Tan, B. and Boyd, J. P.: 1998, Davydov soliton collisions, Physics Letters A 240, 282“286.

Double Fourier series for nonlinear initial value problem.

Tan, B. and Boyd, J. P.: 1999, Coupled-mode envelope solitary waves in a pair of cubic

Schroedinger equations with cross modulation, Nonlinearity. Submitted.

Tan, C. S.: 1985, Accurate solution of three-dimensional Poisson™s equation in cylindrical

coordinates by expansion in Chebyshev polynomials, Journal of Computational Physics

59, 81“95.

Tang, T.: 1993, The Hermite spectral method for Gaussian-type functions, SIAM Journal of

Scienti¬c Computing 14, 594“606.

Taylor, M.: 1995, Cubature for the sphere and the discrete spherical harmonic transform,

SIAM Journal of Numerical Analysis 32, 667“670.

Taylor, M. A. and Wingate, B. A.: 1999, The Fekete collocation points for triangular spectral

elements, SIAM Journal of Numerical Analysis. Submitted.

BIBLIOGRAPHY 657

Taylor, M., Tribbia, J. and Iskandarani, M.: 1997, The spectral element method for the shal-

low water equations on the sphere, Journal of Computational Physics 130(1), 92“108.

Taylor, T. D., Hirsh, R. S. and Nadworny, M. M.: 1981, FFT versus conjugate gradient

method for solutions of ¬‚ow equations by pseudospectral methods, in H. Viviand

(ed.), Proceedings of the 4th GAMM Conference on Numerical Methods in Fluid Mechanics,

Vieweg, Braunschweig, pp. 311“325.

Taylor, T. D., Hirsh, R. S. and Nadworny, M. M.: 1984, Comparison of FFT, direct inversion,

and conjugate gradient methods for use in pseudospectral methods, Computers and

Fluids 12, 1“9. Examines the penalty for using a nonstandard grid such that neither the

FFT nor matrix multiplication transform are applicable, and one must invert matrices

to calculate coef¬cients of f(x) from grid point values of f(x).

T´ mam, R.: 1989, Attractors for the Navier-Stokes equations: localization and approxi-

e

mation, Journal of the Faculty of Science of the University of Tokyo, Sec. Ia, Mathematics

36, 629“647.

T´ mam, R.: 1991a, New emerging methods in numerical analysis; applications to ¬‚uid

e

mechanics, in M. Gunzberger and N. Nicolaides (eds), Computational Fluid Dynamics

” Trend and Advances, Cambridge University Press, Cambridge.

T´ mam, R.: 1991b, Stability analysis of the nonlinear Galerkin method, Mathematics of Com-

e

putation 57, 477“503.

T´ mam, R.: 1992, General methods for approximating inertial manifolds. applications to

e

computing, in D. S. Broomhead and A. Iserles (eds), The Dynamics of Numerics and the

Numerics of Dynamics, Clarendon Press, Oxford, Oxford, pp. 1“21.

Temperton, C.: 1975, Algorithms for the solutions of cyclic tridiagonal systems, Journal of

Computational Physics 19, 317“323.

Temperton, C.: 1983a, Fast mixed-radix real Fourier transforms, Journal of Computational

Physics 52, 340“350.

Temperton, C.: 1983b, Self-sorting mixed-radix fast Fourier transforms, Journal of Computa-

tional Physics 52, 1“23.

Temperton, C.: 1985, Implementation of a self-sorting in-place prime factor FFT algorithm,

Journal of Computational Physics 58, 283“299.

Temperton, C.: 1992, A generalized prime factor FFT algorithm for any N=(2**p) (3**q)

(5**r), SIAM J. Sci. Stat. Comput. 13, 676“686.

Temperton, C. and Staniforth, A.: 1987, An ef¬cient two-time-level semi-Lagrangian

semi-implicit integration scheme, Quarterly Journal of the Royal Meteorological Society

113, 1025“1039.

Thompson, J., Warsi, Z. and Mastin, C.: 1985, Numerical Grid Generation, North-Holland,

New York. Not spectral.

Thompson, W. D.: 1917, Growth and Form, Cambridge University Press, Cambridge. Not

spectral, 600 pp.

Thuburn, J.: 1997, A PV-based shallow-water model on a hexagonal-icosahedral grid, Mon.

Wea. Rev. 125, 2328“2347. Non-spectral alternative for ¬‚ows on the surface of a sphere.

BIBLIOGRAPHY

658

Trefethen, L. N. and Bau, III, D.: 1997, Numerical Linear Algebra, Society for Industrial and

Applied Mathematics (SIAM), Philadelphia.

Tribbia, J. J.: 1984a, Modons in spherical geometry, Geophys. Astrophys. Fluid Dyn. 30, 131“

168. Spherical harmonics in a rotated coordinate system are used to construct nonlin-

ear Rossby waves.

Tribbia, J. J.: 1984b, A simple scheme for high-order nonlinear normal initialization,

Monthly Weather Review 112, 278“284. Iterative scheme for initialization onto the slow

manifold, illustrated by Hough-Hermite spectral basis.

Tse, K. L. and Chasnov, J. R.: 1998, A Fourier-Hermite pseudospectral method for pene-

trative convection, Journal of Computational Physics 142(2), 489“505. Two Fourier coor-

dinates plus Hermite functions in the unbounded vertical; motion is con¬ned to an

unstable layer far from any boundaries.

Tuckerman, L.: 1989, Divergence-free velocity ¬eld in nonperiodic geometries, Journal of

Computational Physics 80, 403“441. Very careful treatment of in¬‚uence matrix method

and spurious pressure modes in computing incompressible ¬‚ow.

Vallis, G. K.: 1985, On the spectral integration of the quasi-geostrophic equations for

doubly-periodic and channel ¬‚ow, Journal of the Atmospheric Sciences 42, 95“99.

Van Kemenade, V. and Deville, M.: 1994a, Preconditioned Chebyshev collocation for non-

Newtonian ¬‚uid ¬‚ows, in C. Bernardi and Y. Maday (eds), Analysis, Algorithms and

Applications of Spectral and High Order Methods for Partial Differential Equations, Se-

lected Papers from the International Conference on Spectral and High Order Methods

(ICOSAHOM ™92), Le Corum, Montpellier, France, 22-26 June 1992, North-Holland,

Amsterdam, pp. 377“386. Also in Finite Elements in Analysis and Design, vo. 16, pp.

237-245.

Van Kemenade, V. and Deville, M. O.: 1994b, Application of spectral elements to viscoelas-

tic creeping ¬‚ows, Journal of Non-Newtonian Fluid Mechanics 51, 277“308. Plane and

cylindrical geometries; Maxwell-B ¬‚uid; comparisons with 4 x 4 SUPG ¬nite elements.

van Loan, C.: 1992, Computational Frameworks for the Fast Fourier Transform, SIAM, Philadel-

phia.

Vandeven, H.: 1991, Family of spectral ¬lters for discontinuous problems, SIAM Journal of

Scienti¬c Computing 6, 159“192.

variational spectral method for the two-dimensional Stokes problem, A.: 1998, S.-r. jun and

y.-h. kwon and s. kang, Computers Math. Applic. 35(4), 1“17.

Verkley, W. T. M.: 1997a, A pseudo-spectral model for two-dimensional incompressible

¬‚ow in a circular basin. I. Mathematical formulation, Journal of Computational Physics

136(1), 100“114. Employs the basis set r|m| Pk

0,|m| ±,β

(2r2 ’ 1) exp(imθ) where Pk is

the usual Jacobi polynomial; shows and exploits the fact that the inverse of Laplace

operator has a banded Galerkin representation with this basis.

Verkley, W. T. M.: 1997b, A pseudo-spectral model for two-dimensional incompressible

¬‚ow in a circular basin. II. Numerical examples, Journal of Computational Physics

136(1), 115“131.

BIBLIOGRAPHY 659

Vermer, J. G. and van Loon, M.: 1994, An evaluation of explicit pseudo-steady-state approx-

imation schemes to stiff ODE systems from chemical kinetics, Journal of Computational

Physics 113, 347“352. The PSSA strategy, widely used in chemical kinetics, is equiva-

lent to the lowest order Nonlinear Galerkin method: the ODEs for the fast components

are replaced by algebraic relations.

Vianna, M. L. and Holvorcem, P. R.: 1992, Integral-equation approach to tropical ocean

dynamics. 1. Theory and computational methods, J. Marine Res. 50(1), 1“31. Boundary-

element algorithm constructed through ingenious summation of slowly converging

Hermite series.

Vinograde, B.: 1967, Linear and Matrix Algebra, D. C. Heath, Boston. Not spectral.

Voigt, R. G., Gottlieb, D. and Hussaini, M. Y. (eds): 1995, Spectral Methods for Partial Differ-

ential Equations, SIAM, Philadelphia.

Vozovoi, L., Israeli, M. and Averbuch, A.: 1996, Analysis and application of the Fourier-

Gegenbauer method to stiff differential equations, SIAM J. Numer. Anal. 33(5), 1844“

1863.

Vozovoi, L., Weill, A. and Israeli, M.: 1997, Spectrally accurate solution of non-periodic

differential equations by the Fourier-Gegenbauer method, SIAM J. Numer. Anal.

34(4), 1451“1471.

Wahba, G.: 1990, Spline Models for Observational Data, Vol. 59 of CBMS-NSF Regional Confer-

ence Series in Applied Mathematics, SIAM, Philadelphia. 169 pp. Fourier and spherical

harmonic splines.

Wang, D.: 1991, Semi-discrete Fourier spectral approximations of in¬nite dimensional

Hamiltonian systems and conservation laws, Computers Mathematics and Applications

21, 63“75.

Ware, A. F.: 1991, A spectral Lagrange-Galerkin method for convection-dominated diffusion prob-

lems, Ph.D. dissertation, Oxford University, Wolfson College. 106 pp.

Ware, A. F.: 1994, A spectral Lagrange-Galerkin method for convection-dominated diffu-

sion 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 Conference on Spectral and High Order Methods (ICOSAHOM ™92),

Le Corum, Montpellier, France, 22-26 June 1992, North-Holland, Amsterdam, pp. 227“

234. Also in Comp. Meths. Appl. Mech. Engrg., 116.

Ware, A. F.: 1998, Fast approximate Fourier transforms for irregularly spaced data, SIAM

Rev. 40(4), 838“856.

Weideman, J. A. C.: 1992, The eigenvalues of Hermite and rational spectral differentiation

matrices, Numerische Mathematik 61, 409“431.

Weideman, J. A. C.: 1994a, Computation of the complex error function, SIAM Journal of

Numerical Analysis 31, 1497“1518. Errata: 1995, 32, 330“331]. These series of rational

functions are useful for complex-valued z.

Weideman, J. A. C.: 1994b, Computing integrals of the complex error function, Proceedings

of Symposia in Applied Mathematics 48, 403“407. Short version of Weideman(1994a).

BIBLIOGRAPHY

660

Weideman, J. A. C.: 1995a, Errata: computation of the complex error function, SIAM Journal

of Numerical Analysis 32, 330“331.

Weideman, J. A. C.: 1995b, Computing the Hilbert Transform on the real line, Mathe-

matics of Computation 64(210), 745“762. The Hilbert transform of a function f is de-

∞

¬ned by H{f }(y) ≡ π P V ’∞ f (x) dx where P V is the principal value of the (sin-

1

x’y

gular) integral. Weideman shows that a set of complex-valued orthogonal functions,

ρn (x) = (l + ix)n /(l ’ ix)n+l , n = 0, ±1, ±2, . . . , are eigenfunctions of the trans-

form. Weideman™s algorithm using these functions is generally more ef¬cient than the

Fourier transform algorithm, H{f }(y) = F ’1 {isign(k)F {f }(k)}. The Hilbert trans-

form is important in many physical applications, notably the Benjamin-Ono model

for solitary waves in deep strati¬ed ¬‚uid.

Weideman, J. A. C. and Cloot, A.: 1990, Spectral methods and mappings for evolution

equations on the in¬nite line, Computer Methods in Applied Mechanics and Engineering

80, 467“481.

Weideman, J. A. C. and James, R. L.: 1992, Pseudospectral methods for the Benjamin-Ono

equation, Advances in Computer Methods for Partial Differential Equations VII pp. 371“

377. Both Fourier series and rational functions.

Welander, P.: 1955, Studies on the general development of motion in a two dimensional

¬‚uid, Tellus 7, 141“156. Not spectral; classic article that shows (i) a square blob of

¬‚uid becomes greatly stretched and deformed even by very smooth ¬‚ow ” becoming

“chaotically” mixed and bounded by a “fractal” curve, though these terms were not

yet invented which (ii) shows the unfeasibility of a fully Lagrangian (as opposed to

semi-Lagrangian) time-marching for ¬‚uids.

Weyl, H.: 1952, Symmetry, Princeton University Press, Princeton, New Jersey. Popular but

rigorous discussion of symmetry in art and nature and its mathematical embodiment

in group theory.

White, Jr., A. B.: 1982, On the numerical solution of initial/boundary value problems in

one space dimension, SIAM Journal of Numerical Analysis 19(4), 683“697. Shocks in

Burgers™ equation are resolved by transforming the spatial coordinate to arclength,

which requires simultaneously integrating an extra equation.

Whittaker, E. T.: 1915, On the functions which are represented by the expansions of the

interpolation theory, Proceedings of the Royal Society Edinburgh 35, 181“194. Pioneering

paper on sinc functions.

Wiin-Nielsen, A.: 1959, On the application of trajectory methods in numerical forecast-

ing, Tellus 11(2), 180“196. Not spectral; earliest application of semi-Lagrangian time-

marching in which the choice of the particles is reinitialized at each time step to those

arriving at the points of the regular Eulerian-coordinates grid at that time level.

Williamson, D.: 1990, Semi-Lagrangian transport in the NMC spectral model, Tellus A

42, 413“428.

Williamson, D. L. and Rasch, P. J.: 1989, Two-dimensional semi-Lagrangian transport with

shape-preserving interpolation, Monthly Weather Review 117, 102“129. Both Cartesian

and Gaussian (spherical) grid; only pure advection is computed.

BIBLIOGRAPHY 661

Wingate, B. A. and Boyd, J. P.: 1996, Spectral element methods on triangles for geophysical

¬‚uid dynamics problems, in A. V. Ilin and L. R. Scott (eds), Proceedings of the Third

International Conference on Spectral and High Order Methods, Houston Journal of Math-

ematics, Houston, Texas, pp. 305“314.

Wingate, B. A. and Taylor, M. A.: 1999a, A Fekete point triangular spectral element method;

application to the shallow water equations, Journal of Computational Physics. Submit-

ted.

Wingate, B. A. and Taylor, M. A.: 1999b, The natural function space for triangular spectral

elements, SIAM Journal of Numerical Analysis. Submitted.

Wragg, A.: 1966, The use of Lanczos-„ methods in the numerical solution of a Stefan prob-

lem, Computer Journal 9, 106“109.