Runge-Kutta order versus stages

This article discusses the relationship between the number of stages and the order of explicit Runge-Kutta (RK) methods for solving ordinary differential equations. It explains that while lower-order RK methods (p≤4) require exactly p stages, higher-order methods (p≥5) require more stages than their order, a result known as the Butcher barrier. The article defines stages as the number of function evaluations per step and order as the local error scaling. It highlights the Dormand-Prince method, which uses 7 stages to achieve 5th order and provides a free 4th order error estimate. The article is relevant to numerical analysis, computational science, and software development for scientific computing.