A Simplified Adaptive Runge-Kutta Method with Competitive Performance for Ordinary Differential Equations

Abstract

This paper presents a simplified and computationally efficient adaptive Runge–Kutta method for solving ordinary differential equations (ODEs). The proposed approach enhances the classical fourth-order Runge–Kutta (RK4) scheme by incorporating an intelligent step-doubling error estimation strategy, enabling reliable adaptive step-size control without relying on embedded Runge–Kutta pairs or complex Butcher tableaus. By comparing one full RK4 step with two half-steps, the method obtains a robust local error estimate that balances accuracy and computational cost while preserving implementation simplicity. The performance of the proposed adaptive RK4 method is rigorously evaluated against well-established solvers, namely RK45, DOP853, and the backward differentiation formula (BDF), as implemented in the SciPy library. Benchmark tests are conducted on three representative problems: the Van der Pol oscillator, a logistic growth model, and a nonlinear oscillator. Numerical results demonstrate that the proposed method consistently achieves high computational efficiency while maintaining accuracy within prescribed tolerances ranging from 10^(-4) to 10^(-6). In particular, the method attains peak efficiencies exceeding 10^4 steps per second across all test cases. These results indicate that the proposed adaptive RK4 algorithm offers a practical and competitive alternative for general-purpose ODE solving, especially in applications where a balance between numerical accuracy, computational efficiency, and algorithmic simplicity is essential.