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Boyd J.P. — Chebyshev and Fourier Spectral Methods
Boyd J.P. — Chebyshev and Fourier Spectral Methods



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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: en

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
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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 $(y \in[\gets \infty, \infty])$      346—353
Unbounded domain, need to pick scaling or map parameter      338 348 369 378
Unbounded domain, rational Chebyshev $TB_n$ $(y \in [\gets \infty, \infty])$      356—361
Unbounded domain, rational Chebyshev $TL_n$ $(y \in [0, \infty])$      369
Unbounded domain, sinc basis $(y \in [\gets \infty, \infty])$      341—346
Unbounded domain, Weideman — Cloot map $(y \in [\gets \infty, \infty])$      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
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