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The Work of David Gottlieb: A Success Story

Published online by Cambridge University Press:  20 August 2015

Bertil Gustafsson
Bertil Gustafsson*
Affiliation:
Division of Scientific Computing, Uppsala University, Box 337, SE-75105 Uppsala, Sweden
*
*Corresponding author.Email:bertil.gustafsson@it.uu.se

Abstract

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This article is a brief survey of David Gottlieb’s extraordinary research career. It is impossible to give a thorough presentation of all his research and the impact of his work, but we shall describe the main contributions and give examples of the results in some of his papers. David was for many years the dominating person in the development of spectral methods, and we devote much of the space in this article to this field.

Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 3 , March 2011 , pp. 481 - 496
Copyright
Copyright © Global Science Press Limited 2011

References

[1]Abarbanel, S. and Gottlieb, D., High order accuracy finite difference algorithm for quasilinear conservation law hyperbolic systems, Math. Comput., 27 (1973), 505–523.Google Scholar
[2]Abarbanel, S. and Gottlieb, D., A mathematical analysis of PML methods, J. Comput. Phys., 134 (1997), 357–363.Google Scholar
[3]Abarbanel, S. and Gottlieb, D., On the construction and analysis of absorbing layers in CEM, Appl. Numer. Math., 27 (1998), 331–340.Google Scholar
[4]Abarbanel, S., Gottlieb, D. and Hesthaven, J., Long time behaviour of the perfectly matched layer equations in computational electromagnetics, J. Sci. Comput., 17 (2002), 405–422.Google Scholar
[5]Abarbanel, S., Gottlieb, D. and Tadmor, E., Spectral methods for discontinuous problems, Numerical Methods for Fluid Dynamics II (Ed: Morton and Baines), Oxford University Press, 1986, 128–153.Google Scholar
[6]Abarbanel, S., Gottlieb, D. and Turkel, E., Difference schemes with fourth order accuaracy for hyperbolic equations I, SIAM J. Appl. Math., 13 (1975), 329–351.Google Scholar
[7]Berenger, J.-P., A perfectly matched layer for the absorbtion of electromagnetic wave, J. Comput. Phys., 114 (1994), 185–200.Google Scholar
[8]Cai, W., Gottlieb, D. and Harten, A., Cell averages Chebyshev methods for hyperbolic problems, Comput. Math. Appl., 24 (1992), 37–49.Google Scholar
[9]Cai, W., Gottlieb, D. and Shu, C. W., Essentially non-oscillatory spectral Fourier methods for shock wave calculations, Math. Comput., 52 (1989), 389–410.Google Scholar
[10]Cai, W., Shu, C. W. and Gottlieb, D., On one sided filters for spectral Fourier approximations of discontinuous functions, SIAM J. Numer. Anal., 29 (1992), 905–916.Google Scholar
[11]Carpenter, M., Gottlieb, D. and Nordström, J., A stable and conservative interface treatment of arbitrary spatial accuracy, J. Comput. Phys., 148 (1999), 341–365.Google Scholar
[12]Dettori, L., Gottlieb, D. and Temam, R., A nonlinear Galerkin method: the two level Fourier collocation case, J. Sci. Comput., 10 (1995), 371–389.Google Scholar
[13]Don, W. S., Gottlieb, D. and Jung, J. H., A multi-domain method for supersonic reactive flows, J. Comput. Phys., 192 (2003), 325–354.Google Scholar
[14]Ditkowski, A. and Gottlieb, D., On the Engquist Majda absorbing boundary conditions for hyperbolic equations, Contemp. Math., 330 (2003), 55–71.CrossRefGoogle Scholar
[15]Eilon, B., Gottlieb, D. and Zwas, G., Numerical stabilizers and computing time for second order accurate schemes, J. Comput. Phys., 9 (1972), 387–397.Google Scholar
[16]Fischer, P. and Gottlieb, D., On the optimal number of subdomains for hyperbolic problems on parallel computers, Int. J. Supercomput. Ap., 11 (1997), 65–76.Google Scholar
[17]Fornberg, B., On high order approximations of hyperbolic differential equations by a Fourier method, Report No. 39, Dept. Comput. Science, Uppsala University, 1972.Google Scholar
[18]Funaro, D., Domain decomposition for pseudo spectral approximations-part one: second order equations in one dimension, Numer. Math., 52 (1988), 329–344.Google Scholar
[19]Funaro, D. and Gottlieb, D., A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations, Math. Comput., 51, (1988), 599-613.Google Scholar
[20]Funaro, D. and Gottlieb, D., Convergence results for pseudospectral approximations of hyperbolic systems by a penalty type boundary treatment, Math. Comput., 57, (1991), 585-596.Google Scholar
[21]Gottlieb, D., Strang type diference schemes for multi-demesional problem, SIAM J. Numer. Anal., 9 (1972), 650–661.Google Scholar
[22]Gottlieb, D., On the stability of Rusanov’s third order scheme, J. Comput. Phys., 15 (1974), 421–426.Google Scholar
[23]Gottlieb, D., The stability of pseudospectral-Chebyshev, Math. Comput., 36 (1981), 107–118.CrossRefGoogle Scholar
[24]Gottlieb, D. and Fischer, P., On the optimal number of subdomains for hyperbolic problems on parallel computers, 11 (1997), 65–76.Google Scholar
[25]Gottlieb, D. and Gustafsson, B., Generalized Du-Fort Frankel methods for parabolic initial boundary value problems, SIAM J. Appl. Math., 29 (1976), 329–351.Google Scholar
[26]Gottlieb, D., Gustafsson, B. and Forsse’n, P., On the direct Fourier method for computer tomography, IEEE Trans. Medic. Imag., 19 (2000), 223–232.Google Scholar
[27]Gottlieb, D., Gustafsson, B., Olsson, P. and Strand, B., On the superconvergence of Galerkin methods for hyperbolic IBVP, SIAM J. Numer. Anal., 33 (1996), 280–290.Google Scholar
[28]Gottlieb, D. and Hirsh, R., Parallel pseudospectral domain decomposition techniques, J. Sci. Comput., 4 (1989), 309–325.Google Scholar
[29]Gottlieb, D. and Min, M. S., Domain decomposition spectral approximations for an eigenvalue problem with a piecewise constant coefficient, SIAM J. Numer. Anal., 43 (2005), 502– 520.Google Scholar
[30]Gottlieb, D. and Wasberg, C. E., Optimal decomposition of the domain in spectral methods for wave like phenomenon, SIAM J. Sci. Comput., 22 (2000), 617–632.Google Scholar
[31]Gottlieb, D. and Lustman, L., The Du-Fort Frankel Chebyshev method for parabolic initial-boundary value problems, Comput. Fluids., 11 (1983), 107–120.Google Scholar
[32]Gottlieb, D., Lustman, L. and Tadmor, E., Stability analysis of spectral methods for hyperbolic initial boundary value problems, SIAM J. Numer. Anal., 24 (1987), 241–256.Google Scholar
[33]Gottlieb, D., Lustman, L. and Tadmor, E., Convergence of spectral methods for hyperbolic initial boundary value problems, SIAM J. Numer. Anal., 24 (1987), 532–537.Google Scholar
[34]Gottlieb, D. and Orszag, S., Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF 26, Philadalphia: SIAM, 1977.Google Scholar
[35]Gottlieb, D., Orszag, S. and Turkel, E., Stability of pseudospectral and finite difference methods for variable coefficient problems, Math. Comput., 37 (1981), 293–305.Google Scholar
[36]Gottlieb, D., Shu, C.-W., Solomonoff, A. and Vandeven, H., On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a non-periodic analytic function using Gegenbauer polynomials, J. Comput. Appl. Math., 43 (1992), 81–98.Google Scholar
[37]Gottlieb, D. and Shu, C.-W., On the Gibbs phenomenon II: resolution properties of the Fourier method for discontinuous waves, Comput. Meth. Appl. Mech. Eng., 116 (1994), 27–37.Google Scholar
[38]Gottlieb, D. and Shu, C.-W., On the Gibbs phenomenon III: recovering exponential accuracy in a sub-interval from a spectral partial sum of a piecewise analytic function using Gegenbauer polynomials, SIAM J. Numer. Anal., 33 (1996), 280–290.CrossRefGoogle Scholar
[39]Gottlieb, D. and Shu, C.-W., On the Gibbs phenomenon IV: recovering exponential accuracy in a sub-interval from a Gegenbauer partial sum of a piecewise analytic function, Math. Comput., 64 (1995), 1081–1095.Google Scholar
[40]Gottlieb, D. and Shu, C.-W., On the Gibbs phenomenon V: recovering exponential accuracy from collocation point values of a piecewise analytic function, Numer. Math., 71 (1995), 511– 526.Google Scholar
[41]Gottlieb, D. and Tadmor, E., Recovering pointwise value of discontinuous data within spectral accuracy, U.S.-Israel Workshop, Progress in Scientific Computing (Ed: Abarbanel, Murman), Birkhauser Boston, 1985, 357–376.Google Scholar
[42]Gottlieb, D. and Tadmor, E., The CFL condition for spectral approximations to hyperbolic initial-boundary value problems, Math. Comput., 56 (1991), 565–588.Google Scholar
[43]Gottlieb, D. and Temam, R., Implementation of the nonlinear Galerkin method with pseudospectral (collocation) discretizations, Appl. Numer. Math., 12 (1993), 119–134.CrossRefGoogle Scholar
[44]Gottlieb, D. and Turkel, E., Phase error and stability of second order methods for hyperbolic problems, J. Comput. Phys., 15 (1974), 251–265.Google Scholar
[45]Gottlieb, D. and Turkel, E., Dissipative Two-Four methods for time dependent problems, Math. Comput., 136 (1976), 703–723.Google Scholar
[46]Gottlieb, D. and Wasberg, C. E., Optimal decomposition of the domain in spectral methods for wave like phenomenon, SIAM J. Sci. Comput., 22 (2000), 617–632.Google Scholar
[47]Gottlieb, S., Gottlieb, D. and Shu, C.-W., Recovering high order accuracy in WENO computations of steady state hyperbolic systems, J. Sci. Comput., 28 (2006), 307–318.Google Scholar
[48]Gustafsson, B., High Order Difference Methods for Time Dependent PDE, Springer Series in Computational Mathematics, Springer, 2008.Google Scholar
[49]Hesthaven, J., Gottlieb, S. and Gottlieb, D., Spectral Methods for Time-Dependent Problems, Cambridge University Press, 2006.Google Scholar
[50]Kirby, M., Yoosibash, Z. and Gottlieb, D., Collocation methods for the solution of the von Karman dynamical nonlinear plate systems, J. Comput. Phys., 200 (2004), 432–461.Google Scholar
[51]Kreiss, H.-O. and Oliger, J., Comparison of accurate methods for the integration of hyperbolic equations, Tellus., 24 (1972), 199–215.Google Scholar
[52]Kreiss, H.-O. and Oliger, J., Methods for the Approximate Solution of Time Dependent Problems, GARP Publication Series, No. 10, 1973.Google Scholar
[53]Orszag, S., Numerical simulation of incompressible flows within simple boundaries, I, Stud. Appl. Math., 50 (1971), 293–327.Google Scholar
[54]Turkel, E., Abarbanel, S. and Gottlieb, D., Multidimensional difference schemes with fourth order accuracy, J. Comput. Phys., 21 (1976), 85–113.Google Scholar