TWO-STEP SECOND-DERIVATIVE BLOCK HYBRID METHODS FOR THE INTEGRATION OF INITIAL VALUE PROBLEMSs

TWO-STEP SECOND-DERIVATIVE BLOCK HYBRID METHODS

Abstract

One-step collocation and multistep collocation have recently emerged as powerful tools for the derivation of numerical methods for ordinary differential equations. The simplicity and the continuous nature of the collocation process have been the main attraction towards this development. In this paper we exploited some of these qualities of collocation to derive continuous block hybrid collocation methods based on collocation at some polynomial nodes inside the symmetric interval of integration and the two end points of the interval for dense output and for application which favor continuous approximations, like stiff and highly oscillatory initial value problem in ordinary differential equations. The analysis of the block hybrid collocation methods show that they are convergent and provide dense output at all interior selected points of integration within the interval of choice. Preliminary numerical computation carried out is an evidence of better performance of the methods compared with some strong property of algebraic stability required for stiff system integrators existing in the literature. Many examples are used to illustrate these properties.