Boundary value methods: The third way between linear multistep and Runge-Kutta methods

It is well known that the approximation of the solutions of ODEs by means of k-step methods transforms a first-order continuous problem in a kth-order discrete one. Such transformation has the undesired effect of introducing spurious, or parasitic, solutions to be kept under control. It is such control which is responsible of the main drawbacks (e.g., the two Dahlquist barriers) of the classical LMF with respect to Runge-Kutta methods. It is, however, less known that the control of the parasitic solutions is much easier if the problem is transformed into an almost equivalent boundary value problem. Starting from such an idea, a new class of multistep methods, called Boundary Value Methods (BVMs), has been proposed and analyzed in the last few years. Of course, they are free of barriers. Moreover, a block version of such methods presents some similarity with Runge-Kutta schemes, although still maintaining the advantages of being linear methods. In this paper, the recent results on the subject are reviewed.