Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On the numerical integration of a class of singular perturbation problems

Abstract

A three-point difference scheme recently proposed in Ref. 1 for the numerical solution of a class of linear, singularly perturbed, two-point boundary-value problems is investigated. The scheme is derived from a first-order approximation to the original problem with a small deviating argument. It is shown here that, in the limit, as the deviating argument tends to zero, the difference scheme converges to a one-sided approximation to the original singularly perturbed equation in conservation form. The limiting scheme is shown to be stable on any uniform grid. Therefore, no advantage arises from using the deviating argument, and the most accurate and efficient results are obtained with the deviation at its zero limit.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Kadalbajoo, M. K., andReddy, Y. N.,Numerical Integration of a Class of Singular Perturbation Problems, Journal of Optimization Theory and Applications, Vol. 5, pp. 441–452, 1986.

  2. 2.

    Osher, S.,Nonlinear Singular Perturbation Problems and One-Sided Difference Schemes, SIAM Journal on Numerical Analysis, Vol. 18, pp. 129–144, 1981.

  3. 3.

    Abrahamsson, L., andOsher, S.,Monotone Difference Schemes for Singular Perturbation Problems, SIAM Journal on Numerical Analysis, Vol. 19, pp. 979–992, 1982.

  4. 4.

    Varga, R.,Matrix Iterative Methods, Prentice-Hall, Englewood Cliffs, New Jersey, 1962.

  5. 5.

    Keller, H. B.,Numerical Solution of Two-Point Boundary-Value Problems, Blaisdell, Waltham, Massachusetts, 1968.

  6. 6.

    Young, D. M., andGregory, R. T.,A Survey of Numerical Mathematics, Vol. 2, Addison-Wesley, Reading, Massachusetts, 1972.

  7. 7.

    Kreiss, H. O., Nichols, N. K., andBrown, D. L.,Numerical Methods for Stiff Two-Point Boundary-Value Problems, SIAM Journal on Numerical Analysis, Vol. 23, pp. 325–368, 1986.

Download references

Author information

Additional information

Communicated by S. M. Roberts

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Nichols, N.K. On the numerical integration of a class of singular perturbation problems. J Optim Theory Appl 60, 439–452 (1989). https://doi.org/10.1007/BF00940347

Download citation

Key Words

  • Ordinary differential equations
  • singular perturbations
  • boundary-value problems
  • finite-difference approximations
  • numerical stability