
Matrices
Inner products and norms
Linear systems
Eigenvalues and eigenvectors
Bibliography
Analysis
Introduction to functional analysis
Approximation theory
Ordinary differential equations
Bibliography
Index

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