A New Eighth Order Runge-Kutta Family Method

In this article, a new family of Runge-Kutta methods of 8th order for solving ordinary differential equations is discovered and depends on the parameters b8 and a10,5. For b8 = 49/180 and a10,5 = 1/9, we find the Cooper-Verner method [1]. We show that the stability region depends only on coefficient a10,5. We compare the stability regions according to the values of a10,5 with respect to the stability region of Cooper-Verner.


Introduction
Since the time of Newton, one of the main problems of mathematicians is the resolution of various differential equations.Practically, the immense quantity of these equations were not resolvable in the analytical aspect.This has led to the development of numerical methods for their resolution.The method of Runge-Kutta, named RK is used to find a good numerical result.Currently, a large number of high-order RK methods are known (5..10) [1,2,4,8,9,10].However, they are not all found.For example, methods of 8 th order depending of several parameters are not presented in the literature.The field of stability has not been fully investigated.In this article we discover new family of order 8 depending on several parameters.This family summarizes the Cooper-Verner method [1].It depends on the coefficients b 8 and a 10,5 .For b 8 = 49/180 and a 10,5 = 1/9, we find the Cooper-Verner method [1].Subsequently, we show that the region of stability of this family depends only on the coefficient a 10,5 and not on b 8 .The study will consist in comparing the regions of stability according to the values of a 10,5 with respect to the stability region from Cooper-Verner.The study will be led by respecting the following plan: in section 2 Presentation of the new family RK8 method, section 3 The stability region, section 4 Comparison of some stability regions, section 5 Conclusion.

Presentation of the New Family RK8 Method
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) and the corresponding initial values.In doing so, let us discretize the interval x 0 ≤ x ≤ x n to be x 0 , x 1 = x 0 + h, x 2 = x 0 + 2h, ..., where h is the fixed step size.With the initial condition y(x 0 ) = y 0 , the unknown function y 1 , y 2 , y 3 , • • • , y n can be solved by using the RK8 method.
The family of 8 th order method is thus obtained by the resolution of the 200 equations with 11 stages (see Appendix A) on Maple.However, the 200 equations are obtained from the successive derivatives of the exact solution up to 8 (see Appendix B).
Lets consider the Butcher tableau of 8 order 11 steps RK method(see Fig. 1):  (3) The numerical solution is given by the formula with We can notice that if b 8 = 49/180 and a 10,5 = 1/9, then we find the method of Cooper-Verner [1].

The Stability Regions
The concept of stability is based on the discrete solution.It gives back account of the actual behavior of the approximate solution for a given practical value, therefore non-zero, of the h step.In real calculation, the errors accumulate.This is particularly evident in the process of solving a differential equation where one progresses step by step from an initial value.There are various stability conditions.First of all the numerical solution must remain limited.This requirement of minimum stability may be insufficient in practice, the bound obtained being often an exponential of the duration which therefore grows infinitely when it increases.We then introduce more stability criteria, demanding that the digital solution reproduce the behavior physical of the exact solution.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 equation model [1,3,5,6]: Using ( 44) and ( 45), we obtain: Which give: Let's put z = hλ.We obtain by Maple:

Comparison of Some Stability Regions
The stability region depends on coefficient a 10,5 .We obtain by Maple different regions according to values of a 10,5 .
For a 10,5 = 1/9, we obtain the stability region of the Cooper-Verner method (see Fig. 2).From the point of view of the values of x, if we choose a 10,5 = 1 10 (see Fig. 3) or a 10,5 = 1 10.9 ( see Fig. 4) or a 10,5 = 1 11.9 ( see Fig. 5), then we see that the stability regions are bigger than the region of Cooper-Verner.
Let's see some stability region of a 10,5 = {0, −1, −10, −100}.(see Fig. 10, Fig. 11, Fig. 12, Fig. 13).We can see that the more a 10,5 < 0, the more the stability region is smaller.We also notice that all the stability regions are smaller than that of the Cooper-Verner one.

Conclusion
A new family of Runge-Kutta method 8 th order is discovered.This family depends one the parameters a 10,5 and b 8 .For a 10,5 = 1/9 and b 8 = 49/180, we find the method of Cooper-Verner.The stability region depends on the value of a 10,5 but not of the b 8 coefficient.If we want to obtain a region of stability greater than that of the Cooper-Verner method, then it is not better to choose negative values of a 10,5 or values greater than 1.You have to choose wisely a 10,5 between 0 and 1. Velagala, & Sindhuja. (2014).Stability analysis of the 4th order Runge-Kutta method in application to colloidal particle interactions.(Master's thesis, University of Illinois, Urbana-Champaign, USA).Retrieved from http://hdl.handle.net/2142/72750 Appendix A The 200 equations of order 8 with s stages ∑ s i, j,k,l,m,n=1 b i a i j a jk a kl a lm a mn c 2 n = 1 20160 ∑ s i, j,k,l,m,n,o=1 b i a i j a jk a kl a lm a mn a no c o =