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Boyd J.P. — Chebyshev and Fourier Spectral Methods
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Название: Chebyshev and Fourier Spectral MethodsАвтор: 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: 2000Количество страниц: 665Добавлена в каталог: 02.04.2005Операции: Положить на полку |
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Adaptive coordinate transformations see coordinate transformations adaptive Algebraic convergence, calculation of the index for ODE solution 43 Algebraic convergence, examples, half-wave rectifier function 22 Algebraic convergence, examples, sawtooth function 21 Algebraic convergence, index of, defined 25 Algebraic manipulation language see symbolic manipulation language and spectral methods Aliasing instability, dealiasing 211—216 Aliasing instability, energy-conserving schemes and 213—216 Aliasing instability, history 210 Aliasing instability, skew-symmetric advection and 213 Aliasing instability, theory 216—218 Aliasing instability, Two-Thirds Rule for Dealiasing 211—213 220 Aliasing, defined 204 Aliasing, equality on the grid 205 Aliasing, two-h waves 206 Arithmurgy 577 Assumption of Equal Errors defined 32 Asymptotic rate of convergence defined 27 Asymptotic spectral coefficients, Chebyshev polynomials 50 Asymptotic spectral coefficients, Fourier 45 Asymptotic spectral coefficients, Table: Rate vs. Singularity Type 60 Asymptotically diagonal, defined 307 Band-limited functions 343 Basis functions, special 446—450 Basis functions,choice of 9 109 Basis recombination see boundary conditions recombination Benjamin — Davis — Ono (BDO) Eq. 209 Bifurcation point 544 boundary conditions 10 Boundary conditions, absorbing 341 Boundary conditions, basis recombination 64 112—115 Boundary conditions, behavioral versus numerical 75 109—111 361 Boundary conditions, boundary-bordering 111—112 Boundary conditions, coordinate transformation of boundary conditions on derivatives 575—576 Boundary conditions, homogenization of boundary conditions 112—114 Boundary conditions, polar coordinates, at origin 383 Boundary layer 57 Boundary layer, Resolution Rule-of-Thumb 59 Boundary layer, symbolic manipulation example 465 Cardinal functions, general orthogonal polynomials 104 Cardinal functions, polynomial, defined 83 Cardinal functions, pseudospectral Chebyshev ODE-solving 115 Cardinal functions, Whittaker see sinc function Chebyshev polynomials of the Second Kind collected identities 499 Chebyshev polynomials, asymptotic coefficients, calculation of 49 Chebyshev polynomials, cardinal functions, Lobatto grid 570 Chebyshev polynomials, cardinal functions, “roots” grid 571 Chebyshev polynomials, collected identities 497—498 Chebyshev polynomials, computation by recurrence 323 Chebyshev polynomials, computation of and its derivatives by trigonometric formulas 324 553 Chebyshev polynomials, convergence theory 46 Chebyshev polynomials, definition-by-mapping 46 Chebyshev polynomials, ellipse of convergence 48 Chebyshev polynomials, recurrence for coefficients of derivative of Chebyshev series 298 Collocation see pseudospectral Compatibility conditions 41 Condition number 142 Continuation method see nonlinear algebraic equations method Convergence domains in the complex plane Theorem 35 Coordinate transformations, adaptive 332—334 Coordinate transformations, complex-plane detour 151—153 Coordinate transformations, for boundary conditions on derivatives 575—576 Coordinate transformations, general one-dimensional theory 325 Coordinate transformations, Kosloff/Tal-Ezer nearly-equispaced Chebyshev method 334—336 Coordinate transformations, spatially periodic 330 Coordinate transformations, sphere-to-sphere maps 409 Coordinate transformations, Tables of derivative transformations 550—552 Coordinate transformations, transfinite interpolation 114 Coordinate transformations, two-dimensional derivative Table 560 Corner singularities 37—41 Corner singularities, one-dimensional mappings for 327—329 Corner singularities, singular basis functions for 446—448 Corner singularities, two-dimensional mappings for 329 Courant — Friedrichs — Lewy (CFL) Computational Instability, defined 173 Courant — Friedrichs — Lewy (CFL) Computational Instability, eigenvalue analysis 227 Courant — Friedrichs — Lewy (CFL) Computational Instability, Rule-of-Thumb 173 Cylindrical coordinates see polar coordinates Darboux’s Principle 19 Darboux’s Principlem, example 33—34 Darboux’s Principlem, theorem 32 Davidenko equation see nonlinear algebraic equations equation Dealiasing see aliasing instability Definitions, algebraic index of convergence 25 Definitions, aliasing 204 Definitions, asymptotic rate of convergence 27 Definitions, band-limited functions 343 Definitions, bifurcation point 544 Definitions, continuation method 537 Definitions, Courant number 275 Definitions, Courant — Friedrichs — Lewy (CFL) Instability 173 Definitions, dense matrix 516 Definitions, discretization error 31 Definitions, domain truncation 339 Definitions, domain truncation error 339 Definitions, eigenvectors/eigenfunctions 128 Definitions, envelope of spectral coefficients 30 Definitions, equiareal resolution 408 Definitions, exponential convergence 25 Definitions, exponential index of convergence 26 Definitions, Foot of a trajectory/departure point 275 Definitions, Four Kinds of Sturm — Liouville Eigen problems 134 Definitions, Frechet derivative 528 Definitions, geometric convergence 26 Definitions, idiot 122 Definitions, infinite order convergence 25 Definitions, interpolation 81 Definitions, interpolation error 31 Definitions, limit point 542 Definitions, linear eigenvalue problem 128 Definitions, Method of Model Functions 31 Definitions, multiply upstream (semi-Lagrangian scheme) 275 Definitions, order of real axis decay 350 Definitions, parity 159 Definitions, patching 480 Definitions, periodicity 20 Definitions, sideband truncation 444 Definitions, sinc expansion of a function 343 Definitions, skyline of a matrix 517 Definitions, slow manifold 232 Definitions, slow manifold initialization 239 Definitions, sparse matrix 516 Definitions, spectral blocking 207 Definitions, spectral convergence 25 Definitions, splitting/fractional steps 253 Definitions, Spurious Eigenvalues 139 Definitions, subgeometric convergence 26 Definitions, supergeometric convergence 26 Definitions, tau-method 473 Definitions, triangular matrix 515 Definitions, truncation error 31 Definitions, two-h waves 206 Direct matrix-solving methods 514—525 Direct matrix-solving methods for constant coefficient ODEs 313 Direct matrix-solving methods for discretizations of separable PDEs 314 Direct matrix-solving methods, banded matrix 518 519 Direct matrix-solving methods, block and bordered matrices 522 Direct matrix-solving methods, block-tridiagonal matrices 521 Direct matrix-solving methods, matrix-of-a-matrix strategies 520—524 Direct matrix-solving methods, Table of Costs: LU, 514 Discretization error, defined 31 Domain truncation see unbounded interval domain Ease of programming 4 Eigenvalue example, Associated Legendre equation 135 Eigenvalue example, Hermite (parabolic cylinder) eigenproblem 131 Eigenvalue example, Laplace’s Tidal Equation 74 eigenvalue problems 74—75 127—158 Eigenvalue problems, Bibliography Table 157 Eigenvalue problems, common mistakes 155—156 Eigenvalue problems, continuous spectrum 135 136 Eigenvalue problems, four kinds of Sturm — Liouville problems, defined 134 Eigenvalue problems, inverse power method 149 Eigenvalue problems, local (one-mode-at-a-time) methods, defined 129 Eigenvalue problems, local versus global methods 149 Eigenvalue problems, power method 145—148 Eigenvalue problems, quartic oscillator in Maple 467 Eigenvalue problems, rejecting inaccurate eigenvalues 137—139 Eigenvalue problems, Rule-of-Thumb 132 Eigenvalue problems, subharmonic bifurcation of FKdV cnoidal wave 468 Eigenvalue problems, Table of “No-Brain” method 127 Eigenvalue problems, “spurious” eigenvalues 139—142 182 Elliptical coordinates, spectral methods in 439—440 Envelope of spectral coefficients, defined 30 Exponential convergence, defined 25 Exponential convergence, index of, defined 25 Exponential convergence, rates of, defined 26 Fast Fourier Transform (FFT), advantages of simultaneous computation of many transforms 193 Fast Fourier Transform (FFT), Bibliography Table 189 Fast Fourier Transform (FFT), perils and advantages of library software 200 Fast Fourier Transform (FFT), Table of costs 193 Fast Fourier Transform (FFT), theory 187—189 Fast Fourier Transform, generalized see transforms (grid/spectral & inverse) Generalized Fast Multipole Methods (FMM) and pseudospectral method 195—198 Fast Multipole Methods (FMM) for associated Legendre transforms 407 Filters/smoothers, aliasing remedy 211—213 Filters/smoothers, spherical harmonic series 418—424 Filters/smoothers, Two-Thirds Rule 212 Finite difference methods, comparison with spectral 6 55 73 Finite difference methods, computational diffusion 225 Finite difference methods, computational dispersion 225 Finite element method comparisons 4 5 Fourier Asymptotic Coefficient Expansion (FACE) 41 Fourier series, calculation of asymptotic coefficients 45 Fourier series, cardinal functions 562—568 Fourier series, coefficient integrals 20 Fourier series, complex form 20 Fourier series, convergence domain 45 Fourier series, Truncation Error Bound Theorem 50 Fractional steps see time-marching Frechet derivative 528 Galerkin method 67—80 Galerkin method, comparison with pseudospectral 95 Galerkin method, relationship with singular perturbation theory 471 Galerkin method, sparse matrices for constant coefficient ODEs 313 Gegenbauer polynomials in spherical harmonics 399—402 Gegenbauer polynomials, collected identities 502—504 Gegenbauer polynomials, convergence theory 54 Geometric convergence, defined 26 Geometric convergence, examples, imbricated Lorentzian 23 Grid point representation see also cardinal functions H/p-finite elements see spectral elements Hasegawa — Mima Eq. see quasi-geostrophic Eq. Hermite functions as infinite interval basis 346—353 Hermite functions, Bibliography Table 347 Hermite functions, collected identities 505—506 Hermite functions, Hille’s Width-of-Convergence Theorem 350 Hermite functions, Rate-of-Convergence Theorem 350 High order derivatives, special problems of 142 Hilbert transforms, spectral algorithms for computing 453 Infinite interval see unbounded domain Infiuence matrix method, spectral elements 488—491 Integrating factor, defined 265 Integrating factor, misuse of 267—269 Integration, numerical see quadrature Integration-by-Parts Coefficient Bound application to differential equation 43 Integration-by-Parts Coefficient Bound application to differential equation, theorem 42 Interpolation error, defined 31 Interpolation, cardinal functions for 98—108 Interpolation, Cauchy Polynomial Interpolation Error Theorem 85 Interpolation, Chebyshev 95 Interpolation, coefficients of interpolant via quadrature 92 Interpolation, divergence of (Runge Phenomenon) 83 Interpolation, halving grid and basis due to parity 165 Interpolation, Lagrangian 82—86 Interpolation, off-grid 198—199 283—286 Interpolation, spatial uniformity of accuracy 101 Interpolation, transfinite 114 Interpolation, trigonometric see trigonometric interpolation Ioakimidis’ root-finding algorithm 452 Iteration for indefinite matrices 318—320 Iteration for nearly-separable PDEs 317 Iteration for nonlinear equations 320—322 Iteration, computing iterates by FFT 298 Iteration, inferior to direct methods for separable PDEs 315 Iteration, Minimum Residual Richardson’s (MRR) 304—306 Iteration, Moral Principles 322 Iteration, multigrid see multigrid Iteration, Nonlinear Richardson 321 Iteration, preconditioned Newton flow 321 Iteration, preconditioning see preconditioning Iteration, Richardson, defined 291 Iteration, Richardson, optimum pseudotimestep for 292 295 Iteration, Stationary One-Step, defined 291 Iteration, when not to iterate 322 Iterations, Delves — Freeman 307—312 Korteweg — deVries (KdV) Eq., cnoidal waves, computation of 531—532 Korteweg — deVries (KdV) Eq., Fourier pseudospectral time-marching code 179 Korteweg — deVries (KdV) Eq., integrating factors for 268 Korteweg — deVries (KdV) Eq., slow manifold of 234 Laguerre functions 353—354 Laguerre functions, Bibliography Table 353 Laguerre functions, collected identities 508 Lanczos economization 451 Least squares method 62 Legendre polynomials, cardinal functions, Lobatto grid 572—574 Legendre polynomials, collected identities 500—501 Legendre polynomials, convergence theory 52—54 Limit point 542 Matrix diagonalization for solving separable PDEs 316 Matrix Multiplication Transform (MMT) 91 108 190—194 Matrix Multiplication Transform (MMT), cost versus FFT 192 Mean Weight Residual (MWR) Method, defined 61 Mercator coordinate, defined 135 Model Functions, Method of 54 Model Functions, Method of, defined 31 Moments, method of 62 Multigrid 301—304 Multigrid for spectral elements 488 Multigrid, Bibliography Table 302 Nonlinear algebraic equations, bifurcation point (for solution branch) 544 Nonlinear algebraic equations, continuation method 536—549 Nonlinear algebraic equations, continuation method, defined 537 Nonlinear algebraic equations, Davidenko equation method 537 Nonlinear algebraic equations, initializing strategies 538—542 Nonlinear algebraic equations, limit point (for solution branch) 542—544 Nonlinear algebraic equations, Newton iteration 526—529 Nonlinear algebraic equations, pseudoarclength continuation 546—549 Nonlinear boundary value problems, Newton — Kantorovich iteration 527—531 Nonlinear eigenvalue problems, KdV cnoidal wave 531 Nonlinear eigenvalue problems, Newton — Kantorovich iteration 531—534 Nonlinear eigenvalue problems, van der Pol Eq. limit cycle 532 Nonlinear Galerkin algorithm, Bibliography Table 244 Nonlinear Galerkin algorithm, defined 243 Nonlinear Galerkin algorithm, weaknesses 245—248 Nonlinearity 13 Orders of convergence, algebraic, defined 25 Orders of convergence, exponential, defined 25 Orders of convergence, graphical interpretation 27—30 Orders of convergence, infinite order, defined 25 Orders of convergence, spectral, defined 25 p-type finite elements see spectral elements Pade approximants 260 Parallel computation 9 Parity see symmetry parity Parity Matrix Multiplication Transform (PMMT) 190—194 Partial summation (multi-dimensional grid-to-spectral transform) 184—187 Periodicity, definition 20 Polar cap, defined 381 Polar coordinates, annular (ring-shaped) domains 390 Polar coordinates, apparent singularities at origin 383 Polar coordinates, Bibliography Table 382
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