Design and Analysis of Some Third Order Explicit Almost Runge-Kutta Methods

Abstract

In this paper, we propose two new explicit Almost Runge-Kutta (ARK) methods, ARK3 (a three stage third order method, i.e., s = p = 3) and ARK34 (a four-stage third-order method, i.e., s = 4, p = 3), for the numerical solution of initial value problems (IVPs). The methods are derived through the application of order and stability conditions normally associated with Runge-Kutta methods; the derived methods are further tested for consistency and stability, a necessary requirement for convergence of any numerical scheme; they are shown to satisfy the criteria for both consistency and stability; hence their convergence is guaranteed. Numerical experiments carried out further justified the efficiency of the methods.