B-spline multistep methods and their continuous extensions

In this paper, starting from a sequence of results which can be traced back to I. J. Schoenberg, we analyze a class of spline collocation methods for the numerical solution of ordinary differential equations (ODEs) with collocation points coinciding with the knots. Such collocation methods are naturally associated to a special class of linear multistep methods, here called B-spline (BS) methods, which are able to generate the spline values at the knots. We prove that, provided the additional conditions are appropriately chosen, such methods are all convergent and A-stable. The convergence property of the BS methods is naturally inherited by the related spline extensions, which, by the way, are easily and safely computable using their B-spline representation.