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iteration Moments, Method of, 62
computing iterates by FFT, 298 multigrid, 301“304
for inde¬nite matrices, 318“320 Bibliography Table, 302
for nearly-separable PDEs, 317 for spectral elements, 488
for nonlinear equations, 320“322
inferior to direct methods for separa- nonlinear algebraic equations
ble PDEs, 315 bifurcation point (for solution branch),
Minimum Residual Richardson™s (MRR), 544
304“306 continuation method, 536“549
Moral Principles, 322 continuation method, de¬ned, 537
multigrid, see multigrid Davidenko equation method, 537
Nonlinear Richardson, 321 initializing strategies, 538“542
preconditioned Newton ¬‚ow, 321 limit point (for solution branch), 542“
preconditioning, see preconditioning 544
Richardson, de¬ned, 291 Newton iteration, 526“529
INDEX
590

pseudoarclength continuation, 546“ as easy way to raise ¬nite differences
549 to spectral accuracy, 301
nonlinear boundary value problems by ¬nite difference matrix, 293“297
Newton-Kantorovich iteration, 527“ by ¬nite element matrices, 301
531 by incomplete LU factorizations of ¬-
nonlinear eigenvalue problems nite difference matrix, 299
KdV cnoidal wave, 531 by small block-and-diagonal Galerkin
matrix, 307
Newton-Kantorovich iteration, 531“
534 for nonlinear problems, 321
van der Pol Eq. limit cycle, 532 pseudoarclength continuation, 546“549
Nonlinear Galerkin algorithm pseudospectral method
Bibliography Table, 244 boundary conditions, see boundary con-
ditions
de¬ned, 243
checking by decrease of coef¬cients,
weaknesses, 245“248
123
nonlinearity, 13
checking by varying trunction N , 123
orders of convergence checking through ¬nite difference resid-
algebraic, de¬ned, 25 ual, 121
exponential, de¬ned, 25 choice of basis functions, 109
graphical interpretation, 27“30 choice of interpolation grid, 116
in¬nite order, de¬ned, 25 common mistakes, 155
spectral, de¬ned, 25 comparisons with ¬nite differences,
see ¬nite difference methods,comparison
p-type ¬nite elements, see spectral elements with spectral
Pad´ approximants, 260
e de¬ned, 62
parallel computation, 9 derivatives, computing, 116
parity, see symmetry, parity Fast Multipole Method (FMM), and,
196
Parity Matrix Multiplication Transform (PMMT),
190“194 halving grid and basis due to parity,
165
partial summation (multi-dimensional grid-
to-spectral transform), 184“187 inferior to Galerkin for constant co-
ef¬cient ODEs, 313
periodicity
de¬nition, 20 slow manifold: Chebyshev differen-
tiation matrix, 236“238
polar cap
special dif¬culties of high order deriva-
de¬ned, 381
tives, 142
polar coordinates
zero phase and amplitude errors in
annular (ring-shaped) domains, 390
Fourier basis, 224
apparent singularities at origin, 383
Bibliography Table, 382
quadrature,spectrally-accurate
Bibliography Table: annular domains,
Clenshaw-Curtis adaptive, 455
391
Gaussian (Gauss-Jacobi) Theorem, 87
Bibliography Table:Unbounded Do-
mechanics of non-Gaussian, 456
main or External to Cylinder, 390
non-Gaussian in¬nite interval, 456, 458
boundary conditions in, 383
of periodic integrands, 457
One-Sided Jacobi polynomial basis for,
387 of singular integrands, 459
parity in radius, 383“385 quasi-geostrophic Eq.
radial basis sets and grids, 385“390 time-marching, 181
spectral methods in, 381“390 Quasi-Sinusoidal Rule-of-Thumb, 54
unbounded domain, 390
preconditioning (of iteration) rational Chebyshev functions SB, 365
INDEX 591

rational Chebyshev functions T B, 356“361 scattering of waves
special basis functions for, 448“450
Bibliography Table, 357
semi-implicit, see time-marching,semi-implicit
collected identities, 507
semi-in¬nite interval, see unbounded do-
eigenvalue example, 131
main
expansions of functions that decay al-
semi-Lagrangian (SL) time-marching
gebraically at in¬nity, 363“366
accuracy improves with increasing time
numerical examples, 366“368
step, 279
Table of derivative-computing formu-
advantages and disadvantages, 271
las, 555
Bibliography Table, 287
Table of Explict Basis Functions, 358
computational diffusion of, 281
rational Chebyshev functions T L, 327, 369“
iteration for departure points, 275
370
methods of characteristics and, 272
collected identities, 509
noninterpolating variants, 281
numerical examples, 370“372
off-grid interpolation for, 283
Table of derivative-computing formu-
three-level SI scheme, 273
las, 557
two-level SI scheme, 280
Table of Explict Basis Functions, 369
separable PDEs
Regularized Long Wave (RLW) Eq.
Haidvogel-Zang direct method, 314
time-marching, 181
Shamrock Principle, 178
Richardson extrapolation, 261
sideband truncation, 443“446
Richardson iteration, see iteration,Richardson,
sinc function
de¬ned
as in¬nite interval basis, 341“346
Robert functions (spherical basis), 434
Bibliography Table, 345
root-¬nding by Chebyshev algorithms, 450“
connection with trigonometric inter-
452
polation, 102
Rule-of-Thumb
de¬ned, 99
Teller™s Law, 54
derivative formulas, 569
Assumption of Equal Errors, 32
expansions in, 343“344
Behavioral Boundary Conditions at
singular basis functions, 330
In¬nity, 362
skew-symmetric advection, 213
Boundary Layer Resolution Require-
slow manifold
ment, 59
de¬ned, 232
CFL Stability Limit: Physics, 173
forced linear oscillator, 233
Dealiasing/Energy-Conserving, 215
initialization onto, 239“243
Eigenvalue, 132
Korteweg-deVries equation, 234
Explicit-Demanding, 230
Lorenz-Krishnamurthy Quintet, 233
Implicit Scheme Forces Physics Slow-
multiple scale perturbation theory and,
down, 230
241
Last Coef¬cient Error Estimate, 51
numerically-induced, 236
Optimizing In¬nite Interval Map Pa-
steady-state (trivial slow manifold),
rameter, 377
233
Penalties of Unbounded Interval, 338
three-part strategy, 249
Quasi-Sinusoidal Resolution Require-
tracking with implicit scheme, 248
ment, 55
weather forecasting, 231“232
Two-Thirds Rule (for dealiasing), 212
spectral blocking
Witch-of-Agnesi Resolution Estimate,
de¬ned, 207
57
delayed blow-up, 210
Runge Phenomenon, see interpolation,divergence
frontogenesis and, 217
of
linear example, 209
sawtooth function, 21 remedies for, 218
INDEX
592

spectral coef¬cients spherical harmonics, see spherical har-
monics
integral for, 305
variable resolution (limited-area) mod-
spectral coef¬cients, computation by ma-
els, 409
trix multiplication, see Matrix Mul-
vector basis functions, 428
tiplication Transform(MMT)
spherical harmonics
spectral convergence
Addition Theorem/Group Property,
de¬ned, 25
408
spectral elements
alternatives to spherical harmonics,
Bibliography Table:Surface of Sphere,
433
438
asymptotic approximations near equa-
cardinal basis (only) gives diagonal
tor, 412
mass matrix, 485
asymptotic approximations near poles,
choice of basis, 486
411
choice of grid, 486
Bibliography Table: Alternatives to
de¬ned, 479
Spherical Harmonics, 435
degree of inter-element continuity, 485
Bibliography Table: comparisons with
in¬‚uence matrix method, 488“491
¬nite differences, 438
matrix inversion, 487
Bibliography Table: Reviews & Model
patching versus variational formal-
Descriptions, 441
ism, 486
comparisons with ¬nite differences,
sectorial elements, 492
438
spherical coordinates, 437
de¬ned, 399
two-dimensional maps, 491
equiareal resolution property, 409
variational formalism, 484“485
PMMT in latitude, 404
weak element-to-element coupling, 481“ reduced grid (near-pole deletions), 405
484
shallow water wave algorithm, 416
spherical coordinates software and libraries, 414
“pole problem” (severe CFL limit), 398 triangular truncation table, 401
Bibliography Table: Gridpoint Meth- triangular truncation, de¬ned, 400
ods, 432 triangular versus rectangular trunca-
Bibliography Table: Legendre trans- tion, 388
forms, 408 spherical projective ¬lter
Bibliography Table: spectral elements, Bibliography Table: Projective Filters,
438 437
Bibliography Table: variable resolu- splitting, see time-marching,splitting
tion, 410 sponge layer, 341
Glatzmaier stellar/mantle convection subgeometric convergence
model, 430 de¬ned, 26
mapping of sphere into a sphere, 409 examples
non-tensor and icosahedral grids, 432 Fourier series, 23
parity factor, 393“397 supergeometric convergence
parity-modi¬ed Fourier series in lat- de¬ned, 26
itude, 434 symbolic manipulation language and spec-
radial coordinate basis & grid, 429 tral methods, 95, 114, 461“472
resolution and unresolved scales, 425“ symbolic manipulation language example,
427 2
Robert basis functions, 434 symbolic manipulation language: Table
slow Legendre transforms in latitude, of Precepts, 463
402“407 symmetry
spectral elements, 437 halving grid and basis due to parity,
INDEX 593

165 Mean-Square Minimization with a Trun-

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