Convergence and stability of boundary value methods for ordinary differential equations

A usual way to approximate the solution of initial value problems for ordinary differential equations is the use of a linear multistep formula, If the formula has k steps, k values are needed to obtain the discrete solution. The continuous problem provides only the initial value. It is customary to impose the additional k - 1 conditions at the successive k - 1 initial points. However, the class of methods obtained in this way suffers from heavy limitations summarized by the two Dahlquist barriers. It is also possible to impose the additional conditions at different grid-points. For example, some conditions can be imposed at the initial points and the remaining ones at the final points. The obtained methods, called boundary value methods (BVMs), do not have barriers whatsoever. In this paper the question of convergence of BVMs is discussed, along with the linear stability theory. Some numerical examples on stiff test problems are also presented.