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Fourier-Padé approximations and filtering for spectral simulations of an incompressible Boussinesq convection problem


Authors: M. S. Min, S. M. Kaber and W. S. Don
Journal: Math. Comp. 76 (2007), 1275-1290
MSC (2000): Primary 41A20, 41A21, 41A25, 65T10, 65T20
DOI: https://doi.org/10.1090/S0025-5718-07-01831-5
Published electronically: February 16, 2007
MathSciNet review: 2299774
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Abstract: 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.


References [Enhancements On Off] (What's this?)

  • 1. G. K. BATCHELOR, The stability of a large gas bubble moving through a liquid, J. Fluid Mech. 184 (1987) 399.
  • 2. R. H. BARTELS AND G. W. STEWART, Algorithm 432, solution of the matrix equation $ Ax + xB = C$, Comm. ACM 15 (1972) 820-826.
  • 3. J. T. BEALE, T. KATO, AND A. MAJDA, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 (1984) 61-66. MR 0763762 (85j:35154)
  • 4. H. CABANNES, Padé Approximants Method and Its Applications to Mechanics, Lecture Notes in Physics, edited by J. Ehlers, K. Hepp, H. A. Weidenm$ \ddot{u}$ller, and J. Zittartz, Springer-Verlag, New York, 1976. MR 0494843 (58:13627)
  • 5. A. DOLD AND B. ECKMANN, Pade Approximation and Its Application, Springer-Verlag, New York, 1979.
  • 6. 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. MR 2005567 (2004k:65233)
  • 7. T. A. DRISCOLL AND B. FORNBERG, A Padé-based algorithm for overcoming the Gibbs phenomenon, Numer. Algorithms 26 (2001), no. 1, 77-92. MR 1827318 (2002b:65007)
  • 8. J. F. GEER, Rational trigonometric approximations using Fourier series partial sums, J. Sci. Comp. 10, no. 3 (1995) 325-356.MR 1361091 (96k:42003)
  • 9. J. GIBBS, Fourier's series, Nature 59 (1898) 200.
  • 10. D. GOTTLIEB, M. Y. HUSSAINI, AND S. A. ORSZAG, Introduction: Theory and applications of spectral methods, in Spectral Methods for Partial Differential Equations, edited by R. Voigt, D. Gottlieb, and M. Y. Hussaini, SIAM, Philadelphia, 1984, pp. 1-54. MR 0758261 (86h:65142)
  • 11. D. GOTTLIEB AND S. A. ORSZAG, Numerical Analysis of Spectral Methods: Theory and Application, CMBS-NSF Regional Conference Series in Applied Mathematics 26, SIAM, Philadelphia, 1977. MR 0520152 (58:24983)
  • 12. D. GOTTLIEB AND C. W. SHU, The Gibbs phenomenon and its resolution, SIAM Review 39 (1997) 644-668.MR 1491051 (98m:42002)
  • 13. 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.
  • 14. A. PUMIR AND E. D. SIGGIA, Development of singular solutions to the axisymmetric Euler equations, Phys. Fluids A 4 (1992) 1472-1491. MR 1167779 (93c:76014)
  • 15. WEINAN E AND C. W. SHU, Small-scale structures in Boussinesq convection, Phys. Fluids 6, no. 1 (1994) 49-58. MR 1252833 (94i:76075)
  • 16. E. B. SAFF AND R. S. VARGRA, Pade and Rational Approximation: Theory and Application, Academic Press, New York, 1977. MR 0458010 (56:16213)
  • 17. H. Z. TANG AND T. TANG, Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal. 41 (2003), 487-515. MR 2004185 (2004f:65143)
  • 18. H. VANDEVEN, Family of spectral filters for discontinuous problems, J. Sci. Compt. 6 (1991) 159-192. MR 1140344 (92k:65006)
  • 19. Z. R. ZHANG, Moving Mesh Methods for Convection-dominated Equations and Nonlinear Conservation Laws, Ph.D. thesis, Hong Kong Baptist University, 2003.

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Additional Information

M. S. Min
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island
Email: msmin@cfm.brown.edu

S. M. Kaber
Affiliation: Laboratoire Jacques-Louis Lions, Université Paris VI, France
Email: kaber@ann.jussieu.fr

W. S. Don
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island
Email: wsdon@cfm.brown.edu

DOI: https://doi.org/10.1090/S0025-5718-07-01831-5
Keywords: Rational approximation, Gibbs phenomenon, Fourier--Pad\'e--Galerkin method, Fourier--Pad\'e collocation, postprocessing
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.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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