Goeken-Johnson Sixth-Order Runge-Kutta Method

In this paper we drive a new sixth-order Runge-Kutta method, depending on the new fifth order Runge-Kutta method of David Goeken and Olin Johnson, the property of this method is that it needs five function evaluations only where the standard method needs six or seven function evaluations, then this method is compared with the new fifth order Runge-Kutta method.


1-Introduction:-
Given y' = f(y), standard Runge-Kutta methods perform multiple evaluations of f(y) in each integration sub-interval as required for a given accuracy. Evaluations of y) y)f(x, (x, y f ' ' y  or higher derivatives are not considered due to the assumption that the calculations involved in these functions exceed those of f. However, y'' can be approximated to sufficient accuracy from past and current evaluations of f to achieve a higher order of accuracy than is available through current functional evaluations alone. In July of 1998 at the ANODE (Auckland Numerical Ordinary Differential Equations) Workshop, the two scientists David Goeken and Olin Johnson introduced a new class of Runge-Kutta methods based on this observation [4], [11]. They presented a third order method which requires only two evaluations of f and a fourth-order method which requires three and fifth order method which requires four. This paper reviews the fifthorder methods and gives the general solution to the equations generated by the sixth-order methods of this new class. Interestingly, these sixth-order methods require only five functional evaluations per step whereas standard Runge-Kutta methods require six or seven. Goeken

2-Third-order method:-
In autonomous form, y and f have n + 1 components with The initial value problem is then written: Most efforts to increase the order of the Runge-Kutta methods have been accomplished by increasing the number of Taylor's series terms used and thus the number of functional evaluations required [10], [5], [12] and [9]. The use of higher order derivative terms has been proposed for stiff problems [16] and [7]. The method adds higher order derivative terms to the Runge-Kutta i k terms (i > 1) to achieve a higher order of accuracy, For more details see [3] [6]. For example, the new third-order method, GJ3, for autonomous systems, is:-[10] . However, they introduce additional terms by assigning:- [4], [10], [12].

3-Fourth-order method:-
Similarly, the fourth-order method, GJ4, for autonomous systems, is:- [10], [12] The Taylor's series expansion of these higher order methods is tedious and error prone. Goeken and Johnson used modern symbolic mathematical packages to expand and then to solve the resulting systems of nonlinear equations that were generated. In this work, they used the symbolic mathematical packages [14], [10], [11], and [7].
The general solution to the system of equations (with 0 a 34  ) has been found with example solutions are shown in:

4-Fifth order method:-
In July of 1998, Goeken and Johnson presented [11] this numerical integration technique at a meeting attended by John Butcher. Using his tree-based approach [5], Butcher suggested a fifth-order method. Since the meeting, his technique has been verified using Taylor's series expansion techniques to determine the general solution for the fifth-order methods. The fifth-order method, GJ5, for autonomous systems, is:- [10], [12] The solution presented by Butcher and verified using the above system of equations is:

5-Sixth order method:-
We drive now the Sixth order, GJ6, for autonomous systems, lets: The sixth order Goeken-Johnson can be able to generate the Taylor's series expansion of the above, and we get the following systems of equations: By using the above systems of equations we get the following systems of general solution:

6-Stability Region of Sixth order method:-
The stability properties of numerical methods of Ordinary Differential equations determined by study the behavior of these methods, by using test problem:- [1], [2], [13]

7-Comparison:-
In this section we compare with the fifth-order method, GJ5, and the sixth-order method, GJ6, and we get the solutions of the two methods have the same results, and the GJ6 is useful for solving any problems of Ordinary Differential equations, In this comparison we use the example

8-Conclusions:-
New third-, fourth-, fifth-, and sixth-order numerical integration techniques inspired by the Runge-Kutta method have been presented. The new methods exploit the use of higher order derivatives, specifically y f . In particular, a technique utilizing an approximation to ' ' y has been presented resulting in a multistep Runge-Kutta method. The proposed methods are more efficient than the standard Runge-Kutta methods for cases:- [4], [10], [12] 1.