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.
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
Abrahamsson, L., Osher, S.: Monotone difference schemes for singularly perturbed problems. SIAM J. Num. Anal.19, 979–992 (1982).
Berger, A. E. et al.: Generalized OCI-schemes for boundary value problems. Math. Comput.35, 695–731 (1980).
Berger, A. E. et al.: An analysis of a uniformly accurate difference method for a singular perturbation problem. Math. Comput.37, 79–94 (1981).
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).
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.
Goering, H. et al.: Singularly perturbed differential equations. Berlin: Akademie-Verlag 1983.
Krätzschmar M.: Iterationsverfahren zur Lösung schwach nichtlinearer elliptischer Randwertaufgaben mit monotoner Lösungseinschließung. Dissertation, TU Dresden 1983.
Leventhal, S.: An operator compact implicit method of exponential type. J. Comput. Ph.46, 138–165 (1983).
Lorenz, J.: Zur Inversmonotonie diskreter Probleme. Num. Math.27, 227–238 (1977).
Lorenz, J.: Nonlinear singular perturbation problems and the Enquist-Osher difference scheme. Report 8115, Nijmegen 1981.
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.
Lorenz, J.: Nonlinear boundary value problems with turning points and properties of difference schemes. Lecture Notes in Math.942, 150–169 (1982).
Lorenz, J.: Numerical solution of a singular perturbation problem with turning points. Lecture Notes in Math. 1027 (1983).
Niijima, K.: A uniformly convergent difference scheme for a semilinear singular perturbation problem. Num. Math.43, 175–198 (1984).
Osher, S.: Nonlinear singular perturbation problems and one sided difference schemes. SIAM J. Num. Anal.18, 129–144 (1981).
Riordan, E.: Singularly perturbed finite element methods. Num. Math.44, 425–434 (1984).
Tobiska, L.: Diskretisierungsverfahren zur Lösung singulär gestörter Randwertprobleme. ZAMM63, 115–123 (1983).
Weiss, R.: An analysis of the box and trapezoidal schemes for linear singularly perturbed boundary value problems. Math. Comput.42, 41–68 (1984).
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Roos, H.G. A second order monotone upwind scheme. Computing 36, 57–67 (1986). https://doi.org/10.1007/BF02238192
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DOI: https://doi.org/10.1007/BF02238192