Boundary Value Methods Solve Initial Value Problem Of ODEs

The approximation of the solutions of ODEs by means of k-step methods transforms a first-order continuous problem in a k^th -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 of the classical LMF with respect to Runge-Kutta methods. However, 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. Of course, they are free of barriers. Moreover, a block of such methods presents some similarity with Runge-Kutta schemes, although still maintaining the advantages of being linear methods, especially, the stepsize variation becomes very simple.In this paper, we in detail introduced the boundary value method, has produced boundary value methods ETRs and GBDF which produce in classical Initial value methods, and has analyzed its stability and convergence theories. Finally we produced several numerical examples, by comparing the boundary value method and the initial value method, see the advantage of BVMs.