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213, 220
49
arithmurgy, 577
cardinal functions, “roots” grid, 571
Assumption of Equal Errors
cardinal functions, Lobatto grid, 570
de¬ned, 32
collected identities, 497“498
asymptotic rate of convergence
computation by recurrence, 323
de¬ned, 27
computation of and its derivatives by
asymptotic spectral coef¬cients
trigonometric formulas, 324, 553
Chebyshev polynomials, 50
convergence theory, 46
Fourier, 45
de¬nition-by-mapping, 46
Table: Rate vs. Singularity Type, 60
ellipse of convergence, 48
asymptotically diagonal, de¬ned, 307
recurrence for coef¬cients of deriva-
tive of Chebyshev series, 298
band-limited functions, 343
Chebyshev polynomials of the Second Kind
basis functions, special, 446“450
collected identities, 499
basis functions,choice of, 9, 109
collocation, see pseudospectral
basis recombination, see boundary condi-
compatibility conditions, 41
tions,basis recombination
Benjamin-Davis-Ono (BDO) Eq., 209 condition number, 142

586
INDEX 587

continuation method, see nonlinear alge- domain truncation error, 339
braic equations,continuation method eigenvectors/eigenfunctions, 128
convergence domains in the complex plane envelope of spectral coef¬cients, 30
equiareal resolution, 408
Theorem, 35
exponential convergence, 25
coordinate transformations
exponential index of convergence, 26
adaptive, 332“334
Foot of a trajectory/departure point,
complex-plane detour, 151“153
275
for boundary conditions on deriva-
Four Kinds of Sturm-Liouville Eigen-
tives, 575“576
problems, 134
general one-dimensional theory, 325
Frechet derivative, 528
Kosloff/Tal-Ezer nearly-equispaced Cheby-
geometric convergence, 26
shev method, 334“336
idiot, 122
spatially periodic, 330
in¬nite order convergence, 25
sphere-to-sphere maps, 409
interpolation, 81
Tables of derivative transformations,
interpolation error, 31
550“552
limit point, 542
trans¬nite interpolation, 114
linear eigenvalue problem, 128
two-dimensional derivative Table, 560
Method of Model Functions, 31
corner singularities, 37“41
multiply upstream (semi-Lagrangian
one-dimensional mappings for, 327“
scheme), 275
329
order of real axis decay, 350
singular basis functions for, 446“448
parity, 159
two-dimensional mappings for, 329
patching, 480
Courant-Friedrichs-Lewy (CFL) Compu-
periodicity, 20
tational Instability
sideband truncation, 444
de¬ned, 173
sinc expansion of a function, 343
eigenvalue analysis, 227
skyline of a matrix, 517
Rule-of-Thumb, 173
slow manifold, 232
cylindrical coordinates, see polar coordi-
slow manifold initialization, 239
nates
sparse matrix, 516
spectral blocking, 207
Darboux™s Principle, 19
spectral convergence, 25
example, 33“34
splitting/fractional steps, 253
theorem, 32
Spurious Eigenvalues, 139
Davidenko equation, see nonlinear alge-
subgeometric convergence, 26
braic equations,Davidenko equa-
supergeometric convergence, 26
tion method
tau-method, 473
dealiasing, see aliasing instability,dealiasing
triangular matrix, 515
De¬nitions
truncation error, 31
algebraic index of convergence, 25
two-h waves, 206
aliasing, 204
direct matrix-solving methods, 514“525
asymptotic rate of convergence, 27
banded matrix, 518, 519
band-limited functions, 343
block and bordered matrices, 522
bifurcation point, 544
block-tridiagonal matrices, 521
continuation method, 537
for constant coef¬cient ODEs, 313
Courant number, 275
for discretizations of separable PDEs,
Courant-Friedrichs-Lewy (CFL) Insta-
314
bility, 173
matrix-of-a-matrix strategies, 520“524
dense matrix, 516
Table of Costs: LU, LLT , QR, 514
discretization error, 31
discretization error
domain truncation, 339
INDEX
588

de¬ned, 31 Fast Multipole Methods (FMM) for asso-
ciated Legendre transforms, 407
domain truncation, see unbounded inter-
val,domain truncation ¬lters/smoothers
aliasing remedy, 211“213
ease of programming, 4 spherical harmonic series, 418“424
eigenvalue example Two-Thirds Rule, 212
Associated Legendre equation, 135 ¬nite difference methods
Hermite (parabolic cylinder) eigen- comparison with spectral, 6, 55, 73
problem, 131 computational diffusion, 225
Laplace™s Tidal Equation, 74 computational dispersion, 225
eigenvalue problems, 74“75, 127“158 ¬nite element method comparisons, 4, 5
“spurious” eigenvalues, 139“142, 182 Fourier Asymptotic Coef¬cient Expansion
Bibliography Table, 157 (FACE), 41
common mistakes, 155“156 Fourier series
continuous spectrum, 135, 136 calculation of asymptotic coef¬cients,
45
four kinds of Sturm-Liouville prob-
lems, de¬ned, 134 cardinal functions, 562“568
inverse power method, 149 coef¬cient integrals, 20
local (one-mode-at-a-time) methods, complex form, 20
de¬ned, 129 convergence domain, 45
local versus global methods, 149 Truncation Error Bound Theorem, 50
power method, 145“148 fractional steps, see time-marching,splitting
quartic oscillator in Maple, 467 Frechet derivative, 528
rejecting inaccurate eigenvalues, 137“
139 Galerkin method, 67“80
Rule-of-Thumb, 132 comparison with pseudospectral, 95
relationship with singular perturba-
subharmonic bifurcation of FKdV cnoidal
tion theory, 471
wave, 468
sparse matrices for constant coef¬cient
Table of “No-Brain” method, 127
ODEs, 313
elliptical coordinates
Gegenbauer polynomials
spectral methods in, 439“440
collected identities, 502“504
envelope of spectral coef¬cients, de¬ned,
convergence theory, 54
30
in spherical harmonics, 399“402
exponential convergence
geometric convergence
de¬ned, 25
de¬ned, 26
index of, de¬ned, 25
examples
rates of, de¬ned, 26
imbricated Lorentzian, 23
Fast Fourier Transform (FFT) grid point representation, see also cardinal
functions
advantages of simultaneous compu-
tation of many transforms, 193
h/p-¬nite elements, see spectral elements
Bibliography Table, 189
Hasegawa-Mima Eq., see quasi-geostrophic
perils and advantages of library soft-
Eq.
ware, 200
Hermite functions
Table of costs, 193
as in¬nite interval basis, 346“353
theory, 187“189
Bibliography Table, 347
Fast Fourier Transform, generalized, see
collected identities, 505“506
transforms (grid/spectral & in-
verse),Generalized FFTs Hille™s Width-of-Convergence Theo-
rem, 350
Fast Multipole Methods (FMM) and pseu-
dospectral method, 195“198 Rate-of-Convergence Theorem, 350
INDEX 589

high order derivatives, special problems Richardson, optimum pseudotimestep
of, 142 for, 292, 295
Stationary One-Step, de¬ned, 291
Hilbert transforms, spectral algorithms for
when not to iterate, 322
computing, 453
iterations
Delves-Freeman, 307“312
in¬nite interval, see unbounded domain
in¬‚uence matrix method
Korteweg-deVries (KdV) Eq.
spectral elements, 488“491
cnoidal waves, computation of, 531“
integrating factor
532
de¬ned, 265
Fourier pseudospectral time-marching
misuse of, 267“269
code, 179
integration, numerical, see quadrature
integrating factors for, 268
Integration-by-Parts Coef¬cient Bound
slow manifold of, 234
application to differential equation,
43
Laguerre functions, 353“354
theorem, 42
Bibliography Table, 353
interpolation
collected identities, 508
cardinal functions for, 98“108
Lanczos economization, 451
Cauchy Polynomial Interpolation Er-
Least Squares Method, 62
ror Theorem, 85
Legendre polynomials
Chebyshev, 95
cardinal functions, Lobatto grid, 572“
spatial uniformity of accuracy, 101
574
coef¬cients of interpolant via quadra-
collected identities, 500“501
ture, 92
convergence theory, 52“54
divergence of (Runge Phenomenon),
limit point, 542
83
halving grid and basis due to parity, matrix diagonalization for solving sepa-
165 rable PDEs, 316
Lagrangian, 82“86 Matrix Multiplication Transform (MMT),
off-grid, 198“199, 283“286 91, 108, 190“194
trans¬nite, 114 cost versus FFT, 192
trigonometric, see trigonometric inter- Mean Weight Residual (MWR) Method
polation de¬ned, 61
interpolation error Mercator coordinate, de¬ned, 135
de¬ned, 31 Model Functions, Method of, 54
Ioakimidis™ root-¬nding algorithm, 452 de¬ned, 31

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