On The Component Analysis and Transformation of An Explicit Fourth – Stage Fourth – Order Runge – Kutta Methods

This work is designed to transform the fourth stage – fourth order explicit Runge-Kutta method with the aim of projecting a new method of implementing it through tree diagram analysis. Efforts will be made to represent the equations derived from the y derivatives and x,y derivatives separately on Butcher’s rooted trees. This idea is derivable from general graphs and combinatorics. Key words: Rooted tree diagram, Transformation, Vertex, explicit, y partial derivatives, x,y partial derivatives, Runge-Kutta Methods, Linear and non- linear equations, Taylor series, Graphs. Keywords: key words, Runge - Kutta, Combinatorics, Derivatives, Transformation, Analysi

On the Stability Analysis of a Geometric Mean Order Runge-Kutta Formula

In the solution of non-stiff initial-value problems, sometimes stepsize is restricted by stability rather than by accuracy. Just as solutions to differential equations evolve with time, so do numerical approximations progress in small time steps. In each step an error is made and it is important to keep these errors small. But the error caused in one time-step may have an effect on the accuracy of later steps. It may be more important to control the buildup of errors as it is to control the size of the errors themselves. In numerical analysis the smallness of the individual errors is called accuracy and the ability to keep the effect of errors under control is called stability. Physical systems give rise to system of ordinary differential equations with widely varying eigenvalues resulting precisely to the concept of stiffness. This paper seek to establish the region of absolute stability of a special 4th Runge-Kutta formula first derived and implemented in Agbeboh et al (2007). Keywords: Absolute Stability, geometric mean, one-third 4th order Runge-Kutta formula and boundary locus