Accuracy–Efficiency Trade-Offs in Euler and Runge–Kutta Fourth-Order Methods: A Comparative Error Analysis

Abstract

Numerical techniques are essential for approximating solutions to ordinary differential equations (ODEs) that lack closed-form solutions. Although Euler's method and the fourth-order Runge–Kutta (RK4) scheme are frequently compared, most studies report accuracy at a single step size, which cannot by itself demonstrate a genuine accuracy–efficiency trade-off. This paper addresses that gap by solving the benchmark initial value problem   over with both methods across five step sizes, tracking absolute error against the number of function evaluations required at each . At , Euler required 10 evaluations and produced 3.187485 (absolute error ), while RK4 required 40 evaluations and produced  3.436559 (absolute error ). Across the full range of step sizes, Euler's error decreases roughly in proportion to , consistent with its first-order global accuracy, whereas RK4's error decreases roughly in proportion to , so RK4 attains errors several orders of magnitude smaller than Euler even after its four-fold higher per-step cost is taken into account. A work-precision diagram of error against function evaluations places RK4's curve consistently below Euler's across every tested , confirming that RK4 dominates Euler over the tested range rather than only at one operating point. Maple 17 code implementing both schemes with an adjustable step-size parameter is provided to support reproduction and extension of these results. These findings give multi-point, quantitative support for preferring RK4 whenever accuracy per function evaluation matters, while Euler remains attractive only when a single, very cheap evaluation is the overriding constraint.