Analysis of discretization methods for ordinary differential equations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1973
|
| Schriftenreihe: | Springer tracts in natural philosophy
23 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVI, 388 S. graph. Darst. |
| ISBN: | 3540060081 0387060081 |
Internformat
MARC
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| 100 | 1 | |a Stetter, Hans J. |d 1930- |e Verfasser |0 (DE-588)108696219 |4 aut | |
| 245 | 1 | 0 | |a Analysis of discretization methods for ordinary differential equations |c Hans J. Stetter |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1973 | |
| 300 | |a XVI, 388 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Springer tracts in natural philosophy |v 23 | |
| 650 | 7 | |a Equations aux différences - Solutions numériques |2 ram | |
| 650 | 7 | |a Equations différentielles - Solutions numériques |2 ram | |
| 650 | 4 | |a Difference equations |x Numerical solutions | |
| 650 | 4 | |a Differential equations |x Numerical solutions | |
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Datensatz im Suchindex
| _version_ | 1804116387371679744 |
|---|---|
| adam_text | Table of Contents
Chapter 1 General Discretization Methods 1
1.1. Basic Definitions 1
1.1.1 Discretization Methods 1
1.1.2 Consistency 5
1.1.3 Convergence 7
1.1.4 Stability 9
1.2 Results Concerning Stability 11
1.2.1 Existence of the Solution of the Discretization 11
1.2.2 The Basic Convergence Theorem 13
1.2.3 Linearization 15
1.2.4 Stability of Neighboring Discretizations 18
1.3 Asymptotic Expansions of the Discretization Errors 21
1.3.1 Asymptotic Expansion of the Local Discretization Error 21
1.3.2 Asymptotic Expansion of the Global Discretization Error .... 25
1.3.3 Asymptotic Expansions in Even Powers of n 29
1.3.4 The Principal Error Terms 30
1.4 Applications of Asymptotic Expansions 33
1.4.1 Richardson Extrapolation 33
1.4.2 Linear Extrapolation 37
1.4.3 Rational Extrapolation 42
1.4.4 Difference Correction 44
1.5 Error Analysis 49
1.5.1 Computing Error 49
1.5.2 Error Estimates 51
1.5.3 Strong Stability 54
1.5.4 Richardson extrapolation and Error Estimation 56
1.5.5 Statistical Analysis of Round off Errors 58
1.6 Practical Aspects 61
Table of Contents XIII
Chapter 2 Forward Step Methods 63
2.1 Preliminaries 63
2.1.1 Initial Value Problems for Ordinary Differential Equations ... 63
2.1.2 Grids 65
2.1.3 Characterization of Forward Step Methods 67
2.1.4 Restricting the Interval 69
2.1.5 Notation 72
2.2 The Meaning of Consistency, Convergence, and Stability
with Forward Step Methods 74
2.2.1 Our Choice of Norms in £„ and E° 74
2.2.2 Other Definitions of Consistency and Convergence 76
2.2.3 Other Definitions of Stability 79
2.2.4 Spijker sNormfor£2 81
2.2.5 Stability of Neighboring Discretizations 84
2.3 Strong Stability of f.s.m 87
2.3.1 Perturbation of IVP1 87
2.3.2 Discretizations of {IVP 1}T 92
2.3.3 Exponential Stability for Difference Equations on [0,oo) 94
2.3.4 Exponential Stability of Neighboring Discretizations 98
2.3.5 Strong Exponential Stability 101
2.3.6 Stability Regions 103
2.3.7 Stiff Systems of Differential Equations 104
Chapter 3 Runge Kutta Methods 107
3.1 RK procedures 107
3.1.1 Characterization 107
3.1.2 Local Solution and Increment Function 110
3.1.3 Elementary Differentials 111
3.1.4 The Expansion of the Local Solution 115
3.1.5 The Exact Increment Function 117
3.2 The Group of RK schemes 120
3.2.1 RK schemes 120
3.2.2 Inverses of RK schemes 125
3.2.3 Equivalent Generating Matrices 127
3.2.4 Explicit and Implicit RK schemes 131
3.2.5 Symmetric RK procedures 134
3.3 RK methods and Their Orders 135
3.3.1 RK methods 135
3.3.2 The Order of Consistency 137
XIV Table of Contents
3.3.3 Construction of High order RK procedures 141
3.3.4 Attainable Order of m stage RK procedures 144
3.3.5 Effective Order of RK schemes 148
3.4 Analysis of the Discretization Error 150
3.4.1 The Principal Error Function 150
3.4.2 Asymptotic Expansion of the Discretization Error 153
3.4.3 The Principal Term of the Global Discretization Error 157
3.4.4 Estimation of the Local Discretization Error 161
3.5 Strong Stability of RK methods 165
3.5.1 Strong Stability for Sufficiently Large n 165
3.5.2 Strong Stability for Arbitrary n 170
3.5.3 Stability Regions of RK methods 174
3.5.4 Use of Stability Regions for General {IVP1}T 178
3.5.5 Suggestion for a General Approach 182
Chapter 4 Linear Multistep Methods 185
4.1 Linear fc step Schemes 185
4.1.1 Characterization 185
4.1.2 The Order of Linear k step Schemes 191
4.1.3 Construction of Linear fc step Schemes of High Order 195
4.2 Uniform Linear fc step Methods 199
4.2.1 Characterization, Consistency 199
4.2.2 Auxiliary Results 203
4.2.3 Stability of Uniform Linear fc step Methods 206
4.2.4 Convergence 210
4.2.5 Highest Obtainable Orders of Convergence 214
4.3 Cyclic Linear fc step Methods 216
4.3.1 Stability of Cyclic Linear fc step Methods 216
4.3.2 The Auxiliary Method 221
4.3.3 Attainable Order of Cyclic Linear Multistep Methods 225
4.4 Asymptotic Expansions 228
4.4.1 The Local Discretization Error 228
4.4.2 Asymptotic Expansion of the Global Discretization Error,
Preparations 232
4.4.3 The Case of No Extraneous Essential Zeros 234
4.4.4 The Case of Extraneous Essential Zeros 240
4.5 Further Analysis of the Discretization Error 245
4.5.1 Weak Stability 245
4.5.2 Smoothing 248
Table of Contents XV
4.5.3 Symmetric Linear fc step Schemes 250
4.5.4 Asymptotic Expansions in Powers of hz 256
4.5.5 Estimation of the Discretization Error 260
4.6 Strong Stability of Linear Multistep Methods 263
4.6.1 Strong Stability for Sufficiently Large n 263
4.6.2 Stability Regions of Linear Multistep Methods 266
4.6.3 Strong Stability for Arbitrary n 270
Chapters Multistage Multistep Methods 272
5.1 General Analysis 272
5.1.1 A General Class of Multistage Multistep Procedures 272
5.1.2 Simple w stage fc step Methods 275
5.1.3 Stability and Convergence of Simple m stage fc step Methods . . . 278
5.2 Predictor corrector Methods 282
5.2.1 Characterization, Subclasses 282
5.2.2 Stability and Order of Predictor corrector Methods 284
5.2.3 Analysis of the Discretization Error 290
5.2.4 Estimation of the Local Discretization Error 294
5.2.5 Estimation of the Global Discretization Error 297
5.3 Predictor corrector Methods with Off step Points 300
5.3.1 Characterization 300
5.3.2 Determination of the Coefficients and Attainable Order 302
5.3.3 Stability of High Order PC methods with Off step Points .... 306
5.4 Cyclic Forward Step Methods 308
5.4.1 Characterization 308
5.4.2 Stability and Error Propagation 311
5.4.3 Primitive /w cyclic fc step Methods 316
5.4.4 General Straight w cyclic £ step Methods 322
5.5 Strong Stability 324
5.5.1 Characteristic Polynomial, Stability Regions 324
5.5.2 Stability Regions of PC methods 326
5.5.3 Stability Regions of Cyclic Methods 329
Chapter 6 Other Discretization Methods for IVP 1 332
6.1 Discretization Methods with Derivatives of/ 332
6.1.1 Recursive Computation of Higher Derivatives of the
Local Solution 332
6.1.2 Power Series Methods 335
XVI Table of Contents
6.1.3 The Perturbation Theory of Groebner Knapp Wanner 336
6.1.4 Groebner Knapp Wanner Methods 339
6.1.5 Runge Kutta Fehlberg Methods 343
6.1.6 Multistep Methods with Higher Derivatives 347
6.2 General Multi value Methods 349
6.2.1 Nordsieck s Approach 349
6.2.2 Nordsieck Predictor corrector Methods 354
6.2.3 Equivalence of Generalized Nordsieck Methods 357
6.2.4 Appraisal of Nordsieck Methods 361
6.3 Extrapolation Methods 362
6.3.1 The Structure of an Extrapolation Method 362
6.3.2 Gragg s Method 364
6.3.3 Strong Stability of OTG 369
6.3.4 The Gragg Bulirsch Stoer Extrapolation Method 372
6.3.5 Extrapolation Methods for Stiff Systems 375
Bibliography 380
Subject Index 385
|
| any_adam_object | 1 |
| author | Stetter, Hans J. 1930- |
| author_GND | (DE-588)108696219 |
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| dewey-raw | 515/.352 |
| dewey-search | 515/.352 |
| dewey-sort | 3515 3352 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T15:37:42Z |
| institution | BVB |
| isbn | 3540060081 0387060081 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001268770 |
| oclc_num | 762238 |
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| physical | XVI, 388 S. graph. Darst. |
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| series | Springer tracts in natural philosophy |
| series2 | Springer tracts in natural philosophy |
| spelling | Stetter, Hans J. 1930- Verfasser (DE-588)108696219 aut Analysis of discretization methods for ordinary differential equations Hans J. Stetter Berlin [u.a.] Springer 1973 XVI, 388 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer tracts in natural philosophy 23 Equations aux différences - Solutions numériques ram Equations différentielles - Solutions numériques ram Difference equations Numerical solutions Differential equations Numerical solutions Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Diskretisierung (DE-588)4012469-1 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 s Diskretisierung (DE-588)4012469-1 s Numerisches Verfahren (DE-588)4128130-5 s 1\p DE-604 DE-604 Springer tracts in natural philosophy 23 (DE-604)BV000692404 23 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001268770&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Stetter, Hans J. 1930- Analysis of discretization methods for ordinary differential equations Springer tracts in natural philosophy Equations aux différences - Solutions numériques ram Equations différentielles - Solutions numériques ram Difference equations Numerical solutions Differential equations Numerical solutions Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Diskretisierung (DE-588)4012469-1 gnd |
| subject_GND | (DE-588)4020929-5 (DE-588)4128130-5 (DE-588)4012469-1 |
| title | Analysis of discretization methods for ordinary differential equations |
| title_auth | Analysis of discretization methods for ordinary differential equations |
| title_exact_search | Analysis of discretization methods for ordinary differential equations |
| title_full | Analysis of discretization methods for ordinary differential equations Hans J. Stetter |
| title_fullStr | Analysis of discretization methods for ordinary differential equations Hans J. Stetter |
| title_full_unstemmed | Analysis of discretization methods for ordinary differential equations Hans J. Stetter |
| title_short | Analysis of discretization methods for ordinary differential equations |
| title_sort | analysis of discretization methods for ordinary differential equations |
| topic | Equations aux différences - Solutions numériques ram Equations différentielles - Solutions numériques ram Difference equations Numerical solutions Differential equations Numerical solutions Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Diskretisierung (DE-588)4012469-1 gnd |
| topic_facet | Equations aux différences - Solutions numériques Equations différentielles - Solutions numériques Difference equations Numerical solutions Differential equations Numerical solutions Gewöhnliche Differentialgleichung Numerisches Verfahren Diskretisierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001268770&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000692404 |
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