Spectral methods in fluid dynamics:
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
Berlin u. a.
Springer
1988
|
| Ausgabe: | Corrected 2. print. |
| Schriftenreihe: | Springer series in computational physics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 567 S. graph. Darst. |
| ISBN: | 3540522050 0387522050 |
Internformat
MARC
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Datensatz im Suchindex
| _version_ | 1804120347074625536 |
|---|---|
| adam_text | Contents
Preface v
Authors’ Affiliations xv
1 Introduction 1
1 1 Historical Background 1
1 2 Some Examples of Spectral Methods 3
121A Fourier Galerkin Method for the Wave Equation 3
122A Chebyshev Collocation Method for the Heat
Equation 7
123A Legendre Tau Method for the Poisson Equation 10
124 Basic Aspects of Galerkin, Tau and Collocation
Methods 12
1 3 The Equations of Fluid Dynamics 13
131 Compressible Navier-Stokes 13
132 Compressible Euler 15
133 Compressible Potential 16
134 Incompressible Flow 17
135 Boundary Layer 18
1 4 Spectral Accuracy for a Two-Dimensional Fluid Calculation 19
1 5 Three-Dimensional Applications in Fluids 25
2 Spectral Approximation 31
2 1 The Fourier System 32
211 The Continuous Fourier Expansion 32
212 The Discrete Fourier Expansion 38
213 Differentiation 42
214 The Gibbs Phenomenon 45
2 2 Orthogonal Polynomials in (— 1,1) 53
221 Sturm-Liouville Problems 53
222 Orthogonal Systems of Polynomials 54
223 Gauss-Type Quadratures and Discrete Polynomial
Transforms 55
2 3 Legendre Polynomials 60
231 Basic Formulas 60
232 Differentiation 62
X
Contents
2 4 Chebyshev Polynomials 65
241 Basic Formulas 65
242 Differentiation 68
2 5 Generalizations 70
251 Jacobi Polynomials 70
252 Mapping 71
253 Semi-Infinite Intervals 72
254 Infinite Intervals 74
3 Fundamentals of Spectral Methods for PDEs 76
3 1 Spectral Projection of the Burgers Equation 76
311 Fourier Galerkin 77
312 Fourier Collocation 78
313 Chebyshev Tau 79
314 Chebyshev Collocation 81
3 2 Convolution Sums 82
321 Pseudospectral Transform Methods 83
322 Aliasing Removal by Padding or Truncation 84
323 Aliasing Removal by Phase Shifts 85
324 Convolution Sums in Chebyshev Methods 86
325 Relation Between Collocation and Pseudospectral
Methods 86
3 3 Boundary Conditions 87
3 4 Coordinate Singularities 90
341 Polar Coordinates 90
342 Spherical Polar Coordinates 91
3 5 Two-Dimensional Mapping 92
4 Temporal Discretization 94
4 1 Introduction 94
4 2 The Eigenvalues of Basic Spectral Operators 96
421 The First-Derivative Operator 96
422 The Second-Derivative Operator 98
4 3 Some Standard Schemes 101
431 Multistep Schemes 101
432 Runge-Kutta Methods 107
4 4 Special Purpose Schemes 110
441 High Resolution Temporal Schemes 110
442 Special Integration Techniques 112
443 Lerat Schemes 113
4 5 Conservation Forms 114
4 6 Aliasing 118
5 Solution Techniques for Implicit Spectral Equations 124
5 1 Direct Methods 125
Contents
xi
511 Fourier Approximations 125
512 Chebyshev Tau Approximations 129
513 Schur-Decomposition and Matrix-Diagonalization___ 133
5 2 Fundamentals of Iterative Methods 137
521 Richardson Iteration 137
522 Preconditioning 139
523 Non-Periodic Problems 144
524 Finite-Element Preconditioning 148
5 3 Conventional Iterative Methods 149
531 Descent Methods for Symmetric, Positive-Definite
Systems 149
532 Descent Methods for Non-Symmetric Problems 155
533 Chebyshev Acceleration 157
5 4 Multidimensional Preconditioning 159
541 Finite-Difference Solvers 159
542 Modified Finite-Difference Preconditioners 160
5 5 Spectral Multigrid Methods 166
551 Model Problem Discussion 166
552 Two-Dimensional Problems 168
553 Interpolation Operators 170
554 Coarse-Grid Operators 172
555 Relaxation Schemes 172
56A Semi-Implicit Method for the Navier-Stokes Equations 174
6 Simple Incompressible Flows 183
6 1 Burgers Equation 183
6 2 Shear Flow Past a Circle 186
6 3 Boundary-Layer Flows 188
6 4 Linear Stability 193
7 Some Algorithms for Unsteady Navier-Stokes Equations 201
7 1 Introduction 201
7 2 Homogeneous Flows 203
721A Spectral Galerkin Solution Technique 203
722 Treatment of the Nonlinear Terms 204
723 Refinements 207
724 Pseudospectral and Collocation Methods 208
7 3 Inhomogeneous Flows 212
731 Coupled Methods 213
732 Splitting Methods 222
733 Galerkin Methods 226
734 Other Confined Flows 228
735 Unbounded Flows 230
736 Aliasing in Transition Calculations 231
Contents
xii
7 4 Flows with Multiple Inhomogeneous Directions 233
741 Choice of Mesh 234
742 Coupled Methods 236
743 Splitting Methods 237
744 Other Methods 238
7 5 Mixed Spectral/Finite-Difference Methods 238
8 Compressible Flow 240
8 1 Introduction 240
8 2 Boundary Conditions for Hyperbolic Problems 242
8 3 Basic Results for Scalar Nonsmooth Problems 246
8 4 Homogeneous Turbulence 252
8 5 Shock-Capturing 255
851 Potential Flow 255
852 Ringleb Flow 259
853 Astrophysical Nozzle 264
8 6 Shock-Fitting 266
8 7 Reacting Flows 273
9 Global Approximation Results 275
9 1 Fourier Approximation 275
911 Inverse Inequalities for Trigonometric Polynomials 275
912 Estimates for the Truncation and Best Approximation
Errors 276
913 Estimates for the Interpolation Error 279
9 2 Sturm-Liouville Expansions 281
921 Regular Sturm-Liouville Problems 282
922 Singular Sturm-Liouville Problems 284
9 3 Discrete Norms 286
9 4 Legendre Approximations 287
941 Inverse Inequalities for Algebraic Polynomials 288
942 Estimates for the Truncation and Best Approximation
Errors 288
943 Estimates for the Interpolation Error 293
9 5 Chebyshev Approximations 294
951 Inverse Inequalities for Polynomials 295
952 Estimates for the Truncation and Best Approximation
Errors 295
953 Estimates for the Interpolation Error 298
954 Proofs of Some Approximation Results 299
9 6 Other Polynomial Approximations 305
961 Jacobi Polynomials 306
962 Laguerre and Hermite Polynomials 306
9 7 Approximation Results in Several Dimensions 307
Contents
xiii
971 Fourier Approximations 307
972 Legendre Approximations 308
973 Chebyshev Approximations 310
974 Blended Fourier and Chebyshev Approximations 311
10 Theory of Stability and Convergence for Spectral Methods 315
10 1 The Three Examples Revisited 315
10 11A Fourier Galerkin Method for the Wave Equation 316
10 12A Chebyshev Collocation Method for the Heat
Equation 317
10 13A Legendre Tau Method for the Poisson Equation 321
10 2 Towards a General Theory 323
10 3 General Formulation of Spectral Approximations to
Linear Steady Problems 325
10 4 Galerkin, Collocation and Tau Methods 329
10 4 1 Galerkin Methods 330
10 4 2 Tau Methods 335
10 4 3 Collocation Methods 344
10 5 General Formulation of Spectral Approximations to
Linear Evolution Equations 353
10 5 1 Conditions for Stability and Convergence: The
Parabolic Case 355
10 5 2 Conditions for Stability and Convergence: The
Hyperbolic Case 362
10 6 The Error Equation 371
11 Steady, Smooth Problems 375
11 1 The Poisson Equation 375
11 1 1 Legendre Methods 376
11 1 2 Chebyshev Methods 377
11 1 3 Other Boundary Value Problems 382
11 2 Advection-Diffusion Equation 383
11 2 1 Linear Advection-Diffusion Equation 383
11 2 2 Steady Burgers Equation 386
11 3 Navier-Stokes Equations 392
11 3 1 Compatibility Conditions Between Velocity and
Pressure 394
11 3 2 Direct Discretization of the Continuity Equation:
The “inf-sup” Condition 397
11 3 3 Discretizations of the Continuity Equation by an
Influence-Matrix Technique: The Kleiser-
Schumann Method 404
11 3 4 Navier-Stokes Equations in Streamfunction
Formulation 406
XIV
Contents
11 4 The Eigenvalues of Some Spectral Operators 407
11 4 1 The Discrete Eigenvalues for Lu = — uxx 407
11 4 2 The Discrete Eigenvalues for Lu — — vuxx + bux 409
11 4 3 The Discrete Eigenvalues for Lu = ux 412
12 Transient, Smooth Problems 415
12 1 Linear Hyperbolic Equations 415
12 1 1 Periodic Boundary Conditions 415
12 1 2 Non-Periodic Boundary Conditions 421
12 1 3 Hyperbolic Systems 427
12 1 4 Spectral Accuracy for Non-Smooth Solutions 430
12 2 Heat Equation 435
12 2 1 Semi-Discrete Approximation 435
12 2 2 Fully Discrete Approximation 437
12 3 Advection-Diffusion Equation 440
12 3 1 Semi-Discrete Approximation 440
12 3 2 Fully Discrete Approximation 441
13 Domain Decomposition Methods 444
13 1 Introduction 444
13 2 Patching Methods 447
13 2 1 Notation 447
13 2 2 Discretization 448
13 2 3 Solution Techniques 454
13 2 4 Examples 456
13 3 Variational Methods 459
13 3 1 Formulation 459
13 3 2 The Spectral-Element Method 461
13 4 The Alternating Schwarz Method 466
13 5 Mathematical Aspects of Domain Decomposition Methods 470
13 5 1 Patching Methods 470
13 5 2 Equivalence Between Patching and Variational
Methods 471
13 6 Some Stability and Convergence Results 473
13 6 1 Patching Methods 473
13 6 2 Variational Methods 475
Appendices 477
A Basic Mathematical Concepts 477
B Fast Fourier Transforms 499
C Jacobi-Gauss-Lobatto Roots 525
References 529
Index
|
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| author_GND | (DE-588)11157434X |
| building | Verbundindex |
| bvnumber | BV006123308 |
| classification_rvk | UF 4000 |
| ctrlnum | (OCoLC)844200680 (DE-599)BVBBV006123308 |
| dewey-full | 515.353 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.353 |
| dewey-search | 515.353 |
| dewey-sort | 3515.353 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| edition | Corrected 2. print. |
| format | Book |
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| id | DE-604.BV006123308 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:40:38Z |
| institution | BVB |
| isbn | 3540522050 0387522050 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-003868594 |
| oclc_num | 844200680 |
| open_access_boolean | |
| owner | DE-703 DE-526 DE-634 DE-188 DE-706 DE-29T |
| owner_facet | DE-703 DE-526 DE-634 DE-188 DE-706 DE-29T |
| physical | XIV, 567 S. graph. Darst. |
| publishDate | 1988 |
| publishDateSearch | 1988 |
| publishDateSort | 1988 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer series in computational physics |
| spelling | Spectral methods in fluid dynamics Claudio Canuto ... Corrected 2. print. Berlin u. a. Springer 1988 XIV, 567 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in computational physics Hydrodynamik (DE-588)4026302-2 gnd rswk-swf Strömungsmechanik (DE-588)4077970-1 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Spektraltheorie (DE-588)4116561-5 gnd rswk-swf Spektralmethode (DE-588)4224817-6 gnd rswk-swf Strömungsmechanik (DE-588)4077970-1 s Partielle Differentialgleichung (DE-588)4044779-0 s Spektralmethode (DE-588)4224817-6 s DE-604 Numerisches Verfahren (DE-588)4128130-5 s 1\p DE-604 Hydrodynamik (DE-588)4026302-2 s Spektraltheorie (DE-588)4116561-5 s 2\p DE-604 Canuto, Claudio 1952- Sonstige (DE-588)11157434X oth HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003868594&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Spectral methods in fluid dynamics Hydrodynamik (DE-588)4026302-2 gnd Strömungsmechanik (DE-588)4077970-1 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Spektraltheorie (DE-588)4116561-5 gnd Spektralmethode (DE-588)4224817-6 gnd |
| subject_GND | (DE-588)4026302-2 (DE-588)4077970-1 (DE-588)4044779-0 (DE-588)4128130-5 (DE-588)4116561-5 (DE-588)4224817-6 |
| title | Spectral methods in fluid dynamics |
| title_auth | Spectral methods in fluid dynamics |
| title_exact_search | Spectral methods in fluid dynamics |
| title_full | Spectral methods in fluid dynamics Claudio Canuto ... |
| title_fullStr | Spectral methods in fluid dynamics Claudio Canuto ... |
| title_full_unstemmed | Spectral methods in fluid dynamics Claudio Canuto ... |
| title_short | Spectral methods in fluid dynamics |
| title_sort | spectral methods in fluid dynamics |
| topic | Hydrodynamik (DE-588)4026302-2 gnd Strömungsmechanik (DE-588)4077970-1 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Spektraltheorie (DE-588)4116561-5 gnd Spektralmethode (DE-588)4224817-6 gnd |
| topic_facet | Hydrodynamik Strömungsmechanik Partielle Differentialgleichung Numerisches Verfahren Spektraltheorie Spektralmethode |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003868594&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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