ORDER OF THE RUNGE–KUTTA METHOD AND EVOLUTION OF THE STABILITY REGION 1

: In this article, we demonstrate through speciﬁc examples that the evolution of the size of the absolute stability regions of Runge–Kutta methods for ordinary diﬀerential equation does not depend on the order of methods.


Introduction
Representations of the stability regions of Runge-Kutta methods are presented in several literatures [1-8, 11, 13].It has been found that the stability region varies according to the order of the method.However, it is not proven in the literature whether or not there is a relation between the evolution of the size of the region of stability and the order of the method.In this article, we demonstrate that the evolution of the size of the stability region does not depend on the order of the methods.For that we exhibit methods whose regions of stability grow according to the order.Subsequently, we give a counter-example where we introduce a new 8 order method [12].We compare the stability region of this new 8 order method with those of certain lower order methods.We show that the stability regions of lower order methods are larger than that of the new 8 order method.The study will be done in accordance with the following plan: in Section 2 we describe some generalities on the stability regions, in Section 3 we present some stability functions, in Section 4 we present the new 8 order method and its stability regions, Section 5 we give a conclusion.

Generalities on the stability regions
Consider a general form of the first-order ODE given below: with the initial condition y(x 0 ) = y 0 for the interval x 0 ≤ x ≤ x n .Here, x is the independent variable, y is the dependent variable, n is the number of point values, and f is the function of the derivation.The goal is to determine the unknown function y(x) whose derivative satisfies (1.1) and the corresponding initial values.In doing so, let us discretize the interval x 0 ≤ x ≤ x n to be where h is the fixed step size.With the initial condition y(x 0 ) = y 0 , the unknown grid function y 1 , y 2 , y 3 , • • • , y n can be calculated by using the Runge-Kutta method of the order 8 (RK8 method).The 8-th order method is thus obtained by the resolution of the 200 equations with 11 stages [12] on Maple.
Lets consider the Butcher tableau of 8 order and 11 steps RK method (see Fig. 1):  The numerical solution is given by the formula with The concept of absolute stability, in its simplest form, is based on the analysis of the behavior, according to the values of the step h, of the numerical solutions of the model equation [9][10][11][12]: Using (1.3) and (1.4), we obtain: which gives: If z = hλ, then the absolute stability region is the set

Presentation of some stability functions
Consider the standard Runge-Kutta methods of orders 1 to 4. When (1.2) and (1.3) are applied to the model problem (1.4), the resulting equations are RK1: RK2: RK3: RK4: The stability regions are shown at the next figure: We can see the evolution of the size of the region of stability as the order of the method increases.
Let's now give a counterexample for which the stability region is very small.

Presentation of the new 8 order method and its stability regions
The family of the 8 th order method is thus obtained by the resolution of the 200 equations with 11 stages [12] on Maple.This method depends on free parameters b 8 and a 10,5 [12].
Some of related coefficients have fixed values, not depending on b 8 and a 10,5 , these coefficients are:  The numerical solution is given by the formula (1.2).The values of k s are given by the formula (1.3).We can notice that if b 8 = 49/180 and a 10,5 = 1/9, then we find the method of Cooper-Verner [1,12].With the help of Maple, the stability function depends on a 10,5 and is given by [12]: The stability region of the new RK8 method for a 10,5 = 10 6 is given by Fig. 3.We can see that the stability region of the new method of order 8 is smaller than 2, 3, 4.There is a decrease in the values of x and y.
For a 10,5 = 10 12 the stability region is the following (see Fig. 4): We can see that the stability region of the new method of order 8 is smaller than those of ordering regions 1, 2, 3, 4.There is a decrease in the values of x and y.We find that the values of x and y have very strongly diminished and the region of stability is very small.

Conclusion
Presumably, by representing the domains of stability of methods of the order of 1, 2, 3, 4, one could assume that the higher the order, the greater the area of stability.However, a new 8 order method is discovered.The stability region of this 8 order method is smaller than that of the regions of orders 2, 3, 4. We can therefore conclude that the evolution of the size of the stability regions of Runge-Kutta methods does not depend on the order of the method.

Figure 2 .
Figure 2. Evolution of the stability region according to the order.

For a 10 , 5
= 9 . . .9 37 times the stability region is is shown at the next figure:

Figure 5 .
Figure 5. Stability region of the new RK8 method.