B‐Spline Linear Multistep Methods and their Continuous Extensions

Abstract

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.