On a continuous block method for the solution of initial value problems.

Abstract

In this paper, we convert a conventional linear multistep method for solving initial value problems into the continuous form. The approach of collocation approximation is adopted in the derivation of the schemes. The continuous schemes so derived are then applied as simultaneous integrators to the solution of the initial value problem (ivp) for first order ordinary differential equations (odes). This implementation strategy is expected to produce results that are more accurate and efficient than those given when applied over overlapping intervals in predictor-corrector mode. In addition to eliminating the rigours associated with predictor-corrector methods, the new block method possesses the self-starting feature of Runge-Kutta methods. Numerical experiments confirm the theoretical expectations.