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. 128
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Marine Science, Springer-Verlag, New York, pp. 404“437. REVIEW.

Haidvogel, D. B. and Zang, T. A.: 1979, The accurate solution of Poisson™s equation by
expansion in Chebyshev polynomials, Journal of Computational Physics 30, 167“180.
Classic paper on separable BVP PDE™s.

Haj, A., Phillips, C. and Delves, L. M.: 1980, The global element method for stationary
advective problems, International Journal of Numerical Methods in Engineering 15, 167“
175.

Hald, O. H.: 1981, Convergence of Fourier methods for Navier-Stokes equations, Journal of
Computational Physics 40, 305“317. Proofs.

Haldenwang, P., Labrosse, G., Abboudi, S. and Deville, M.: 1984, Chebyshev 3-d spec-
tral and 2-d pseudospectral solvers for the Helmholtz equation, Journal of Computa-
tional Physics 55, 115“128. Comparison of three methods for constant coef¬cient PDE:
diagonalization in two dimensions plus solution of tridiagonal systems in the third
(Haidvogel and Zang, 1979), diagonalization in all three dimensions, and an itera-
tion preconditioned by a fast direct ¬nite difference solver for a separable system of
cost O(N 3 log2 (N )). The HZ algorithm was fastest, but loses about ¬ve digits due to
roundoff for N as small as 64.

Haltiner, G. J. and Williams, R. T.: 1980, Numerical Prediction and Dynamic Meteorology, sec-
ond edn, John Wiley. Chapter on spherical harmonics.
BIBLIOGRAPHY 623

Han, H. C., Schultz, W. W., Boyd, J. P. and Schumack, M. R.: 1999, The ¬‚ow in a elliptical
journal bearing by the spectral element method for a rotating and translating shaft.
To appear. The movement of the spinning central shaft was calculated simultaneously
with the ¬‚ow.

Hardiker, V.: 1997, A global numerical weather prediction model with variable resolution,
Monthly Weather Review 125(1), 59“73. Conformal mapping is used to make at T-83
spherical harmonics model as effective as a T-170 uniform resolution code for mod-
elling hurricanes.

Harding, R. C.: 1968, Response of a one-dimensional Vlasov plasma to external electric
¬elds, Physics of Fluids 11(10), 2233“2240. Fourier for space coordinate, Hermite for
velocity coordinate, ¬nite difference for time. Little about numerics except that sev-
eral hundred Hermite modes were needed because of problems resolved by later im-
provements (Holloway, 1996a,b).

Hargittai, I. and Hargittai, M.: 1994, Symmetry: A Unifying Concept, Shelter Publications,
Bolinas, California. Good popularization of symmetry in art and nature.

Haugen, J. E. and Machenhauer, B.: 1993, A spectral limited-area model formulation
with time-dependent boundary conditions applied to the shallow-water equations,
Monthly Weather Review 121, 2618“2630.

Haupt, S. E. and Boyd, J. P.: 1988, Modeling nonlinear resonance: A modi¬cation to Stokes™
perturbation expansion, Wave Motion 10, 83“98. Fourier basis for nonlinear eigen-
problem. Analysis of the relationship between perturbation theory and the Galerkin
numerical algorithm.

Haupt, S. E. and Boyd, J. P.: 1991, Double cnoidal waves of the Korteweg-deVries equation:
The boundary value approach, Physica D 50, 117“134. Two-dimensional Fourier basis
for a nonlinear problem with two eigenparameters.

Haurwitz, B.: 1940, The motion of atmospheric disturbances on the spherical earth, J. Ma-
rine Res. 3, 254“267. Shows spherical harmonics are travelling wave solutions of the
shallow water equation, now known as “Rossby-Haurwitz” waves.

Healy, D. M., Rockmore, D. N., Kostelec, P. and Moore, S. S. B.: 1999, FFTs for the 2-sphere
” Improvements and variations, Adv. Appl. Math. Submitted. Fast spherical harmonic
transforms.

Heikes, R. and Randall, D. A.: 1995a, Numerical integration of the shallow-water equations
on a twisted icosahedral grid. Part I: Basic design and result of tests., Monthly Weather
Review 123, 1862“1880. Non-spectral alternative to spherical harmonics.

Heikes, R. and Randall, D. A.: 1995b, Numerical integration of the shallow-water equa-
tions on a twisted icosahedral grid. Part II: A detailed description of the grid and an
analysis of numerical accuracy., Monthly Weather Review 123, 1881“1887. Non-spectral
alternative to spherical harmonics.

Heinrichs, W.: 1987, Kollokationsverfahren und Mehrgittermethoden bei elliptschen Randwertauf-
gaben, Vol. 168 of GMD-Bericht Nr., Oldenbourg-Verlag, Oldenbourg. Doctoral thesis.

Heinrichs, W.: 1988a, Line relaxation for spectral multigrid, Journal of Computational Physics
77, 166“182.
BIBLIOGRAPHY
624

Heinrichs, W.: 1988b, Multigrid methods for combined ¬nite difference and Fourier prob-
lems, Journal of Computational Physics 78, 424“436.

Heinrichs, W.: 1988c, Collocation and full multigrid methods, Applied Mathematics and Com-
putation 28, 35“45.

Heinrichs, W.: 1989a, Improved condition number for spectral methods, Mathematics of
Computation 53, 103“119.

Heinrichs, W.: 1989b, Spectral methods with sparse matrices, Numerische Mathematik 56, 25“
41.

¨
Heinrichs, W.: 1989c, Konvergenzaussagen fur Kollokationsverfahren bei elliptschen
Randwertaufgaben, Numerische Mathematik 54, 619“637.

Heinrichs, W.: 1990, Algebraic spectral multigrid methods, Computer Methods in Applied
Mechanics and Engineering 80, 281“289.

Heinrichs, W.: 1991a, A 3D spectral multigrid method, Applied Mathematics and Computation
41, 117“128.

Heinrichs, W.: 1991b, Stabilization techniques for spectral methods, Journal of Scienti¬c
Computing 6(1), 1“19. Basis functions that satisfy homogeneous boundary conditions
reduce condition number.

Heinrichs, W.: 1991c, A stabilized treatment of the biharmonic operator with spectral meth-
ods, SIAM Journal of Scienti¬c and Statistical Computing 12, 1162“1172.

Heinrichs, W.: 1992a, A spectral multigrid method for the Stokes problem in stream-
function formulation, Journal of Computational Physics 102, 310“318.

Heinrichs, W.: 1992b, A stabilized multidomain approach for singular perturbation meth-
ods, Journal of Scienti¬c Computing 7, 95“127.

Heinrichs, W.: 1992c, Strong convergence estimates for pseudospectral methods, Applica-
tions of Mathematics 37(6), 401“417.

Heinrichs, W.: 1992d, Spectral projective Newton-methods for quasilinear elliptic bound-
ary value problems, Calcolo 29, 33“48.

Heinrichs, W.: 1993a, Distributive relaxations for the spectral Stokes operator, Journal of
Scienti¬c Computing 8, 389“398.

Heinrichs, W.: 1993b, Splitting techniques for the pseudospectral approximation of the
unsteady Stokes equation, SIAM Journal of Numerical Analysis 30, 19“39.

Heinrichs, W.: 1993c, Spectral multi-grid techniques for the Navier-Stokes equations, Com-
puter Methods in Applied Mechanics and Engineering 106, 297“314.

Heinrichs, W.: 1993d, Spectral multigrid methods for the reformulated Stokes equations,
Journal of Computational Physics 107(2), 213“224.

Heinrichs, W.: 1993e, Domain decomposition for fourth order problems, SIAM Journal of
Numerical Analysis 30(2), 435“453.

Heinrichs, W.: 1993f, Finite element preconditioning for spectral multigrid methods, Ap-
plied Mathematics and Computation 59(1), 19“40.
BIBLIOGRAPHY 625

Heinrichs, W.: 1993g, Spectral multigrid methods for domain decomposition problems us-
ing patching techniques, Applied Mathematics and Computation 59(2), 165“176.

Heinrichs, W.: 1993h, Defect correction for convection dominated ¬‚ow, in F.-K. Hebeker,
R. Rannacher and G. Wittum (eds), Proceedings of the International Workshop on Nu-
merical Methods for the Navier-Stokes Equations, number 47 in Notes on Numerical Fluid
Mechanics, Heidelberg, pp. 111“121.

Heinrichs, W.: 1994a, Spectral methods for singular perturbation problems, Applications of
Mathematics 39(3), 161“188.

Heinrichs, W.: 1994b, Ef¬cient iterative solution of spectral systems for the Navier-Stokes equa-
tions, Vol. 2 of Wissenschaftliche Schriftenreihe Mathematik, Verlag Dr. Koster, Berlin. Ha-
bilitationshrift, 109 pp.

Heinrichs, W.: 1994c, Spectral viscosity for convection dominated ¬‚ow, Journal of Scienti¬c
Computing 9(2), 137“147.

Heinrichs, W.: 1994d, Defection correction for the advection-diffusion equation, Computer
Methods in Applied Mechanics and Engineering 119, 191“197.

Heinrichs, W.: 1996, Defect correction for convection dominated ¬‚ow, SIAM Journal of Sci-
enti¬c Computing 17(5), 1082“1091.

Heinrichs, W.: 1998a, Spectral collocation on triangular elements, J. Comput. Phys.
145(2), 743“757.

Heinrichs, W.: 1998b, Splitting techniques for the unsteady Stokes equations, SIAM J. Nu-
mer. Anal. 35(4), 1646“1662. Third order time accurate Uzawa scheme with a single,
unstaggered grid for in a pseudospectral approach.

Held, I. M. and Suarez, M. J.: 1994, A proposal for the intercomparison of the dynamical
cores of atmospheric general circulation models, Bull. Amer. Meteor. Soc. 75, 1825“1830.
Comparison of 20-level models:, leapfrog time-stepping T 63 spherical harmonics ver-
sus 2d order ¬nite difference with G72[144 grid points around the equator]. There “is
an impressive degree of agreement between the two models” but “the climate of both
models are sensitive to resolution and have not yet converged at the resolution pre-
sented here”. This article is unusual in claiming similar results for low order and high
order methods for complex ¬‚ows at the same resolution.

Hendry, J. A. and Delves, L. M.: 1979, The global element method applied to a harmonic
mixed boundary value problem, Journal of Computational Physics 33, 33“44.

Hendry, J. A., Delves, L. M. and Mohamed, J.: 1982, Iterative solution of the global element
equations, Computer Methods in Applied Mechanics and Engineering 35, 271“283.

Herbst, B. M. and Ablowitz, M.: 1992, Numerical homoclinic instabilities in the sine-
Gordon equation, Quaestiones Mathematicae 15, 345“363.

Herbst, B. M. and Ablowitz, M.: 1993, Numerical chaos, symplectic integrators, and expo-
nentially small splitting distances, Journal of Computational Physics 105, 122“132.

Herring, J. R., Orszag, S. A., Kraichnan, R. H. and Fox, D. G.: 1974, Decay of two-
dimensional homogeneous turbulence, Journal of Fluid Mechanics 66, 417“444.
BIBLIOGRAPHY
626

Hesthaven, J. S.: 1998a, From electrostatics to almost optimal nodal sets for polynomial
interpolation in a simplex, SIAM Journal of Numerical Analysis 35(2), 655“676. Good
distribution of points for a non-tensor product grid in a triangular subdomain. An
alternative is given by Taylor&Wingate(1999).

Hesthaven, J. S.: 1998b, Integration preconditioning of pseudospectral operators. I. Basic
linear operators, SIAM J. Numer. Anal. 35(2), 1571“1593.

Hesthaven, J. S.: 1999, A stable penalty method for the compressible Navier-Stokes equa-
tions. III. Multidimensional domain decomposition schemes, SIAM Journal of Scienti¬c
Computing 20(1), 62“93.

Hesthaven, J. S. and Gottlieb, D.: 1996, A stable penalty method for the compressible
Navier-Stokes equations. I. Open boundary conditions., SIAM Journal of Scienti¬c Com-
puting 17(3), 579“612.

Hesthaven, J. S. and Gottlieb, D.: 1999, Stable spectral methods for conservation laws on
triangles with unstructured grids, Comput. Meths. Appl. Mech. Engin. In press.

Higgins, J. R.: 1977, Completeness and Basis Properties of Sets of Special Functions, Cambridge
University Press, New York. This slim book is a very readable introduction to basis
sets, but has no numerical content. Pp. 59-64 are a good discussion of complex-valued
rational Chebyshev functions.

Hille, E.: 1939, Contributions to the theory of Hermitian series, Duke Math. Journal 5, 875“
936. This and the next three papers prove Hermite function convergence theorems.

Hille, E.: 1940a, Contributions to the theory of Hermitian series. II. The representation
problem, Transaction of the American Mathematical Society 47, 80“94.

Hille, E.: 1940b, A class of differential operators of in¬nite order, I, Duke Mathematical Jour-
nal 7, 458“495.

Hille, E.: 1961, Sur les fonctions analytiques d´ ¬nies par des s´ ries d™hermite, Journal of
e e
Mathematiques Pures Appliques 40, 335“342.

Ho, L.-W. and Patera, A. T.: 1990, A Legendre spectral element method for simulation of
unsteady incompressible viscous free-surface ¬‚ows, 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. 355“366. Also in Comput. Meths. Appl. Mech. Engrg., vol. 80, with the same page
numbers.

Ho, L. W. and Patera, A. T.: 1991, Variational formulation of three-dimensional viscous free-
surface ¬‚ows: natural imposition of surface tension boundary conditions, International
Journal for Numerical Methods in Fluids 13, 691“698.

Ho, L.-W. and Rønquist, E. M.: 1994, Spectral element solution of steady incompressible
viscous free-surface ¬‚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. 347“368. Also in Finite Elements in Analysis and Design, vol. 16, pp.
207-229.
BIBLIOGRAPHY 627

Ho, L.-W., Maday, Y., Patera, A. T. and Rønquist, E. M.: 1990, A high-order Lagrangian-
decoupling method for the incompressible Navier-Stokes equations, 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. 65“90.
Hogan, T. F. and Rosmond, T. E.: 1991, The description of the Navy operational global
atmospheric prediction systems™s spectral forecast model, Monthly Weather Review
119, 1786“1815.
Holloway, J. P.: 1996a, Spectral velocity discretizations for the Vlasov-Maxwell equations,
Transport Theory and Statistical Physics 25, 1“32.
Holloway, J. P.: 1996b, Hamiltonian spectral methods, Journal of Computational Physics
129(1), 121“133. Fourier-Hermite spectral methods.
Holly, Jr., F. M. and Preissman, A.: 1977, Accurate evaluation of transport in two dimen-

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. 128
( 136 .)



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