Time dependent problems and difference methods:
Time dependent problems frequently pose challenges in areas of science and engineering dealing with numerical analysis, scientific computation, mathematical models, and most importantly - numerical experiments intended to analyze physical behavior and test design. Time Dependent Problems and Differe...
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY [u.a.]
Wiley
1995
|
| Series: | Pure and applied mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | Time dependent problems frequently pose challenges in areas of science and engineering dealing with numerical analysis, scientific computation, mathematical models, and most importantly - numerical experiments intended to analyze physical behavior and test design. Time Dependent Problems and Difference Methods addresses these various industrial considerations in a pragmatic and detailed manner, giving special attention to time dependent problems in its coverage of the derivation and analysis of numerical methods for computational approximations to Partial Differential Equations (PDEs). The authors draw on their own interests and combined extensive experience in applied mathematics and computer science to bring about this practical and useful guide. They provide complete discussions of the pertinent theorems and back them up with examples and illustrations For physical scientists, engineers, or anyone who uses numerical experiments to test designs or to predict and investigate physical phenomena, this invaluable guide is destined to become a constant companion. Time Dependent Problems and Difference Methods is also extremely useful to numerical analysts, mathematical modelers, and graduate students of applied mathematics and scientific computations |
| Physical Description: | XI, 642 S. graph. Darst. |
| ISBN: | 0471507342 |
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| 520 | |a For physical scientists, engineers, or anyone who uses numerical experiments to test designs or to predict and investigate physical phenomena, this invaluable guide is destined to become a constant companion. Time Dependent Problems and Difference Methods is also extremely useful to numerical analysts, mathematical modelers, and graduate students of applied mathematics and scientific computations | ||
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Record in the Search Index
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| adam_text | CONTENTS
Preface ix
Acknowledgments xiii
PART I Problems with Periodic Solutions 1
1. Fourier Series and Trigonometric Interpolation 3
1.1 Some results from the theory of Fourier series, 3
1.2 Periodic gridfunctions and difference operators, 17
1.3 Trigonometric interpolation, 24
1.4 The S operator for differentiation, 30
1.5 Generalizations, 33
Bibliographic Notes, 37
2. Model Equations 38
2.1 First order wave equation, convergence, and stability, 38
2.2 The Leap frog scheme, 50
2.3 Implicit methods, 55
2.4 Truncation error, 59
2.5 Heat equation, 61
2.6 Convection diffusion equation, 70
2.7 Higher order equations, 74
2.8 Generalization to several space dimensions, 76
Bibliographic Notes, 78
3. Higher Order Accuracy 80
3.1 Efficiency of higher order accurate difference
approximations, 80
3.2 Fourier method, 95
Bibliographic Notes, 104
4. Well Posed Problems 106
4.1 Well posedness, 106
v
vi CONTENTS
4.2 Scalar differential equations with constant coefficients in
one space dimension, 113
4.3 First order systems with constant coefficients in one space
dimension, 116
4.4 Parabolic systems with constant coefficients in one space
dimension, 122
4.5 General systems with constant coefficients, 127
4.6 Semibounded operators with variable coefficients, 134
4.7 The solution operator and Duhamel s principle, 142
4.8 Generalized solutions, 149
4.9 Well posedness of nonlinear problems, 152
Bibliographic Notes, 156
5. Stability and Convergence for Numerical Approximations
of Linear and Nonlinear Problems 157
5.1 Stability and convergence, 157
5.2 Stability for approximations with constant coefficients,
171
5.3 Approximations with variable coefficients: The energy
method, 182
5.4 Splitting methods, 195
5.5 Stability for nonlinear problems, 201
Bibliographic Notes, 210
6. Hyperbolic Equations and Numerical Methods 211
6.1 Systems with constant coefficients in one space
dimension, 211
6.2 Systems with variable coefficients in one space
dimension, 214
6.3 Systems with constant coefficients in several space
dimensions, 218
6.4 Systems with variable coefficients in several space
dimensions, 221
6.5 Approximations with constant coefficients, 222
6.6 Approximations with variable coefficients, 235
6.7 The method of lines, 238
6.8 The finite volume method, 254
6.9 The Fourier method, 262
Bibliographic Notes, 267
7. Parabolic Equations and Numerical Methods 270
7.1 General parabolic systems, 270
7.2 Stability for difference and Fourier methods, 275
CONTENTS vii
7.3 Difference approximations in several space
dimensions, 282
Bibliographic Notes, 289
8. Problems with Discontinuous Solutions 290
8.1 Difference methods for linear hyperbolic equations, 290
8.2 Method of characteristics, 297
8.3 Method of characteristics in several space dimensions,
304
8.4 Method of characteristics on a regular grid, 305
8.5 Regularization using viscosity, 313
8.6 The inviscid Burgers equations, 316
8.7 The viscous Burgers equation and traveling waves, 320
8.8 Numerical methods for scalar equations based on
regularization, 328
8.9 Regularization for systems of equations, 336
8.10 High resolution methods, 345
Bibliographic Notes, 355
PART II Initial Boundary Value Problems 357
9. The Energy Method for Initial Boundary Value Problems 359
9.1 Characteristics and boundary conditions for hyperbolic
systems in one space dimension, 359
9.2 Energy estimates for hyperbolic systems in one space
dimension, 368
9.3 Energy estimates for parabolic differential equations in
one space dimension, 375
9.4 Well posed problems, 381
9.5 Semibounded operators, 385
9.6 Quarter space problems in more than one space
dimension, 390
Bibliographic Notes, 397
10. The Laplace Transform Method for Initial Boundary Value
Problems 398
10.1 Solution of hyperbolic systems, 398
10.2 Solution of parabolic problems, 404
10.3 Generalized well posedness, 410
10.4 Systems with constant coefficients in one space
dimension, 419
10.5 Hyperbolic systems with constant coefficients in several
space dimensions, 427
vin CONTENTS
10.6 Parabolic systems in more than one space dimension, 440
10.7 Systems with variable coefficients in general
domains, 443
Bibliographic Notes, 444
11. The Energy Method for Difference Approximations 445
11.1 Hyperbolic problems, 445
11.2 Parabolic differential equations, 458
11.3 Stability, consistency, and order of accuracy, 465
11.4 Higher order approximations, 471
11.5 Several space dimensions, 484
Bibliographic Notes, 491
Appendix 11, 492
12. The Laplace Transform Method for Difference
Approximations 496
12.1 Necessary conditions for stability, 496
12.2 Sufficient conditions for stability, 506
12.3 A fourth order accurate approximation for hyperbolic
differential equations, 524
12.4 Stability in the generalized sense for hyperbolic
systems, 526
12.5 An example that does not satisfy the Kreiss condition
but is stable in the generalized sense, 538
12.6 Parabolic equations, 552
12.7 The convergence rate, 567
Bibliographic Notes, 575
13. The Laplace Transform Method for Fully Discrete
Approximations: Normal Mode Analysis 576
13.1 General theory for approximations of hyperbolic
systems, 576
13.2 The method of lines and generalized stability, 599
13.3 Several space dimensions, 611
13.4 Domains with irregular boundaries and overlapping
grids, 615
Bibliographic Notes, 620
Appendix A.I Results from Linear Algebra, 622
Appendix A.2 Laplace Transform, 625
Appendix A.3 Iterative Methods, 629
References, 633
Index 639
|
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| spelling | Gustafsson, Bertil 1930- Verfasser (DE-588)133993825 aut Time dependent problems and difference methods Bertil Gustafsson ; Heinz-Otto Kreiss ; Joseph Oliger New York, NY [u.a.] Wiley 1995 XI, 642 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics Time dependent problems frequently pose challenges in areas of science and engineering dealing with numerical analysis, scientific computation, mathematical models, and most importantly - numerical experiments intended to analyze physical behavior and test design. Time Dependent Problems and Difference Methods addresses these various industrial considerations in a pragmatic and detailed manner, giving special attention to time dependent problems in its coverage of the derivation and analysis of numerical methods for computational approximations to Partial Differential Equations (PDEs). The authors draw on their own interests and combined extensive experience in applied mathematics and computer science to bring about this practical and useful guide. They provide complete discussions of the pertinent theorems and back them up with examples and illustrations For physical scientists, engineers, or anyone who uses numerical experiments to test designs or to predict and investigate physical phenomena, this invaluable guide is destined to become a constant companion. Time Dependent Problems and Difference Methods is also extremely useful to numerical analysts, mathematical modelers, and graduate students of applied mathematics and scientific computations Numerieke methoden gtt Partiële differentiaalvergelijkingen gtt Problèmes aux dérivées partielles - Solutions numériques ram Differential equations, Partial Numerical solutions Evolutionsgleichung (DE-588)4129061-6 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Differenzenverfahren (DE-588)4134362-1 gnd rswk-swf Evolutionsgleichung (DE-588)4129061-6 s Differenzenverfahren (DE-588)4134362-1 s DE-604 Partielle Differentialgleichung (DE-588)4044779-0 s Kreiss, Heinz-Otto Verfasser (DE-588)122643615 aut Oliger, Joseph Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007143879&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gustafsson, Bertil 1930- Kreiss, Heinz-Otto Oliger, Joseph Time dependent problems and difference methods Numerieke methoden gtt Partiële differentiaalvergelijkingen gtt Problèmes aux dérivées partielles - Solutions numériques ram Differential equations, Partial Numerical solutions Evolutionsgleichung (DE-588)4129061-6 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Differenzenverfahren (DE-588)4134362-1 gnd |
| subject_GND | (DE-588)4129061-6 (DE-588)4044779-0 (DE-588)4134362-1 |
| title | Time dependent problems and difference methods |
| title_auth | Time dependent problems and difference methods |
| title_exact_search | Time dependent problems and difference methods |
| title_full | Time dependent problems and difference methods Bertil Gustafsson ; Heinz-Otto Kreiss ; Joseph Oliger |
| title_fullStr | Time dependent problems and difference methods Bertil Gustafsson ; Heinz-Otto Kreiss ; Joseph Oliger |
| title_full_unstemmed | Time dependent problems and difference methods Bertil Gustafsson ; Heinz-Otto Kreiss ; Joseph Oliger |
| title_short | Time dependent problems and difference methods |
| title_sort | time dependent problems and difference methods |
| topic | Numerieke methoden gtt Partiële differentiaalvergelijkingen gtt Problèmes aux dérivées partielles - Solutions numériques ram Differential equations, Partial Numerical solutions Evolutionsgleichung (DE-588)4129061-6 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Differenzenverfahren (DE-588)4134362-1 gnd |
| topic_facet | Numerieke methoden Partiële differentiaalvergelijkingen Problèmes aux dérivées partielles - Solutions numériques Differential equations, Partial Numerical solutions Evolutionsgleichung Partielle Differentialgleichung Differenzenverfahren |
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