Highly Stable Super-Implicit Hybrid Methods for Special Second Order IVPs

Abstract

Abstract:The idea of symmetric super-implicit linear multi-step methods (SSILMMs) necessitates the use of not just past and present solution values of the ordinary differential equations (ODEs), but also, future values of the solution. Such methods have been proposed recently for the numerical solution of second-order ODEs. One technique to obtain more accurate integration process is to construct linear multi-step methods with hybrid points employing future solution values. In this regard, we construct families of Stӧrmer-Cowell type hybrid SSILMMs having higher order than that of the symmetric super-implicit method recently proposed for the same step number using the Taylors series approach. The newly derived hybrid SSILMMs are p-stable with accurate results when tested on some special second order IVPs.

A Family of Stiffly Stable Second Derivative Block Methods for Initial Value Problems in Ordinary Differential Equations

“…This leads into the large field general linear methods Hairer and Wanner [14]. However, the super-future points could be likened to the super-implicit points presented in Ibrahim and Ikhile [16], Ibrahim and Ikhile [17] respectively for the case of special second-order linear multistep methods. Several authors have considered the second and higher derivatives methods because of the existence of A-stable higher multiderivative schemes, see for example, Enright [10], Ismail and Ibrahim [18], Kumleng and Sirisena [20].…”