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

Abdulrahman Ndanusa,Khadeejah James Audu

doi:10.4236/am.2016.71002

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.

Highlights

According to [1] the s-stage Runge-Kutta method for solving the initial value problem=y′ f= ( x, y), y ( x0 ) y0, (1) is defined by s∑ yn+=1 yn + h biki, (2) i=1 where s∑ ki = f xn + cih, yn + h aijk j, i =1, 2, s, (3) j =1How to cite this paper: Ndanusa, A. and Audu, K.J. (2016) Design and Analysis of Some Third Order Explicit Almost RungeKutta Methods
Almost Runge-Kutta (ARK) methods are a special class of RK methods that arose out of the quest to develop efficient and accurate methods that have advantages over the traditional methods by retaining the simple stability function of RK methods, allowing minimal information to be passed between steps and adjusting stepsize
Since the introduction of ARK methods in by [2], other researchers who have made their input toward the development of this method include [3]-[7]

Summary

Introduction

According to [1] the s-stage Runge-Kutta method for solving the initial value problem. (2016) Design and Analysis of Some Third Order Explicit Almost RungeKutta Methods. Alternative forms of the above equations are:. The two forms of Equations (2) and (5) are equivalent by making the interpretation ki = f ( xn + cih,Yi ), i = 1, , s (7). Where Yi is the inner stages that tend to estimate the solutions at some points; s is the number of stages and ci is the points where the function f is computed for a step. ARK methods are a special class of RK methods that arose out of the quest to develop efficient and accurate methods that have advantages over the traditional methods by retaining the simple stability function of RK methods, allowing minimal information to be passed between steps and adjusting stepsize . Since the introduction of ARK methods in by [2], other researchers who have made their input toward the development of this method include [3]-[7]

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Convergence Analysis

Numerical Examples

Conclusion