Fifth-order five-stage explicit Almost Runge-Kutta methods

Abstract

Two new Almost Runge-Kutta methods of orders five with five stages are introduced through a judicious and careful choice of free parameters to the general form of deriving order five methods. Appropriate convergence analysis established that the methods are consistent and stable, hence their convergence. The proposed methods, christened ARK5a and ARK5b, were implemented using sample initial value problems and the results compared with those of some existing ARK methods of the same order. Experimental results revealed that the methods are more efficient and effective than the existing methods by producing lesser errors and lesser computational rigour.