Abstract.Iterated Defect Correction (IDeC)-methods based on the implicit Ruler scheme are shown to have a fixed point. This fixed point coincides with the solution of certain implicit multi-stage l=~unge-Kutta methods (equivalent to polynomial collocation). Sufficient conditions for the convergence of the iterates to the fixed point are given for linear problems. These results indlca~ that for a large variety of general nomlinear stiff problems, fixed-point.convergence can be expected, and moreover they indicate that the rate of convergence to the fixed point is very high for very stiff problems. Thus the proposed methods combine the high orders and the high accuracy of multistage-methods with the low computational effort of singlestage methods.
Summary. In Part I of this paper we present a method for the numerical solution of two-point boundary value problems, give results concerning the asymptotic behaviour (h -+0) of this method and we indicate the ideas behind the proofs. In Part II, which will appear shortly, these results are proved rigorously.
~~Let us discuss first of all a method which has been given by Zadunaisky [5] for the estimation of the global discretization error of the numerical solution of differential equations with classical discretization methods. Consider the following boundary value problem (BVP) :y" (t) = ! (t, y (t));(1)
y(a)=A,y(b)=B;with the exact solution z (t) defined on [a, b], --oo < a < b < + oo. Let ]13 ( [a, b] • 0. (2c)Although the results hold under weaker differentiability conditions, we assume, for simplicity, the existence of all derivatives. Let us assume that our BVP (1) is solved using a classicaldiscretization method, which yields on the grid ~h--B=O;
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