RungeKutta

Runge-Kutta methods are a family of iterative techniques for approximating solutions to ordinary differential equations, especially initial value problems of the form y' = f(t, y) with a given y(t0) = y0. They achieve higher accuracy by combining multiple slope evaluations within a single step to estimate the solution at the next time point.

In a typical Runge-Kutta step, several intermediate slopes k1, k2, ..., sk are calculated based on weighted evaluations of f at strategically chosen times and intermediate values. The next value y_{n+1} is formed as a weighted sum of these slopes. The method can be described compactly by a Butcher tableau, which encodes the stage times, coefficients, and weights. Methods can be explicit, where all k_i are computed directly from previously known stages, or implicit, where k_i must be found by solving equations that include the unknowns themselves. Implicit Runge-Kutta methods are often used for stiff problems because of their favorable stability properties.

Common examples include RK2 (the midpoint or improved Euler methods) and RK4, which are widely used due to a favorable balance of accuracy and computational cost. Many implementations also provide embedded Runge-Kutta pairs, which share the same stages but use different weights to estimate local truncation error and adapt the step size accordingly.

Applications of Runge-Kutta methods span physics, engineering, biology, and beyond, wherever accurate solution of initial value problems is required. While explicit RK methods are efficient for non-stiff systems, implicit RK methods and other high-order variants are favored for stiff equations, offering improved stability at the expense of solving nonlinear stage equations. The development and use of Runge-Kutta methods continue to be central in numerical analysis and scientific computing.