ADAPTIVE TIME-STEPPING FOR RUNGE-KUTTA METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS

Abstract

Ordinary Differential Equations (ODEs) are usually used in numerous fields especially in solving the modelling problem. Numerical methods are one of the vital mathematical tools to solve the ODEs that appear in various modelling problems by determining the approximation solution close to the in exact solution if it exists. Runge-Kutta methods (RK) are the numerical methods used to integrate the ODEs by applying multistage methods at the midpoint of an interval which can efficiently produce a more accurate result or small magnitude of error. We proposed Runge-Kutta methods (RK) to solve the 1st_ order nonlinear stiff ODEs. The RK methods used in this research are known as the RK-2, RK-4, and RK-5 methods. We proved the existence and uniqueness of the ODEs before we solved it numerically. We also proved the absolute-stability of the RK methods to determine the overall stability of these methods. We found two suitable test cases which are the standard test problem and manufactured solution. We proved that by combining the adaptive step size with RK methods can result in more efficient computation. We implemented the 2nd_, 4th_ and 5th_ order of RK methods with step size adaptively algorithm to solve the test problem and manufactured solution via Octave programming language. The resulting numerical error and the stability of each method can be studied. We compared our results using several error plots versus the Central Processing Unit (CPU) time required to compute a given nonlinear 1st_ order stiff ODE problem. In a conclusion, RK methods which combine with the adaptive step size can result in more efficient computation and accuracy compare with the fixed step size RK methods.