Canuto / Hussaini / Quarteroni Spectral Methods

Evolution to Complex Geometries and Applications to Fluid Dynamics
1. Auflage 2007
ISBN: 978-3-540-30728-0
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 596 Seiten

Reihe: Scientific Computation

Following up the seminal Spectral Methods in Fluid Dynamics, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics contains an extensive survey of the essential algorithmic and theoretical aspects of spectral methods for complex geometries. These types of spectral methods were only just emerging at the time the earlier book was published. The discussion of spectral algorithms for linear and nonlinear fluid dynamics stability analyses is greatly expanded. The chapter on spectral algorithms for incompressible flow focuses on algorithms that have proven most useful in practice, has much greater coverage of algorithms for two or more non-periodic directions, and shows how to treat outflow boundaries. Material on spectral methods for compressible flow emphasizes boundary conditions for hyperbolic systems, algorithms for simulation of homogeneous turbulence, and improved methods for shock fitting. This book is a companion to Spectral Methods: Fundamentals in Single Domains.

Autoren/Hrsg.

Canuto, Claudio

Hussaini, M. Yousuff

Quarteroni, Alfio

Zang, Thomas A.

Inhaltsverzeichnis

1;Preface;5
2;Contents;9
3;List of Figures;17
4;List of Tables;22
5;Contents of the Companion Book Spectral Methods – Fundamentals in Single Domains;23
6;1. Fundamentals of Fluid Dynamics;29
6.1;1.1 Introduction;29
6.2;1.2 Fluid Dynamics Background;29
6.3;1.3 Compressible Fluid Dynamics Equations;35
6.4;1.4 Incompressible Fluid Dynamics Equations;49
6.5;1.5 Linear Stability of Parallel Flows;55
6.6;1.6 Stability Equations for Nonparallel Flows;64
7;2. Single-Domain Algorithms and Applications for Stability Analysis;67
7.1;2.1 Introduction;67
7.2;2.2 Boundary-Layer Flows;69
7.3;2.3 Linear Stability of Incompressible Parallel Flows;80
7.4;2.4 Linear Stability of Compressible Parallel Flows;92
7.5;2.5 Nonparallel Linear Stability;97
7.6;2.6 Transient Growth Analysis;100
7.7;2.7 Nonlinear Stability;103
8;3. Single-Domain Algorithms and Applications for Incompressible Flows;111
8.1;3.1 Introduction;111
8.2;3.2 Conservation Properties and Time-Discretization;114
8.3;3.3 Homogeneous Flows;126
8.4;3.4 Flows with One Inhomogeneous Direction;149
8.5;3.5 Flows with Multiple Inhomogeneous Directions;175
8.6;3.6 Outflow Boundary Conditions;187
8.7;3.7 Analysis of Spectral Methods for Incompressible Flows;190
9;4. Single-Domain Algorithms and Applications for Compressible Flows;214
9.1;4.1 Introduction;214
9.2;4.2 Boundary Treatment for Hyperbolic Systems;214
9.3;4.3 Boundary Treatment for the Euler Equations;230
9.4;4.4 High-Frequency Control;235
9.5;4.5 Homogeneous Turbulence;238
9.6;4.6 Smooth, Inhomogeneous Flows;245
9.7;4.7 Shock Fitting;253
9.8;4.8 Shock Capturing;260
10;5. Discretization Strategies for Spectral Methods in Complex Domains;263
10.1;5.1 Introduction;263
10.2;5.2 The Spectral Element Method (SEM) in 1D;265
10.3;5.3 SEM for Multidimensional Problems;271
10.4;5.4 Analysis of SEM and SEM-NI Approximations;283
10.5;5.5 Some Numerical Results for the SEM- NI Approximations;299
10.6;5.6 SEM for Stokes and Navier–Stokes Equations;304
10.7;5.7 The Mortar Element Method (MEM);315
10.8;5.8 The Spectral Discontinuous Galerkin Method ( SDGM) in 1D;326
10.9;5.9 SDGM for Multidimensional Problems;342
10.10;5.10 SDGM for Diffusion Equations;349
10.11;5.11 Analysis of SDGM;352
10.12;5.12 SDGM for Euler and Navier–Stokes Equations;358
10.13;5.13 The Patching Method;365
10.14;5.14 3D Applications in Complex Geometries;378
11;6. Solution Strategies for Spectral Methods in Complex Domains;384
11.1;6.1 Introduction;384
11.2;6.2 On Domain Decomposition Preconditioners;384
11.3;6.3 (Overlapping) Schwarz Alternating Methods;389
11.4;6.4 Schur Complement Iterative Methods;410
11.5;6.5 Solution Algorithms for Patching Collocation Methods;427
12;7. General Algorithms for Incompressible Navier– Stokes Equations;432
12.1;7.1 Introduction;432
12.2;7.2 High-Order Fractional-Step Methods;434
12.3;7.3 Solution of the Algebraic System Associated with the Generalized Stokes Problem;440
12.4;7.4 Algebraic Factorization Methods;450
13;8. Spectral Methods Primer;459
13.1;8.1 The Fourier System;459
13.2;8.2 General Jacobi Polynomials in the Interval (- 1, 1);469
13.3;8.3 Chebyshev Polynomials;475
13.4;8.4 Legendre Polynomials;479
13.5;8.5 Modal and Nodal Boundary-Adapted Bases on the Interval;482
13.6;8.6 Orthogonal Systems in Unbounded Domains;484
13.7;8.7 Multidimensional Expansions;486
13.8;8.8 Mappings;492
13.9;8.9 Basic Spectral Discretization Methods;502
14;Appendix A. Basic Mathematical Concepts;512
14.1;A.1 Hilbert and Banach Spaces;512
14.2;A.2 The Cauchy-Schwarz Inequality;514
14.3;A.3 The Lax-Milgram Theorem;515
14.4;A.4 Dense Subspace of a Normed Space;515
14.5;A.5 The Spaces Cm(O), m = 0;516
14.6;A.6 The Spaces Lp(O), 1 = p = +8;516
14.7;A.7 Infinitely Differentiable Functions and Distributions;517
14.8;A.8 Sobolev Spaces and Sobolev Norms;519
14.9;A.9 The Sobolev Inequality;524
14.10;A.10 The Poincar´ e Inequality;524
15;Appendix B. Fast Fourier Transforms;525
16;Appendix C. Iterative Methods for Linear Systems;531
16.1;C.1 A Gentle Approach to Iterative Methods;531
16.2;C.2 Descent Methods for Symmetric Problems;535
16.3;C.3 Krylov Methods for Nonsymmetric Problems;540
17;Appendix D. Time Discretizations;547
17.1;D.1 Notation and Stability Definitions;547
17.2;D.2 Standard ODE Methods;550
17.3;D.3 Low-Storage Schemes;557
18;Appendix E. Supplementary Material;559
18.1;E.1 Numerical Solution of the Generalized Eigenvalue Problem;559
18.2;E.2 Tau Correction for the Kleiser–Schumann Method;561
18.3;E.3 The Piola Transform;563
19;References;566
20;Index;607