. 121
( 136 .)


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
(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,
Power Series with De¬nite Parity, 161
linear differential equation, 476
Shannon-Whittaker Sampling Theo-
Taylor-Green vortex, 166
rem, 343
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

Crank-Nicholson, fourth order, 260 preconditioning odd derivatives, 296
Explicit-Demanding Rule-of-Thumb,
unbounded domain
behavioral versus numerical bound-
hybrid grid point/Galerkin algorithm,
ary conditions, 361“363
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
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
splitting, basic theory, 252“254
rational Chebyshev T Ln (y ∈ [0, ∞]),
splitting, boundary dif¬culties, 256
splitting, high order for noncommut-
sinc basis (y ∈ [’∞, ∞]), 341“346
ing operators, 262
Weideman-Cloot map(y ∈ [’∞, ∞]),
trapezoidal implicit scheme, see time-
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),
truncation error
de¬ned, 31
two-h waves, 206
Two-Thirds Rule, see aliasing instability,Two-
Thirds Rule for Dealiasing
Chebyshev and Fourier Spectral Methods

Second Edition

John P. Boyd

University of Michigan
Ann Arbor, Michigan 48109-2143
email: jpboyd@engin.umich.edu



Abadi, M. H. A. and Ortiz, E. L.: 1991, A tau method based on non-uniform space-time
elements for the numerical simulation of solitons, Computers Math. Applic. 22(9), 7“19.
Abarbanel, S. and Gottlieb, D.: 1985, Information content in spectral calculations, in E. M.
Murman and S. S. Abarbanel (eds), Progress and Supercomputing in Computational Fluid
Dynamics, number 6 in Proc. U. S.-Israel Workshop, Birkhauser, Boston, pp. 345“356.
Abarbanel, S., Gottlieb, D. and Carpenter, M. H.: 1996, On the removal of boundary errors
caused by Runge-Kutta integration of nonlinear partial differential equations, SIAM
Journal of Scienti¬c Computing 17(3), 777“782. Not spectral, but essential in optimizing
Runge-Kutta methods with Chebyshev or other high order spatial discretizations.
Abarbanel, S., Gottlieb, D. and Tadmor, E.: 1986, Spectral methods for discontinuous prob-
lems, in K. W. Morton and M. J. Baines (eds), Numerical Methods for Fluid Dynamics, II,
Oxford University Press, Oxford, pp. 129“153.
Abe, K. and Inoue, O.: 1980, Fourier expansion solution of the Korteweg-deVries equation,
Journal of Computational Physics 34, 202“210.
Abramowitz, M. and Stegun, I. A.: 1965, Handbook of Mathematical Functions, Dover, New
Alpert, B. A. and Rokhlin, V.: 1991, A fast algorithm for the evaluation of Legendre expan-
sions, SIAM Journal of Scienti¬c Computing 12, 158“179. Provides a fast Legendre-to-
Chebyshev transform.
Amon, C. H. and Patera, A. T.: 1989, Numerical calculation of stable three-dimensional
tertiary states in grooved-channel ¬‚ow, Physics of Fluids A 1, 2005“2009.
Anagnostou, G., Maday, Y. and Patera, A. T.: 1991, Numerical simulation of incompress-
ible ¬‚uid ¬‚ows, Concurrency-Practice and Experience 3(6), 667“685. Review: high order
operator-integration factors methods, sliding mesh spectral mortar-element, and par-
allel computing.
Anderson, C. and Dahleh, M. D.: 1996, Rapid computation of the discrete Fourier trans-
form, SIAM Journal of Scienti¬c Computing 17, 913“919. Off-grid interpolation to irreg-
ularly spaced points by using local Taylor expansions at a cost of about 8 FFTs.
Anderson, D. L. T.: 1973, An ocean model using parabolic cylinder functions, World Mete-
orological Organization, Geneva. Model of equatorial ocean where Hermite functions
are the linearized normal modes; discontinued.
Aoyagi, A.: 1995, Nonlinear leapfrog instability for Fornberg™s pattern, Journal of Computa-
tional Physics 120, 316“322. Aliasing instability.


Arakawa, A.: 1966, Computational design for long-term numerical integration of the equa-
tions of ¬‚uid motion: Two dimensional incompressible ¬‚ow, Journal of Computational
Physics 1, 119“143. Energy-conserving ¬nite differences schemes.

Armstrong, T. P.: 1967, Numerical studies of nonlinear Vlasov equation, Physics of Fluids
10, 1269“1280. Fourier-Galerkin in the space dimension, Hermite-Galerkin in the ve-
locity coordinate. Slow convergence of the Hermite series, a dif¬culty not fully over-
come until variable scaling introduced in Holloway, 1996a,b.

Arnold, V. I.: 1983, Geometrical Methods in the Theory of Ordinary Differential Equations,
Springer-Verlag, New York, chapter 17, pp. 110“111. The algorithm of “counting the
noses” to approximate irrational numbers by rational fractions.

Augenbaum, J.: 1989, An adaptive pseudospectral method for discontinuous problems,
Applied Numerical Mathematics 5, 459“480.

Augenbaum, J. M.: 1990, Multidomain adaptive pseudospectral methods for acoustic wave
propagation in discontinuous media, in D. Lee, A. Cakmak and R. Vichnevetsky (eds),
Computational Acoustics-Seismo-Ocean Acoustics and Modeling, number 3 in Computa-
tional Acoustics, Elsevier/North-Holland, Amsterdam, pp. 19“40.

Augenbaum, J. M.: 1992, The pseudospectral method for limited-area elastic wave calcu-
lations, in W. E. Fitzgibbon and M. F. Wheeler (eds), Computational Methods in Geo-
sciences, Frontiers in Applied Mathematics, SIAM, Philadelphia. Chebyshev method
needs only about half the resolution in each coordinate compared to fourth order dif-

Averbuch, A., Ioffe, L., Israeli, M. and Vozovoi, L.: 1998a, Two-dimensional parallel solver
for the solution of navier-stokes equations with constant and variable coef¬cients us-
ing ADI on cells, Parallel Comput. 24(5“6), 673“699. Local Fourier basis for nonperiodic
problems, here with domain decomposition.

Averbuch, A., Israeli, M. and Vozovoi, L.: 1995, Parallel implementation of nonlinear evo-
lution problems using parabolic domain decomposition, Parallel Comput. 21(7), 1151“
1183. Fourier method for nonperiodic boundary conditions; multidomain.

Averbuch, A., Israeli, M. and Vozovoi, L.: 1998b, A fast poisson solver of arbitrary order
accuracy in rectangular regions, SIAM J. Sci. Comput. 19(3), 933“952. Fourier method
for nonperiodic boundary conditions with subtractions to handle Gibbs™ Phenomenon
and corner singularities.

Averbuch, A., Vozovoi, L. and Israeli, M.: 1997, On a fast direct elliptic solver by a modi¬ed
fourier method, Numer. Algorithms 15(3“4), 287“313. Fourier method for nonperiodic
boundary conditions with subtractions to handle Gibbs™ Phenomenon and corner sin-

Babolian, E. and Delves, L. M.: 1979, An augmented Galerkin method for ¬rst kind Fred-
holm equations, Journal of the Institute for Mathematics and its Applications 24, 151“174.
Galerkin basis is Chebyshev polynomials.

Babolian, E. and Delves, L. M.: 1981, A fast Galerkin method for linear integro-differential
equations, IMA Journal of Numerical Analysis 1, 193“213. Chebyshev series.

Baer, F.: 1977, Adjustment of initial conditions required to suppress gravity oscillations in
non-linear ¬‚ows, Contributions to Atmospheric Physics 50, 350“366.

Baer, F. and Tribbia, J. J.: 1977, On complete ¬ltering of gravity modes through nonlinear
initialization, Monthly Weather Review 105, 1536“1539.

Bain, M.: 1978, Convergence theorems for Hermite functions and Jacobi polynomials, Jour-
nal of the Institute for Mathematics and its Applications 21, 379“386.

Baker, Jr., G. A.: 1975, Essentials of Pad´ Approximants, Academic Press, New York. 220 pp.


. 121
( 136 .)