Parallel multigrid waveform relaxation for parabolic problems:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Stuttgart
Teubner
1993
|
| Schriftenreihe: | Teubner Skripten zur Numerik
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 237 - 247 |
| Beschreibung: | 253 S. graph. Darst. |
| ISBN: | 3519027178 |
Internformat
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| 100 | 1 | |a Vandewalle, Stefan |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Parallel multigrid waveform relaxation for parabolic problems |c Stefan Vandewalle |
| 264 | 1 | |a Stuttgart |b Teubner |c 1993 | |
| 300 | |a 253 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Teubner Skripten zur Numerik | |
| 500 | |a Literaturverz. S. 237 - 247 | ||
| 650 | 7 | |a Equations différentielles paraboliques - Solutions numériques - Informatique |2 ram | |
| 650 | 7 | |a Multigrid-methoden |2 gtt | |
| 650 | 7 | |a Méthodes multigrilles (Analyse numérique) |2 ram | |
| 650 | 7 | |a Parallélisme (Informatique) |2 ram | |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Differential equations, Parabolic |x Numerical solutions |x Data processing | |
| 650 | 4 | |a Multigrid methods (Numerical analysis) | |
| 650 | 4 | |a Parallel processing (Electronic computers) | |
| 650 | 0 | 7 | |a Parabolische Differentialgleichung |0 (DE-588)4173245-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Mehrgitterverfahren |0 (DE-588)4038376-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Waveform-Relaxation |0 (DE-588)4320091-6 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
Preface 5
Table of contents 7
Notations 11
1 Introduction 15
1.1 Numerical simulation and parallel processing 15
1.2 The simulation of time dependent processes 17
1.2.1 The serial bottleneck of time marching 18
1.2.2 Accelerating the time marching process 19
1.3 Outline 21
2 Waveform Relaxation Methods 23
2.1 Introduction 23
2.2 Waveform relaxation: basic ideas 25
2.3 A classification of waveform methods 28
2.4 General convergence results 30
2.4.1 The contraction mapping principle 30
2.4.2 Waveform relaxation as a contraction map 31
2.4.3 On the order of accuracy 33
2.5 Convergence analysis for linear systems 35
2.5.1 Functional analysis preliminaries 35
2.5.2 Waveform relaxation for linear systems 37
2.5.3 Continuous time convergence results 38
2.5.4 Discrete time convergence results 40
2.6 Waveform relaxation acceleration techniques 45
2.7 Some concluding remarks 48
3 Waveform Relaxation Methods for Initial Boundary Value Problems 49
3.1 Introduction and notations 49
3.2 Standard waveform relaxation 50
3.2.1 A model problem: the two dimensional heat equation 51
3.2.2 Standard waveform relaxation methods 51
3.2.3 Some convergence results 52
3.2.4 Numerical experiments 54
8 CONTENTS
3.2.5 Further remarks 59
3.3 Linear multigrid acceleration 59
3.3.1 The multigrid principle 59
3.3.2 Linear multigrid waveform relaxation 64
3.4 Convergence analysis 68
3.4.1 Continuous time convergence analysis 68
3.4.2 Discrete time convergence analysis 70
3.5 Experimental results 73
3.6 Nonlinear multigrid waveform relaxation 78
3.7 A multigrid method on a space time grid * 80
3.8 Concluding remarks 82
4 Waveform Relaxation for Solving Time Periodic Problems 83
4.1 Introduction 83
4.2 Standard time periodic PDE solvers 85
4.3 Time periodic waveform relaxation 88
4.4 Analysis of the continuous time iteration 90
4.4.1 The linear model problem 90
4.4.2 The time periodic waveform iteration 92
4.4.3 The time periodic integral operator 93
4.4.4 Convergence of the time periodic waveform relaxation 95
4.4.5 The existence of convergent and divergent splittings 99
4.5 Analysis of the discrete time iteration 100
4.5.1 Discretization and solution of the model problem 100
4.5.2 Reformulation as a linear algebra problem 102
4.5.3 The discrete time time periodic waveform iteration 104
4.5.4 Convergence of the discrete time iteration 105
4.5.5 The spectral picture 107
4.6 Multigrid acceleration 110
4.6.1 Introduction 110
4.6.2 Time periodic multigrid waveform relaxation 110
4.6.3 Analysis of the continuous time iteration 112
4.6.4 Analysis of the discrete time iteration 114
4.6.5 Numerical example 115
4.7 Autonomous time periodic problems 116
4.7.1 Introduction 116
4.7.2 Shooting with coarse grid Jacobian approximation 118
4.7.3 The use of waveform relaxation within shooting 120
4.7.4 A numerical example: the Brusselator 120
5 A Short Introduction to Parallel Computers and Parallel Computingl23
5.1 Introduction 123
5.2 Classification of parallel computers 124
5.3 The hypercube topology 126
5.3.1 Definition and properties 126
5.3.2 Binary reflected Gray codes 128
CONTENTS 9
5.3.3 Topology embedding onto the hypercube 128
5.4 The Intel iPSC/2 hypercube multiprocessor 130
5.4.1 System overview 130
5.4.2 Some computation benchmarks 131
5.5 Parallel performance parameters 132
6 Parallel Implementation of Standard Parabolic Marching Schemes 135
6.1 Introduction 135
6.2 Problem class and discretization 136
6.2.1 Problem class 136
6.2.2 Discretization 137
6.2.3 The numerical kernels 139
6.3 Parallel implementation: preliminaries 140
6.3.1 The parallel computer model 140
6.3.2 The grid partitioning approach 141
6.4 The explicit update step 142
6.5 The multigrid solver 144
6.5.1 Introduction 144
6.5.2 Parallelizing the multigrid components 146
6.5.3 Agglomeration/deagglomeration strategies 150
6.6 The tridiagonal systems solver 154
6.6.1 Introduction 154
6.6.2 Substructured Gaussian elimination 155
6.6.3 Solution of the intermediate system 157
6.7 Timing results on the Intel hypercube 165
6.7.1 The explicit update step 165
6.7.2 The multigrid solver 166
6.7.3 The tridiagonal system solver 170
6.8 Numerical examples 174
6.8.1 Introduction 174
6.8.2 Test problem one 175
6.8.3 Test problem two 176
6.8.4 Test problem three 178
6.9 Concluding remarks 178
7 Computational Complexity of Multigrid Waveform Relaxation 183
7.1 Introduction 183
7.2 Arithmetic complexity 184
7.2.1 Initial boundary value problems 184
7.2.2 . Time periodic problems 187
7.3 Parallel implementation 190
7.3.1 Grid partitioning 190
7.3.2 Communication complexity 191
7.4 Vectorization 193
7.5 Concluding remarks 194
10 CONTENTS
8 Case Studies 195
8.1 Introduction 195
8.2 Programming considerations 196
8.3 Representation of the results 197
8.4 Linear initial boundary value problems 198
8.4.1 Example 1 198
8.4.2 Example 2 204
8.5 Nonlinear initial boundary value problems 207
8.5.1 Example 3 207
8.5.2 Example 4 210
8.6 Linear time periodic problems 214
8.6.1 Example 5 214
8.6.2 Example 6 217
8.7 Example 7: a nonlinear periodic system 220
8.8 Further remarks, limits of applicability 222
9 Concluding Remarks and Suggestions for Future Research 225
A Discretization and Stencils 231
Bibliography 237
Index 249
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T16:49:45Z |
| institution | BVB |
| isbn | 3519027178 |
| language | English |
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| spelling | Vandewalle, Stefan Verfasser aut Parallel multigrid waveform relaxation for parabolic problems Stefan Vandewalle Stuttgart Teubner 1993 253 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Teubner Skripten zur Numerik Literaturverz. S. 237 - 247 Equations différentielles paraboliques - Solutions numériques - Informatique ram Multigrid-methoden gtt Méthodes multigrilles (Analyse numérique) ram Parallélisme (Informatique) ram Datenverarbeitung Differential equations, Parabolic Numerical solutions Data processing Multigrid methods (Numerical analysis) Parallel processing (Electronic computers) Parabolische Differentialgleichung (DE-588)4173245-5 gnd rswk-swf Mehrgitterverfahren (DE-588)4038376-3 gnd rswk-swf Waveform-Relaxation (DE-588)4320091-6 gnd rswk-swf Parallelrechner (DE-588)4173280-7 gnd rswk-swf Parabolische Differentialgleichung (DE-588)4173245-5 s Mehrgitterverfahren (DE-588)4038376-3 s Waveform-Relaxation (DE-588)4320091-6 s Parallelrechner (DE-588)4173280-7 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004246645&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Vandewalle, Stefan Parallel multigrid waveform relaxation for parabolic problems Equations différentielles paraboliques - Solutions numériques - Informatique ram Multigrid-methoden gtt Méthodes multigrilles (Analyse numérique) ram Parallélisme (Informatique) ram Datenverarbeitung Differential equations, Parabolic Numerical solutions Data processing Multigrid methods (Numerical analysis) Parallel processing (Electronic computers) Parabolische Differentialgleichung (DE-588)4173245-5 gnd Mehrgitterverfahren (DE-588)4038376-3 gnd Waveform-Relaxation (DE-588)4320091-6 gnd Parallelrechner (DE-588)4173280-7 gnd |
| subject_GND | (DE-588)4173245-5 (DE-588)4038376-3 (DE-588)4320091-6 (DE-588)4173280-7 |
| title | Parallel multigrid waveform relaxation for parabolic problems |
| title_auth | Parallel multigrid waveform relaxation for parabolic problems |
| title_exact_search | Parallel multigrid waveform relaxation for parabolic problems |
| title_full | Parallel multigrid waveform relaxation for parabolic problems Stefan Vandewalle |
| title_fullStr | Parallel multigrid waveform relaxation for parabolic problems Stefan Vandewalle |
| title_full_unstemmed | Parallel multigrid waveform relaxation for parabolic problems Stefan Vandewalle |
| title_short | Parallel multigrid waveform relaxation for parabolic problems |
| title_sort | parallel multigrid waveform relaxation for parabolic problems |
| topic | Equations différentielles paraboliques - Solutions numériques - Informatique ram Multigrid-methoden gtt Méthodes multigrilles (Analyse numérique) ram Parallélisme (Informatique) ram Datenverarbeitung Differential equations, Parabolic Numerical solutions Data processing Multigrid methods (Numerical analysis) Parallel processing (Electronic computers) Parabolische Differentialgleichung (DE-588)4173245-5 gnd Mehrgitterverfahren (DE-588)4038376-3 gnd Waveform-Relaxation (DE-588)4320091-6 gnd Parallelrechner (DE-588)4173280-7 gnd |
| topic_facet | Equations différentielles paraboliques - Solutions numériques - Informatique Multigrid-methoden Méthodes multigrilles (Analyse numérique) Parallélisme (Informatique) Datenverarbeitung Differential equations, Parabolic Numerical solutions Data processing Multigrid methods (Numerical analysis) Parallel processing (Electronic computers) Parabolische Differentialgleichung Mehrgitterverfahren Waveform-Relaxation Parallelrechner |
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