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teorology.

Fjørtoft, R.: 1955, On a numerical method of integrating the barotropic vorticity equation,
Tellus 7, 462“480. Not spectral; early use of Lagrangian coordinates in numerical me-
teorology.

Fla, T.: 1992, A numerical energy conserving method for the DNLS equation, Journal of
Computational Physics 101, 71“79. Fourier pseudospectral scheme for the Derivative-
Nonlinear Schroedinger equation.

Flatau, P., Boyd, J. P. and Cotton, W. R.: 1987, Symbolic algebra in applied mathemat-
ics and geophysical ¬‚uid dynamics ” REDUCE examples, Technical report, Colorado
State University, Department of Atmospheric Science, Fort Collins, CO 80523. Some
examples of spectral methods in REDUCE.

Flyer, N.: 1998, Asymptotic upper bounds for the coef¬cients in the Chebyshev series ex-
pansion for a general order integral of a function, Math. Comput. 67(224), 1601“1616.

Foias, C., Jolly, M. S., Kevrekidis, I. G. and Titi, E. S.: 1991, Dissipativity of numerical
schemes, Nonlinearity 4, 591“613. Blow-up of various schemes including Nonlinear
Galerkin.

Foias, C., Jolly, M. S., Kevrekidis, I. G. and Titi, E. S.: 1994, On some dissipative fully
discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation, Physics
Letters A 186, 87“96.

Foias, C., Manley, O. and T´ mam, R.: 1988, Modelling of the interaction of small and large
e
eddies in two dimensional turbulent ¬‚ows, Rairo-Math. Mod. Numer. Anal. 22(1), 93“
118. First paper on approximate inertial manifolds, which led to rediscovery of Non-
linear Galerkin method.

Forbes, J. M. and Garrett, H. B.: 1976, Solar diurnal tide in the thermosphere, Journal of the
Atmospheric Sciences 33, 2226“2241.

Fornberg, B.: 1975, On a Fourier method for the integration of hyperbolic equations, SIAM
Journal of Numerical Analysis 12, 509“528.

Fornberg, B.: 1977, A numerical study of 2-d turbulence, Journal of Computer Physics 25, 1“
31.

Fornberg, B.: 1978, Pseudospectral calculations on 2-d turbulence and nonlinear waves,
SIAM-AMS Proceedings 11, 1“18.

Fornberg, B.: 1987, The pseudospectral method: Comparisons with ¬nite differences for
the elastic wave equation, Geophysics 52, 483“501.
BIBLIOGRAPHY
614

Fornberg, B.: 1988a, The pseudospectral method: accurate representation of interfaces in
elastic wave calculations, Geophysics 53, 625“637.

Fornberg, B.: 1988b, Generation of ¬nite difference formulas on arbitrarily spaced grids,
Mathematics of Computation 51, 699“706. Simple recursion to compute ¬nite difference
weights. This is helpful for ¬nite difference preconditioning of spectral methods; the
¬nite difference problem must be solved on the same unevenly spaced grid as used
by the Chebyshev or Legendre pseudospectral scheme.

Fornberg, B.: 1990a, High order ¬nite differences and the pseudospectral method on stag-
gered grids, SIAM Journal of Numerical Analysis 27, 904“918.

Fornberg, B.: 1990b, An improved pseudospectral method for initial boundary value prob-
lems, Journal of Computational Physics 91, 381“397. Uses additional boundary condi-
tions, derived from the differential equation, to greatly reduce the largest eigenval-
ues of Chebyshev differentiation matrices, greatly reducing the “stiffness” of time-
dependent problems and allowing a longer time step.

Fornberg, B.: 1992, Fast generation of weights in ¬nite difference formulas, in G. D. Byrne
and W. E. Schiesser (eds), Recent Developments in Numerical Methods and Software for
QDEs/DAEs/PDEs, World Scienti¬c, Singapore, pp. 97“123.

Fornberg, B.: 1995, A pseudospectral approach for polar and spherical geometries, SIAM
Journal of Scienti¬c Computing 16, 1071“1081. Double Fourier series on latitude and
longitude grid with strong ¬ltering of large zonal wavenumbers near the poles to
avoid CFL instability.

Fornberg, B.: 1996, A Practical Guide to Pseudospectral Methods, Cambridge University Press,
New York.

Fornberg, B.: 1998, Calculation of weights in ¬nite difference formulas, SIAM Rev.
40(3), 685“691. Arbitrary order ¬nite differences, which in the limit of order equal
to the number of grid points gives a pseudospectral approximation.

Fornberg, B. and Merrill, D.: 1997, Comparison of ¬nite difference and pseudospectral
methods for convective ¬‚ow over a sphere, Geophys. Res. Lett. 24, 3245“3248. Sec-
ond and fourth order ¬nite differences and double Fourier series, all on a latitude-
longitude grid with strong high latitude ¬ltering of large zonal wavenumbers, are
compared with spherical harmonics for a problem whose solution is very smooth.
Equal accuracy for both spectral methods, but the double Fourier algorithm is faster.

Fornberg, B. and Sloan, D.: 1994, A review of pseudospectral methods for solving partial
differential equations, in A. Iserles (ed.), Acta Numerica, Cambridge University Press,
New York, pp. 203“267.

Fornberg, B. and Whitham, G. B.: 1978, A numerical and theoretical study of certain nonlin-
ear wave phenomena, Philosophical Transactions of the Royal Society of London 289, 373“
404. Develops ef¬cient Fourier pseudospectral method for Korteweg“deVries and re-
lated wave equations; the linear terms, which have constant coef¬cient, are integrated
in time exactly.

Foster, I. T. and Worley, P. H.: 1997, Parallel algorithms for the spectral transform method,
SIAM J. Sci. Comput. 18(3), 806“837. Comparison of methods for spherical harmonics
transforms for the nonlinear shallow water wave equations on various parallel archi-
tectures.
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Fox, D. G. and Orszag, S. A.: 1973, Pseudospectral approximation to two-dimensional tur-
bulence, Journal of Computational Physics 11, 612“619.

Fox, L.: 1962, Chebyshev methods for ordinary differential equations, Computer Journal
4, 318“331.

Fox, L. and Parker, I. B.: 1968, Chebyshev Polynomials in Numerical Analysis, Oxford Uni-
versity Press, London. Very readable and still good for general background, but the
Clenshaw-type recursive algorithms for solving differential and integral equations are
no longer popular except for special applications.

Fox, L., Hayes, L. and Mayers, D. F.: 1973, The double eigenvalue problem, in J. C. P. Miller
(ed.), Numerical Analysis, Academic Press, pp. 93“112. Chebyshev solution of a 2d
order ODE which has three boundary conditions imposed. Solutions exist because
the problem has two independent eigenparameters.

Francis, P. E.: 1972, The possible use of Laguerre polynomials for representing the vertical
structure of numerical models of the atmosphere, Quart. J. Roy. Met. Soc. 98, 662“667.
See also the correspondence by B. J. Hoskins and Francis, QJRMS 99, 571“572. The
conclusion is that Laguerre polynomials demand a very short timestep, and are there-
fore useless in this context.

Francken, P., Deville, M. O. and Mund, E. H.: 1990, On the spectrum of the iteration opera-
tor associated to the ¬nite element preconditioning of Chebyshev collocation calcula-
tions, Computer Methods in Applied Mechanics and Engineering 80, 295“304.

Fraser, W. and Wilson, M. W.: 1966, Remarks on the Clenshaw-Curtis quadrature scheme,
SIAM Review 8, 322“327.

Froes Bunchaft, M. E.: 1997, Some extensions of the Lanczos-Ortiz theory of canonical
polynomials in the Tau method, Mathematics of Computation 66(218), 609“621. Extends
and simpli¬es the canonical polynomial tau method; no numerical illustrations.

Frutos, J. and Sanz-Serna, J. M.: 1992, An easily implementable fourth-order method for
the time integration of wave problems, Journal of Computational Physics 103, 160“168.

Frutos, J. et al.: 1990, A Hamiltonian explicit algorithm with spectral accuracy for the ˜good™
Boussinesq equation, Computer Methods in Applied Mechanics and Engineering 80, 417“
423.

Funaro, D.: 1986, A multidomain spectral approximation of elliptic equations, Methods for
Partial Differential Equations 2, 187“205.

Funaro, D.: 1987a, Some results about the spectrum of the Chebyshev differencing oper-
ator, in E. L. Ortiz (ed.), Numerical Approximations of P. D. E., Part III, North-Holland,
Amsterdam, pp. 271“284.

Funaro, D.: 1987b, A preconditioning matrix for the Chebyshev differencing operator,
SIAM Journal of Numerical Analysis 24, 1024“1031.

Funaro, D.: 1988a, Computing the inverse of the Chebyshev collocation derivative matrix,
SIAM Journal of Scienti¬c and Statistical Computing 9, 1050“1058.

Funaro, D.: 1988b, Domain decomposition methods for pseudo spectral approximations.
Part I. Second order equations in one dimension, Numerische Mathematik 52, 329“344.
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Funaro, D.: 1990a, Computational aspects of pseudospectral Laguerre approximations,
Appl. Numer. Math. 6(6), 447“457.

Funaro, D.: 1990b, Convergence analysis for pseudospectral multidomain approximations
of linear advection equations, IMA J. Numer. Anal. 10(1), 63“74.

Funaro, D.: 1990c, A variational formulation for the Chebyshev pseudospectral approxi-
mation of Neumann problems, SIAM J. Numer. Anal. 27(3), 695“703.

Funaro, D.: 1991, Pseudospectral approximation of a PDE de¬ned on a triangle, Appl. Math.
Comput. 42, 121“138. Subdivides each triangle into three quadrilaterals.

Funaro, D.: 1992a, Approximation by the Legendre collocation method of a model problem
in electrophysiology, Journal of Computational and Applied Mathematics 43, 261“271.

Funaro, D.: 1992b, Polynomial Approximation of Differential Equations, Springer-Verlag, New
York. 313 pp.

Funaro, D.: 1993a, A new scheme for the approximation of advection-diffusion equations
by collocation, SIAM Journal of Numerical Analysis 30(6), 1664“1676.

Funaro, D.: 1993b, FORTRAN routines for spectral methods, Report 891, I. A. N.-C. N. R.,
Pavia, Italy. The 82 FORTRAN routines plus manual are available at the Web address
ftp:/ftp.ian.pv.cnr.it/pub/splib.

Funaro, D.: 1994a, A fast solver for elliptic boundary-value problems in the square, 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 Interna-
tional Conference on Spectral and High Order Methods (ICOSAHOM ™92), Le Corum,
Montpellier, France, 22-26 June 1992, North-Holland, Amsterdam, pp. 253“256. Also
in Comp. Meths. Appl. Mech. Engrg., 116.

Funaro, D.: 1994b, Spectral elements in the approximation of boundary-value-problems in
complex geometries, Applied Numerical Mathematics 15(2), 201“205.

Funaro, D.: 1997a, Some remarks about the collocation method on a modi¬ed Legendre
grid, Computers and Mathematics with Applications 33, 95“103.

Funaro, D.: 1997b, Spectral Elements for Transport-Dominated Equations, Vol. 1 of Lecture Notes
in Computational Science and Engineering, Springer-Verlag, Heidelberg. 200 pp.

Funaro, D. and Gottlieb, D.: 1988, A new method of imposing boundary conditions in
pseudospectral approximations of hyperbolic equations, Mathematics of Computation
51, 599“613.

Funaro, D. and Gottlieb, D.: 1991, Convergence results for pseudospectral approximations
of hyperbolic systems by a penalty-type boundary treatment, Mathematics of Computa-
tion 57, 585“596.

Funaro, D. and Heinrichs, W.: 1990, Some results about the pseudospectral approximation
of one dimensional fourth order problems, Numerische Mathematik 58, 399“418.

Funaro, D. and Kavian, O.: 1991, Approximation of some diffusion evolutions equations in
unbounded domains by Hermite functions, Mathematics of Computation 57, 597“619.
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Funaro, D. and Rothman, E.: 1989, Preconditioning matrices for the pseudospectral ap-
proximations of ¬rst-order operators, in T. J. Chung and G. R. Karr (eds), Finite Ele-
ments Analysis in Fluids, UAH Press, Huntsville, Alabama, pp. 1458“1463.

Funaro, D. and Russo, A.: 1993, Approximation of advection-diffusion problems by a mod-
i¬ed Legendre grid, in K. Morgan, E. Onate, J. Periaux, J. Peraire and O. C. Zienkiewicz
(eds), Finite Elements in Fluids, New Trends and Applications, Pineridge Press, Battersea,
England, pp. 1311“1318.

Funaro, D., Quarteroni, A. and Zanolli, P.: 1988, An iterative procedure with interface
relaxation for domain decomposition methods, SIAM Journal of Numerical Analysis
25, 1213“1236.

Funaro, O. C. D. and Kavian, O.: 1990, Laguerre spectral approximations of elliptic prob-
lems in exterior domains, Computer Methods in Applied Mechanics and Engineering
80, 451“458.

Gad-el-Hak, M., Davis, S. H., McMurray, J. T. and Orszag, S. A.: 1984, On the stability of
the decelerating laminar boundary layer, Journal of Fluid Mechanics 138, 297“323.

Garba, A.: 1998, A mixed spectral/wavelet method for the solution of Stokes problem, J.
Comput. Phys. 145(1), 297“315. Chebyshev polynomials in the non-periodic coordinate,
Daubechies wavelets in the spatially periodic coordinate.

Garc´
ia-Archilla, B.: 1995, Some practical experience with the time integration of dissipative
equations, Journal of Computational Physics 122, 25“29. Kuramoto-Sivashinsky equa-
tion is used to compare Nonlinear Galerkin and standard Galerkin methods; ¬nds
that standard stiff ODE methods are very effective, more so than Nonlinear Galerkin
schemes.

Garc´
ia-Archilla, B.: 1996, A spectral method for the equal width equation, Journal of Com-
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Garc´
ia-Archilla, B. and de Frutos, J.: 1995, Time integration of the non-linear Galerkin
method, IMA Journal of Numerical Analysis 15, 221“244. “The results show that for
these problems [Kuramoto-Sivashinsky and reaction-diffusion equations], the non-
linear Galerkin method is not competitive with either pure spectral Galerkin or pseu-
dospectral discretizations”, from the abstract.

Gardner, D. R., Trogdon, S. A. and Douglass, R. W.: 1989, A modi¬ed tau spectral method
that eliminates spurious eigenvalues, Journal of Computational Physics 80, 137“167.

Gary, J. and Helgason, R.: 1970, A matrix method for ordinary differential equation eigen-
value problems, Journal of Computational Physics 5, 169“187. Shows that the QR and
QZ matrix eigensolvers make high order discretizations especially advantageous.

Gautheir, S., Guillard, H., Lumpp, T., Mal´ , J., Peyret, R. and Renaud, F.: 1996, A spectral

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