Abstract
In this paper the general classV of spline-collocation methods presented by Mülthei is investigated. The methods ofV approximate solutions of first order initial value problems. ClassV contains as subclass the methods of so-called multivalue type, and in particular contains the generalized singly-implicit methods treated by Butcher.
Any multivalue type representativeU εV yields a matrix valued function Ω corresponding toU, which characterizes the region of absolute stability ofU. If a sequence (U(δ)) of multivalue type representatives ofV tending to some singlevalue type representative\(\tilde U\) εV is considered, it can easily be seen by the structure of Ω, that the sequence of the greatest eigenvalues of the Ω(.,δ) tends to the stability function\(\tilde R\) corresponding to\(\tilde U\). This fact allows one to construct one-parameter families of A-stable methods of multivalue type.
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Fuchs, P.M. A-stable spline-collocation methods of multivalue type. BIT 29, 295–310 (1989). https://doi.org/10.1007/BF01952684
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DOI: https://doi.org/10.1007/BF01952684