Runge–Kutta methods

Summary

Runge–Kutta methods are one-step methods for the numerical solution of an initial value problem $y' = f(t, y)$, $y(t_0) = y_0$. Each step evaluates $f$ at several intermediate points, the stages, and combines these slopes to advance the solution. They reach high order without derivatives of $f$, and they include the Euler method, the classical fourth-order method, and implicit methods for stiff equations. A method is described by its coefficients, collected in a Butcher tableau.