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cated Series, 305
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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