Boundary Value Methods for the Numer ica l Approx imat ion of Ordinary Differential Equat ions

Abstract. Many numerical methods for the approximation ofordinary differential equations (ODEs) are obtained by using Linear Multistep Formulae (LMF). Such methods, however, in their usual implementa-tion suffer of heavy theoretical limitations, summarized by the two well known Dahlquist barriers. For this reason, Runge-Kutta schemes have become more popular than LMF, in the last twenty years. This situ-ation has recently changed, with the introduction of Boundary Value Methods (BVMs), which are methods till based on LMF. Their main feature consists in approximating a given continuous initial value prob-lem (IVP) by means of a discrete boundary value problem (BVP). Such use allows to avoid order barriers for stable methods. Moreover, BVMs provide several families of methods, which make them very flexible and computationMly efficient. In particular, we shall see that they allow a natural implementation f efficient mesh selection strategies. 1 In t roduct ion Usually, the solution of an initial value ODE problem