Runge-Kutta Methods

Runge-Kutta methods are a family of explicit numerical methods for solving ODEs of the form $\dot{y}=g(t,y)$. They generalise the Explicit Euler method by evaluating $g$ at multiple intermediate points within each step.

The general form is

$$y_{i+1}=y_i+dt\cdot \varphi (t_i,y_i,dt)$$

where $\varphi$ is the increment function, which depends on the specific method. Higher-order Runge-Kutta methods achieve smaller Global truncation error by choosing $\varphi$ to match more terms of the Taylor expansion of the exact solution.

Examples

Method	Order	Notes
Explicit Euler method	1	$\varphi =g(t_i,y_i)$
Midpoint method	2	Single intermediate evaluation
Ralston method	2	Minimises LTE bound
Fourth order Runge-Kutta	4	Four stage evaluations

All explicit Runge-Kutta methods are conditionally stable - see Stability of a method.

Explicit Euler method | Midpoint method | Ralston method | Fourth order Runge-Kutta | Order of a method | Global truncation error | Stability of a method