A first course in the numerical analysis of differential equations

Başlık:

Yazar:

Iserles, A.

ISBN:

9780521734905

Ek Yazar:

Iserles, A.

Edition:

2nd ed.

Yayım Bilgisi:

Cambridge ; New York : Cambridge University Press, 2010.

Fiziksel Tanım:

xviii, 459 p. : ill. ; 25 cm.

Series:

Cambridge texts in applied mathematics

Series Title:

Cambridge texts in applied mathematics

Konu Terimleri:

Diferensiyel denklemler -- Sayısal çözümler.

Differential equations -- Numerical solutions.

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Library	Materyal Türü	Barkod	Yer Numarası	Durum
Searching... Pamukkale Merkez Kütüphanesi	Kitap	0061291	QA371S481 2010	Searching... Unknown

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 second 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.

Reviews (1)

Choice Review

Their considerable intrinsic interest notwithstanding, differential equations simply demand one's attention outright as a matter of continual practical necessity. In that mathematicians often have no choice but to solve them approximately using high-speed computation, one practical necessity begets another. Surprisingly then, the practical business of numerical analysis demands more than all the rigor of pure mathematics. Delicate error estimates merely provide the basic principles for having faith in one's calculations, but some applications demand better than what merely works in principle; those adept in numerical analysis must know the theorems and top that off with application-specific street smarts. Iserles (applied mathematics and theoretical physics, Univ. of Cambridge, UK) pitches his subject with the gruff zeal of a martial arts instructor, waxing alternately spiritual and worldly, passing along the secrets of survival itself to naive aspirants facing a dangerous world. In particular, he embraces a numerical analysis as an intellectual discipline and bemoans the culture of cookbooks reflected in the existing literature. Three new chapters grace this second edition coming 14 years after the first (1st ed., 1996); they cover geometric numerical integration, spectral methods, and conjugate gradients. Summing Up: Recommended. Upper-division undergraduate through professional collections. D. V. Feldman University of New Hampshire

Preface to the second edition	p. ix
Preface to the first edition	p. xiii
Flowchart of contents	p. xix
I Ordinary differential equations	p. 1
1 Euler's method and beyond	p. 3
1.1 Ordinary differential equations and the Lipschitz condition	p. 3
1.2 Euler's method	p. 4
1.3 The trapezoidal rule	p. 8
1.4 The theta method	p. 13
Comments and bibliography	p. 15
Exercises	p. 16
2 Multistep methods	p. 19
2.1 The Adams method	p. 19
2.2 Order and convergence of multistep methods	p. 21
2.3 Backward differentiation formulae	p. 26
Comments and bibliography	p. 28
Exercises	p. 31
3 Runge-Kutta methods	p. 33
3.1 Gaussian quadrature	p. 33
3.2 Explicit Runge-Kutta schemes	p. 38
3.3 Implicit Runge-Kutta schemes	p. 41
3.4 Collocation and IRK methods	p. 43
Comments and bibliography	p. 48
Exercises	p. 50
4 Stiff equations	p. 53
4.1 What are stiff ODEs?	p. 53
4.2 The linear stability domain and A-stability	p. 56
4.3 A-stability of Runge-Kutta methods	p. 59
4.4 A-stability of multistep methods	p. 63
Comments and bibliography	p. 68
Exercises	p. 70
5 Geometric numerical integration	p. 73
5.1 Between quality and quantity	p. 73
5.2 Monotone equations and algebraic stability	p. 77
5.3 From quadratic invariants to orthogonal flows	p. 83
5.4 Hamiltonian systems	p. 87
Comments and bibliography	p. 95
Exercises	p. 99
6 Error control	p. 105
6.1 Numerical software vs. numerical mathematics	p. 105
6.2 The Milne device	p. 107
6.3 Embedded Runge-Kutta methods	p. 113
Comments and bibliography	p. 119
Exercises	p. 121
7 Nonlinear algebraic systems	p. 123
7.1 Functional iteration	p. 123
7.2 The Newton-Raphson algorithm and its modification	p. 127
7.3 Starting and stopping the iteration	p. 130
Comments and bibliography	p. 132
Exercises	p. 133
II The Poisson equation	p. 137
8 Finite difference schemes	p. 139
8.1 Finite differences	p. 139
8.2 The five-point formula for ∇ 2 u = f	p. 147
8.3 Higher-order methods for ∇ 2 u = f	p. 158
Comments and bibliography	p. 163
Exercises	p. 166
9 The finite element method	p. 171
9.1 Two-point boundary value problems	p. 171
9.2 A synopsis of FEM theory	p. 184
9.3 The Poisson equation	p. 192
Comments and bibliography	p. 200
Exercises	p. 201
10 Spectral methods	p. 205
10.1 Sparse matrices vs. small matrices	p. 205
10.2 The algebra of Fourier expansions	p. 211
10.3 The fast Fourier transform	p. 214
10.4 Second-order elliptic PDEs	p. 219
10.5 Chebyshev methods	p. 222
Comments and bibliography	p. 225
Exercises	p. 230
11 Gaussian elimination for sparse linear equations	p. 233
11.1 Banded systems	p. 233
11.2 Graphs of matrices and perfect Cholesky factorization	p. 238
Comments and bibliography	p. 243
Exercises	p. 246
12 Classical iterative methods for sparse linear equations	p. 251
12.1 Linear one-step stationary schemes	p. 251
12.2 Classical iterative methods	p. 259
12.3 Convergence of successive over-relaxation	p. 270
12.4 The Poisson equation	p. 281
Comments and bibliography	p. 286
Exercises	p. 288
13 Multigrid techniques	p. 291
13.1 In lieu of a justification	p. 291
13.2 The basic multigrid technique	p. 298
13.3 The full multigrid technique	p. 302
13.4 Poisson by multigrid	p. 303
Comments and bibliography	p. 307
Exercises	p. 308
14 Conjugate gradients	p. 309
14.1 Steepest, but slow, descent	p. 309
14.2 The method of conjugate gradients	p. 312
14.3 Krylov subspaces and preconditioners	p. 317
14.4 Poisson by conjugate gradients	p. 323
Comments and bibliography	p. 325
Exercises	p. 327
15 Fast Poisson solvers	p. 331
15.1 TST matrices and the Hockney method	p. 331
15.2 Fast Poisson solver in a disc	p. 336
Comments and bibliography	p. 342
Exercises	p. 344
III Partial differential equations of evolution	p. 347
16 The diffusion equation	p. 349
16.1 A simple numerical method	p. 349
16.2 Order, stability and convergence	p. 355
16.3 Numerical schemes for the diffusion equation	p. 362
16.4 Stability analysis I: Eigenvalue techniques	p. 368
16.5 Stability analysis II: Fourier techniques	p. 372
16.6 Splitting	p. 378
Comments and bibliography	p. 381
Exercises	p. 383
17 Hyperbolic equations	p. 387
17.1 Why the advection equation?	p. 387
17.2 Finite differences for the advection equation	p. 394
17.3 The energy method	p. 403
17.4 The wave equation	p. 407
17.5 The Burgers equation	p. 413
Comments and bibliography	p. 418
Exercises	p. 422
Appendix Bluffer's guide to useful mathematics	p. 427
A.1 Linear algebra	p. 428
A.1.1 Vector spaces	p. 428
A.1.2 Matrices	p. 429
A.1.3 Inner products and norms	p. 432
A.1.4 Linear systems	p. 434
A.1.5 Eigenvalues and eigenvectors	p. 437
Bibliography	p. 439
A.2 Analysis	p. 439
A.2.1 Introduction to functional analysis	p. 439
A.2.2 Approximation theory	p. 442
A.2.3 Ordinary differential equations	p. 445
Bibliography	p. 446
Index	p. 447

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