multi-dimensional, 166

Orthogonality under the Discrete In-

parity, 159“165

ner Product, 90

rotation and dihedral groups, 165

Parity Decomposition, 163

Parity Matrix Multiplication Transform

tau-method

(PMMT), 190

Bibliography Table, 477

Parity of Basis Functions, 160

canonical polynomials, 478

Parity of the Powers of x, 161

de¬ned, 473

Polar Coordinates: Parity in Radius,

for approximating a rational function,

383

474

Power Series with De¬nite Parity, 161

linear differential equation, 476

Shannon-Whittaker Sampling Theo-

Taylor-Green vortex, 166

rem, 343

Theorems

Singularities of the Solution to a Lin-

Asymptotic Equality of Coef¬cients ear ODE, 36

if Singularities Match, 32

Strip of Convergence, Fourier Series,

Cauchy Interpolation Error, 85 45

Chebyshev Asymptotic Rate of Con- Symmetry Properties of an ODE, 164

vergence, 49 Trapezoidal Rule Error for Periodic

Chebyshev Ellipse-of-Convergence, 48 Integrands, 457

Chebyshev Interpolation and its Er- Trigonometric Interpolation, 93

ror Bound, 95 Three-Halves Rule, see Two-Thirds Rule

Chebyshev Minimal Amplitude, 85 time-dependent problems, 15, 16

Chebyshev Truncation, 47 time-marching

Convergence Domain in Complex Plane, (Adams-Moulton 2d order) implicit

35 scheme, see time-marching,Crank-

Darboux™s Principle:Singularities Con- Nicholson

trol Convergence, 32 A-stable property, 229

Differentiation and Parity, 162 Adams-Bashforth scheme, 3rd order

Elliptical Coordinates: Parity in Quasi- (AB3), 173

Radial Coordinate, 440 Adams-Bashforth, 2d order (AB2), 174

Fourier Interpolation Error, 94 Adams-Bashforth/Crank-Nicholson (AB3CN)

Fourier Truncation Error Bound, 50 semi-implicit scheme, 229

Gaussian Quadrature (Gauss-Jacobi Adams-Moulton 1st order implicit scheme,

Integration), 87 see time-marching,Backwards Eu-

Hermite Rate-of-Convergence, 350 ler

Hille™s Hermite Width-of-Convergence, adaptive Runge-Kutta (RK45), 174

350 Alternating-Direction Implicit (ADI),

Inner Product for Spectral Coef¬cients, 261

66 Backward Differentiation (BDF) im-

Integration-by-Parts Coef¬cient Bound, plicit schemes, 228

42 Backwards Euler (BE) implicit scheme,

Interpolation by Quadrature, 92 228

Legendre Rate of Convergence, 52 Bibliography Table: Fourier basis, multi-

LU Decomposition of a Banded Ma- dimensional, 181

trix, 518 Bibliography Table: Fourier basis, one-

Matrices Whose Elements Are Matri- dimensional, 180

ces, 520 consistency of schemes, 258

Matrices Whose Elements Depend on Crank-Nicholson (CN) implicit scheme,

a Parameter, 466 228

INDEX

594

Crank-Nicholson, fourth order, 260 preconditioning odd derivatives, 296

Explicit-Demanding Rule-of-Thumb,

unbounded domain

231

behavioral versus numerical bound-

hybrid grid point/Galerkin algorithm,

ary conditions, 361“363

176

comparison of logarithmic, algebra and

Implicit Scheme Forces Physics Slow-

exponential maps, 355

down Rule-of-Thumb, 230

domain truncation, 326, 339“340

implicit schemes, 228“229

functions that decay algebraically at

implicitly-implicit problems, 181

in¬nity, 363“366

KdV Eq. example (Fourier basis), 179

functions with non-decaying oscilla-

leapfrog, 174

tions, 372“374

operator theory of, 259

Hermite basis (y ∈ [’∞, ∞]), 346“

Runge-Kutta, 4th order (RK4), 173

353

semi-implicit, de¬ned, 229

need to pick scaling or map parame-

semi-Lagrangian (SL), see semi-Lagrangian

ter, 338, 348, 369, 378

splitting for diffusion, 255“256

rational Chebyshev T Bn (y ∈ [’∞, ∞]),

splitting for ¬‚uid mechanics, 263

356“361

splitting, basic theory, 252“254

rational Chebyshev T Ln (y ∈ [0, ∞]),

splitting, boundary dif¬culties, 256

369

splitting, high order for noncommut-

sinc basis (y ∈ [’∞, ∞]), 341“346

ing operators, 262

Weideman-Cloot map(y ∈ [’∞, ∞]),

trapezoidal implicit scheme, see time-

374“377

marching,Crank-Nicholson

toroidal coordinates

Van der Pol equation, 532“534

basis functions for, 392

de¬ned, 381 weak form of a differential equation, 68

illustrated, 391 weakly nonlocal solitary waves

trans¬nite interpolation, 114 special basis functions for, 450

transforms (grid/spectral & inverse) weather forecasting, numerical

Fast Fourier Transform (FFT), see Fast Lorenz-Krishnamurthy Quintet, 233

Fourier Transform multiple scales perturbation & itera-

Generalized FFTs, 195“198 tion, 243

Bibliography Table, 196 slow manifold, 231“232

Matrix Multiplication Transform (MMT),

see Matrix Multiplication Trans-

form, 190

partial summation, 184“187

to points off the grid, 198“199

triangular truncation, see spherical harmon-

ics,triangular truncation

trigonometric interpolation, 93

cardinal functions, 101“104

in¬nite series for interpolant coef¬-

cients, 93

solving differential equations (BVP),

103

truncation error

de¬ned, 31

two-h waves, 206

Two-Thirds Rule, see aliasing instability,Two-

Thirds Rule for Dealiasing

References

for

Chebyshev and Fourier Spectral Methods

Second Edition

John P. Boyd

University of Michigan

Ann Arbor, Michigan 48109-2143

email: jpboyd@engin.umich.edu

http://www-personal.engin.umich.edu/∼jpboyd/

2000

2

590

,

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e