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, Volume 36, Issue 1–2, pp 57–67 | Cite as

A second order monotone upwind scheme

  • H. -G. Roos
Contributed Papers

Abstract

We analyze a special finite difference scheme of upwind type for an ordinary singularly perturbed nonlinear boundary value problem. In particular we prove the uniqueness and monotone dependence upon the right hand sides of the discrete solutions and the second order accuracy in the global domain.

AMS Subject Classification

65L Key words Convection-diffusion upwind finite differencing singular perturbation 

Ein monotones upwind-Schema zweiter Ordnung

Zusammenfassung

Wir analysieren ein spezielles upwind-Differenzenschema für ein gewöhnliches, nichtlineares, singulär gestörtes Randwertproblem. Es wird insbesondere gezeigt, daß die Lösung des diskreten Problems eindeutig ist sowie monoton von der rechten Seite abhängt. Im globalen Gebiet ist die Methode von zweiter Ordnung.

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References

  1. [1]
    Abrahamsson, L., Osher, S.: Monotone difference schemes for singularly perturbed problems. SIAM J. Num. Anal.19, 979–992 (1982).Google Scholar
  2. [2]
    Berger, A. E. et al.: Generalized OCI-schemes for boundary value problems. Math. Comput.35, 695–731 (1980).Google Scholar
  3. [3]
    Berger, A. E. et al.: An analysis of a uniformly accurate difference method for a singular perturbation problem. Math. Comput.37, 79–94 (1981).Google Scholar
  4. [4]
    Berger, A. E. et al.: A priori estimates and analysis of a numerical method for a turning point problem. Math. Comput.42, 465–492 (1984).Google Scholar
  5. [5]
    Doolan, E. P., Miller, J. J. H., Schilders, W. H. A.: Uniform Numerical Methods for Problems with Initial and Boundary Layers. Dublin: Boole Press 1980.Google Scholar
  6. [6]
    Goering, H. et al.: Singularly perturbed differential equations. Berlin: Akademie-Verlag 1983.Google Scholar
  7. [7]
    Krätzschmar M.: Iterationsverfahren zur Lösung schwach nichtlinearer elliptischer Randwertaufgaben mit monotoner Lösungseinschließung. Dissertation, TU Dresden 1983.Google Scholar
  8. [8]
    Leventhal, S.: An operator compact implicit method of exponential type. J. Comput. Ph.46, 138–165 (1983).Google Scholar
  9. [9]
    Lorenz, J.: Zur Inversmonotonie diskreter Probleme. Num. Math.27, 227–238 (1977).Google Scholar
  10. [10]
    Lorenz, J.: Nonlinear singular perturbation problems and the Enquist-Osher difference scheme. Report 8115, Nijmegen 1981.Google Scholar
  11. [11]
    Lorenz, J.: Stability and consistency analysis of difference methods for singular perturbation problems. In: Analytical and Numerical Approaches to Asymptotic Problems in Analysis, pp. 141–156. Amsterdam 1981.Google Scholar
  12. [12]
    Lorenz, J.: Nonlinear boundary value problems with turning points and properties of difference schemes. Lecture Notes in Math.942, 150–169 (1982).Google Scholar
  13. [13]
    Lorenz, J.: Numerical solution of a singular perturbation problem with turning points. Lecture Notes in Math. 1027 (1983).Google Scholar
  14. [14]
    Niijima, K.: A uniformly convergent difference scheme for a semilinear singular perturbation problem. Num. Math.43, 175–198 (1984).Google Scholar
  15. [15]
    Osher, S.: Nonlinear singular perturbation problems and one sided difference schemes. SIAM J. Num. Anal.18, 129–144 (1981).Google Scholar
  16. [16]
    Riordan, E.: Singularly perturbed finite element methods. Num. Math.44, 425–434 (1984).Google Scholar
  17. [17]
    Tobiska, L.: Diskretisierungsverfahren zur Lösung singulär gestörter Randwertprobleme. ZAMM63, 115–123 (1983).Google Scholar
  18. [18]
    Weiss, R.: An analysis of the box and trapezoidal schemes for linear singularly perturbed boundary value problems. Math. Comput.42, 41–68 (1984).Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • H. -G. Roos
    • 1
  1. 1.Sektion MathematikTechnische UniversitätDresdenGerman Democratic Republic

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