Application of Adaptive Step Size Runge-Kutta Method in Solving Ordinary Differential Equations

Abstract

To address the imbalance between accuracy and efficiency in traditional fixed-step Runge-Kutta methods when dealing with nonlinear ordinary differential equations, this paper proposes an adaptive-step Runge-Kutta method. This method is based on the framework of the fourth-order Runge-Kutta method and dynamically adjusts the step size through local error estimation. It combines an expansion coefficient optimization strategy, reducing the step size when the error exceeds the limit and expanding it when the error is redundant, achieving sensitive feedback of the step size within the error limit. Compared with the fixed-step method, the adaptive-step method can more effectively avoid error accumulation and maintain extremely low error throughout the process of solving initial value problems of ordinary differential equations, effectively balancing computational accuracy and efficiency. Through numerical experiments, the effectiveness and practicality of this method have been verified. The research in this paper provides a balanced solution for initial value problems of ordinary differential equations that combines accuracy retention and computational economy.