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Numerical Mathematics and Advanced Applications pp 97–118Cite as

Highly Oscillatory Quadrature: The Story so Far

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Abstract

The last few years have witnessed substantive developments in the computation of highly oscillatory integrals in one or more dimensions. The availability of new asymptotic expansions and a Stokes-type theorem allow for a comprehensive analysis of a number of old (although enhanced) and new quadrature techniques: the asymptotic, Filon-type and Levin-type methods. All these methods share the surprising property that their accuracy increases with growing oscillation.

Keywords

  • Asymptotic Expansion
  • Simplicial Complex
  • Asymptotic Method
  • Quadrature Method
  • Gaussian Quadrature

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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  • DOI: 10.1007/978-3-540-34288-5_6
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References

  1. Cochran, J. A., Hinds, E. W.: Eigensystems associated with the complexsymmetric kernels of laser theory. SIAM J. Appld Maths, 26, 776–786 (1974).

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. Degani, I., Schi., J.: RCMS: Right correction Magnus series approach for integration of linear ordinary differential equations with highly oscillatory terms, Technical report, Weizmann Institute of Science.(2003)

    Google Scholar 

  3. Filon, L. N. G.: On a quadrature formula for trigonometric integrals. Proc. Royal Soc. Edinburgh,49, 38–47 (1928).

    Google Scholar 

  4. Flinn, E. A.: A modification of Filon's method of numerical integration. J ACM,7, 181–184 (1960).

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. Huybrechs, D., Vandewalle, S., On the evalution of highly oscillatory integrals by analytic continuation, Technical report, Katholieke Universiteit Leuven, (2005).

    Google Scholar 

  6. Iserles, A.: On the global error of discretization methods for highly-oscillatory ordinary differential equations. BIT, 42, 561–599 (2002).

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. Iserles, A.: On the method of Neumann series for highly oscillatory equations. BIT, 44, 473–488 (2004).

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. Iserles, A., Nørsett, S. P.: Effcient quadrature of highly oscillatory integrals using derivatives. Proc. Royal Soc. A, 461, 1383–1399 (2005a).

    CrossRef  MATH  Google Scholar 

  9. Iserles, A., Nørsett, S. P.: On quadrature methods for highly oscillatory integrals and their implementation. BIT.(2005 b) To appear.

    Google Scholar 

  10. Iserles, A., Nørsett, S. P.: Quadrature methods for multivariate highly oscillatory integrals using derivatives. Maths Comp. (2006) To appear.

    Google Scholar 

  11. Levin, D.: Procedure for computing one- and two-dimensional integrals of functions with rapid irregular oscillations. Maths Comp., 38, 531–538 (1982).

    CrossRef  MATH  Google Scholar 

  12. Lorenz, K., Jahnke, T., Lubich, C.: Adiabatic integrators for highly oscillatory second order linear differential equations with time-varying eigendecomposition. BIT, 45, 91–115 (2005).

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. Luke, Y. L.: On the computation of oscillatory integrals. Proc. Cambridge Phil. Soc., 50, 269–277 (1954).

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. Munkres, J. R., Analysis on Manifolds, Addison-Wesley, Reading, MA (1991).

    MATH  Google Scholar 

  15. Olver, F. W. J., Asymptotics and Special Functions, Academic Press, New York (1974).

    Google Scholar 

  16. Olver, S.: Moment-free numerical integration of highly oscillatory functions. IMA J. Num. Anal. (2005a) To appear.

    Google Scholar 

  17. Olver, S., Multivariate Levin-type method, Technical Report TBD, DAMTP, University of Cambridge (2005b).

    Google Scholar 

  18. Stein, E.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ (1993).

    MATH  Google Scholar 

  19. Ursell, F.: Integral equations with a rapidly oscillating kernel. J. London Math. Soc., 44, 449–459 (1969).

    CrossRef  MATH  MathSciNet  Google Scholar 

  20. Xiang, S.: On quadrature of Bessel transformations. J. Comput. Appld Maths,177, 231–239 (2005).

    CrossRef  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, CB3 0WA, Cambridge, United Kingdom

    A. Iserles

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    S.P. Nørsett

  3. Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, CB3 0WA, Cambridge, United Kingdom

    S. Olver

Authors
  1. A. Iserles
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  2. S.P. Nørsett
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  3. S. Olver
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Editor information

Editors and Affiliations

  1. Departamento de Matemática Aplicada Facultade de Matemáticas, Universidade de Santiago de Compostela, 15706, Santiago de Compostela, Spain

    Alfredo Bermúdez de Castro, Dolores Gómez & Peregrina Quintela,  & 

  2. Departamento de Matématica Aplicada, Escola Politécnica Superior, 27002, Lugo, Spain

    Pilar Salgado

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Iserles, A., Nørsett, S., Olver, S. (2006). Highly Oscillatory Quadrature: The Story so Far. In: de Castro, A.B., Gómez, D., Quintela, P., Salgado, P. (eds) Numerical Mathematics and Advanced Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-34288-5_6

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