A class of stabilized extended one-step methods for the numerical solution of ODEs

To overcome the “order barrier” imposed by A-stability on linear multistep methods (LMMs), Usmani and Agarwal [1] had constructed a third order A-stabilized extended one-step method by coupling two LMMs. In the present paper, we introduce a class of extended one-step methods generalizing the method of Usmani and Agarwal. Methods of this class of orders three and four, which are A- and/or L-stable, have been given previously in [2,3]. It is natural to ask the maximum attainable order for methods of this class, which are also A-stable or L-stable. The purpose of this paper is to show that the maximum attainable order of a method in this class is five. We derive fifth order extended one-step methods, and show the existence of sub-families of these methods which are A-stable or L-stable; these methods are illustrated by considering two numerical examples.