An Investigative Comparison of Higher-Order Runge-Kutta Techniques for Resolving First-Order Differential Equations

In the context of solving first-order ordinary differential equations (ODEs), this paper thoroughly compares various higher-order Runge-Kutta methods. Reviewing the effectiveness, precision, and practicality of several Runge-Kutta schemes and highlighting their usage in numerical approximation is the main goal of the research. The study explores traditional approaches, including the fifth-order, six-stage Runge-Kutta (RK56), the sixth-order, seven-stage Runge-Kutta (RK67), and the seventh-order, nine-stage Runge-Kutta (RK79), with the goal of offering a comprehensive comprehension of their individual advantages and disadvantages. In order to help academics and practitioners choose the best approach based on the features of the problem, comparative benchmarks are constructed, utilizing both theoretical underpinnings and real-world implementations. Robustness evaluations and sensitivity analysis complement the comparison research by illuminating how flexible these techniques are in various context. The results of this study provide important new understandings of how higher-order Runge-Kutta methods function and provide a thorough manual for applying them to solve first-order differential problems in a variety of scientific and engineering fields. The study’s examination of three higher order Runge-Kutta algorithms reveals that the RK56 is more effective at solving first order ODEs.