Spectral Methods (eBook)

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