Advertisement

Numerische Mathematik

, Volume 50, Issue 4, pp 483–501 | Cite as

A uniform scheme for the singularly perturbed Riccati equation

  • M. J. OReilly
On the Numerical Solution of the First Biharmonic Boundary Value Problem

Summary

We consider the singularly perturbed Riccati equation
$$\varepsilon u'(x) + a(x)u^2 (x) + b(x)u(x) + c(x) = 0, u(0) = A$$
and present a new finite difference scheme, which is exact when the coeffcientsa, b andc are constant. We show that, under certain restrictions, the solution,u i h , of this difference scheme converges, uniformly in ε, tou(xi), the solution of the differential equation on the uniform meshxi=ih. Numerical results are presented.

Subject Classifications

AMS (MOS): Primary 65L05 65G99 Secondary 34A10 34E15 CR: G1.7 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abrahamsson, L., Osher, S.: Monotone difference schemes for singular perturbation problems. SIAM J. Numer. Anal.19, 979–992 (1982)Google Scholar
  2. 2.
    Berger, A.E.: A conservative uniformly accurate difference methods for a singular perturbation problem in conservation form. SIAM J. Numer. Anal. (to appear)Google Scholar
  3. 3.
    Carroll, J.: Exponentially fitted one-step methods for the numerical integration of some stiff initial value problems. PhD Thesis, Trinity College, Dublin 1983Google Scholar
  4. 4.
    Doolan, E.P., Miller, J.J.H., Schilders, W.H.A.: Uniform numerical methods for problems with initial and boundary layers. Dublin: Boole Press 1980Google Scholar
  5. 5.
    Farrell, P.A.: Uniformly convergent difference schemes for singularly perturbed turning and non-turning point problems. PhD Thesis. Trinity College, Dublin 1983Google Scholar
  6. 6.
    Gear, C.W.: Numerical initial value problems for ordinary differential equations. Englewood Cliffs, N.J.: Prentice Hall 1971Google Scholar
  7. 7.
    Henrici, P.: Discrete variable methods in ordinary differential equations. New York: John Wiley 1962Google Scholar
  8. 8.
    Kellogg, R.B., Shubin, G.R., Stephens, A.B.: Uniqueness and the cell Reynolds number. SIAM J. Numer. Anal.17, 733–739 (1980)Google Scholar
  9. 9.
    Lorenz, J.: Nonlinear singular perturbation problems and the Engquist-Osher difference sheme. University of Nijmegen, Report8115 (1981)Google Scholar
  10. 10.
    Niijima, K.: A uniformly convergent difference scheme for a semilinear singular perturbation problem. Numer. Math.43, 175–198 (1984)Google Scholar
  11. 11.
    Niijima, K.: On a difference scheme of exponential type for a nonlinear singular perturbation problem. Research report RMC 58-12. Fukuoka: Kyushu University 1983Google Scholar
  12. 12.
    O'Reilly, M.J.: On uniformly convergent finite difference methods for non-linear singular perturbation problems. PhD Thesis, Trinity College, Dublin 1983Google Scholar
  13. 13.
    O'Reilly, M.J.: A uniformly convergent finite difference scheme for the singularly perturbed Riccati equation. Proc. BAIL III (J.J.H. Miller, ed.), pp. 265–270, Dublin Boole Press 1984Google Scholar
  14. 14.
    Osher, S.: Nonlinear singular perturbation problems and one-sided difference schemes. SIAM J. Numer. Anal.18, 129–144 (1981)Google Scholar
  15. 15.
    Stynes, M., O'Riordan, E.:L 1 andL uniform convergence of a difference scheme for a semilinear singular perturbation problem. Numers. Math. (to appear)Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • M. J. OReilly
    • 1
  1. 1.Regional Technical CollegeDundalkIreland

Personalised recommendations