Computer aided analysis of difference schemes for partial differential equations:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Wiley
1996
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XI, 458 S. |
| ISBN: | 0471129461 |
Internformat
MARC
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| 100 | 1 | |a Ganža, Viktor G. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Computer aided analysis of difference schemes for partial differential equations |c Victor G. Ganzha ; E. V. Vorozhtsov |
| 246 | 1 | 3 | |a Computer-aided analysis of difference schemes for partial differential equations |
| 264 | 1 | |a New York [u.a.] |b Wiley |c 1996 | |
| 300 | |a XI, 458 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Differential equations |x Use of |x Computers | |
| 650 | 4 | |a Differential equations, Partial |x Numerical solutions |x Data processing | |
| 650 | 4 | |a Finite differences |x Data processing | |
| 650 | 0 | 7 | |a Differenzenverfahren |0 (DE-588)4134362-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Computeralgebra |0 (DE-588)4010449-7 |2 gnd |9 rswk-swf |
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| 689 | 1 | |5 DE-604 | |
| 700 | 1 | |a Vorožcov, Evgenij V. |d 1946- |e Verfasser |0 (DE-588)121730611 |4 aut | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007214387&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-007214387 | |
Datensatz im Suchindex
| _version_ | 1812633930250059776 |
|---|---|
| adam_text |
Contents
Preface jx
1 The Necessary Basics from the Stability Theory of Difference
Schemes and Polynomials 1
1.1 Preliminary Discussion of Stability and Approximation 1
1.2 Computer Algebra Systems 5
1.3 A Brief Review of the Contents of Chapters 8
1.4 Stability, Approximation, and Convergence 12
1.5 A Survey of Methods for the Stability Analysis of Difference
Schemes 18
1.5.1 Von Neumann Stability Analysis 32
1.5.2 Differential Approximation Method 37
1.5.3 Method of Frozen Coefficients 40
1.6 Algebraic Criteria for Localization of Polynomial Zeros 42
1.6.1 Similarity and Dimensional Considerations 42
1.6.2 Lienard Chipart Criterion 46
1.6.3 Generalized Routh Hurwitz Problem for the
Characteristic Polynomial 53
1.7 Determination of the Maximal Time Step from Stability
Analysis Results 56
1.7.1 The Use of the Least Squares Method 57
1.7.2 A Method Based on the Requirement of a Constant
Volume of a Cell of a Spatial Computing Mesh 59
1.7.3 The Use of the Tables of the Coordinates of Points of
Stability Region Boundaries 59
1.8 On the Choice of Nondimensional Complexes 61
1.9 Bibliographical Notes 63
1.9.1 Historical Note on Stability Theories 63
1.9.2 Application of Algebraic Criteria to Stability Analyses 64
1.9.3 Use of Computer Algebra for the Automation of Certain
Stages of the Stability Analyses 66
References 67
2 Symbolic Numerical Method for the Stability Investigation of
Difference Schemes on a Computer 77
2.1 General Structure of the Symbolic Numerical Method 77
v
vi CONTENTS
2.2 The Case of Diagonalizable Amplification Matrices 79
2.3 Scheme Checker 80
2.4 Symbolic Stages of the Method 85
2.5 Generation of a FORTRAN Program by Computer Algebra 88
2.6 Computation of the Coordinates of Points of a Stability Region
Boundary 94
2.6.1 Use of the Bisection Method 94
2.6.2 Automatic Determination of the Number of Spectral
Grid Points 96
2.7 Improved Accuracy of Numerical Results 98
2.7.1 Scaling in the Routh Algorithm 99
2.7.2 Scaling in the Routh Hurwitz Algorithm 104
2.8 Examples of Stability Analyses of Difference Schemes for
Equations of Hyperbolic Type 108
2.8.1 Two Step Richtmyer's Form of the Lax Wendroff Scheme 108
2.8.2 MacCormack Scheme for the Two Dimensional
Advection Equation 111
2.8.3 Jameson's Schemes 113
2.9 Stability Analysis of the MacCormack Scheme for Two
Dimensional Euler Equations 120
2.10 Stability Analysis of the MacCormack Scheme for Three
Dimensional Euler Equations 131
2.11 Examples of Stability Analyses of Difference Schemes for
Navier Stokes Equations 137
2.11.1 A Family of Schemes for One Dimensional Navier
Stokes Equations 138
2.11.2 Difference Schemes on Curvilinear Grids 141
References 156
3 Application of Optimization Methods to the Stability Analysis of
Difference Schemes 161
3.1 Formulation of a Search for Stability Region Boundaries of
Difference Schemes in Terms of Optimization Theory 161
3.1.1 The Case of One Nondimensional Complex 161
3.1.2 The Case of Many Nondimensional Complexes 164
3.2 Symbolic Computation of Algebraic Expressions 168
3.3 Numerical Realization of the Optimization Method 171
3.4 Some Practical Applications 173
3.4.1 Dick's Scheme for the One Dimensional Advection
Equation 173
3.4.2 Explicit Implicit Scheme for Equations of the Shallow
Water Theory 174
3.4.3 The Family of Schemes S". for the One Dimensional
Advection Diffusion Equation 176
3.4.4 An Explicit Scheme for Equations of Elasticity Theory 178
CONTENTS vii
3.4.5 Richtmyer's Scheme for Three Dimensional Euler
Equations 181
References 196
4 Stability Analysis of Difference Schemes by Catastrophe Theory
Methods 199
4.1 Ideas Underlying Catastrophe Theory 200
4.2 Reduction of the von Neumann Analysis to a Canonical
Problem of Catastrophe Theory 205
4.3 Numerical Determination of a Segment of the Stability
Region Boundary 207
4.4 Direct Use of the Resultant for the Determination of
Boundary Points 216
4.5 Automatic Generation of FORTRAN Subroutines 220
4.6 Some Practical Applications 221
4.6.1 Family of Schemes for One Dimensional Advection
Equation 221
4.6.2 Monocyclic MacCormack Schemes 224
4.6.3 The Two Cycle MacCormack Scheme 228
4.6.4 The MacCormack Scheme for One Dimensional
Euler Equations 231
4.6.5 Scheme with Upwind Differencing for the Two
Dimensional Advection Equation 233
4.6.6 The Lomax and Pulliam Scheme for the One
Dimensional Advection Diffusion Equation 235
References 237
5 Construction of Multiply Connected Stability Regions of Difference
Schemes by Computer Algebra and Pattern Recognition 240
5.1 General Organizational Scheme of a Process for Detection of
Stability Regions of Difference Schemes 243
5.2 Detection of Boundaries by Digital Image Segmentation 245
5.2.1 Segmentation of Two Dimensional Images 246
5.2.2 Segmentation of Three Dimensional Images 256
5.3 Tracing Contour Segments 260
5.4 Refinement of Boundary Point Positions 267
5.5 Extraction of Singular Points 272
5.6 Some Practical Applications 277
5.6.1 The Case of Ordinary Differential Equations 278
5.6.2 The Case of Partial Differential Equations 288
References 294
6 Maximally Stable Difference Schemes 299
6.1 Basic Definitions 303
viii CONTENTS
6.2 Stability and Accuracy Functional 305
6.3 A Search Algorithm for Maximally Stable Difference Schemes
and Its Computer Implementation 309
6.4 Application to One Dimensional Problems 312
6.5 Jameson's Scheme for the Two Dimensional Advection
Equation 324
6.6 Peyret Taylor's Family of Schemes for the Two Dimensional
Advection Diffusion Equation 327
References 342
7 Stability Analysis of Nonlinear Difference Schemes 347
7.1 Theoretical Background 348
7.2 The Case of One Spatial Variable 357
7.3 The Case of Two Spatial Variables 371
7.4 Formulation of the Optimization Problem 377
7.5 Symbolic Stages 380
7.6 Numerical Realization of the Optimization Method 383
7.7 Two Methods for Determining the Stability Region's Boundary 383
7.7.1 A Modified Yesipov's Method 383
7.7.2 The Use of the Digital Pattern Recognition 386
7.8 Computational Examples 386
7.9 Bibliographic Notes 393
7.10 Concluding Remarks 398
References 399
8 Symbolic Computation of Differential Approximations 403
8.1 Basic Definitions 404
8.2 Symbolic Algorithm for the Scalar Equation Case 407
8.3 Symbolic Computation of Differential Approximations of
Schemes with Fractional Steps 417
8.4 Local Approximation Study of Difference Operators on
Nonorthogonal Curvilinear Spatial Grids 422
8.4.1 Preliminary Discussion 422
8.4.2 Description of the Algorithm 428
8.4.3 Description of the Program 432
8.5 Differential Approximation and Stability of Difference Schemes 435
References 441
Appendix A. Gas Dynamic Matrices 446
Appendix B. REDUCE Program for Scheme (4.6.19) 451
Index 453 |
| any_adam_object | 1 |
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| institution | BVB |
| isbn | 0471129461 |
| language | English |
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| physical | XI, 458 S. |
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| spelling | Ganža, Viktor G. Verfasser aut Computer aided analysis of difference schemes for partial differential equations Victor G. Ganzha ; E. V. Vorozhtsov Computer-aided analysis of difference schemes for partial differential equations New York [u.a.] Wiley 1996 XI, 458 S. txt rdacontent n rdamedia nc rdacarrier Datenverarbeitung Differential equations Use of Computers Differential equations, Partial Numerical solutions Data processing Finite differences Data processing Differenzenverfahren (DE-588)4134362-1 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Computeralgebra (DE-588)4010449-7 gnd rswk-swf Wissenschaftliches Rechnen (DE-588)4338507-2 gnd rswk-swf Differenzenverfahren (DE-588)4134362-1 s Wissenschaftliches Rechnen (DE-588)4338507-2 s DE-604 Partielle Differentialgleichung (DE-588)4044779-0 s Computeralgebra (DE-588)4010449-7 s Vorožcov, Evgenij V. 1946- Verfasser (DE-588)121730611 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007214387&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ganža, Viktor G. Vorožcov, Evgenij V. 1946- Computer aided analysis of difference schemes for partial differential equations Datenverarbeitung Differential equations Use of Computers Differential equations, Partial Numerical solutions Data processing Finite differences Data processing Differenzenverfahren (DE-588)4134362-1 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Computeralgebra (DE-588)4010449-7 gnd Wissenschaftliches Rechnen (DE-588)4338507-2 gnd |
| subject_GND | (DE-588)4134362-1 (DE-588)4044779-0 (DE-588)4010449-7 (DE-588)4338507-2 |
| title | Computer aided analysis of difference schemes for partial differential equations |
| title_alt | Computer-aided analysis of difference schemes for partial differential equations |
| title_auth | Computer aided analysis of difference schemes for partial differential equations |
| title_exact_search | Computer aided analysis of difference schemes for partial differential equations |
| title_full | Computer aided analysis of difference schemes for partial differential equations Victor G. Ganzha ; E. V. Vorozhtsov |
| title_fullStr | Computer aided analysis of difference schemes for partial differential equations Victor G. Ganzha ; E. V. Vorozhtsov |
| title_full_unstemmed | Computer aided analysis of difference schemes for partial differential equations Victor G. Ganzha ; E. V. Vorozhtsov |
| title_short | Computer aided analysis of difference schemes for partial differential equations |
| title_sort | computer aided analysis of difference schemes for partial differential equations |
| topic | Datenverarbeitung Differential equations Use of Computers Differential equations, Partial Numerical solutions Data processing Finite differences Data processing Differenzenverfahren (DE-588)4134362-1 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Computeralgebra (DE-588)4010449-7 gnd Wissenschaftliches Rechnen (DE-588)4338507-2 gnd |
| topic_facet | Datenverarbeitung Differential equations Use of Computers Differential equations, Partial Numerical solutions Data processing Finite differences Data processing Differenzenverfahren Partielle Differentialgleichung Computeralgebra Wissenschaftliches Rechnen |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007214387&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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