Predictor–corrector methods of high order for numerical integration of initial value problems

Abstract

Two tenth order implicit linear multistep methods are derived after applying appropriate order conditions to the Taylor series approach in the derivation of linear multistep methods. Each of the derived schemes is further combined with an Adams – Bashforth scheme of order ten to form two separate predictor – corrector pairs for numerical integration of initial value problems of ordinary differential equations. A tenth order Runge – Kutta method is further employed in order to generate the necessary starting values typical of linear multistep methods. The derived schemes are proven to be convergent by satisfying both consistency and zero – stability requirements. Numerical examples are further carried out to ascertain their efficiency and effectiveness.