RungeKuttaperheet

RungeKuttaperheet, or Runge–Kutta families, are a broad class of numerical methods for solving initial-value problems for ordinary differential equations. They approximate the solution by combining several slope evaluations within each integration step, yielding different orders of accuracy depending on the method.

The methods are named after Carl Runge and Wilhelm Kutta, who introduced the ideas in the early 20th century. The term covers both explicit formulas, which compute slopes directly, and implicit formulas, which require solving equations at each step and are more stable for stiff problems.

Common members include the classical fourth‑order Runge–Kutta (RK4), a widely used explicit method, and various embedded pairs such as Runge–Kutta–Fehlberg (RKF45) that provide adaptive step size control. Other families include Dormand–Prince, Cash–Karp, and related implicit RK schemes such as Gauss, Radau, and Lobatto. These variants differ in order, stability properties, and whether they are explicit or implicit, influencing their suitability for non-stiff versus stiff problems.

RungeKuttaperheet are central to numerical analysis and applied computation because they offer a flexible framework for approximating ODE solutions with controllable accuracy. They are implemented in many scientific computing libraries and software, often with adaptive step size and error estimation to balance efficiency and precision.