Авторизация
Поиск по указателям
Boyd J.P. — Chebyshev and Fourier Spectral Methods
Обсудите книгу на научном форуме
Нашли опечатку? Выделите ее мышкой и нажмите Ctrl+Enter
Название: Chebyshev and Fourier Spectral Methods
Автор: Boyd J.P.
Аннотация: Completely revised text focuses on use of spectral methods to solve boundary value, eigenvalue, and time-dependent problems, but also covers Hermite, Laguerre, rational Chebyshev, sinc, and spherical harmonic functions, as well as cardinal functions, linear eigenvalue problems, matrix-solving methods, coordinate transformations, methods for unbounded intervals, spherical and cylindrical geometry, and much more. 7 Appendices. Glossary. Bibliography. Index. Over 160 text figures.
Язык:
Рубрика: Математика /Анализ /Продвинутый анализ /
Статус предметного указателя: Готов указатель с номерами страниц
ed2k: ed2k stats
Год издания: 2000
Количество страниц: 665
Добавлена в каталог: 02.04.2005
Операции: Положить на полку |
Скопировать ссылку для форума | Скопировать ID
Предметный указатель
Polar coordinates, Bibliography Table: annular domains 391
Polar coordinates, Bibliography Table:Unbounded Domain or External to Cylinder 390
Polar coordinates, boundary conditions in 383
Polar coordinates, One-Sided Jacobi polynomial basis for 387
Polar coordinates, parity in radius 383—385
Polar coordinates, radial basis sets and grids 385—390
Polar coordinates, spectral methods in 381—390
Polar coordinates, unbounded domain 390
Power Series with Definite Parity 161
Preconditioning (of iteration) as easy way to raise finite differences to spectral accuracy 301
Preconditioning (of iteration) by finite difference matrix 293—297
Preconditioning (of iteration) by finite element matrices 301
Preconditioning (of iteration) by incomplete LU factorizations of finite difference matrix 299
Preconditioning (of iteration) by small block-and-diagonal Galerkin matrix 307
Preconditioning (of iteration) for nonlinear problems 321
Pseudoarclength continuation 546—549
Pseudospectral method, boundary conditions see boundary conditions
Pseudospectral method, checking by decrease of coefficients 123
Pseudospectral method, checking by varying trunction N 123
Pseudospectral method, checking through finite difference residual 121
Pseudospectral method, choice of basis functions 109
Pseudospectral method, choice of interpolation grid 116
Pseudospectral method, common mistakes 155
Pseudospectral method, comparisons with finite differences see finite difference methods with
Pseudospectral method, defined 62
Pseudospectral method, derivatives, computing 116
Pseudospectral method, Fast Multipole Method (FMM), and 196
Pseudospectral method, halving grid and basis due to parity 165
Pseudospectral method, inferior to Galerkin for constant coefficient ODEs 313
Pseudospectral method, slow manifold: Chebyshev differentiation matrix 236—238
Pseudospectral method, special difficulties of high order derivatives 142
Pseudospectral method, zero phase and amplitude errors in Fourier basis 224
Quadrature,spectrally-accurate of periodic integrands 457
Quadrature,spectrally-accurate of singular integrands 459
Quadrature,spectrally-accurate, Clenshaw — Curtis adaptive 455
Quadrature,spectrally-accurate, Gaussian (Gauss — Jacobi) Theorem 87
Quadrature,spectrally-accurate, mechanics of non-Gaussian 456
Quadrature,spectrally-accurate, non-Gaussian infinite interval 456 458
Quasi-geostrophic Eq., time-marching 181
Quasi-Sinusoidal Rule-of-Thumb 54
Rational Chebyshev functions SB 365
Rational Chebyshev functions TB 356—361
Rational Chebyshev functions TB, Bibliography Table 357
Rational Chebyshev functions TB, collected identities 507
Rational Chebyshev functions TB, eigenvalue example 131
Rational Chebyshev functions TB, expansions of functions that decay algebraically at infinity 363—366
Rational Chebyshev functions TB, numerical examples 366—368
Rational Chebyshev functions TB, Table of derivative-computing formulas 555
Rational Chebyshev functions TB, Table of Explict Basis Functions 358
Rational Chebyshev functions TL 327 369—370
Rational Chebyshev functions TL, collected identities 509
Rational Chebyshev functions TL, numerical examples 370—372
Rational Chebyshev functions TL, Table of derivative-computing formulas 557
Rational Chebyshev functions TL, Table of Explict Basis Functions 369
Regularized Long Wave (RLW) Eq., time-marching 181
Richardson extrapolation 261
Richardson iteration see iteration Richardson defined
Robert functions (spherical basis) 434
Root-finding by Chebyshev algorithms 450—452
Rule-of-Thumb, Assumption of Equal Errors 32
Rule-of-Thumb, Behavioral Boundary Conditions at Infinity 362
Rule-of-Thumb, Boundary Layer Resolution Requirement 59
Rule-of-Thumb, CFL Stability Limit: Physics 173
Rule-of-Thumb, Dealiasing/Energy-Conserving 215
Rule-of-Thumb, Eigenvalue 132
Rule-of-Thumb, Explicit-Demanding 230
Rule-of-Thumb, Implicit Scheme Forces Physics Slowdown 230
Rule-of-Thumb, Last Coefficient Error Estimate 51
Rule-of-Thumb, Optimizing Infinite Interval Map Parameter 377
Rule-of-Thumb, Penalties of Unbounded Interval 338
Rule-of-Thumb, Quasi-Sinusoidal Resolution Requirement 55
Rule-of-Thumb, Teller’s Law 54
Rule-of-Thumb, Two-Thirds Rule (for dealiasing) 212
Rule-of-Thumb, Witch-of-Agnesi Resolution Estimate 57
Runge phenomenon see interpolation divergence
Sawtooth function 21
Scattering of waves, special basis functions for 448—450
Semi-implicit see time-marching semi-implicit
Semi-infinite interval see unbounded domain
Semi-Lagrangian (SL) time-marching, accuracy improves with increasing time step 279
Semi-Lagrangian (SL) time-marching, advantages and disadvantages 271
Semi-Lagrangian (SL) time-marching, Bibliography Table 287
Semi-Lagrangian (SL) time-marching, computational diffusion of 281
Semi-Lagrangian (SL) time-marching, iteration for departure points 275
Semi-Lagrangian (SL) time-marching, methods of characteristics and 272
Semi-Lagrangian (SL) time-marching, noninterpolating variants 281
Semi-Lagrangian (SL) time-marching, off-grid interpolation for 283
Semi-Lagrangian (SL) time-marching, three-level SI scheme 273
Semi-Lagrangian (SL) time-marching, two-level SI scheme 280
Separable PDEs, Haidvogel — Zang direct method 314
Shamrock Principle 178
Sideband truncation 443—446
Sinc function as infinite interval basis 341—346
Sinc function, Bibliography Table 345
Sinc function, connection with trigonometric interpolation 102
Sinc function, defined 99
Sinc function, derivative formulas 569
Sinc function, expansions in 343—344
Singular basis functions 330
Skew-symmetric advection 213
Slow manifold, defined 232
Slow manifold, forced linear oscillator 233
Slow manifold, initialization onto 239—243
Slow manifold, Korteweg — deVries equation 234
Slow manifold, Lorenz — Krishnamurthy Quintet 233
Slow manifold, multiple scale perturbation theory and 241
Slow manifold, numerically-induced 236
Slow manifold, steady-state (trivial slow manifold) 233
Slow manifold, three-part strategy 249
Slow manifold, tracking with implicit scheme 248
Slow manifold, weather forecasting 231—232
Spectral blocking, defined 207
Spectral blocking, delayed blow-up 210
Spectral blocking, frontogenesis and 217
Spectral blocking, linear example 209
Spectral blocking, remedies for 218
Spectral coefficients, computation by matrix multiplication see Matrix Multiplication Transform(MMT)
Spectral coefficients, integral for 305
Spectral convergence, defined 25
Spectral elements, Bibliography Table:Surface of Sphere 438
Spectral elements, cardinal basis (only) gives diagonal mass matrix 485
Spectral elements, choice of basis 486
Spectral elements, choice of grid 486
Spectral elements, defined 479
Spectral elements, degree of inter-element continuity 485
Spectral elements, influence matrix method 488—491
Spectral elements, matrix inversion 487
Spectral elements, patching versus variational formalism 486
Spectral elements, sectorial elements 492
Spectral elements, spherical coordinates 437
Spectral elements, two-dimensional maps 491
Spectral elements, variational formalism 484—485
Spectral elements, weak element-to-element coupling 481—484
Spherical coordinates, Bibliography Table: Gridpoint Methods 432
Spherical coordinates, Bibliography Table: Legendre transforms 408
Spherical coordinates, Bibliography Table: spectral elements 438
Spherical coordinates, Bibliography Table: variable resolution 410
Spherical coordinates, Glatzmaier stellar/mantle convection model 430
Spherical coordinates, mapping of sphere into a sphere 409
Spherical coordinates, non-tensor and icosahedral grids 432
Spherical coordinates, parity factor 393—397
Spherical coordinates, parity-modified Fourier series in latitude 434
Spherical coordinates, radial coordinate basis & grid 429
Spherical coordinates, resolution and unresolved scales 425—427
Spherical coordinates, Robert basis functions 434
Spherical coordinates, slow Legendre transforms in latitude 402—407
Spherical coordinates, spectral elements 437
Spherical coordinates, spherical harmonics see spherical harmonics
Spherical coordinates, variable resolution (limited-area) models 409
Spherical coordinates, vector basis functions 428
Spherical coordinates, “pole problem” (severe CFL limit) 398
Spherical harmonics, Addition Theorem/Group Property 408
Spherical harmonics, alternatives to spherical harmonics 433
Spherical harmonics, asymptotic approximations near equator 412
Spherical harmonics, asymptotic approximations near poles 411
Spherical harmonics, Bibliography Table: Alternatives to Spherical Harmonics 435
Spherical harmonics, Bibliography Table: comparisons with finite differences 438
Spherical harmonics, Bibliography Table: Reviews & Model Descriptions 441
Spherical harmonics, comparisons with finite differences 438
Spherical harmonics, defined 399
Spherical harmonics, equiareal resolution property 409
Spherical harmonics, PMMT in latitude 404
Spherical harmonics, reduced grid (near-pole deletions) 405
Spherical harmonics, shallow water wave algorithm 416
Spherical harmonics, software and libraries 414
Spherical harmonics, triangular truncation table 401
Spherical harmonics, triangular truncation, defined 400
Spherical harmonics, triangular versus rectangular truncation 388
Spherical projective filter, Bibliography Table: Projective Filters 437
Splitting see time-marching
Sponge layer 341
Subgeometric convergence, defined 26
Subgeometric convergence, examples, Fourier series 23
Supergeometric convergence, defined 26
Symbolic manipulation language and spectral methods 95 114 461—472
Symbolic manipulation language example 2
Symbolic manipulation language: Table of Precepts 463
Symmetry, halving grid and basis due to parity 165
Symmetry, multi-dimensional 166
Symmetry, parity 159—165
Symmetry, rotation and dihedral groups 165
Tau-method, Bibliography Table 477
Tau-method, canonical polynomials 478
Tau-method, defined 473
Tau-method, for approximating a rational function 474
Tau-method, linear differential equation 476
Taylor — Green vortex 166
Theorems, Asymptotic Equality of Coefficients if Singularities Match 32
Theorems, Cauchy Interpolation Error 85
Theorems, Chebyshev Asymptotic Rate of Convergence 49
Theorems, Chebyshev Ellipse-of-Convergence 48
Theorems, Chebyshev Interpolation and its Error Bound 95
Theorems, Chebyshev Minimal Amplitude 85
Theorems, Chebyshev Truncation 47
Theorems, Convergence Domain in Complex Plane 35
Theorems, Darboux’s Principle:Singularities Control Convergence 32
Theorems, Differentiation and Parity 162
Theorems, Elliptical Coordinates: Parity in Quasi-Radial Coordinate 440
Theorems, Fourier Interpolation Error 94
Theorems, Fourier Truncation Error Bound 50
Theorems, Gaussian Quadrature (Gauss — Jacobi Integration) 87
Theorems, Hermite Rate-of-Convergence 350
Theorems, Hille’s Hermite Width-of-Convergence 350
Theorems, Inner Product for Spectral Coefficients 66
Theorems, Integration-by-Parts Coefficient Bound 42
Theorems, Interpolation by Quadrature 92
Theorems, Legendre Rate of Convergence 52
Theorems, LU Decomposition of a Banded Matrix 518
Theorems, Matrices Whose Elements Are Matrices 520
Theorems, Matrices Whose Elements Depend on a Parameter 466
Theorems, Mean-Square Minimization with a Truncated Series 305
Theorems, Orthogonality under the Discrete Inner Product 90
Theorems, Parity Decomposition 163
Theorems, Parity Matrix Multiplication Transform (PMMT) 190
Theorems, Parity of Basis Functions 160
Theorems, Parity of the Powers of x 161
Theorems, Polar Coordinates: Parity in Radius 383
Theorems, Shannon-Whittaker Sampling Theorem 343
Theorems, Singularities of the Solution to a Linear ODE 36
Theorems, Strip of Convergence, Fourier Series 45
Theorems, Symmetry Properties of an ODE 164
Theorems, Trapezoidal Rule Error for Periodic Integrands 457
Theorems, Trigonometric Interpolation 93
Three-Halves Rule see Two-Thirds Rule
Time-dependent problems 15 16
Time-marching, (Adams — Moulton 2d order) implicit scheme see time-marching Crank
Time-marching, A-stable property 229
Time-marching, Adams — Bashforth scheme, 3rd order (AB3) 173
Time-marching, Adams — Bashforth, 2d order (AB2) 174
Time-marching, Adams — Bashforth/Crank — Nicholson (AB3CN) semi-implicit scheme 229
Time-marching, Adams — Moulton 1st order implicit scheme see time-marching Backwards
Time-marching, adaptive Runge — Kutta (RK45) 174
Time-marching, Alternating-Direction Implicit (ADI) 261
Time-marching, Backward Differentiation (BDF) implicit schemes 228
Time-marching, Backwards Euler (BE) implicit scheme 228
Time-marching, Bibliography Table: Fourier basis, multidimensional 181
Time-marching, Bibliography Table: Fourier basis, one-dimensional 180
Time-marching, consistency of schemes 258
Time-marching, Crank — Nicholson (CN) implicit scheme 228
Time-marching, Crank — Nicholson, fourth order 260
Time-marching, Explicit-Demanding Rule-of-Thumb 231
Time-marching, hybrid grid point/Galerkin algorithm 176
Time-marching, Implicit Scheme Forces Physics Slowdown Rule-of-Thumb 230
Time-marching, implicit schemes 228—229
Time-marching, implicitly-implicit problems 181
Time-marching, KdV Eq. example (Fourier basis) 179
Time-marching, leapfrog 174
Time-marching, operator theory of 259
Time-marching, Runge — Kutta, 4th order (RK4) 173
Time-marching, semi-implicit, defined 229
Time-marching, semi-Lagrangian (SL) see semi-Lagrangian
Time-marching, splitting for diffusion 255—256
Time-marching, splitting for fluid mechanics 263
Time-marching, splitting, basic theory 252—254
Time-marching, splitting, boundary difficulties 256
Time-marching, splitting, high order for noncommuting operators 262
Time-marching, trapezoidal implicit scheme see time-marching Crank
Toroidal coordinates, basis functions for 392
Toroidal coordinates, defined 381
Toroidal coordinates, illustrated 391
Transfinite interpolation 114
transform 190
Transforms (grid/spectral & inverse) to points off the grid 198—199
Transforms (grid/spectral & inverse), Bibliography Table 196
Transforms (grid/spectral & inverse), Fast Fourier Transform (FFT) see Fast Fourier Transform
Transforms (grid/spectral & inverse), Generalized FFTs 195—198
Transforms (grid/spectral & inverse), Matrix Multiplication Transform (MMT) see Matrix Multiplication
Transforms (grid/spectral & inverse), partial summation 184—187
Triangular truncation see spherical harmonics triangular
Trigonometric interpolation 93
Trigonometric interpolation, cardinal functions 101—104
Trigonometric interpolation, infinite series for interpolant coefficients 93
Trigonometric interpolation, solving differential equations (BVP) 103
Truncation error, defined 31
Two-h waves 206
Two-Thirds Rule see aliasing instability Rule
Two-Thirds Rule, preconditioning odd derivatives 296
Unbounded domain, behavioral versus numerical boundary conditions 361—363
Unbounded domain, comparison of logarithmic, algebra and exponential maps 355
Unbounded domain, domain truncation 326 339—340
Unbounded domain, functions that decay algebraically at infinity 363—366
Unbounded domain, functions with non-decaying oscillations 372—374
Unbounded domain, Hermite basis 346—353
Unbounded domain, need to pick scaling or map parameter 338 348 369 378
Unbounded domain, rational Chebyshev 356—361
Unbounded domain, rational Chebyshev 369
Unbounded domain, sinc basis 341—346
Unbounded domain, Weideman — Cloot map 374—377
van der Pol equation 532—534
Weak form of a differential equation 68
Weakly nonlocal solitary waves, special basis functions for 450
Weather forecasting, numerical, Lorenz — Krishnamurthy Quintet 233
Weather forecasting, numerical, multiple scales perturbation & iteration 243
Weather forecasting, numerical, slow manifold 231—232
Реклама