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Carcione, J. M.: 1996, A 2-D Chebyshev differential operator for the elastic wave equation,
Computer Methods in Applied Mechanics and Engineering 130, 33“45. Adaptive grid map-
ping using the Kosloff/Tal-Ezer map (for longer time step), the Augenbaum map (to
resolve narrow features) and different maps on different subdomains for maximum

Carpenter, M. H.: 1996, Spectral methods on arbitrary grids, Journal of Computational Physics
129(1), 74“86. Differentiation is performed using one grid of points while the equation
is collocated on a different grid. Generalization of Don and Gottlieb(1994).

Carpenter, R. L., Droegemeier, K. K., Woodward, P. R. and Hane, C. E.: 1990, Application of
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real arithmetic. This algorithm was lost in mid-century until rediscovered, in com-
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forms devised by E. T. Whittaker [today, we would call them “spreadsheets”] and the
mechanical harmonic analyzer of Michaelson and Stratton (1898) which empirically
discovered the Gibbs™ phenomenon.

Chan, T. and Kerkhoven, T.: 1985, Fourier methods with extended stability intervals for the
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Chan, T. F.: 1984, Newton-like pseudo-arclength methods for computing simple turning
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Chaouche, A. M.: 1990, A collocation method basedon an in¬‚uence matrix technique for
axisymmetric ¬‚ows in an annulus, Rech. A´rosp. 1990-5, 1“13.

Chaouche, A., Randriamampianina, A. and Bontoux, P.: 1990, A collocation method based
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Chapman, S. and Lindzen, R. S.: 1970, Atmospheric Tides, D. Reidel, Dordrecht, Holland.
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Chen, H. B.: 1993a, On the instability of a full non-parallel ¬‚ow ” Kovasznay ¬‚ow, Inter-
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Chen, S., Doolen, G. D., Kraichnan, R. H. and She, Z.-S.: 1993, On statistical correlation
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Chen, X.-S.: 1993b, The aliased and dealiased spectral models of the shallow-water equa-
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(Rossby-Haurwitz waves) and realistic conditions (observational data from FGGE). In
all cases, the aliased models are no better than aliased models with the same number
of points after ¬ltering (the dealiased code uses (3/2)N points to compute N modes);
the aliased models are always considerably worse than dealiased models when com-
pared on the basis of the same number of collocation points (the dealiased code has
a smaller number of modes than the aliased code after the dealiasing ¬ltering is ap-
plied). One of the dealiased methods is novel in that it uses the Walsh Hadamard
transform instead of the usual FFT. The WT transform method is a little cheaper than
the FFT for the same accuracy, but all of the dealiased codes are 1. 7 to 2 times more
expensive than the aliasing codes with the same number of grid points. but unfortu-
nately no more accurate.

Christov, C. I.: 1982, A complete orthonormal system in L2 (’∞, ∞) space, SIAM Journal of
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Christov, C. I. and Bekyarov, K. L.: 1990, A Fourier-series method for solving soliton prob-
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Chu, M. T.: 1988, On the continuous realization of iterative processes, SIAM Review 30, 375“
387. Differential equations in pseudotime as models for Newton™s and other iterations.

Cividini, A. and Zampieri, E.: 1997, Nonlinear stress analysis problems by spectral collo-
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series, Proceedings of the Cambridge Philosophical Society 53, 134“149.

Clenshaw, C. W. and Curtis, A. R.: 1960, A method for numerical integration on an auto-
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vorticity formulation for ¬‚ows with two nonperiodic directions, J. Comput. Phys.
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unbounded domains, Applied Mathematics Letters 4, 23“27.

Cloot, A. and Weideman, J. A. C.: 1990, Spectral methods and mappings for evolution
equations on the in¬nite line, Computer Methods in Applied Mechanics and Engineering
80, 467“481.

Cloot, A. and Weideman, J. A. C.: 1992, An adaptive algorithm for spectral computations
on unbounded domains, Journal of Computational Physics 102, 398“406.

Cloot, A., Herbst, B. M. and Weideman, J. A. C.: 1990, A numerical study of the cubic-
quintic Schrodinger equation, Journal of Computational Physics 86, 127“146.

Concus, P. and Golub, G. H.: 1973, Use of fast direct methods for the ef¬cient numerical
solution of nonseparable elliptic equations, SIAM Journal of Numerical Analysis. Not
spectral, but shows how a fast direct method, either spectral or otherwise, can be used
to ef¬ciently solve nonseparable elliptic equations.

Cooley, J. W. and Tukey, J. W.: 1965, An algorithm for the machine calculation of complex
Fourier series, Mathematics of Computation 19, 297“301.

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the spurious resonant response of semi-Lagrangian schemes to orographic forcing,
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variable resolution: Application to the shallow-water equations, Quarterly Journal of
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Courtier, P. and Naughton, M.: 1994, A polar problem in the reduced Gaussian grid, Quart.
J. Roy. Met. Soc. 120, 1389“1407.

Courtier, P., Freydier, C., Geleyn, J. F., Rabier, F. and Rochas, M.: 1991, The Arpege project at
M´ t´ o-France., Proceedings of Numerical Methods in Atmospheric Models, European Cen-
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Shin¬eld Park, Reading, United Kingdom, pp. 193“231.

Coutsias, E. A., Hagstrom, T. and Torres, D.: 1996, An ef¬cient spectral method for ordinary
differential equations with rational function coef¬cients, Mathematics of Computation
65, 611“635. Obtained banded matrix representations of one-dimensional differential
operators for all standard polynomial basis sets, assuming that the coef¬cients of the
operators are restricted to polynomials or rational functions. These banded matrices
can be inverted in O(N ) operations.

Couzy, W. and Deville, M.: 1994, Iterative solution technique for spectral-element pressure
operators at high Reynolds number, in S. Wagner et al. (eds), Proceedings of the Second
European Computational Fluid Dynamics Conference, Stuttgart, Germany, pp. 613“618.

Craik, A. D. D.: 1985, Wave Interactions and Fluid Flows, Cambridge University Press, New
York. Not spectral, but a good description of weakly nonlinear waves and resonant
triad and four-wave interactions.

Curchitser, E. N., Iskandarani, M. and Haidvogel, D. B.: 1998, A spectral element solu-
tion of the shallow-water equations on multiprocessor computers, J. Atmos. Oceanic
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modes, Monthly Weather Review 108, 100“110.

Daley, R.: 1981, Normal mode initialization, Reviews of Geophysics and Space Physics
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cation: Some veri¬cation statistics, Atmosphere-Ocean 16, 187“196.

Davis, P. J.: 1975, Interpolation and Approximation, Dover Publications, New York. 200 pp.

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Dawkins, P. T., Dunbar, S. R. and Douglass, R. W.: 1998, The origin and nature of spurious
eigenvalues in the spectral tau method, J. Comput. Phys. 147(2), 441“462. Theoretical
paper proving the existence of two spurious eigenvalues for the exemplary problem
uxxxx = »uxx which are larger than N 4 . The cure and numerical experiments are also

de Veronico, M. C., Funaro, D. and Reali, G. C.: 1994, A novel numerical technique to
investigate nonlinear guided waves: approximation of the Nonlinear Schroedinger
equation by nonperiodic pseudospectral methods, Numerical Methods for Partial Dif-
ferential Equations 10(6), 667“675. Domain decomposition with Legendre polynomials
on interior domains and Laguerre polynomials on the exterior elements, which extend
to in¬nity.

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equation, using a spectral multigrid method, in W. F. Ballhaus and M. Y. Hussaini
(eds), Advances in Fluid Dynamics, Springer-Verlag, New York, pp. 25“35.

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modeling for complex geometry ¬‚ows: Application to grooved channels and circular
cylinders, Physics of Fluids A 3, 2337“2354. Spectral elements; also derivation and
application of a four-mode model using “empirical eigenfunctions” as the basis (also
known as “proper orthogonal decomposition”.

Debussche, A., Dubois, T. and Temam, R.: 1995, The Nonlinear Galerkin method: A mul-
tiscale method applied to the simulation of homogeneous turbulent ¬‚ows, Theoretical
and Computational Fluid Dynamics 7, 279“315.

Decker, D. W. and Keller, H. B.: 1980, Path following near bifurcation, Communications in
Pure and Applied Mathematics 34, 149“175.
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Methods for O. D. E.s, Clarendon Press, Oxford University Press, Oxford, pp. 269“281.
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of the Institute for Mathematics and its Applications 20, 173“184.
Delves, L. M.: 1977b, On the solution of the linear equation arising from Galerkin methods,
Journal of the Institute for Mathematics and its Applications 20, 163“171.
Delves, L. M.: 1977c, A linear equation solver for Galerkin and least squares methods, J.
Comp. 20, 371“374.
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solution of Fredholm integral equations, Journal of the Institute for Mathematics and its
Applications 23, 139“166.
Delves, L. M., Abd-Elal, L. F. and Hendry, J. A.: 1981a, A set of modules for the solution of
integral equations, Comp. J. 24, 184“190.
Delves, L. M. and Hall, C. A.: 1979, An implicit matching procedure for global element
calculations, Journal of the Institute for Mathematics and its Applications 23, 223“234.
Delves, L. M. and Mead, K. O.: 1971, On the convergence rates of variational methods.
I. Asymptotically diagonal systems, Mathematics of Computation 25, 699“716. Theory
Delves, L. M. and Phillips, C.: 1980, A fast implementation of the global element method,
Journal of the Institute for Mathematics and its Applications 25, 177“197.
Delves, L. M., McKerrell, A. and Henry, J. A.: 1981b, A note on Chebyshev methods for the
solution of partial differential equations, Journal of Computational Physics 41, 444“452.
Delves, L. M., McKerrell, A. and Peters, S. A.: 1986, Performance of GEM2 on the ELLPACK
problem population, International Journal for Numerical Methods in Engineering 23, 229“
Delves, L. N. and Freeman, T. N.: 1981, Analysis of Global Expansion Methods: Weakly Asymp-
totically Diagonal Systems, Academic Press, New York. 275 pp. Mostly theory with
only a handful of elementary examples, but the preconditioned “Delves-Freeman”
iteration is very interesting.
Demaret, P. and Deville, M. O.: 1989, Chebyshev pseudospectral solution of the Stokes
equation using ¬nite element preconditioning, Journal of Computational Physics 83, 463“
Demaret, P. and Deville, M. O.: 1991, Chebyshev collocations solutions of the Navier-
Stokes equations using multi-domain decomposition and ¬nite element precondition-
ing, Journal of Computational Physics 95, 359“386.
Demaret, P., Deville, M. O. and Schneidesch, C.: 1989, Thermal convection solutions by
Chebyshev pseudospectral multi-domain decomposition and ¬nite element precon-
ditioning, Applied Numerical Mathematics 6, 107“121. Nonlinear steady ¬‚ows through
nested iterations: outer Newton/inner Richardson.

D´ qu´ Piedelievre, M.: 1995, High resolution climate model over Europe, Climate Dynamics
11, 321“339.

Dettori, L., Gottlieb, D. and T´ mam, R.: 1995, Nonlinear Galerkin method: the two-level
Fourier collocation case, Journal of Scienti¬c Computing 10, 371“.


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