Modified Euler methods for differential inclusions

F. Lempio: Modified Euler methods for differential inclusions
in: Set-Valued Analysis and Differential Inclusions. A Collection of Papers resulting from a Workshop held in Pamporovo, Bulgaria, September 17-21, 1990, A. B. Kurzhanski, V. M. Veliov (eds.)
Progr. Systems Control Theory 16, Birkhäuser Verlag, Boston-Basel-Berlin, 1993, 131 - 148

MR Nummer: 1269810
Zentralblattnummer: 0788.34008
Mathematics Subject Classification Code: 34A45 (34A60 65L05 65L99)


Classical Euler method and simple modifications like the method of Euler-Cauchy, improved Euler method and implicit midpoint rule are discussed with regard to the approximate solution of differential inclusions.

Numerical tests suggest first order convergence of Euler's method at least for specially structured right-hand sides even if the usual Lipschitz condition does not hold. The basic idea of the proof of this convergence property is sketched using a strengthened one-sided Lipschitz condition.

Other reduction for methods which are of higher order for single-valued sufficiently smooth right-hand sides is exemplified numerically for improved Euler method and implicit midpoint rule. Typical advantages of implicit midpoint rule are discussed.


1. Introduction and Preliminaries
2. Euler Method
3. Modified Euler Methods
3.1 Method of Euler-Cauchy
3.2 Improved Euler Method
3.3 Implicit Midpoint Rule

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