A first course in the numerical analysis of differential equations

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A first course in the numerical analysis of differential equations / Arieh Iserles

版:: 2nd ed
資料種別:: 図書
出版情報:: Cambridge : Cambridge University Press, 2009
形態:: xviii, 459 p. ; 25 cm
シリーズ名:: Cambridge texts in applied mathematics <BA03474162>
著者名:: Iserles, A. <DA01762581>
ISBN:: 9780521734905 [0521734908] (: pbk)
書誌ID:: BA87972377

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注記:

Includes bibliographical reference and index

主題:

Differential equations--Numerical solutions

言語:

英語

目次情報:

Preface to the second edition

Preface to the first edition

Flowchart of contents

Ordinary differential equations / I：

Euler's method and beyond / 1：

Ordinary differential equations and the Lipschitz condition / 1.1：

Euler's method / 1.2：

The trapezoidal rule / 1.3：

The theta method / 1.4：

Comments and bibliography

Exercises

Multistep methods / 2：

The Adams method / 2.1：

Order and convergence of multistep methods / 2.2：

Backward differentiation formulae / 2.3：

Runge-Kutta methods / 3：

Gaussian quadrature / 3.1：

Explicit Runge-Kutta schemes / 3.2：

Implicit Runge-Kutta schemes / 3.3：

Collocation and IRK methods / 3.4：

Stiff equations / 4：

What are stiff ODEs? / 4.1：

The linear stability domain and A-stability / 4.2：

A-stability of Runge-Kutta methods / 4.3：

A-stability of multistep methods / 4.4：

Geometric numerical integration / 5：

Between quality and quantity / 5.1：

Monotone equations and algebraic stability / 5.2：

From quadratic invariants to orthogonal flows / 5.3：

Hamiltonian systems / 5.4：

Error control / 6：

Numerical software vs. numerical mathematics / 6.1：

The Milne device / 6.2：

Embedded Runge-Kutta methods / 6.3：

Nonlinear algebraic systems / 7：

Functional iteration / 7.1：

The Newton-Raphson algorithm and its modification / 7.2：

Starting and stopping the iteration / 7.3：

The Poisson equation / II：

Finite difference schemes / 8：

Finite differences / 8.1：

The finite element method / 8.2：

Two-point boundary value problems / 9.1：

A synopsis of FEM theory / 9.2：

Spectral methods / 9.3：

Sparse matrices vs. small matrices / 10.1：

The algebra of Fourier expansions / 10.2：

The fast Fourier transform / 10.3：

Second-order elliptic PDEs / 10.4：

Chebyshev methods / 10.5：

Gaussian elimination for sparse linear equations / 11：

Banded systems / 11.1：

Graphs of matrices and perfect Cholesky factorization / 11.2：

Classical iterative methods for sparse linear equations / 12：

Linear one-step stationary schemes / 12.1：

Classical iterative methods / 12.2：

Convergence of successive over-relaxation / 12.3：

Multigrid techniques / 12.4：

In lieu of a justification / 13.1：

The basic multigrid technique / 13.2：

The full multigrid technique / 13.3：

Poisson by multigrid / 13.4：

Conjugate gradients / 14：

Steepest, but slow, descent / 14.1：

The method of conjugate gradients / 14.2：

Krylov subspaces and preconditioners / 14.3：

Poisson by conjugate gradients / 14.4：

Fast Poisson solvers / 15：

TST matrices and the Hockney method / 15.1：

Fast Poisson solver in a disc / 15.2：

Partial differential equations of evolution / III：

The diffusion equation / 16：

A simple numerical method / 16.1：

Order, stability and convergence / 16.2：

Numerical schemes for the diffusion equation / 16.3：

Stability analysis I: Eigenvalue techniques / 16.4：

Stability analysis II: Fourier techniques / 16.5：

Splitting / 16.6：

Hyperbolic equations / 17：

Why the advection equation? / 17.1：

Finite differences for the advection equation / 17.2：

The energy method / 17.3：

The wave equation / 17.4：

The Burgers equation / 17.5：

Appendix Bluffer's guide to useful mathematics

Linear algebra / A.1：

Vector spaces / A.1.1：

Matrices / A.1.2：

Inner products and norms / A.1.3：

Linear systems / A.1.4：

Eigenvalues and eigenvectors / A.1.5：

Bibliography

Analysis / A.2：

Introduction to functional analysis / A.2.1：

Approximation theory / A.2.2：

Index / A.2.3：

Preface to the second edition

Preface to the first edition

Flowchart of contents

Ordinary differential equations / I：

Euler's method and beyond / 1：

Ordinary differential equations and the Lipschitz condition / 1.1：

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