Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Google eBook)

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Springer Science & Business Media, Jun 30, 2007 - Mathematics - 628 pages
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Spectral methods, particularly in their multidomain version, have become firmly established as a mainstream tool for scientific and engineering computation. While retaining the tight integration between the theoretical and practical aspects of spectral methods that was the hallmark of their 1988 book, Canuto et al. now incorporate the many improvements in the algorithms and the theory of spectral methods that have been made since then. This second new treatment, Evolution to Complex Geometries and Applications to Fluid Dynamics, provides an extensive overview of the essential algorithmic and theoretical aspects of spectral methods for complex geometries, in addition to detailed discussions of spectral algorithms for fluid dynamics in simple and complex geometries. Modern strategies for constructing spectral approximations in complex domains, such as spectral elements, mortar elements, and discontinuous Galerkin methods, as well as patching collocation, are introduced, analyzed, and demonstrated by means of numerous numerical examples. Representative simulations from continuum mechanics are also shown. Efficient domain decomposition preconditioners (of both Schwarz and Schur type) that are amenable to parallel implementation are surveyed. The discussion of spectral algorithms for fluid dynamics in single domains focuses on proven algorithms for the boundary-layer equations, linear and nonlinear stability analyses, incompressible Navier-Stokes problems, and both inviscid and viscous compressible flows. An overview of the modern approach to computing incompressible flows in general geometries using high-order, spectral discretizations is also provided. The recent companion book Fundamentals in Single Domains discusses the fundamentals of the approximation of solutions to ordinary and partial differential equations on single domains by expansions in smooth, global basis functions. The essential concepts and formulas from this book are included in the current text for the reader’s convenience.
  

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Contents

Fundamentals of Fluid Dynamics
1
121 Phases of Matter
2
122 Thermodynamic Relationships
3
123 Historical Perspective
6
13 Compressible Fluid Dynamics Equations
7
131 Compressible NavierStokes Equations
8
132 Nondimensionalization
12
133 NavierStokes Equations with Turbulence Models
13
562 Stability and Convergence Analysis
283
Proof of the Global Infsup Condition
285
563 Numerical Results
286
57 The Mortar Element Method MEM
289
571 Formulation of MEM
290
572 Algebraic Aspects of MEM
294
573 Analysis of MEM
296
574 Other Applications
299

134 Compressible Euler Equations
17
136 Compressible BoundaryLayer Equations
19
137 Compressible Stokes Limit
20
138 Low Mach Number Compressible Limit
21
142 Incompressible NavierStokes Equations with Turbulence Models
22
143 VorticityStreamfunction Equations
25
144 VorticityVelocity Equations
26
145 Incompressible BoundaryLayer Equations
27
151 Incompressible Linear Stability
29
152 Compressible Linear Stability
31
16 Stability Equations for Nonparallel Flows
36
SingleDomain Algorithms and Applications for Stability Analysis
39
22 BoundaryLayer Flows
41
222 Compressible BoundaryLayer Flows
48
23 Linear Stability of Incompressible Parallel Flows
52
232 Numerical Examples for Plane Poiseuille Flow
57
233 Some Other Incompressible Linear Stability Problems
61
24 Linear Stability of Compressible Parallel Flows
64
25 Nonparallel Linear Stability
69
252 TwoDimensional Global Stability Analysis
71
26 Transient Growth Analysis
72
27 Nonlinear Stability
75
272 Secondary Instability Theory
77
273 Nonlinear Parabolized Stability Equations
81
SingleDomain Algorithms and Applications for Incompressible Flows
83
32 Conservation Properties and TimeDiscretization
86
The Rotation Form
88
The SkewSymmetric Form
90
Convection and Divergence Forms
92
323 Coupled Methods
93
324 Splitting Methods
95
325 Other Integration Methods
96
33 Homogeneous Flows
98
332 Dealiasing Using Transform Methods
99
333 Pseudospectral and Collocation Methods
103
334 Rogallo Transformation for Homogeneous Turbulence
106
335 LargeEddy Simulation of Isotropic Turbulence
108
Stability Accuracy and Aliasing
110
34 Flows with One Inhomogeneous Direction
121
341 Coupled Methods
123
KleiserSchumann Algorithm
124
Normal VelocityNormal Vorticity Algorithms
127
342 Galerkin Methods Using DivergenceFree Bases
131
343 Splitting Methods
133
ZangHussaini Algorithm
135
344 Other Confined Flows
138
345 Unbounded Flows
140
FreeShearLayer Flows
142
Accuracy
144
35 Flows with Multiple Inhomogeneous Directions
147
351 The Choice of Spatial Discretization in a Cavity
149
352 The Choice of Spatial Discretization on a Reference Domain
157
36 Outflow Boundary Conditions
159
362 Buffer Domains
161
37 Analysis of Spectral Methods for Incompressible Flows
162
371 Compatibility Conditions Between Velocity and Pressure
165
The Infsup Condition
168
General Theory
169
373 Specific Applications
172
Numerical Results
177
Extensions
178
374 The Infsup Condition and the Pressure Operator
179
The KleiserSchumann Method
183
4 SingleDomain Methods for Compressible Flows
186
421 Characteristic Compatibility Conditions
188
The Characteristic Compatibility Method CCM
189
CCM for a General 1D System
192
CCM for the Collocation Method
193
CCM for a General Multidimensional System
195
References and Outlook
196
422 Boundary Treatment for Linear Systems in Weak Formulations
197
423 Spectral Accuracy and Conservation
199
424 Analysis of Spectral Methods for Symmetric Hyperbolic Systems
200
43 Boundary Treatment for the Euler Equations
203
44 HighFrequency Control
208
45 Homogeneous Turbulence
211
452 Representative Applications
214
46 Smooth Inhomogeneous Flows
218
462 NavierStokes Equations
221
463 Numerical Example
224
47 Shock Fitting
226
48 Shock Capturing
233
5 Discretization Strategies
237
52 The Spectral Element Method SEM in 1D
239
522 Construction of SEM Basis Functions
241
523 SEMNI and its Collocation Interpretation
243
53 SEM for Multidimensional Problems
245
532 Construction of SEM Basis Functions
247
533 SEM and SEMNI Formulations
250
534 Algebraic Aspects of SEM and SEMNI
252
535 FiniteElement Preconditioning of SEMNI Matrices
253
54 Analysis of SEM and SEMNI Approximations
257
A Priori Error Analysis for SEM
258
A Priori Analysis for SEMNI
260
A Posteriori Error Analysis
261
542 Multidimensional Analysis
263
A Priori Error Analysis
264
A Posteriori Error Analysis
265
543 Some Proofs
268
55 Some Numerical Results for the SEMNI Approximations
273
552 Eigenfunction Approximation
275
56 SEM for Stokes and NavierStokes Equations
278
561 SEM and SEMNI Formulations
279
58 The Spectral Discontinuous Galerkin Method SDGM in 1D
300
581 Linear Advection Problems in ID
301
582 Linear Hyperbolic Systems in 1D
303
583 TimeDependent Problems
308
584 Nonlinear Conservation Laws in 1D
313
59 SDGM for Multidimensional Problems
316
591 Multidimensional Formulation
317
Nonlinear Conservation Laws
319
592 The Mortar Technique for Geometrical Nonconformities
321
510 SDGM for Diffusion Equations
323
511 Analysis of SDGM
326
512 SDGM for Euler and NavierStokes Equations
332
TimeDiscretizations
333
Numerical Examples
334
Shock Tracking
337
513 The Patching Method
339
5132 Comparison of Patching and SEMNI
343
5133 Collocation Methods for the Euler Equations
345
Collocation Using a Nonstaggered Grid
346
Collocation Using a Staggered Grid
348
Multidomain Shock Fitting
350
514 3D Applications in Complex Geometries
352
Application to Compressible Flow
353
Application to Thermoelasticity
354
Structural Dynamics Analysis of the Roman Colosseum
356
6 Solution Strategies for Spectral Methods in Complex Domains
358
63 Overlapping Schwarz Alternating Methods
364
631 Algebraic Form of Schwarz Methods for FiniteElement Discretization
367
632 The Schwarz Method as an Algebraic Preconditioner
370
633 Additive Schwarz Preconditioners for HighOrder Methods
373
634 FEMSEM Spectral Equivalence
380
635 Analysis of Schwarz Methods
381
636 A General Theoretical Framework for the Analysis of DD Iterations
383
64 Schur Complement Iterative Methods
385
642 Properties of the SteklovPoincareOperator
387
644 DD Preconditioners for the Schur Complement Matrix
393
645 Preconditioners for the Stiffness Matrix Derived from Preconditioners for the Schur Complement Matrix
398
65 Solution Algorithms for Patching Collocation Methods
402
7 General Algorithms for Incompressible NavierStokes Equations
407
72 HighOrder FractionalStep Methods
409
73 Solution of the Algebraic System Associated with the Generalized Stokes Problem
415
731 Preconditioners for the Generalized Stokes Matrix A
416
732 Conditioning and Preconditioning for the Pressure Schur Complement Matrix
419
733 Domain Decomposition Preconditioners for the Stokes and NavierStokes Equations
421
74 Algebraic Factorization Methods
425
742 Numerical Results for Yosida Schemes
428
743 Preconditioners for the Approximate Pressure Schur Complement
430
8 Spectral Methods Primer
435
The Fourier Series
436
Decay of the Fourier Coefficients
437
Discrete Fourier Expansion and Interpolation
438
Aliasing
440
Differentiation
441
Gibbs Phenomenon and Filtering
443
82 General Jacobi Polynomials in the Interval 11
445
The Jacobi Series Truncation and Projection
447
GaussType Quadrature Formulas and Discrete Inner Products
448
Discrete Polynomial Transform and Interpolation
449
Differentiation
451
Quadrature Formulas and Discrete Transforms
453
Differentiation
454
84 Legendre Polynomials
455
Quadrature Formulas and Discrete Norms
457
85 Modal and Nodal BoundaryAdapted Bases on the Interval
458
86 Orthogonal Systems in Unbounded Domains
460
Hermite Polynomials and Hermite Functions
461
87 Multidimensional Expansions
462
871 TensorProduct Expansions
463
872 Expansions on Simplicial Domains
465
88 Mappings
468
881 Finite Intervals
469
882 SemiInfinite Intervals
471
883 The Real Line
473
884 Multidimensional Mappings on Finite Domains
475
89 Basic Spectral Discretization Methods
478
891 Tau Method
479
892 Collocation Method
481
893 Galerkin Method
482
894 Galerkin with Numerical Integration GNI Method
484
895 Other Boundary Conditions
485
Appendix A Basic Mathematical Concepts
488
A2 The CauchySchwarz Inequality
491
A3 The LaxMilgram Theorem
492
A5 The Spaces Cm𝜴 m 0
493
A7 Infinitely Differentiable Functions and Distributions
494
A8 Sobolev Spaces and Sobolev Norms
496
A9 The Sobolev Inequality
501
Appendix B Fast Fourier Transforms
503
Appendix C Iterative Methods for Linear Systems
509
C2 Descent Methods for Symmetric Problems
513
C3 Krylov Methods for Nonsymmetric Problems
518
Appendix D Time Discretizations
525
D2 Standard ODE Methods
528
D21 Leap Frog Method
529
D23 AdamsMoulton Methods
531
D24 BackwardsDifference Formulas
533
D25 RungeKutta Methods
534
D3 LowStorage Schemes
535
Appendix E Supplementary Material
537
E2 Tau Correction for the KleiserSchumann Method
539
E3 The Piola Transform
541
References
544
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