Boundary Value Methods for the numerical approximation of ordinary differential equations

Abstract

Many numerical methods for the approximation of ordinary differential equations (ODEs) are obtained by using Linear Multistep Formulae (LMF). Such methods, however, in their usual implementation 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 situation has recently changed, with the introduction of Boundary Value Methods (BVMs), which are methods still based on LMF. Their main feature consists in approximating a given continuous initial value problem (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 computationally efficient. In particular, we shall see that they allow a natural implementation of efficient mesh selection strategies.