On the stability of spline-collocation methods of multivalue type

Abstract

Abstract.In this paper the general class V of spline-collocation methods for first order systems of ordinary differential equations is investigated. The methods can in part be regarded as so-called multivalue methods. This type contains the generalized singly-implicit methods treated by Butcher.It is shown here, how any multivalue type representative of V yields a matrix valued function ~u, for the characterization of stability at infinity. It is shown in particular, that the structure of t/, allows us to construct infinity-stable methods by an appropriate choice of the collocation points.

The stability function for multistep collocation methods

“…In the case of multistep methods, we cannot apply Noorsett's results directly because the solution to (2) is not necessarily a simple rational function as it is in the one-step case. However, we will see in Sect.…”

“…2. We have seen that the stability function can be written in the form (9) which is of the desired form (2). The goal is now to find the polynomials in z which are the coefficients of Y,-i, i=0 ..... k in (9).…”

“…Butcher, [1], constructs multistep collocation methods in Nordsieck form, and Fuchs, [2], gives linear stability results for general spline collocation methods of which Butcher's methods is a special case. Since the spline collocation methods are not equivalent to our multistep collocation methods Fuchs' results cannot be applied directly.…”