Iterative splitting methods for differential equations:
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boca Raton
CRC Press
2011
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| Schriftenreihe: | Chapman & Hall/CRC numerical analysis and scientific computing
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references (p. 285-300) and index |
| Beschreibung: | XXIII, 303 S. graph. Darst. |
| ISBN: | 9781439869826 |
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| 245 | 1 | 0 | |a Iterative splitting methods for differential equations |c Juergen Geiser |
| 264 | 1 | |a Boca Raton |b CRC Press |c 2011 | |
| 300 | |a XXIII, 303 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Chapman & Hall/CRC numerical analysis and scientific computing | |
| 500 | |a Includes bibliographical references (p. 285-300) and index | ||
| 650 | 4 | |a Evolution equations |x Numerical solutions | |
| 650 | 4 | |a Iterative methods (Mathematics) | |
| 650 | 0 | 7 | |a Iteration |0 (DE-588)4123457-1 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
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| adam_text | Titel: Iterative splitting methods for differential equations
Autor: Geiser, Jürgen
Jahr: 2011
Iterative Splitting
Methods for
Differential
Equations
Juergen Geiser
C) CRC Press
J Taylor Francis Croup
Boca Raton London New York
CRC Press is an imprint of the
Taylor fit Francis Group, an informa business
A CHAPMAN HALL BOOK
Contents
Preface vii
Introduction 1
1 Model Problems 7
1.1 Related Models for Decomposition 8
1.2 Examples in Real-Life Applications 9
1.2.1 Waste Disposal 9
1.2.2 Elastic Wave Propagation 11
1.2.3 Deposition Models: CVD (Chemical Vapor Deposition)
Processes 13
1.2.3.1 Standard Transport Model 14
1.2.3.2 Flow Field 16
1.2.3.3 Multiphase Model: Mobile and Immobile Zones
18
1.2.4 Navier-Stokes Molecular Dynamics: Coupling Continu¬
ous and Discrete Problems 20
1.2.4.1 Mathematical Model 21
2 Iterative Decomposition of Ordinary Differential Equations 25
2.1 Historical Overview 25
2.2 Decomposition Ideas 26
2.2.1 Physical Decomposition 27
2.2.1.1 Direct Decoupling Method 27
2.2.1.2 Decoupling Method Based on Numerical Meth¬
ods 28
2.2.2 Mathematical Decomposition 29
2.3 Introduction to Classical Splitting Methods 30
2.3.1 Classical Formulation of Splitting Methods 31.
2.3.2 Sequential Splitting Method 31
2.3.3 Symmetrical Weighted Sequential Splitting Methods .
33
2.3.4 Strang-Marchuk Splitting Method 34
2.3.5 Higher-Order Splitting Methods 35
2.4 Iterative Splitting Method 37
2.4.0.1 Iterative Splitting Method (Algorithm) . . .
39
2.5 Consistency Analysis of the Iterative Splitting Method . . .
40
ix
X
40
2.5.1 Local Error Analysis -
2.5.2 Increasing the Order of Accuracy with Improved Initial
Functions and Consistency Analysis
2.6 Stability Analysis of the Iterative Splitting Method for Bounded
Operators
2.6.1 Time Integration Methods
48
2.6.1.1 Case 1: Alternating A and B ^8
2.6.1.2 Case 2: A is Stiff and B is Nonstiff 50
2.6.2 Time Discretization Methods 52
2.6.2.1 Runge-Kutta Methods 53
2.6.2.2 BDF Methods 54
3 Decomposition Methods for Partial Differential Equations 59
3.1 Iterative Schemes for Unbounded Operators f O
3.1.1 Iterative Splitting Schemes ( 1
3.1.2 One-Stage Iterative Splitting Method for ^4-bounded
Operators 62
3.1.2.1 Consistency Analysis ( 3
3.1.2.2 Stability Analysis OG
3.1.3 Two-Stage Iterative Schemes for Operators Generating
an Analytical Semigroup t 7
3.1.3.1 Consistency analysis ( 8
3.1.3.2 Stability Analysis 72
3.1.4 Some Examples for One-Stage and Two-Stage Iterative
Operator Splitting Schemes 73
3.1.4.1 One-Stage Iterative Scheme 73
3.1.4.2 Two-Stage Iterative Scheme 75
4 Computation of the Iterative Splitting Methods: Algorithmic
Part 77
4.1 Exponential Runge-Kutta Methods to Compute Iterative Split¬
ting Schemes 77
4.2 Matrix Exponentials to Compute Iterative Splitting Schemes 79
4.2.1 Derivation of the Formula ,S()
4.3 Algorithms ,Xl
4.3.1 Two-Side Scheme H2
4.3.2 One-Side Scheme (Alternative Notation with Commu¬
tators) 32
5 Extensions of Iterative Splitting Schemes 85
5.1 Embedded Spatial Discretization Methods SO
5.1.1 Balancing of Time and Spatial Discretization
5.1.2 Spatial Discretization Schemes with Dimensional
Splitting 87
5.1.2.1 The Lax-Wendroff Scheme in One Dimension 87
xi
5.1.2.2 Generalization to Two Dimensions 90
5.1.2.3 Dimensional Splitting 91
5.1.2.4 Advection-Diffusion Splitting 92
5.2 Domain Decomposition Methods Based on Iterative Operator
Splitting Methods 93
5.2.1 Combined Time-Space Iterative Splitting Method . . .
93
5.2.2 Nonoverlapping Time-Space Iterative Splitting Method 95
5.2.3 Overlapping Time-Space Iterative Splitting Method .
96
5.2.4 Error Analysis and Convergence of Combined Method 97
5.3 Successive Approximation for Time-Dependent Operators . .
101
5.3.1 Algorithm for Successive Approximation 102
Numerical Experiments 105
6.1 Introduction 105
6.2 Benchmark Problems 1: Introduction 108
6.2.1 Introduction Problem 1: Starting Conditions 108
6.2.2 Introduction Problem 2: Stiffness of Matrices 112
6.2.3 Introduction Problem 3: Nonsplitting and Splitting . .
115
6.2.4 Introduction Problem 4: System of ODEs with Stiff and
Nonstiff Cases 117
6.2.4.1 First Experiment: Linear ODE with Nonstiff
Parameters 118
6.2.4.2 Second Experiment: Linear ODE with Stiff
Parameters 121
6.2.5 Introduction Problem 5: Linear Partial Differential
Equation 122
6.2.6 Introduction Problem 6: Nonlinear Ordinary Differen¬
tial Equation 124
6.2.7 Introduction Problem 7: Coupling Convection-Diffusion
and Reaction Equations with Separate Codes 126
6.2.7.1 First Experiment: Linear Reaction 128
6.2.7.2 Second Experiment: Nonlinear Reaction . . .
129
6.3 Benchmark Problems 2: Comparison with Standard Splitting-
Methods 130
6.3.1 Comparison Problem 1: Iterative Splitting Method with
Improved Time Discretization Methods 130
6.3.1.1 First Experiment: Heat Equation 131
6.3.1.2 Secotid Experiment: Anisotropic Equation with
Time-Dependent Reaction 133
6.3.2 Comparison Problem 2: Iterative Splitting Method Com¬
pared to Standard Splitting Methods 138
6.3.2.1 First Experiment: Convection-Diffusion Equa¬
tion Split into its Spatial Dimensions ....
139
xii
6.3.2.2 Second Experiment: Convection-Diffusion Equa¬
tion Split into Operators 144
6.4 Benchmark Problems 3: Extensions to Iterative Splitting
Methods 149
6.4.1 Extension Problem 1: Spatial Decomposition Methods (Clas¬
sical and Iterative Splitting Schemes) 149
6.4.1.1 First Experiment: One-Dimensional Convection-
Diffusion-Reaction Equation 150
6.4.1.2 Second Experiment: Two-Dimensional Convection-
Diffusion-Reaction Equation 154
6.4.1.3 Third Experiment: Three-Dimensional Convection-
Diffusion-Reaction Equation 155
6.4.1.4 Fourth Experiment: Time-Dependent Diffu¬
sion Equation 157
6.4.2 Extension Problem 2: Hyperbolic Equations 159
6.4.2.1 First Experiment: Elastic Wave Propagation
with Noniterative Splitting Methods 160
6.4.2.2 Second Experiment: Elastic Wave Propaga¬
tion with Iterative Splitting Methods ....
170
6.4.3 Extension Problem 3: Nonlinear Partial Differential Equa¬
tions 174
6.4.3.1 First Experiment: Burgers Equation 174
6.4.3.2 Second Experiment: Mixed Convection-Diffusion
and Burgers Equation 177
6.4.3.3 Third Experiment: Momentum Equation (Molec¬
ular Flow) 179
6.4.4 Extension Problem 4: Coupled Equations 186
6.4.5 First Example: Matrix Problem 186
6.4.6 Second Experiment: 10 x 10 Matrix 186
6.4.7 Third Example: Commutator Problem 187
6.4.8 Two-Phase Example 189
6.5 Real-Life Applications 202
6.5.1 Waste Disposal: Transport and Reaction of Radioactive
Contaminants 202
6.5.1.1 Two-Dimensional Model of Waste Disposal .
203
6.5.1.2 Three-Dimensional Model of Waste Disposal 205
6.5.2 Elastic Wave Propagation 209
6.5.2.1 Real-Life Application of Elastic Wave Propa¬
gation 209
6.5.2.2 Basic Numerical Methods 209
6.5.2.3 Fourth-Order Splitting Method 211
6.5.2.4 Initial Values and Boundary Conditions . . .
212
6.5.2.5 Test Example of the 2D Wave Equation .. .
212
6.5.2.6 Singular Forcing Terms 213
6.5.2.7 Computational Cost of the Splitting Method 217
i
i
xiii
6.5.2.8 A Three-Dimensional Splitting Method ...
217
6.5.2.9 Test Example of the 3D Wave Equation .. .
219
6.5.3 CVD Apparatus: Optimization of a Deposition
Problem 221
6.5.3.1 Mathematical Model 223
6.5.3.2 Parameters of the Two-Dimensional Simula¬
tions 225
6.5.3.3 Experiments with the Plasma Reactor (Two-
Dimensional) 227
6.5.3.4 Parameters of the Three-Dimensional Simula¬
tions 239
6.5.3.5 Experiments with the Plasma Reactor (Three-
Dimensional) 241
6.5.4 Complex Flow Phenomena: Navier-Stokes and Molecu¬
lar Dynamics 246
6.5.4.1 Mathematical Model 246
6.5.4.2 Implicit Dual-Time Stepping Method for Time-
Dependent Flows 248
6.5.4.3 Spatial Discretization of Equations for Micro-
and Macroscales 248
6.5.4.4 Dual-Time Stepping Method 249
6.5.4.5 Time-Dependent Channel Flow Simulation
.
249
6.5.4.6 Numerical Experiments: Splitting Methods for
Coupled Micro-Macro System of Equations .
250
6.6 Conclusion to Numerical Experiments: Discussion of Some
Delicate Problems 256
7 Summary and Perspectives 259
8 Software Tools 261
8.1 Software Package Unstructured Grids (UG) 261
8.1.1 Rough Structuring of the Software Packages 261
8.1.2 UG Software Toolbox 262
8.1.3 UG Concept 263
8.1.4 Software Package d3f 265
8.1.5 Equations in d3f 265
8.1.6 Structure of d3f 266
8.2 Software Package r3t 267
8.2.1 Equation in r3t 267
8.2.2 Taskofr3t 267
8.2.3 Conception of r3t 268
8.2.4 Application of r3t 269
8.2.5 Coupling Concept of r3t 270
8.3 Solving PDEs Using FIDOS 270
8.3.1 PDEs Treated 271
xiv
8.3.1.1 Wave Equation 271
8.3.1.2 Viscous Burgers Equation 272
8.3.1.3 Mixed Convection-Diffusion and Burgers
Equation 272
8.3.1.4 Momentum Equation 272
8.3.1.5 Diffusion Equation 273
8.3.2 Methods 273
8.3.2.1 ADI Method 273
8.3.2.2 LOD Method 274
8.3.3 Iterative Operator Splitting Methods 274
8.3.3.1 Standard IOS Method 275
8.3.3.2 Coupled 77-IOS Method 275
8.3.4 Eigenvalue Methods 275
8.3.5 Numerical Examples 276
Appendix 281
List of Abbreviations 281
Symbols 282
General Notations 284
Bibliography 285
Index 301
|
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| discipline | Mathematik |
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| isbn | 9781439869826 |
| language | English |
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| spelling | Geiser, Jürgen Verfasser (DE-588)128742402 aut Iterative splitting methods for differential equations Juergen Geiser Boca Raton CRC Press 2011 XXIII, 303 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Chapman & Hall/CRC numerical analysis and scientific computing Includes bibliographical references (p. 285-300) and index Evolution equations Numerical solutions Iterative methods (Mathematics) Iteration (DE-588)4123457-1 gnd rswk-swf Iteration (DE-588)4123457-1 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025773764&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Geiser, Jürgen Iterative splitting methods for differential equations Evolution equations Numerical solutions Iterative methods (Mathematics) Iteration (DE-588)4123457-1 gnd |
| subject_GND | (DE-588)4123457-1 |
| title | Iterative splitting methods for differential equations |
| title_auth | Iterative splitting methods for differential equations |
| title_exact_search | Iterative splitting methods for differential equations |
| title_full | Iterative splitting methods for differential equations Juergen Geiser |
| title_fullStr | Iterative splitting methods for differential equations Juergen Geiser |
| title_full_unstemmed | Iterative splitting methods for differential equations Juergen Geiser |
| title_short | Iterative splitting methods for differential equations |
| title_sort | iterative splitting methods for differential equations |
| topic | Evolution equations Numerical solutions Iterative methods (Mathematics) Iteration (DE-588)4123457-1 gnd |
| topic_facet | Evolution equations Numerical solutions Iterative methods (Mathematics) Iteration |
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