Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Fourier–Padé approximations and filtering for spectral simulations of an incompressible Boussinesq convection problem
HTML articles powered by AMS MathViewer

by M. S. Min, S. M. Kaber and W. S. Don PDF
Math. Comp. 76 (2007), 1275-1290 Request permission


In this paper, we present rational approximations based on Fourier series representation. For periodic piecewise analytic functions, the well-known Gibbs phenomenon hampers the convergence of the standard Fourier method. Here, for a given set of the Fourier coefficients from a periodic piecewise analytic function, we define Fourier–Padé–Galerkin and Fourier–Padé collocation methods by expressing the coefficients for the rational approximations using the Fourier data. We show that those methods converge exponentially in the smooth region and successfully reduce the Gibbs oscillations as the degrees of the denominators and the numerators of the Padé approximants increase. Numerical results are demonstrated in several examples. The collocation method is applied as a postprocessing step to the standard pseudospectral simulations for the one-dimensional inviscid Burgers’ equation and the two-dimensional incompressible inviscid Boussinesq convection flow.
  • G. K. Batchelor, The stability of a large gas bubble moving through a liquid, J. Fluid Mech. 184 (1987) 399.
  • R. H. Bartels and G. W. Stewart, Algorithm 432, solution of the matrix equation $Ax + xB = C$, Comm. ACM 15 (1972) 820–826.
  • J. T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Comm. Math. Phys. 94 (1984), no. 1, 61–66. MR 763762
  • H. Cabannes (ed.), Padé approximants method and its applications to mechanics, Lecture Notes in Physics, Vol. 47, Springer-Verlag, Berlin-New York, 1976. MR 0494843
  • A. Dold and B. Eckmann, Pade Approximation and Its Application, Springer-Verlag, New York, 1979.
  • L. Emmel, S. M. Kaber, and Y. Maday, Padé-Jacobi filtering for spectral approximations of discontinuous solutions, Numer. Algorithms 33 (2003), no. 1-4, 251–264. International Conference on Numerical Algorithms, Vol. I (Marrakesh, 2001). MR 2005567, DOI 10.1023/A:1025572207222
  • Tobin A. Driscoll and Bengt Fornberg, A Padé-based algorithm for overcoming the Gibbs phenomenon, Numer. Algorithms 26 (2001), no. 1, 77–92. MR 1827318, DOI 10.1023/A:1016648530648
  • James F. Geer, Rational trigonometric approximations using Fourier series partial sums, J. Sci. Comput. 10 (1995), no. 3, 325–356. MR 1361091, DOI 10.1007/BF02091779
  • J. Gibbs, Fourier’s series, Nature 59 (1898) 200.
  • David Gottlieb, M. Yousuff Hussaini, and Steven A. Orszag, Theory and applications of spectral methods, Spectral methods for partial differential equations (Hampton, Va., 1982) SIAM, Philadelphia, PA, 1984, pp. 1–54. MR 758261
  • David Gottlieb and Steven A. Orszag, Numerical analysis of spectral methods: theory and applications, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. MR 0520152
  • David Gottlieb and Chi-Wang Shu, On the Gibbs phenomenon and its resolution, SIAM Rev. 39 (1997), no. 4, 644–668. MR 1491051, DOI 10.1137/S0036144596301390
  • H. Padé, Mémoire sur les développements en fractions continues de la fonction exponentielle puvant servir d’introduction $\grave {a}$ la théorie des fractions continues algébriques, Ann. Fac. Sci. de l’Ec. Norm. Sup. 16 (1899) 395–436.
  • Alain Pumir and Eric D. Siggia, Development of singular solutions to the axisymmetric Euler equations, Phys. Fluids A 4 (1992), no. 7, 1472–1491. MR 1167779, DOI 10.1063/1.858422
  • Weinan E and Chi-Wang Shu, Small-scale structures in Boussinesq convection, Phys. Fluids 6 (1994), no. 1, 49–58. MR 1252833, DOI 10.1063/1.868044
  • E. B. Saff and Richard S. Varga (eds.), Padé and rational approximation, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. Theory and applications. MR 0458010
  • Huazhong Tang and Tao Tang, Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal. 41 (2003), no. 2, 487–515. MR 2004185, DOI 10.1137/S003614290138437X
  • Hervé Vandeven, Family of spectral filters for discontinuous problems, J. Sci. Comput. 6 (1991), no. 2, 159–192. MR 1140344, DOI 10.1007/BF01062118
  • Z. R. Zhang, Moving Mesh Methods for Convection-dominated Equations and Nonlinear Conservation Laws, Ph.D. thesis, Hong Kong Baptist University, 2003.
Similar Articles
Additional Information
  • M. S. Min
  • Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island
  • Email:
  • S. M. Kaber
  • Affiliation: Laboratoire Jacques-Louis Lions, Université Paris VI, France
  • Email:
  • W. S. Don
  • Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island
  • Email:
  • Received by editor(s): June 3, 2003
  • Received by editor(s) in revised form: July 7, 2004
  • Published electronically: February 16, 2007
  • Additional Notes: This research was supported by Grant AFOSR F49620-02-1-0113.
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 76 (2007), 1275-1290
  • MSC (2000): Primary 41A20, 41A21, 41A25, 65T10, 65T20
  • DOI:
  • MathSciNet review: 2299774