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Deville, M.: 1990, Chebyshev collocation solutions of ¬‚ow problems, in C. Canuto and
A. Quarteroni (eds), Spectral and High Order Methods for Partial Differential Equations:
Proceedings of the ICOSAHOM ™89 Conference in Como, Italy, North-Holland/Elsevier,
Amsterdam, pp. 27“38. Also in Comput. Meths. Appl. Mech. Engrg., vol. 80.

Deville, M. and Labrosse, G.: 1982, An algorithm for the evaluation of multi-dimensional
(direct and inverse) discrete Chebyshev transform, Journal of Computational and Applied
Mathematics 8, 293“304.

Deville, M. and Mund, E.: 1984, On a mixed one step/Chebyshev pseudospectral tech-
nique for the integration of parabolic problems using ¬nite element preconditioning,
in C. Brezinski, A. Draux, A. P. Magnus, P. Maroni and A. Ronveaux (eds), Polynomes
Orthogonaux et Applications: Proceedings of the Laguerre Symposium at Bar-le-Duc, num-
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article is in English; employs unusual implicit time-marching which is exact (instead
of second order) for time integration of the two slowest-decaying diffusion eigen-
modes.

Deville, M. and Mund, E.: 1985, Chebyshev pseudospectral solution of second-order el-
liptic equations with ¬nite element pre-conditioning, Journal of Computational Physics
60, 517“533.

Deville, M. and Mund, E.: 1991, Finite element preconditioning of collocation schemes for
advection-diffusion equations, in R. Beauwens and P. de Groen (eds), Proceedings of the
IMACS International Symposium on Iterative Methods in Linear Algebra, IMACS, North-
Holland, Amsterdam, pp. 181“190.

Deville, M., Haldenwang, P. and Labrosse, G.: 1981, Comparison of time integration (¬nite
difference and spectral)) for the nonlinear Burgers™ equation, in H. Viviond (ed.), Pro-
ceedings of the 4th GAMMConference on Nuemrical Methods in Fluid Mechanics, Vieweg,
Braunschweig.

Deville, M., Kleiser, L. and Montigny-Rannou, F.: 1984, Pressure and time treatment for
Chebyshev spectral solution of a Stokes problem, Internat. J. Numer. Meth. Fluids
4, 1149“1163. Backward Euler is raised by Richardson extrapolation to a second or-
der time-marching. Four different schemes including penalty method, splitting, in¬‚u-
ence matrix and Morchoisne™s space-time pseudospectral scheme. Good discussion of
compatibility conditions on the initial condition.

Deville, M. O. and Mund, E. H.: 1990, Finite-element preconditioning for pseudospec-
tral solutions of elliptic problems, SIAM Journal of Scienti¬c and Statistical Computing
12, 311“342.

Deville, M. O. and Mund, E. H.: 1992, Fourier analysis of ¬nite element preconditioned
collocation schemes, SIAM Journal of Scienti¬c and Statistical Computing 13(2), 596“610.

Deville, M. O., Mund, E. H. and Van Kemenade, V.: 1994, Preconditioned Chebyshev col-
location methods and triangular ¬nite elements, in C. Bernardi and Y. Maday (eds),
BIBLIOGRAPHY 609

Analysis, Algorithms and Applications of Spectral and High Order Methods for Partial Dif-
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Devulder, C. and Marion, M.: 1992, A class of numerical algorithms for large time integra-
tion: the nonlinear Galerkin methods, SIAM Journal of Numerical Analysis 29(2), 462“
483. Many theorems.
+
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Dimitropoulus, C. and Beris, A. N.: 1998, Ef¬cient pseudospectral ¬‚ow simulations in mod-
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doconformal orthogonal curvilinear coordinates in two dimensions, Fourier in one
coordinate and Chebyshev in the other.
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in C. Canuto and A. Quarteroni (eds), Spectral and High Order Methods for Partial Dif-
ferential Equations: Proceedings of the ICOSAHOM ™89 Conference in Como, Italy, North-
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Chebyshev collocation and a mapping technique, SIAM Journal of Scienti¬c Computing
18(4), 1040“1055. The Kosloff/Tal-Ezer mapping is used to reduce roundoff error and
allow a larger timestep.
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sion problems based on combining the method of characteristics with ¬nite element
or ¬nite difference procedures, SIAM Journal of Numerical Analysis 19, 871“885. Not
spectral; invention of a semi-Lagrangian scheme.
Drake, J., Foster, I., Michalakes, J., Toonen, B. and Worley, P.: 1995, Design and performance
of a scalable parallel community climate model, Parallel Comput. 21(10), 1571“1591.
Parallel version, PCCM2, of the CCM2 spherical harmonics/vertical ¬nite difference
climate model. Performance on the IBM SP2 and Intel Paragon.
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610

Driscoll, J., Healy, Jr., D. M. and Rockmore, D.: 1997, Fast discrete polynomial transform
with applications to data analysis on distance transitive graphs, SIAM J. Comput.
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nomials; more ef¬cient than Orszag(1986).

Driscoll, T. A. and Fornberg, B.: 1998, A block pseudospectral method for Maxwell™s equa-
tions. I. One-dimensional case, Journal of Computational Physics 140(1), 47“65. Employ
a domain decomposition scheme in which ¬ctitious points beyond the domain walls
are used, in a sort of grid overlapping scheme, to allow a much more uniform separa-
tion between grid points than in a standard pseudospectral algorithm. This allows a
relatively long time step at the cost of much increased domain-to-domain communi-
cation compared to the standard domain decomposition method. They generalize the
scheme so that it works well even when there are discontinuous changes in material
properties at domain walls.

Dubois, T., Jauberteau, F. and T´ mam, R.: 1990, The nonlinear Galerkin method for the two
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and three dimensional Navier-Stokes equations, in K. W. Morton (ed.), Proceedings of
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Verlag, New York, pp. 117“120.

Dubois, T., Jauberteau, F. and T´ mam, R.: 1998, Incremental unknowns, multilevel meth-
e
ods and the numerical simulation of turbulence, Comput. Meth. Appl. M. 159, 123“189.

Durran, D. R.: 1991, The third-order Adams-Bashforth method: an attractive alternative to
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Comput. 14, 1368“1393.

Dutt, A. and Rokhlin, V.: 1995, Fast Fourier Transforms for nonequispaced data, II, Applied
and Computational Harmonic Analysis 2, 85“110.

D™yakonov, E. G.: 1961, An iteration method for solving systems of ¬nite difference equa-
tions, Doklady Akademiia Nauk SSSR 138, 522“525. Not spectral; iteration precondi-
tioned by separable PDE.

Dym, H. and McKean, H. P.: 1972, Fourier Series and Integrals, Academic Press, New York.
129 pp.

Eggert, N., Jarratt, M. and Lund, J.: 1987, Sinc function computation of Sturm-Liouville
problems, Journal of Computational Physics 69, 209“229.

Ehrenstein, U. and Peyret, R.: 1989, A Chebyshev collocation method for the Navier-Stokes
equations with application to double-diffusive convection, International Journal for Nu-
merical Methods in Fluids 9, 427“452. Semi-implicit time integration with in¬‚uence ma-
trix method.

Eisen, H. and Heinrichs, W.: 1992, A new method of stabilization for singular perturba-
tion problems with spectral methods, SIAM Journal of Numerical Analysis 29, 107“122.
Shows that basis functions which vanish at boundaries are much better conditioned
than the Chebyshev polynomials from which these basis functions are formed.

Eisen, H., Heinrichs, W. and Witch, K.: 1991, Spectral collocation methods and polar coor-
dinate singularities, Journal of Computational Physics 96, 241“257.
BIBLIOGRAPHY 611

Eisenstat, S. C., Elman, H. C. and Schultz, M. H.: 1983, Variational iterative methods for
nonsymmetric systems of linear equations, SIAM J. Numer. Anal. 20, 345“357.
El-Daou, M. K. and Ortiz, E. L.: 1993, Error analysis of the tau method: dependence of
the error on the degree and on the length of the interval of approximation, Computers
Math. Applic. 25(7), 33“45.
El-Daou, M. K. and Ortiz, E. L.: 1994a, A recursive formulation of collocation in terms of
canonical polynomials, Computing 52, 177“202.
El-Daou, M. K. and Ortiz, E. L.: 1994b, The weighting subspaces of collocation and the Tau
method, in J. D. Brown, M. T. Chu, D. C. Ellison and R. J. Plemmons (eds), Proceedings
of the Cornelius Lanczos International Centenary Conference, Society for Industrial and
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PA.
El-Daou, M. K. and Ortiz, E. L.: 1997, The uniform convergence of the Tau method for
singularly perturbed problems, Applied Mathematics Letters 10(2), 91“94. Existence and
stability of the algorithm is proved, independent of the perturbation parameter .
El-Daou, M. K., Ortiz, E. L. and Samara, H.: 1993, A uni¬ed approach to the tau method
and Chebyshev series expansion techniques, Computers Math. Applic. 25(3), 73“82.
Eliassen, E. and Machenhauer, B.: 1974, On spectral representation of the vertical vari-
ation of the meteorological ¬elds in numerical integration of a primitive equation
model, GARP WGNE Report 7, World Meteorological Organization, Geneva, Switzer-
land. Legendre polynomials in the vertical.
Elliott, D. and Stenger, F.: 1984, Sinc method of solution of singular integral equations, in
A. Gerasoulis and R. Vichnevetsky (eds), Numerical Solution of Singular Integral Equa-
tions, IMACS.
Ellsaesser, H. W.: 1966, Evaluation of spectral versus grid point methods of hemispheric
numerical weather prediction, J. Appl. Meteor. 5, 246“262. Fig. 12 is a good illustra-
tion of late onset of spectral blocking due to violation of the CFL criterion after the
advecting ¬‚ow has intensi¬ed from its initial maximum.
Engelmann, F., Feix, M., Minardi, E. and Oxenius, J.: 1963, Nonlinear effects from vlasov™s
equation, The Physics of Fluids 6(2), 266“275. Low order Hermite series (N = 2, 3) for
velocity coordinate; such low truncations are found to suppress instabilities that occur
in some parameter regions for the full equations.
Erlebacher, G., Zang, T. A. and Hussaini, M. Y.: 1987, Spectral multigrid methods for the
numerical simulation of turbulence, in S. McCormick and K. Stuben (eds), Multigrid
Methods, Marcel Dekker, New York, pp. 177“194.
Errico, R. M.: 1984, The dynamic balance of a general circulation model, Monthly Weather
Review 112, 2439“2454. Not spectral, but evaluation of the usefulness of slow manifold
concept in a complicated model.
Errico, R. M.: 1989, The degree of Machenauer balance in a climate model, Monthly Weather
Review 112, 2723“2733. Test of slow manifold ideas of Machenauer™s NG(1) approxi-
mation in a global hydrodynamics-with-physics model.
Falqu´ s, A. and Iranzo, V.: 1992, Edge waves on a longshore shear ¬‚ow, Physics of Fluids
e
pp. 2169“2190. Rational Chebyshev and Laguerre on semi-in¬nite domain.
BIBLIOGRAPHY
612

Finlayson, B. A.: 1973, The Method of Weighted Residuals and Variational Principles, Academic,
New York. 412 pp. Many good examples of low order pseudospectral methods,
dubbed “orthogonal collocation” here, and mostly drawn from chemical engineer-
ing and ¬‚uid dynamics. Mostly Legendre and Gegenbauer polynomials rather than
Chebyshev and no mention of the FFT.

Fischer, P. F.: 1990, Analysis and application of a parallel spectral element method for the
solution of the Navier-Stokes equations, Computer Methods in Applied Mechanics and
Engineering 80(1“3), 483“491.

Fischer, P. F.: 1994a, Domain decomposition methods for large scale parallel Navier-Stokes
calculations, in A. Quarteroni (ed.), Proceedings of the Sixth International Conference on
Domain Decomposition Methods for Partial Differential Equations, Como, Italy, AMS, Prov-
idence.

Fischer, P. F.: 1994b, Parallel domain decomposition for incompressible ¬‚uid dynamics,
Contemp. Math. 157, 313.

Fischer, P. F.: 1997, An overlapping Schwarz method for spectral element solution of the
incompressible Navier-Stokes equations, J. Comput. Phys. 133, 84“101.

Fischer, P. F.: 1998, Projection techniques for iterative solution of ax=b with successive
right-hand sides, Comput. Meth. Appl. M. 163(1“4), 193“204. Not spectral, but useful
for semi-implicit time marching.

Fischer, P. F. and Gottlieb, D.: 1997, On the optimal number of subdomains for hyperbolic
problems on parallel computers, International Journal of Supercomputing and Appl. High
Performance Computing 11, 65“76. Spectral elements.

Fischer, P. F. and Patera, A. T.: 1989, Parallel spectral element methods for the incompress-
ible Navier-Stokes equations, in J. H. Kane and A. D. Carlson (eds), Solution of Super
Large Problems in Computational Mechanics, Plenum, New York.

Fischer, P. F. and Patera, A. T.: 1991, Parallel spectral element solution of the Stokes prob-
lem, Journal of Computational Physics 92(2), 380“421.

Fischer, P. F. and Patera, A. T.: 1992, Parallel spectral element solutions of eddy-promoter
channel ¬‚ow, Proceedings of the European Research Community on Flow Turbulence and
Combustion Workshop, Laussane, Switzerland, Cambridge University Press, Cambridge.

Fischer, P. F. and Patera, A. T.: 1994, Parallel simulation of viscous incompressible ¬‚ows,
Annual Reviews of Fluid Mechanics 26, 483“527. REVIEW.

Fischer, P. F. and Rønquist, E. M.: 1994, Spectral element methods for large scale parallel
Navier-Stokes calculations, 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. 69“76. Also in Comput. Methods. Appl. Mech. Engrg., vol. 116.

Fischer, P. F., Ho, L.-W., Karniadakis, G. E., Rønquist, E. M. and Patera, A. T.: 1988a, Recent
advances in parallel spectral element simulation of unsteady incompressible ¬‚ows,
Computers and Structures 30, 217“231.
BIBLIOGRAPHY 613

Fischer, P. F., Rønquist, E. M. and Patera, A. T.: 1989, Parallel spectral element methods
for viscous ¬‚ow, in G. Carey (ed.), Parallel Supercomputing: Methods, Algorithms and
Applications, Wiley, New York, pp. 223“238.

Fischer, P., Rønquist, E. M., Dewey, D. and Patera, A. T.: 1988b, Spectral element methods:
Algorithms and architectures, in R. Glowinski, G. Golub, G. Meurant and J. Periaux
(eds), Proceedings of the First International Conference on Domain Decomposition Methods
for Partial Differential Equations, SIAM, SIAM, Philadelphia, pp. 173“197.

Fjørtoft, R.: 1952, On a numerical method of integrating the barotropic vorticity equation,
Tellus 4, 179“194. Not spectral; early use of Lagrangian coordinates in numerical me-

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