On Fourth and Fifth Order Explicit Almost Runge – Kutta Methods

Abstract

Almost Runge – Kutta (ARK) methods are a special class of general linear methods that have close affinity to Runge – Kutta methods for the numerical integration of initial value problems of ordinary differential equations, but with some advantages over the classical Runge – Kutta methods. In this research work, we exploit the order, annihilation and Runge – Kutta stability conditions normally associated with Runge – Kutta methods to derive two new explicit Almost Runge – Kutta methods of orders four (ARK4) and five (ARK5)respectively. The methods were tested for consistency and stability and proven to satisfy the criteria for both; hence their convergence is guaranteed. In order to demonstrate the efficiency and reliability of the proposed methods, some initial value problems were solved with the methods. Numerical results revealed that the proposed methods exhibited numerical errors within acceptable limits when compared with the exact solutions. More so, the methods performed better than some existing Almost Runge – Kutta methods of equal standing.