Summary. The paper introduces a new semi-implicit extrapolation methodespecially designed for the numerical solution of stiff systems of ordinary differential equations. The existence of a quadratic asymptotic expansion in terms of the stepsize is shown. Moreover, the new discretization is analyzed in the light of well-known stability models. O. IntroductionFor many years, the numerical solution of stiff ordinary differential equations (ODEs) has been an active field of research motivated by challenging real life applications. The presently most effective algorithm for this type of problems seems to be the BDF method of Gear [10] in the advanced version due to Hindmarsh [13]. As this method is based on an implicit discretization, it requires the iterative solution of a nonlinear algebraic system per each integration step. In view of this feature, recent attention has focussed on so-called semiimplicit methods, which require the solution of just one linear system per each step (or stage). Examples of this latter kind are the Rosenbrock-Wanner methods (see e.g. Kaps and Rentrop [15]) or the new extrapolation method to be introduced in the present paper.The latter method is based on a new type of discretization, which is called the semi-implicit mid-point rule, as it has been derived from the well-known explicit mid-point rule (see Gragg [11,12], and Bulirsch and Stoer [2]). In Sect. 1, this semi-implicit mid-point discretization is shown to permit a quadratic asymptotic expansion in terms of the stepsize h. This expansion and certain additional stability considerations (see Sect. 2) back the expectation that the new method should be useful for solving stiff problems. Details of the algorithmic realization are described in Sect. 3. Finally, in Sect. 4, numerical

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