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Superconvergence for multistep collocation. (English) Zbl 0658.65063
In a k-step collocation method for $$y'=f(x,y)$$, $$y(x_ 0)=y_ 0$$, the collocation polynomial for a subinterval $$[x_ n,x_{n+1}]$$ (given by two consecutive mesh points) is generated by also using previously computer approximations at the mesh points $$x_{n-k+1},...,x_ n$$. This paper is concerned with the question of the attainable order p of k-step collocation methods using m distinct collocation points in each subinterval $$[x_ n,x_{n+1}]:$$ one has $$p=2m+k-1$$, and it is shown that there exist $$\left( \begin{matrix} m+k-1\\ k-1\end{matrix} \right)$$ different sets of “multistep Gaussian” collocation points for which this order is obtained. These sets are given explicit for $$m=2,3$$, and $$k=1,2,3$$. Forthcoming papers will deal with error estimation techniques and stability properties (for variable stepsize) of such multistep collocation methods.
Reviewer: H.Brunner

##### MSC:
 65L05 Numerical methods for initial value problems 34A34 Nonlinear ordinary differential equations and systems, general theory
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##### References:
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