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

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.

The two forms of Equations ( 2) and ( 5) are equivalent by making the interpretation ( ) where i Y is the inner stages that tend to estimate the solutions at some points; s is the number of stages and i c 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 easily.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].

Method ARK3 (s = p = 3)
The general third order three stages Almost Runge-Kutta scheme is of the form: We represent the abscissa vector The order conditions for order three ARK schemes are derived through the standard rooted tree approach used for Runge-Kutta methods [8].
The conditions of Runge-Kutta stability for 3 rd order, 3 stages are: ( ) ( ) ( ) where ( ) Acquiring order 2 estimation with respect to 2 nd scaled derivative for the 3 rd outgoing solution, we need: From Equation ( 12) we have, Solving Equation ( 9) we obtain ( ) ( ) And from Equation (11), we obtain ( ) Evaluating both sides of Equation (10) we obtain This implies that Thus Equation (13) becomes

Method ARK34 (s = 4, p = 3)
The third order four stages scheme has the general form: Its stability function is expressed as The order conditions are derived using the standard rooted tree approach used for Runge-Kutta methods [8].
The i α values are obtained by expanding ( ) There is also the additional condition ( )( ) ( ) From Equation (30) we obtain Evaluating the stability matrix of a four stage third order method, we arrive at ( ) where Tr is the trace of a matrix and

BA U A e A e c Ae A e c Ac
A e A c Ae A c Ac And it follows that: We introduce Thus from Equation ( 46) we arrived at And from Equations ( 32) and ( 33) we obtain respectively ( ) ( ) Further simplification produces the following results ( ) ( )
λ λ λ = Applying Cayley-Hamilton theorem to matrix V ( ) Problem (72) is solved using the proposed ARK34 method.The results are obtained and compared with similar ARK34 methods of [3] and [5] respectively and presented in Figure 1.From Figure 1 it is evident that our Proposed ARK34 method performed better than the methods of [3] and [5] since it exhibits lesser error than the errors of the existing methods.

Conclusion
Two ARK methods are proposed, ARK3 ( ) and ARK34 ( ) . The methods have been proven to be consistent and stable, thereby guaranteeing their convergence.This is further illustrated by comparing the performance of one of the methods with other methods of similar order.The proposed method ARK34 is shown to perform better than the existing methods.

and 2 c
are required for an order three scheme.Thus T 8 calculating the members of the U matrix we obtain the a scheme for method 3 s p = = .

4 L
be assumed to be the free parameters, where 1 − is the error coefficient comparable to the bushy tree.From Equations (25)-(27) together with Equation (34