Accuracy–Efficiency Comparison of Fixed-Step and Adaptive Runge–Kutta Methods for Ordinary Differential Equations

Ordinary differential equations arise in diverse scientific and engineering contexts, yet exact analytic solutions are often unavailable. This study evaluates the accuracy and efficiency of three explicit time-integration schemes—Euler’s method, classical fourth-order Runge–Kutta (RK4), and an embedded error–controlled adaptive RK4—on representative linear, nonlinear, and oscillatory test problems. Linear decay, logistic growth, and the Van der Pol oscillator were solved numerically with multiple step sizes and compared against exact or high-accuracy reference solutions. Tabulated results and log–log error plots confirm first-order convergence for Euler and fourth-order convergence for RK4. Adaptive RK4 maintained RK4-level accuracy while reducing the number of accepted steps by concentrating effort where local error increased, achieving modest computational savings without loss of precision. These findings underscore the advantage of higher-order and adaptive methods for reliable integration of smooth initial-value problems and provide clear guidance for solver selection in routine scientific computing.