Abstract
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.
Highlights
Representations of the stability regions of Runge–Kutta methods are presented in several literatures [1,2,3,4,5,6,7,8, 11, 13]
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
Summary
Representations of the stability regions of Runge–Kutta methods are presented in several literatures [1,2,3,4,5,6,7,8, 11, 13]. It has been found that the stability region varies according to the order of the method. 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. We demonstrate that the evolution of the size of the stability region does not depend on the order of the methods. 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
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