Spectral methods. Evolution to complex geometries and applications to fluid dynamics / ed. by Canuto C.G., Hussaini M.Y., Quarteroni A., Zang T.A. - Berlin, Heidelberg: Springer-Verlag GmbH., 2007. - 596 p. - (Scientific computation). - ISSN 1434-8322; ISBN 978-540-30727-3  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / Contents

1. Fundamentals of Fluid Dynamics .............................. 1 1.1. Introduction .......................................... 1 1.2. Fluid Dynamics Background ............................. 1 1.2.1. Phases of Matter .............................. 2 1.2.2. Thermodynamic Relationships ................... 3 1.2.3. Historical Perspective ........................ 6 1.3. Compressible Fluid Dynamics Equations ................. 7 1.3.1. Compressible Navier-Stokes Equations .......... 8 1.3.2. Nondimensionalization ........................ 12 1.3.3. Navier-Stokes Equations with Turbulence Models ....................................... 13 1.3.4. Compressible Euler Equations ................. 17 1.3.5. Compressible Potential Equation .............. 17 1.3.6. Compressible Boundary-Layer Equations ........ 19 1.3.7. Compressible Stokes Limit .................... 20 1.3.8. Low Mach Number Compressible Limit ........... 21 1.4. Incompressible Fluid Dynamics Equations .............. 21 1.4.1. Incompressible Navier-Stokes Equations ....... 21 1.4.2. Incompressible Navier-Stokes Equations with Turbulence Models ....................... 22 1.4.3. Vorticity-Streamfunction Equations ........... 25 1.4.4. Vorticity-Velocity Equations ................. 26 1.4.5. Incompressible Boundary-Layer Equations ...... 27 1.5. Linear Stability of Parallel Flows ................... 27 1.5.1. Incompressible Linear Stability .............. 29 1.5.2. Compressible Linear Stability ................ 31 1.6. Stability Equations for Nonparallel Flows ............ 36 2. Single-Domain Methods for Stability Analysis ............... 39 2.1. Introduction ......................................... 39 2.2. Boundary-Layer Flows ................................. 41 2.2.1. Incompressible Boundary-Layer Flows .......... 41 2.2.2. Compressible Boundary-Layer Flows ............ 48 2.3. Linear Stability of Incompressible Parallel Flows .... 52 2.3.1. Spectral Approximations for Plane Poiseuille Flow .............................. 52 2.3.2. Numerical Examples for Plane Poiseuille Flow ......................................... 57 2.3.3. Some Other Incompressible Linear Stability Problems ..................................... 61 2.4. Linear Stability of Compressible Parallel Flows ...... 64 2.5. Nonparallel Linear Stability ......................... 69 2.5.1. Linear Parabolized Stability Equations ....... 69 2.5.2. Two-Dimensional Global Stability Analysis .... 71 2.6. Transient Growth Analysis ............................ 72 2.7. Nonlinear Stability .................................. 75 2.7.1. Quasi-Steady Finite-Amplitude Solutions ...... 75 2.7.2. Secondary Instability Theory ................. 77 2.7.3. Nonlinear Parabolized Stability Equations .... 81 3. Single-Domain Methods for Incompressible Flows ............. 83 3.1. Introduction ......................................... 83 3.2. Conservation Properties and Time-Discretization ...... 86 3.2.1. Conservation Properties ...................... 86 The Rotation Form ............................ 88 The Skew-Symmetric Form ...................... 90 Convection and Divergence Forms .............. 92 3.2.2. General Guidelines for Time-Discretization ... 92 3.2.3. Coupled Methods .............................. 93 Fully Implicit Schemes ....................... 93 Semi-Implicit Schemes ........................ 93 3.2.4. Splitting Methods ............................ 95 3.2.5. Other Integration Methods .................... 96 Operator Integration Factors ................. 96 Characteristics Methods ...................... 97 3.3. Homogeneous Flows .................................... 98 3.3.1. Fourier Galerkin Approximation for Isotropic Turbulence ......................... 98 3.3.2. De-aliasing Using Transform Methods .......... 99 3.3.3. Pseudospectral and Collocation Methods ...... 103 3.3.4. Rogallo Transformation for Homogeneous Turbulence .................................. 106 3.3.5. Large-Eddy Simulation of Isotropic Turbulence .................................. 108 3.3.6. The Taylor-Green Vortex Example: Stability, Accuracy and Aliasing ............ 110 3.4. Flows with One Inhomogeneous Direction .............. 121 3.4.1. Coupled Methods ............................. 123 Kleiser-Schumann Algorithm .................. 124 Normal Velocity-Normal Vorticity Algorithms .................................. 127 3.4.2. Galerkin Methods Using Divergence-Free Bases ....................................... 131 3.4.3. Splitting Methods ........................... 133 Chebyshev Staggered Grid .................... 133 Zang-Hussaini Algorithm ..................... 135 3.4.4. Other Confined Flows ........................ 138 3.4.5. Unbounded Flows ............................. 140 Flat-Plate Boundary-Layer Flows ............. 140 Free-Shear-Layer Flows ...................... 142 3.4.6. A Numerical Example: Accuracy ............... 144 3.5. Flows with Multiple Inhomogeneous Directions ........ 147 3.5.1. The Choice of Spatial Discretization in a Cavity ...................................... 149 3.5.2. The Choice of Spatial Discretization on a Reference Domain ............................ 157 3.6. Outflow Boundary Conditions ......................... 159 3.6.1. Fringe Regions .............................. 159 3.6.2. Buffer Domains .............................. 161 3.7. Analysis of Spectral Methods for Incompressible Flows ............................................... 162 3.7.1. Compatibility Conditions Between Velocity and Pressure ................................ 165 3.7.2. Direct Discretization of the Continuity Equation: The Inf-sup Condition ............. 168 General Theory .............................. 169 3.7.3. Specific Applications ....................... 172 Numerical Results ........................... 177 Extensions .................................. 178 3.7.4. The Inf-sup Condition and the Pressure Operator .................................... 179 3.7.5. Discretizations of the Continuity Equation by an Influence-Matrix Technique: The Kleiser-Schumann Method ..................... 183 4. Single-Domain Methods for Compressible Flows .............. 187 4.1. Introduction ........................................ 187 4.2. Boundary Treatment for Hyperbolic Systems ........... 187 4.2.1. Characteristic Compatibility Conditions ..... 188 An Example of Unstable Treatment ............ 188 The Characteristic Compatibility Method (CCM) ....................................... 189 CCM for a General ID System ................. 192 CCM for the Collocation Method .............. 193 CCM for a General Multidimensional System ...................................... 195 References and Outlook ...................... 196 4.2.2. Boundary Treatment for Linear Systems in Weak Formulations ........................... 197 4.2.3. Spectral Accuracy and Conservation .......... 199 4.2.4. Analysis of Spectral Methods for Symmetric Hyperbolic Systems .......................... 200 4.3. Boundary Treatment for the Euler Equations .......... 203 4.4. High-Frequency Control .............................. 208 4.5. Homogeneous Turbulence .............................. 211 4.5.1. Algorithmic Considerations .................. 211 4.5.2. Representative Applications ................. 214 4.6. Smooth, Inhomogeneous Flows ......................... 218 4.6.1. Euler Equations ............................. 218 4.6.2. Navier-Stokes Equations ..................... 221 4.6.3. Numerical Example ........................... 224 4.7. Shock Fitting ....................................... 226 4.8. Shock Capturing ..................................... 233 5. Multidomain Discretizations ............................... 237 5.1. Introduction ........................................ 237 5.2. The Spectral Element Method (SEM) in ID ............. 239 5.2.1. SEM Formulation ............................. 239 5.2.2. Construction of SEM Basis Functions ......... 241 5.2.3. SEM-NI and its Collocation Interpretation ... 243 5.3. SEM for Multidimensional Problems ................... 245 5.3.1. Construction of SEM Function Spaces ......... 245 5.3.2. Construction of SEM Basis Functions ......... 247 5.3.3. SEM and SEM-NI Formulations ................. 250 5.3.4. Algebraic Aspects of SEM and SEM-NI ......... 252 5.3.5. Finite-Element Preconditioning of SEM-NI Matrices .................................... 253 5.4. Analysis of SEM and SEM-NI Approximations ........... 257 5.4.1. One-Dimensional Analysis .................... 257 A Priori Error Analysis for SEM ............. 258 A Priori Analysis for SEM-NI ................ 260 A Posteriori Error Analysis ................. 261 5.4.2. Multidimensional Analysis ................... 263 A Priori Error Analysis ..................... 264 A Posteriori Error Analysis ................. 265 5.4.3. Some Proofs ................................. 268 5.5. Some Numerical Results for the SEM-NI Approximations ...................................... 273 5.5.1. Error Decay vs. N and h ..................... 273 5.5.2. Eigenfunction Approximation ................. 275 5.6. SEM for Stokes and Navier-Stokes Equations .......... 278 5.6.1. SEM and SEM-NI Formulations ................. 279 5.6.2. Stability and Convergence Analysis .......... 283 Proof of the Global Inf-sup Condition ....... 285 5.6.3. Numerical Results ........................... 286 5.7. The Mortar Element Method (MEM) ..................... 289 5.7.1. Formulation of MEM .......................... 290 5.7.2. Algebraic Aspects of MEM .................... 294 5.7.3. Analysis of MEM ............................. 296 5.7.4. Other Applications .......................... 299 5.8. The Spectral Discontinuous Galerkin Method (SDGM) in ID ............................................... 300 5.8.1. Linear Advection Problems in ID ............. 301 5.8.2. Linear Hyperbolic Systems in ID ............. 303 5.8.3. Time-Dependent Problems ..................... 308 5.8.4 Nonlinear Conservation Laws in ID ............. 313 5.9. SDGM for Multidimensional Problems .................. 316 5.9.1. Multidimensional Formulation ................ 317 Linear Problems ............................. 317 Nonlinear Conservation Laws ................. 319 5.9.2. The Mortar Technique for Geometrical Nonconformities ............................. 321 5.10. SDGM for Diffusion Equations ........................ 323 5.11. Analysis of SDGM .................................... 326 5.12. SDGM for Euler and Navier-Stokes Equations .......... 332 Approximate Numerical Fluxes ........................ 332 Time-Discretizations ................................ 333 Numerical Examples .................................. 334 Shock Tracking ...................................... 337 5.13. The Patching Method ................................. 339 5.13.1. Formulation of Patching Methods ............. 339 5.13.2. Comparison of Patching and SEM-NI ........... 343 5.13.3. Collocation Methods for the Euler Equations ................................... 345 Collocation Using a Nonstaggered Grid ....... 346 Collocation Using a Staggered Grid .......... 348 Multidomain Shock Fitting ................... 350 5.14. 3D Applications in Complex Geometries ............... 352 5.14.1. The Spectral Element Method: Application to Incompressible Flow ...................... 352 5.14.2. The Spectral Discontinuous Galerkin Method: Application to Compressible Flow .... 353 5.14.3. The Spectral Element Method: Application to Thermoelasticity ......................... 354 5.14.4. The Spectral Element Method: Structural Dynamics Analysis of the Roman Colosseum .... 356 6. Multidomain Solution Strategies ........................... 359 6.1. Introduction ........................................ 359 6.2. On Domain Decomposition Preconditioners ............. 359 6.3. (Overlapping) Schwarz Alternating Methods ........... 364 6.3.1. Algebraic Form of Schwarz Methods for Finite-Element Discretization ............... 367 6.3.2. The Schwarz Method as an Algebraic Preconditioner .............................. 370 6.3.3. Additive Schwarz Preconditioners for High-Order Methods .......................... 373 6.3.4. FEM-SEM Spectral Equivalence ................ 380 6.3.5. Analysis of Schwarz Methods ................. 381 6.3.6. A General Theoretical Framework for the Analysis of DD Iterations ................... 383 6.4. Schur Complement Iterative Methods .................. 385 6.4.1. The Steklov-Poincare Interface Problem ...... 385 6.4.2. Properties of the Steklov-Poincare Operator .................................... 387 6.4.3. The Schur Complement Matrix ................. 387 6.4.4. DD Preconditioners for the Schur Complement Matrix............................ 393 6.4.5. Preconditioners for the Stiffness Matrix Derived from Preconditioners for the Schur Complement Matrix ........................... 398 6.5. Solution Algorithms for Patching Collocation Methods ............................................. 402 7. Incompressible Flows in Complex Domains ................... 407 7.1. Introduction ........................................ 407 7.2. High-Order Fractional-Step Methods .................. 409 7.3. Solution of Generalized Stokes System ............... 415 7.3.1. Preconditioners for the Generalized Stokes Matrix .................................... 416 7.3.2. Conditioning and Preconditioning for the Pressure Schur Complement Matrix ............ 419 7.3.3. Domain Decomposition Preconditioners for the Stokes and Navier-Stokes Equations ...... 421 7.4. Algebraic Factorization Methods ..................... 425 7.4.1. Chorin-Temam and Yosida Algebraic Factorization Methods ....................... 425 7.4.2. Numerical Results for Yosida Schemes ........ 428 7.4.3. Preconditioners for the Approximate Pressure Schur Complement ................... 430 8. Spectral Methods Primer ................................... 435 8.1. The Fourier System .................................. 435 The Fourier Series .................................. 436 Truncation and Projection ........................... 436 Decay of the Fourier Coefficients ................... 437 Discrete Fourier Expansion and Interpolation ........ 438 Aliasing ............................................ 440 Differentiation ..................................... 441 Gibbs Phenomenon and Filtering ...................... 443 8.2. General Jacobi Polynomials in the Interval (—1,1) ... 445 The Jacobi Series. Truncation and Projection ........ 447 Gauss-Type Quadrature Formulas and Discrete Inner Products ............................................ 448 Discrete Polynomial Transform and Interpolation ..... 449 Differentiation ..................................... 451 8.3. Chebyshev Polynomials ............................... 451 Quadrature Formulas and Discrete Transforms ......... 453 Differentiation ..................................... 454 8.4. Legendre Polynomials ................................ 455 Quadrature Formulas and Discrete Norms .............. 457 Differentiation ..................................... 457 8.5. Modal and Nodal Boundary-Adapted Bases on the Interval ............................................ 458 8.6. Orthogonal Systems in Unbounded Domains ............. 460 Laguerre Polynomials and Laguerre Functions ......... 460 Hermite Polynomials and Hermite Functions ........... 461 8.7. Multidimensional Expansions ......................... 462 8.7.1. Tensor-Product Expansions ................... 463 8.7.2. Expansions on Simplicial Domains ............ 465 Collapsed Coordinates and Warped Tensor- Product Expansions .......................... 465 Non- Tensor-Product Expansions .............. 468 8.8. Mappings ............................................ 468 8.8.1. Finite Intervals ............................ 469 8.8.2. Semi-Infinite Intervals ..................... 471 8.8.3. The Real Line ............................... 473 8.8.4. Multidimensional Mappings on Finite Domains ..................................... 475 8.9. Basic Spectral Discretization Methods ............... 478 8.9.1. Tau Method .................................. 479 8.9.2. Collocation Method .......................... 481 8.9.3. Galerkin Method ............................. 482 8.9.4. Galerkin with Numerical Integration (G-NI) Method ...................................... 484 8.9.5. Other Boundary Conditions ................... 485 Appendix A. Basic Mathematical Concepts ....................... 489 A.l. Hilbert and Banach Spaces ............................... 489 A.2. The Cauchy-Schwarz Inequality ........................... 491 A.3. The Lax-Milgram Theorem ................................. 492 A.4. Dense Subspace of a Normed Space ........................ 492 A.5. The Spaces .................................. 493 A.6. The Spaces .............................. 493 A.7. Infinitely Differentiable Functions and Distributions ... 494 A.8. Sobolev Spaces and Sobolev Norms ........................ 496 A.9. The Sobolev Inequality .................................. 501 A.10. The Poincare Inequality ................................. 501 Appendix B. Fast Fourier Transforms ........................... 503 Appendix C. Iterative Methods for Linear Systems .............. 509 C.1. A Gentle Approach to Iterative Methods .................. 509 C.2. Descent Methods for Symmetric Problems .................. 513 C.3. Krylov Methods for Nonsymmetric Problems ................ 518 Appendix D. Time Discretizations .............................. 525 D.1. Notation and Stability Definitions ...................... 525 D.2. Standard ODE Methods .................................... 528 D.2.1. Leap Prog Method ................................. 529 D.2.2. Adams-Bashforth Methods .......................... 529 D.2.3. Adams-Moulton Methods ............................ 531 D.2.4. Backwards-Difference Formulas .................... 533 D.2.5. Runge-Kutta Methods .............................. 534 D.3. Low-Storage Schemes ...................................... 535 Appendix E. Supplementary Material ............................ 537 E.1. Numerical Solution of the Generalized Eigenvalue Problem .................................................. 537 E.2. Tau Correction for the Kleiser-Schumann Method ........... 539 E.3. The Piola Transform ...................................... 541 References .................................................... 544