Development and Investigation of Runge- Kutta Coefficients Dependent Stability Polynomial

Abstract

Simulation of reliable solutions of nonlinear engineering problems by means of stable numerical algorithms is a frequent and acceptable practice. This study focuses the development and investigation of Range-Kutta coefficients dependent stability polynomial for the second, third and fourth orders Runge-Kutta schemes. The development utilized matrix inversion operation procedure that involves determinant and cofactors computation of relevant matrix. The validation was made referencing the standard result of [I] and extended to several cases. The resulting polynomials obtained consist of combination of the scheme coefficients with increasing power of time step that follows a rhyme pattern. The validation test case result agreed perfectly with test standard result. Selected studied version of different schemes shows wide variation in the shape of stability curve and region bounded. It is interesting to note that the popular second, third andrfourth order schemes have stability curve that bounded large? region than their respective counterpart. It is concluded that the study results can be utilized as reliable platform for stability analysis for different versions of the second, third and fourth order schemes.