A comparative study of Taylor method, fourth order Runge-Kutta method and Runge-Kutta Fehlberg method to solve ordinary differential equations

This paper mainly presents the Taylor method, Runge-Kutta fourth-order (RK4) method and Runge-Kutta Fehlberg (RKF) method for solving initial value problem (IVP) for ordinary differential equations (ODE). These problems can be effectively addressed using any of the three proposed methods, which have demonstrated high efficiency and practical suitability. Two differential equations model which describe the physical situation are chosen; Newton’s cooling law and the spring mass damper system. Numerical comparisons between the Taylor method, RK4 and RKF have been presented. For Newton’s cooling law problem, the performance and the computational effort of these methods have been compared. In order to verify the accuracy, we compare numerical solutions with the exact solution in the spring mass damper system problem. The step size needs to be decreased to achieve higher accuracy in the solution. The resulting value indicates that RKF and RK4 are the most efficient for solving the ODE in terms of convergence and accuracy, respectively. Meanwhile, Taylor Methods is still compatible but needs more iterations to converge. In the spring mass damper system problem, the Taylor Method diverges.