Skip to main content

Quadrature formulas for functions with a boundary-layer component

Abstract

Quadrature formulas for one-variable functions with a boundary-layer component are constructed and studied. It is assumed that the integrand can be represented as the sum of a regular and a boundary-layer component, the latter having high gradients that reduce the accuracy of classical quadrature formulas, such as the trapezoidal and Simpson rules. The formulas are modified so that their error is independent of the gradients of the boundary-layer component. Results of numerical experiments are presented.

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

References

  1. L. V. Kantorovich, “Approximate Evaluation of Several Types of Definite Integrals and Some Other Applications of the Method of Isolating Singularities,” Mat. Sb. 41, 235–244 (1943).

    Google Scholar 

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

    MATH  Google Scholar 

  3. N. S. Bakhvalov, Numerical Methods (Nauka, Moscow, 1975) [in Russian].

    Google Scholar 

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

    Google Scholar 

  5. J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems (World Scientific, Singapore, 1996).

    MATH  Google Scholar 

  6. R. B. Kellogg and A. Tsan, “Analysis of Some Difference Approximations for a Singular Perturbation Problems without Turning Points,” Math. Comput. 32, 1025–1039 (1978).

    MathSciNet  MATH  Article  Google Scholar 

  7. A. I. Zadorin, “Method of Interpolation for a Boundary Layer Problem,” Sib. Zh. Vychisl. Mat. 10, 267–275 (2007).

    Google Scholar 

  8. A. I. Zadorin, “Interpolation Method for a Function with a Singular Component,” Lect. Notes Comput. Sci. 5434, 612–619 (2009).

    Article  Google Scholar 

  9. 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 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. I. Zadorin.

Additional information

Original Russian Text © A.I. Zadorin, N.A. Zadorin, 2011, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2011, Vol. 51, No. 11, pp. 1952–1962.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Zadorin, A.I., Zadorin, N.A. Quadrature formulas for functions with a boundary-layer component. Comput. Math. and Math. Phys. 51, 1837–1846 (2011). https://doi.org/10.1134/S0965542511110157

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542511110157

Keywords

  • one-variable function
  • boundary-layer component
  • high gradients
  • definite integral
  • non-polynomial interpolation
  • quadrature rule
  • error estimate