Cover......Page 1
Half-title......Page 3
Series-title......Page 6
Title......Page 7
Copyright......Page 8
Dedication......Page 9
Contents......Page 11
List of Figures......Page 19
Preface......Page 27
1.1 Viscous Fluid Flows......Page 31
1.2 Mass Conservation......Page 33
1.3.1 Linear Momentum......Page 35
1.4 Energy Conservation......Page 36
1.5 Thermodynamics and Constitutive Equations......Page 37
1.6.1 Isothermal Incompressible Flow......Page 38
1.6.2 Thermal Convection: The Boussinesq Approximation......Page 39
1.6.3 Boundary and Initial Conditions......Page 40
1.7 Dimensional Analysis and Reduced Equations......Page 41
1.8 Vorticity Equation......Page 45
1.9 Simplified Models......Page 46
1.10 Turbulence and Challenges......Page 47
1.11.1 Hardware Issues......Page 52
1.11.2 Software Issues......Page 54
1.11.3 Algorithms......Page 56
1.11.4 Advantages of High-Order Methods......Page 58
2 Approximation Methods for Elliptic Problems......Page 63
2.1.1 Variational Functionals......Page 64
2.1.2 Boundary Conditions......Page 69
2.1.3 Sobolev Spaces and the Lax-Milgram Theorem......Page 70
Lemma 2.1 (Poincaré–Friedrichs inequality)......Page 72
2.2 An Approximation Framework......Page 76
Galerkin Method......Page 77
Petrov–Galerkin Methods......Page 80
2.2.2 Collocation Approximation......Page 81
2.3 Finite-Element Methods......Page 83
Global Mesh and Set of Basis Functions......Page 84
Stiffness and Mass Matrices......Page 86
2.3.2 The p-Version of Finite Elements......Page 90
2.4 Spectral-Element Methods......Page 92
2.5.1 Orthogonal Collocation in a Monodomain......Page 97
2.5.2 Orthogonal Collocation in a Multidomain......Page 99
2.6 Error Estimation......Page 101
2.7 Solution Techniques......Page 103
Matrix Norms and Spectral Radius......Page 104
The Condition Number......Page 106
The Conditioning of Spectral Schemes......Page 107
2.7.2 Basic Iterative Methods......Page 111
Convergence Results......Page 112
2.7.3 Preconditioning Schemes of High-Order Methods......Page 113
2.7.4 Iterative Methods Based on Projection......Page 116
2.8 A Numerical Example......Page 122
3.1 Introduction......Page 128
3.2 Time Discretization Schemes......Page 129
3.2.1 Linear Multistep Methods......Page 130
Main Theoretical Concepts......Page 132
Adams–Bashforth and Adams–Moulton Schemes......Page 135
Backward Differencing Schemes......Page 139
3.2.2 Predictor–Corrector Methods......Page 140
3.2.3 Runge–Kutta Methods......Page 143
Explicit Runge–Kutta Schemes......Page 144
Implicit Runge–Kutta Schemes......Page 147
3.3 Splitting Methods......Page 149
3.3.1 The Operator-Integration-Factor Splitting Method......Page 151
3.3.2 OIFS Example: The BDF3/RK4 Scheme......Page 153
3.4 The Parabolic Case: Unsteady Diffusion......Page 154
3.4.1 Spatial Discretization......Page 156
3.4.2 Time Advancement......Page 157
3.5 The Hyperbolic Case: Linear Convection......Page 159
3.5.1 Spatial Discretization......Page 160
3.5.2 Eigenvalues of the Discrete Problem and CFL Number......Page 161
3.5.3 Example of Temporal and Spatial Accuracy......Page 165
3.6 Steady Advection–Diffusion Problems......Page 167
3.6.1 Spectral Elements and Bubble Stabilization......Page 168
3.6.2 Collocation and Staggered Grids......Page 171
3.7 Unsteady Advection–Diffusion Problems......Page 175
3.7.1 Spatial Discretization......Page 176
3.7.3 Outflow Conditions and Filter-Based Stabilization......Page 179
3.8.1 Space and Time Discretization......Page 181
Spectral-Element Method......Page 182
3.8.2 Numerical Results......Page 183
3.9 The OIFS Method and Subcycling......Page 185
3.10 Taylor–Galerkin Time Integration......Page 188
3.10.1 Nonlinear Pure Advection......Page 189
3.10.2 Taylor–Galerkin and OIFS Methods......Page 191
4.2 Tensor Products......Page 192
Operator Evaluation......Page 197
4.3 Elliptic Problems......Page 199
4.3.1 Weak Formulation and Sobolev Spaces......Page 200
4.3.2 A Constant-Coefficient Case......Page 202
4.3.3 The Variable-Coefficient Case......Page 207
4.4 Deformed Geometries......Page 208
4.4.1 Generation of Geometric Deformation......Page 213
4.4.2 Surface Integrals and Robin Boundary Conditions......Page 216
4.5 Spectral-Element Discretizations......Page 218
4.5.1 Continuity and Direct Stiffness Summation......Page 221
4.5.2 Spectral–Element Operators......Page 224
4.5.3 Inhomogeneous Dirichlet Problems......Page 226
4.5.4 Iterative Solution Techniques......Page 227
4.5.5 Two-Dimensional Examples......Page 228
4.6.1 The Diffusion Case......Page 232
Inhomogeneous Neumann Boundary Conditions......Page 239
The Multidomain Approach......Page 242
4.6.2 The Advection–Diffusion Case......Page 246
4.7 Parabolic Problems......Page 250
4.7.1 Time-Dependent Projection......Page 252
4.7.2 Other Diffusion Systems......Page 254
4.8 Hyperbolic Problems......Page 256
A Two-Dimensional Example......Page 259
4.9 Unsteady Advection–Diffusion Problems......Page 260
4.10 Further Reading......Page 262
5.1 Steady Velocity–Pressure Formulation......Page 264
5.2.1 The Weak Formulation......Page 266
5.2.2 The Spectral-Element Method......Page 268
Staggered Spectral Elements......Page 270
Collocative Spectral Elements......Page 272
Legendre Single-Grid Collocation......Page 275
Chebyshev Single-Grid Collocation......Page 277
Legendre Staggered-Grid Collocation......Page 280
5.3.1 Spectral-Element Methods and Uzawa Algorithm......Page 283
5.3.2 Collocation Methods......Page 287
5.4.1 General Considerations......Page 289
5.4.2 The Green’s-Function Method......Page 290
5.4.3 Implementation......Page 293
5.5 Divergence-Free Bases......Page 294
5.6 Stabilization of the PN–PN Approximation by Bubble Functions......Page 299
5.7 hp-Methods for Stokes Problems......Page 302
5.8 Steady Navier–Stokes Equations......Page 303
5.8.1 Weak Formulation......Page 304
5.8.2 Collocation Approximation of the Navier–Stokes Equations......Page 305
5.8.3 Solution Algorithms: Iterative and Newton Methods......Page 307
Square-Cavity Problem......Page 308
Grooved Channel......Page 310
Wannier–Stokes Flow......Page 312
Kovasznay Flow......Page 313
Grooved Channel......Page 315
Cooled Hot Cylinder......Page 317
5.10 Complements and Engineering Considerations......Page 318
6.1 Unsteady Velocity–Pressure Formulation......Page 321
6.2.1 The Weak Formulation......Page 323
6.2.2 Uzawa Algorithm......Page 325
6.2.3 Splitting and Decoupling Algorithms......Page 326
6.3 Pressure Preconditioning......Page 330
6.4.1 Weak Formulation......Page 333
6.4.2 Advection Treatment......Page 334
6.5 Projection Methods......Page 339
6.5.1 Fractional-Step Method......Page 340
6.5.2 Pressure Correction Method......Page 343
6.6 Stabilizing Unsteady Flows......Page 345
6.7 Arbitrary Lagrangian–Eulerian Formulation and Free-Surface Flows......Page 348
6.7.1 ALE Formulation......Page 349
6.7.2 Free-Surface Conditions......Page 350
6.7.3 Variational Formulation of Free-Surface Flows......Page 352
6.7.4 Space and Time Discretization......Page 355
6.8.1 Extrusion from a Die......Page 356
6.8.2 Vortex-Sheet Roll-Up......Page 357
6.9 Further Reading and Engineering Considerations......Page 359
7.1 Introduction......Page 363
The Continuous Presentation......Page 364
The Discrete Presentation......Page 367
7.2.2 Overlapping Schwarz Procedures......Page 368
Discrete Formulation......Page 370
Schwarz Preconditioners......Page 372
A Brief Analysis......Page 373
Two-Level Preconditioners......Page 374
7.2.3 Schwarz Preconditioners for High-Order Methods......Page 375
7.2.4 Spectral-Element Multigrid......Page 378
7.3 The Mortar Element Method......Page 382
7.3.1 Elliptic Problems......Page 387
7.3.2 Implementation......Page 388
7.3.3 Steady Stokes Problems......Page 393
Flow Around an Impeller......Page 395
Resonator Cavity Flow......Page 396
Clearance-Gap Glow......Page 397
7.4 Adaptivity and Singularity Treatment......Page 398
7.4.1 Coupling between Finite and Spectral Elements......Page 399
7.4.2 Singularity Treatment......Page 400
7.4.3 Triangular and Tetrahedral Elements......Page 401
Modal Bases......Page 402
Differentiation......Page 405
Nodal Bases......Page 406
Spectral Error Estimator......Page 407
7.5 Further Reading......Page 408
8.1 Introduction......Page 409
8.2 Serial Architectures......Page 410
8.2.1 Pipelining......Page 411
8.2.2 Memory, Bandwidth, and Caches......Page 412
8.3 Tensor-Product Operator Evaluation......Page 414
8.3.1 Tensor-Product Evaluation......Page 415
8.3.2 Other Operations......Page 420
8.4 Parallel Programming......Page 421
8.4.1 Communication Characteristics......Page 423
8.4.2 Vector Reductions......Page 426
8.5.1 Data Distribution and Operator Evaluation......Page 429
8.5.2 Direct Stiffness Summation......Page 431
8.5.3 Domain Partitioning......Page 436
8.5.4 Coarse-Grid Solves......Page 437
8.6.1 Hairpin Vortices......Page 438
8.6.2 Driven Cavity......Page 440
8.6.3 Backward-Facing Step......Page 442
8.7 Further Reading......Page 446
A.1.1 Definition......Page 447
A.1.2 Open Set, Closed Set, Neighborhood......Page 448
A.1.4 Mapping, Domain, Range, Continuity......Page 449
A.1.5 Convergence, Completeness, Completion Process......Page 450
A.2.1 Definition......Page 451
A.2.2 Banach Spaces......Page 452
A.3.2 The Inverse Operator......Page 453
A.3.3 Bounded Operators, Compact Operators......Page 454
A.3.5 The Fréchet Derivative of an Operator......Page 455
Examples of Fréchet Derivatives......Page 456
A.4.1 Definition......Page 457
Examples of Hilbert Spaces......Page 458
A.4.3 Cauchy–Schwarz Inequality......Page 459
A.4.4 The Riesz Representation......Page 460
A.4.5 Orthogonality, Orthogonal Projection......Page 461
A.4.6 Separable Hilbert Spaces, Basis......Page 462
Example: Orthogonal Polynomials......Page 463
Test Functions......Page 464
Distribution......Page 465
Examples of Distributions......Page 466
Translation and Change in Scale......Page 468
Derivatives of Distributions......Page 469
Examples of Distributional Derivatives......Page 470
B.1 Systems of Orthogonal Polynomials......Page 472
B.1.1 Eigensolutions of Sturm–Liouville Problems......Page 474
B.1.2 The Legendre Polynomials......Page 475
B.1.3 The Chebyshev Polynomials......Page 477
B.2.1 Fundamental Theorems......Page 478
B.2.2 Gaussian Rules Based on Legendre Polynomials......Page 480
B.2.3 Gaussian Rules Based on Chebyshev Polynomials......Page 481
B.2.4 Discrete Inner Products and Norms......Page 482
B.3.1 Preliminaries......Page 484
B.3.2 Discrete Spectral Transforms......Page 485
Finite Legendre and Chebyshev Expansions......Page 488
The Gauss–Lobatto–Jacobi Collocation Derivatives......Page 490
First-Order Legendre and Chebyshev Collocation Derivatives......Page 492
B.3.4 Estimates for Truncation and Interpolation Errors......Page 494
Bibliography......Page 497
Index......Page 519