Skip to main content

Modification of the Euler quadrature formula for functions with a boundary-layer component

Computational Mathematics and Mathematical Physics volume 54, pages 1489–1498 (2014)Cite this article

Abstract

The Euler quadrature formula for the numerical integration of functions with a boundary-layer component on a uniform grid is investigated. If the function under study has a rapidly growing component, the error can be significant. A uniformly accurate quadrature formula is constructed by modifying the Hermite interpolation formula so that the resulting one is exact for the boundary-layer component. An analogue of the Euler formula that is exact for the boundary-layer component is constructed. It is proved that the resulting composite quadrature formula is third-order accurate in space uniformly with respect to the boundary-layer component and its derivatives.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. I. S. Berezin and N. P. Zhidkov, Computing Methods (Pergamon, Oxford, 1965; Nauka, Moscow, 1966).

    MATH  Google Scholar 

  2. N. S. Bakhvalov, Numerical Methods: Analysis, Algebra, Ordinary Differential Equations (Nauka, Moscow, 1975; Mir, Moscow, 1977).

    Google Scholar 

  3. S. M. Nikol’skii, Quadrature Formulas (Nauka, Moscow, 1972) [in Russian].

    Google Scholar 

  4. A. I. Zadorin and N. A. Zadorin, “Euler quadrature formula for functions with a boundary-layer component on a piecewise uniform grid,” Sib. Elektron. Mat. Izv. 10, 491–503 (2013).

    Google Scholar 

  5. G. I. Shishkin, Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations (Ural Otd. Ross. Akad. Nauk, Yekaterinburg, 1992) [in Russian].

    Google Scholar 

  6. J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions (World Scientific, Singapore, 2012).

    Book  Google Scholar 

  7. A. I. Zadorin and N. A. Zadorin, “Quadrature formulas for functions with a boundary-layer component,” Comput. Math. Math. Phys. 51, 1837–1846 (2011).

    MathSciNet  Article  Google Scholar 

  8. A. I. Zadorin and N. A. Zadorin, “Spline interpolation on a uniform grid for functions with a boundary-layer component,” Comput. Math. Math. Phys. 50, 211–223 (2010).

    MathSciNet  Article  Google Scholar 

  9. A. I. Zadorin and N. A. Zadorin, “Interpolation formula for functions with a boundary layer component and its application to derivatives calculation,” Sib. Elektron. Mat. Izv. 9, 445–455 (2012).

    MathSciNet  Google Scholar 

  10. A. I. Zadorin and N. A. Zadorin, “An analogue of Newton-Cotes formula with four nodes for a function with a boundary-layer component,” Sib. Zh. Vychisl. Mat. 16(4), 313–323 (2013).

    MathSciNet  MATH  Google Scholar 

  11. A. Zadorin and N. Zadorin, “Quadrature formula with five nodes for functions with a boundary layer component,” Lect. Notes Comput. Sci. 8236, 540–546 (2013).

    Article  Google Scholar 

  12. V. P. Il’in, Numerical Analysis (Inst. Vychisl. Mat. Mat. Geofiz. Sib. Otd. Ross. Akad. Nauk, Novosibirsk, 2004) [in Russian].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, ul. Pevtsova 13, Omsk, 644043, Russia

    A. I. Zadorin

Authors
  1. A. I. Zadorin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. I. Zadorin.

Additional information

Original Russian Text © A.I. Zadorin, 2014, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2014, Vol. 54, No. 10, pp. 1547–1556.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zadorin, A.I. Modification of the Euler quadrature formula for functions with a boundary-layer component. Comput. Math. and Math. Phys. 54, 1489–1498 (2014). https://doi.org/10.1134/S0965542514100078

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542514100078

Keywords

  • quadrature formula for definite integrals
  • function with a boundary-layer component
  • modified Euler quadrature formula
  • Hermite polynomial
  • modification
  • error estimate for quadrature formulas