A first course in the numerical analysis of differential equations:
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| Format: | Buch |
| Sprache: | English |
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Cambridge
Cambridge University Press
[2009]
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| Ausgabe: | Second Edition |
| Schriftenreihe: | Cambridge texts in applied mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke. |
| Beschreibung: | xviii, 459 Seiten Illustrationen |
| ISBN: | 9780521734905 |
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| 100 | 1 | |a Iserles, Arieh |d 1947- |0 (DE-588)172163641 |4 aut | |
| 245 | 1 | 0 | |a A first course in the numerical analysis of differential equations |c Arieh Iserles, Department of Applied Mathematics and Theoretical Physics, University of Cambridge |
| 250 | |a Second Edition | ||
| 264 | 1 | |a Cambridge |b Cambridge University Press |c [2009] | |
| 264 | 4 | |c © 2009 | |
| 300 | |a xviii, 459 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Cambridge texts in applied mathematics | |
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Datensatz im Suchindex
| _version_ | 1804138891597316096 |
|---|---|
| adam_text | Titel: A first course in the numerical analysis of differential equations
Autor: Iserles, Arieh
Jahr: 2009
Contents
Preface to the second edition page ix
Preface to the first edition xiii
Flowchart of contents xix
I Ordinary differential equations 1
1 Euler s method and beyond 3
1.1 Ordinary differential equations and the Lipschitz condition ...... 3
1.2 Euler s method............................... 4
1.3 The trapezoidal rule............................ 8
1.4 The theta method ............................. 13
Comments and bibliography........................... 15
Exercises ..................................... 16
2 Multistep methods 19
2.1 The Adams method ............................ 19
2.2 Order and convergence of multistep methods .............. 21
2.3 Backward differentiation formulae..................... 26
Comments and bibliography........................... 28
Exercises ..................................... 31
3 Runge-Kutta methods 33
3.1 Gaussian quadrature............................ 33
3.2 Explicit Runge-Kutta schemes...................... 38
3.3 Implicit Runge-Kutta schemes...................... 41
3.4 Collocation and IRK methods....................... 43
Comments and bibliography........................... 48
Exercises ..................................... 50
4 Stiff equations 53
4.1 What are stiff ODEs? ........................... 53
4.2 The linear stability domain and A-stability............... 56
4.3 A-stability of Runge-Kutta methods................... 59
vi Contents
4.4 A-stability of multistep methods..................... 63
Comments and bibliography........................... 68
Exercises ..................................... 70
5 Geometric numerical integration 73
5.1 Between quality and quantity....................... 73
5.2 Monotone equations and algebraic stability............... 77
5.3 From quadratic invariants to orthogonal flows.............. 83
5.4 Hamiltonian systems............................ 87
Comments and bibliography........................... 95
Exercises ..................................... 99
6 Error control 105
6.1 Numerical software vs. numerical mathematics............. 105
6.2 The Milne device.............................. 107
6.3 Embedded Runge-Kutta methods .................... 113
Comments and bibliography........................... 119
Exercises ..................................... 121
7 Nonlinear algebraic systems 123
7.1 Functional iteration ............................ 123
7.2 The Newton-Raphson algorithm and its
modification................................. 127
7.3 Starting and stopping the iteration.................... 130
Comments and bibliography........................... 132
Exercises ..................................... 133
II The Poisson equation 137
8 Finite difference schemes 139
8.1 Finite differences.............................. 139
8.2 The five-point formula for V2m = ƒ.................... 147
8.3 Higher-order methods for V2m = ƒ.................... 158
Comments and bibliography........................... 163
Exercises ..................................... 166
9 The finite element method 171
9.1 Two-point boundary value problems................... 171
9.2 A synopsis of FEM theory......................... 184
9.3 The Poisson equation............................ 192
Comments and bibliography........................... 200
Exercises ..................................... 201
Contents vii
10 Spectral methods 205
10.1 Sparse matrices vs. small matrices.................... 205
10.2 The algebra of Fourier expansions .................... 211
10.3 The fast Fourier transform......................... 214
10.4 Second-order elliptic PDEs ........................ 219
10.5 Chebyshev methods ............................ 222
Comments and bibliography........................... 225
Exercises ..................................... 230
11 Gaussian elimination for sparse linear equations 233
11.1 Banded systems............................... 233
11.2 Graphs of matrices and perfect Cholesky
factorization................................. 238
Comments and bibliography........................... 243
Exercises ..................................... 246
12 Classical iterative methods for sparse linear equations 251
12.1 Linear one-step stationary schemes.................... 251
12.2 Classical iterative methods ........................ 259
12.3 Convergence of successive over-relaxation ................ 270
12.4 The Poisson equation............................ 281
Comments and bibliography........................... 286
Exercises ..................................... 288
13 Multigrid techniques 291
13.1 In lieu of a justification........................... 291
13.2 The basic multigrid technique....................... 298
13.3 The full multigrid technique........................ 302
13.4 Poisson by multigrid............................ 303
Comments and bibliography........................... 307
Exercises ..................................... 308
14 Conjugate gradients 309
14.1 Steepest, but slow, descent ........................ 309
14.2 The method of conjugate gradients.................... 312
14.3 Krylov subspaces and preconditioners .................. 317
14.4 Poisson by conjugate gradients...................... 323
Comments and bibliography........................... 325
Exercises ..................................... 327
15 Fast Poisson solvers 331
15.1 TST matrices and the Hockney method................. 331
15.2 Fast Poisson solver in a disc........................ 336
Comments and bibliography........................... 342
Exercises ..................................... 344
viii Contents
III Partial differential equations of evolution 347
16 The diffusion equation 349
16.1 A simple numerical method........................ 349
16.2 Order, stability and convergence..................... 355
16.3 Numerical schemes for the diffusion equation.............. 362
16.4 Stability analysis I: Eigenvalue techniques................ 368
16.5 Stability analysis II: Fourier techniques ................. 372
16.6 Splitting................................... 378
Comments and bibliography........................... 381
Exercises ..................................... 383
17 Hyperbolic equations 387
17.1 Why the advection equation?....................... 387
17.2 Finite differences for the advection equation............... 394
17.3 The energy method............................. 403
17.4 The wave equation............................. 407
17.5 The Burgers equation........................... 413
Comments and bibliography........................... 418
Exercises ..................................... 422
Appendix Bluffer s guide to useful mathematics 427
A.I Linear algebra................................ 428
A.I.I Vector spaces............................ 428
A.1.2 Matrices............................... 429
A.1.3 Inner products and norms..................... 432
A.1.4 Linear systems........................... 434
A.1.5 Eigenvalues and eigenvectors................... 437
Bibliography ................................ 439
A.2 Analysis................................... 439
A.2.1 Introduction to functional analysis................ 439
A.2.2 Approximation theory....................... 442
A.2.3 Ordinary differential equations.................. 445
Bibliography ................................ 446
Index 447
|
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| discipline | Mathematik |
| edition | Second Edition |
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| spelling | Iserles, Arieh 1947- (DE-588)172163641 aut A first course in the numerical analysis of differential equations Arieh Iserles, Department of Applied Mathematics and Theoretical Physics, University of Cambridge Second Edition Cambridge Cambridge University Press [2009] © 2009 xviii, 459 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Cambridge texts in applied mathematics Hier auch später erschienene, unveränderte Nachdrucke. Differential equations Numerical solutions Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 s Numerische Mathematik (DE-588)4042805-9 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017363872&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Iserles, Arieh 1947- A first course in the numerical analysis of differential equations Differential equations Numerical solutions Differentialgleichung (DE-588)4012249-9 gnd Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4012249-9 (DE-588)4042805-9 |
| title | A first course in the numerical analysis of differential equations |
| title_auth | A first course in the numerical analysis of differential equations |
| title_exact_search | A first course in the numerical analysis of differential equations |
| title_full | A first course in the numerical analysis of differential equations Arieh Iserles, Department of Applied Mathematics and Theoretical Physics, University of Cambridge |
| title_fullStr | A first course in the numerical analysis of differential equations Arieh Iserles, Department of Applied Mathematics and Theoretical Physics, University of Cambridge |
| title_full_unstemmed | A first course in the numerical analysis of differential equations Arieh Iserles, Department of Applied Mathematics and Theoretical Physics, University of Cambridge |
| title_short | A first course in the numerical analysis of differential equations |
| title_sort | a first course in the numerical analysis of differential equations |
| topic | Differential equations Numerical solutions Differentialgleichung (DE-588)4012249-9 gnd Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Differential equations Numerical solutions Differentialgleichung Numerische Mathematik |
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