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mic decay, Advances in Applied Mathematics 9, 358“363.

Boyd, J. P.: 1988c, The superiority of Fourier domain truncation to Chebyshev domain
truncation for solving problems on an in¬nite interval, Journal of Scienti¬c Computing
3, 109“120.

Boyd, J. P.: 1989a, Chebyshev and Fourier Spectral Methods, Springer-Verlag, New York. 792

Boyd, J. P.: 1989b, New directions in solitons and nonlinear periodic waves: Polycnoidal
waves, imbricated solitons, weakly non-local solitary waves and numerical boundary
value algorithms, in T.-Y. Wu and J. W. Hutchinson (eds), Advances in Applied Mechan-
ics, number 27 in Advances in Applied Mechanics, Academic Press, New York, pp. 1“82.

Boyd, J. P.: 1989c, Periodic solutions generated by superposition of solitary waves for the
quarticly nonlinear Korteweg-de Vries equation, ZAMP 40, 940“944. Imbrication of
solitary wave generates good approximate periodic solutions.

Boyd, J. P.: 1989d, The asymptotic Chebyshev coef¬cients for functions with logarithmic
endpoint singularities, Applied Mathematics and Computation 29, 49“67.

Boyd, J. P.: 1989e, Non-local equatorial solitary waves, in J. C. J. Nihoul and B. M. Jamart
(eds), Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence: Proc. 20th Liege
Coll. on Hydrodynamics, Elsevier, Amsterdam, pp. 103“112.

Boyd, J. P.: 1990a, The orthogonal rational functions of Higgins and Christov and Cheby-
shev polynomials, Journal of Approximation Theory 61, 98“103.

Boyd, J. P.: 1990b, A numerical calculation of a weakly non-local solitary wave: the φ4
breather, Nonlinearity 3, 177“195.

Boyd, J. P.: 1990c, The envelope of the error for Chebyshev and Fourier interpolation, Jour-
nal of Scienti¬c Computing 5, 311“363.

Boyd, J. P.: 1990d, A Chebyshev/radiation function pseudospectral method for wave scat-
tering, Computers in Physics 4, 83“85.

Boyd, J. P.: 1991a, A comparison of numerical and analytical methods for the reduced wave
equation with multiple spatial scales, Applied Numerical Mathematics 7, 453“479.

Boyd, J. P.: 1991b, Monopolar and dipolar vortex solitons in two space dimensions, Wave
Motion 57, 223“243.

Boyd, J. P.: 1991c, Nonlinear equatorial waves, in A. R. Osborne (ed.), Nonlinear Topics of
Ocean Physics: Fermi Summer School, Course LIX, North-Holland, Amsterdam, pp. 51“

Boyd, J. P.: 1991d, Weakly nonlocal solitary waves, in A. R. Osborne (ed.), Nonlinear Topics of
Ocean Physics: Fermi Summer School, Course LIX, North-Holland, Amsterdam, pp. 527“
Boyd, J. P.: 1991e, Weakly nonlocal solitons for capillary-gravity waves: Fifth-degree
Korteweg-de Vries equation, Physica D 48, 129“146.
Boyd, J. P.: 1991f, Sum-accelerated pseudospectral methods: The Euler-accelerated sinc
algorithm, Applied Numerical Mathematics 7, 287“296.
Boyd, J. P.: 1992a, The arctan/tan and Kepler-Burger mappings for periodic solutions with
a shock, front, or internal boundary layer, Journal of Computational Physics 98, 181“193.
Numerical trick which is useful for solitary waves and cnoidal waves.
Boyd, J. P.: 1992b, The energy spectrum of fronts: The time evolution of shocks in Burgers™
equation, Journal of the Atmospheric Sciences 49, 128“139.
Boyd, J. P.: 1992c, Multipole expansions and pseudospectral cardinal functions: A new
generalization of the Fast Fourier Transform, Journal of Computational Physics 102, 184“
Boyd, J. P.: 1992d, A fast algorithm for Chebyshev and Fourier interpolation onto an irreg-
ular grid, Journal of Computational Physics 103, 243“257.
Boyd, J. P.: 1992e, Defeating the Runge phenomenon for equispaced polynomial interpola-
tion via Tikhonov regularization, Applied Mathematics Letters 5, 57“59.
Boyd, J. P.: 1993, Chebyshev and Legendre spectral methods in algebraic manipulation
languages, Journal of Symbolic Computing 16, 377“399.
Boyd, J. P.: 1994a, Hyperviscous shock layers and diffusion zones: Monotonicity, spectral
viscosity, and pseudospectral methods for high order differential equations, Journal of
Scienti¬c Computing 9, 81“106.
Boyd, J. P.: 1994b, The rate of convergence of Fourier coef¬cients for entire functions of in¬-
nite order with application to the Weideman-Cloot sinh-mapping for pseudospectral
computations on an in¬nite interval, Journal of Computational Physics 110, 360“372.
Boyd, J. P.: 1994c, The slow manifold of a ¬ve mode model, Journal of the Atmospheric Sci-
ences 51, 1057“1064.
Boyd, J. P.: 1994d, Nonlocal modons on the beta-plane, Geophysical and Astrophysical Fluid
Dynamics 75, 163“182.
Boyd, J. P.: 1994e, Time-marching on the slow manifold: The relationship between the
nonlinear Galerkin method and implicit timestepping algorithms, Applied Mathematics
Letters 7, 95“99.
Boyd, J. P.: 1994f, Sum-accelerated pseudospectral methods: Finite differences and sech-
weighted differences, Computer Methods in Applied Mechanics and Engineering 116, 1“11.
Boyd, J. P.: 1995a, Weakly nonlocal envelope solitary waves: Numerical calculations for the
Klein-Gordon (φ4 ) equation, Wave Motion 21, 311“330.
Boyd, J. P.: 1995b, A hyperasymptotic perturbative method for computing the radiation
coef¬cient for weakly nonlocal solitary waves, Journal of Computational Physics 120, 15“

Boyd, J. P.: 1995c, Eight de¬nitions of the slow manifold: Seiches, pseudoseiches and expo-
nential smallness, Dynamics of Atmospheres and Oceans 22, 49“75.

Boyd, J. P.: 1995d, A lag-averaged generalization of Euler™s method for accelerating series,
Applied Mathematics and Computation 72, 146“166.

Boyd, J. P.: 1995e, A Chebyshev polynomial interval-searching method (“Lanczos econo-
mization”) for solving a nonlinear equation with application to the nonlinear eigen-
value problem, Journal of Computational Physics 118, 1“8.

Boyd, J. P.: 1995f, Multiple precision pseudospectral computations of the radiation coef-
¬cient for weakly nonlocal solitary waves: Fifth-Order Korteweg-deVries equation,
Computers in Physics 9, 324“334.

Boyd, J. P.: 1996a, Asymptotic Chebyshev coef¬cients for two functions with very rapidly
or very slowly divergent power series about one endpoint, Applied Mathematics Letters
9(2), 11“15.

Boyd, J. P.: 1996b, Traps and snares in eigenvalue calculations with application to pseu-
dospectral computations of ocean tides in a basin bounded by meridians, Journal of
Computational Physics 126, 11“20. Corrigendum, 136, no. 1, 227-228 (1997).

Boyd, J. P.: 1996c, Numerical computations of a nearly singular nonlinear equation: Weakly
nonlocal bound states of solitons for the Fifth-Order Korteweg-deVries equation, Jour-
nal of Computational Physics 124, 55“70.

Boyd, J. P.: 1996d, The Erfc-Log ¬lter and the asymptotics of the Vandeven and Euler se-
quence accelerations, in A. V. Ilin and L. R. Scott (eds), Proceedings of the Third Interna-
tional Conference on Spectral and High Order Methods, Houston Journal of Mathematics,
Houston, Texas, pp. 267“276.

Boyd, J. P.: 1997a, Pad´ approximant algorithm for solving nonlinear ODE boundary value
broblems on an unbounded domain, Computers and Physics 11(3), 299“303.

Boyd, J. P.: 1997b, Pseudospectral/Delves-Freeman computations of the radiation coef¬-
cient for weakly nonlocal solitary waves of the Third Order Nonlinear Schroedinger
Equation and their relation to hyperasymptotic perturbation theory, Journal of Compu-
tational Physics 138, 665“694.

Boyd, J. P.: 1997c, The periodic generalization of Camassa-Holm “peakons”: An exact
superposition of solitary waves, Applied Mathematics and Computation 81(2), 173“187.
Classical solitons.

Boyd, J. P.: 1997d, Construction of Lighthill™s unitary functions: The imbricate series of
unity, Applied Mathematics and Computation 86, 1“10.

Boyd, J. P.: 1998a, Radiative decay of weakly nonlocal solitary waves, Wave Motion 27, 211“

Boyd, J. P.: 1998b, Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics: Gener-
alized Solitons and Hyperasymptotic Perturbation Theory, Vol. 442 of Mathematics and Its
Applications, Kluwer, Amsterdam. 608 pp. Three chapters on spectral methods, solv-
ing spectral-discretized nonlinear equations, and special methods for nonlocal solitary
waves and other phenomena that radiate to spatial in¬nity.

Boyd, J. P.: 1998c, High order models for the nonlinear shallow water wave equations on
the equatorial beta-plane with application to Kelvin wave frontogenesis, Dynamics of
Atmospheres and Oceans 28(2), 69“91.

Boyd, J. P.: 1998d, Two comments on ¬ltering, J. Comput. Phys. 143(1), 283“288. Shows
how to apply ¬lters or sum acceleration methods to spectral series so as to preserve
the boundary conditions. Also explains why additional boundary conditions are not
needed: high order ¬ltering operators can be interpreted as powers of the Legendre
or Chebyshev differential operator, which is singular at the boundaries.

Boyd, J. P.: 1998e, Global approximations to the principal real-valued branch of the Lambert
W-function, Appl. Math. Lett. 11(6), 27“31.

Boyd, J. P.: 1999a, Chebyshev and Fourier Spectral Methods, 2d edn, Dover, Mineola, New
York. 665 pp. Heavily revised and updated second edition of Boyd(1989).

Boyd, J. P.: 1999b, A numerical comparison of seven grids for polynomial interpolation on
the interval, Comput. Math. Appl. Submitted.

Boyd, J. P.: 1999c, The Blasius function in the complex plane, J. Experimental Math. In press.

Boyd, J. P.: 2001, Essays on Chebyshev and Fourier Spectral Methods. To appear.

Boyd, J. P. and Christidis, Z. D.: 1982, Low wavenumber instability on the equatorial beta-
plane, Geophysical Research Letters 9, 769“772.

Boyd, J. P. and Christidis, Z. D.: 1983, Instability on the equatorial beta-plane, in J. Nihoul
(ed.), Hydrodynamics of the Equatorial Ocean, Elsevier, Amsterdam, pp. 339“351.

Boyd, J. P. and Christidis, Z. D.: 1987, The continuous spectrum of equatorial Rossby waves
in a shear ¬‚ow, Dynamics of Atmospheres and Oceans 11, 139“151.

Boyd, J. P. and Flyer, N.: 1999, Compatibility conditions for time-dependent partial differ-
ential equations and the the rate of convergence of Chebyshev and Fourier spectral
methods, Comput. Meths. Appl. Mech. Engrg. In press.

Boyd, J. P. and Haupt, S. E.: 1991, Polycnoidal waves: Spatially periodic generalizations of
multiple solitary waves, in A. R. Osborne (ed.), Nonlinear Topics of Ocean Physics: Fermi
Summer School, Course LIX, North-Holland, Amsterdam, pp. 827“856.

Boyd, J. P. and Ma, H.: 1990, Numerical study of elliptical modons by a spectral method,
Journal of Fluid Mechanics 221, 597“611.

Boyd, J. P. and Natarov, A.: 1998, A Sturm-Liouville eigenproblem of the Fourth Kind:
A critical latitude with equatorial trapping, Stud. Appl. Math. 101, 433“455. Rational
TB calculation of logarithmically singular eigenproblems along an in¬nite interval
parallel to the real axis, but shifted from it by a constant.

Boyd, J. P. and Tan, B.: 1998, Vortex crystals and non-existence of non-axisymmetric soli-
tary waves in the Flierl-Petviashvili equation, Chaos, Solitons and Fractals 9, 2007“2021.
Double Fourier algorithm for a generalized, two-dimensional quasi-geostrophic equa-

Boyd, J. P. and Tan, B.: 1999, Composite bound states of wide and narrow envelope soli-
tons in the Coupled Schroedinger equations through matched asymptotic expansions,
Nonlinearity. Submitted.

Brachet, M. E., Meiron, D. I., Orszag, S. A., Nickel, B. G., Morf, R. H. and Frisch, U.: 1983,
Small-scale structure of the Taylor-Green vortex, Journal of Fluid Mechanics 130, 411“
452. Three-dimensional Fourier computations with exploitation of 64-fold symmetry.
Brandt, A., Fulton, S. R. and Taylor, G. D.: 1985, Improved spectral multigrid methods for
periodic elliptic problems, Journal of Computational Physics 58, 96“112.
Braverman, E., Israeli, M., Averbuch, A. and Vozovoi, L.: 1998, A fast 3D Poisson solver
of arbitrary order accuracy, J. Comput. Phys. 144(1), 109“136. Fourier spectral method
for nonperiodic boundary conditions with subtractions to improve convergence and
remedy corner singularities.
Brenier, B., Roux, B. and Bontoux, P.: 1986, Comparaison des m´ thodes tau-Chebyshev et
Galerkin dans l™´ tude de stabilit´ des mouvements de convection naturelle. Probl` me
e e e
des valeurs propres parasites, Journal de M´chanique th´orique et appliqu´e 5, 95“119.
e e e
Bridger, A. F. C. and Stevens, D. E.: 1980, Long atmospheric waves and the polar-plane
approximation to the earth™s spherical geometry, Journal of the Atmospheric Sciences
37, 534“544.
Briggs, W. L.: 1987, A Multigrid Tutorial, SIAM, Philadelphia. 88 pp.
Briggs, W. L. and Henson, V. E.: 1995, The DFT: An Owner™s Manual for the Discrete Fourier
Transform, Society for Industrial and Applied Mathematics, Philadelphia.
Briggs, W. L., Newell, A. C. and Sarie, T.: 1981, The mechanism by which many partial dif-
ference equations destabilize, in H. Haken (ed.), Chaos and Order in Nature, Springer-
Verlag, New York, pp. 269“273. Aliasing instability.
Brown, D. L. and Minion, M. L.: 1995, Performance of under-resolved two-dimensional
¬‚ow simulations, Journal of Computational Physics 122, 165“183.
Brown, J. D., Chu, M. T., Ellison, D. C. and Plemmons, R. J. (eds): 1994, Proceedings of the
Cornelius Lanczos International Centenary Conference, Society for Industrial and Applied
Mathematics, Philadelphia. Collection; many articles on spectral methods.
Browning, G. L., Hack, J. J. and Swarztrauber, P. N.: 1988, A comparison of three numeri-
cal methods for solving differential equations on the sphere, Monthly Weather Review
117, 1058“1075. Spherical harmonics versus fourth and sixth order ¬nite differences.
Cai, W., Gottlieb, D. and Harten, A.: 1992a, Cell-averaging Chebyshev methods for hyper-
bolic problems, Comput. & Math. Applics. 24, 37“49.
Cai, W., Gottlieb, D. and Shu, C.: 1989, Essentially nonoscillatory spectral Fourier methods
for shock wave calculation, Mathematics of Computation 52(186), 389“410.
Cai, W., Gottlieb, D. and Shu, C. W.: 1992b, On one-sided ¬lters for spectral Fourier approx-
imation of discontinuous functions, SIAM Journal of Numerical Analysis 29, 905“916.
Cain, A. B., Ferziger, J. H. and Reynolds, W. C.: 1984, Discrete orthogonal function expan-
sions for non-uniform grids using the Fast Fourier transform, Journal of Computational
Physics 56, 272“286. Use Fourier series with a mapping which is equivalent to rational
Calahan, D. A.: 1967, Numerical solution of linear systems with widely separated time con-
stants, Proceedings IEEE 55, 2016“2017. Not spectral; higher order Pad´ time marching

Canuto, C. and Funaro, D.: 1988, The Schwarz algorithm for spectral methods, SIAM Jour-
nal of Numerical Analysis 25, 24“40.

Canuto, C. and Quarteroni, A.: 1985, Preconditioned minimal residual methods for Cheby-
shev spectral calculations, Journal of Computational Physics 60, 315“337.

Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A.: 1988, Spectral Methods for Fluid
Dynamics, Springer-Verlag, New York. Classic text, 556 pp. Very comprehensive and
readable, now in paperback.

Carcione, J. M.: 1994, Boundary conditions for wave propagation problems, in C. Bernardi
and Y. Maday (eds), Analysis, Algorithms and Applications of Spectral and High Order
Methods for Partial Differential Equations, Selected Papers from the International Con-
ference on Spectral and High Order Methods (ICOSAHOM ™92), Le Corum, Montpel-


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