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-