Graphical Interpretation of the Slopes Used in the Derivation of Classical Fourth Order Runge-Kutta (RK4) Formula

Abstract

Many practical issues in science and engineering are formulated by ordinary differential equations (ODE) that require their own numerical solution. There are a variety of numerical approaches, e.g. the Euler method, the modified Euler method, the Heun’s method, the Adam-Bashforth method, and so on, that exist in the context of numerical analysis. Amongst them, the classical fourth order Runge-Kutta (RK4) technique is the most reliable and most used. The objective of this paper is twofold. The first goal is to derive the value of different parameters in the formulation of the fourth order Runge-Kutta method, and the second goal is to give details of the geometrical interpretation of this method, principally explaining the role of the increment parameters in the formula. The whole discussion will facilitate perception of the key mechanism of the Runge-Kutta method.