BIT Numerical Mathematics

, Volume 46, Issue 3, pp 549–566 | Cite as

On the computation of highly oscillatory multivariate integrals with stationary points



We consider two types of highly oscillatory bivariate integrals with a nondegenerate stationary point. In each case we produce an asymptotic expansion and two kinds of quadrature algorithms: an asymptotic method and a Filon-type method. Our results emphasize the crucial role played by the behaviour at the stationary point and by the geometry of the boundary of the underlying domain.

Key words

numerical quadrature asymptotic methods high oscillation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. Dahlquist, On summation formulas due to Plana, Lindelöf and Abel, BIT, 37 (1997), pp. 256–295.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    G. Dahlquist and A. Björck, Numerical Methods, Dover, Mineola, NY, 2003.Google Scholar
  3. 3.
    D. Huybrechs and S. Vandewalle, On the evaluation of highly oscillatory integrals by analytic continuation, SIAM J. Numer. Anal. (2006). To appear.Google Scholar
  4. 4.
    A. Iserles and S. P. Nørsett, On quadrature methods for highly oscillatory integrals and their implementation, BIT, 44 (2004), pp. 755–772.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    A. Iserles and S. P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. A, 461 (2005), pp. 1383–1399.MATHCrossRefGoogle Scholar
  6. 6.
    A. Iserles and S. P. Nørsett, Quadrature methods for multivariate highly oscillatory integrals using derivatives, Math. Comput., 75 (2006), pp. 1233–1258.MATHCrossRefGoogle Scholar
  7. 7.
    A. Iserles, S. P. Nørsett, and S. Olver, Highly oscillatory quadrature: The story so far, in Proceedings of ENuMath 2005, Santiago de Compostela, Springer, Berlin, 2006. To appear.Google Scholar
  8. 8.
    S. Olver, Moment-free numerical integration of highly oscillatory functions, IMA J. Numer. Anal., 26 (2006), pp. 213–227.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    S. Olver, On the quadrature of multivariate highly oscillatory integrals over non-polytope domains, Numer. Math. (2006). To appear.Google Scholar
  10. 10.
    E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUnited Kingdom
  2. 2.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

Personalised recommendations