Journal of Mathematical Sciences

, Volume 195, Issue 6, pp 865–872 | Cite as

Difference Scheme on a Uniform Grid for the Singularly Perturbed Cauchy Problem

  • A. I. Zadorin
  • S. V. Tikhovskaya

We consider the Cauchy problem for a singularly perturbed second order ordinary differential equation. Based on the maximum principle, we estimate the solution and its derivatives. We construct an exponential fitted scheme generalizing the well-known scheme due to A. M. Il’in. We also prove the uniform convergence with the first order accuracy and illustrate the results by numerical experiments. Bibliography: 8 titles.


Boundary Layer Cauchy Problem Maximum Principle Uniform Convergence Uniform Grid 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N. S. Bakhvalov, “The optimization of methods of solving boundary value problems with a boundary layer” [in Russian], Zh. Vychisl. Mat. Mat. Fiz. 9, No. 4, 841–859 (1969); English transl.: U.S.S.R. Comput. Math. Math. Phys. 9, No. 4, 139–166 (1971).Google Scholar
  2. 2.
    G. I. Shishkin, Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations [in Russian], Ekaterinoburg (1992).Google Scholar
  3. 3.
    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 (1996).Google Scholar
  4. 4.
    A. M. Il’in, “Difference scheme for a differential equation with a small parameter affecting the highest derivative” [in Russian], Mat. Zametki 6, No. 2, 237–248 (1969); English transl.: Math. Notes 6, No. 2, 596–602 (1969).Google Scholar
  5. 5.
    E. P. Doolan, J. J. H. Miller, and W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers Boole Press, Dublin (1980).Google Scholar
  6. 6.
    A. A. Samarskii, The Theory of Difference Schemes [in Russian], Nauka, Moscow (1983); English transl.: Marcel Dekker, New York (2001).Google Scholar
  7. 7.
    R. B. Kellog and A. Tsan, “Analysis of some difference approximations for a singular perturbation problem without turning points,” Math. Comput. 32, No. 144, 1025–1039 (1978).CrossRefGoogle Scholar
  8. 8.
    G. I. Marchuk and V. V. Shaidurov, Increasing the Accuracy of Solutions of Difference Schemes [in Russian], Nauka, Moscow (1979); English transl.: Difference Methods and Their Extrapolations, Springer, New York etc. (1983).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Omsk Branch of Sobolev Institute of Mathematics 13OmskRussia

Personalised recommendations