Modification of the Fourth Order Runge Kutta Method Based on the Contra Harmonic Average

Purpose of the StudyThis study aims to modify the fourth-order Runge-Kutta method based on the contra harmonic mean. The research discusses the theoretical modification of the fourth-order Runge-Kutta method. MethodologyThe research was conducted using a literature study method. The study begins by introducing the general form of the Runge-Kutta method up to the nth order. This general form is then specialized to the fourth order. Additionally, the concept of the contra harmonic mean is introduced. After obtaining the general form of the fourth-order Runge-Kutta method and the contra harmonic mean, these two general forms are modified to derive a new formula. Main FindingsBased on the results, the modified fourth-order Runge-Kutta method has the following equation form:yi+1 = yi + (h/4) * [(k1^2 + k2^2) / (k1 + k2) + 2 * (k2^2 + k3^2) / (k2 + k3) + (k3^2 + k4^2) / (k3 + k4)],with an error of order O(h^5). Numerical simulations demonstrate that the modified method provides better results compared to the original fourth-order Runge-Kutta method. Novelty/Originality of this StudyThe numerical simulations using the RKKCM method show improved accuracy compared to the unmodified fourth-order Runge-Kutta method, highlighting the innovation and contribution of this study.