List of Runge–Kutta methods

Runge–Kutta methods are methods for the numerical solution of the ordinary differential equation

which take the form

Each method listed on this page is defined by its Butcher tableau, which puts the coefficients of the method in a table as follows:

The explicit methods are those where the matrix ${\displaystyle [a_{ij}]}$ is lower triangular.

The Euler method is first order. The lack of stability and accuracy limits its popularity mainly to use as a simple introductory example of a numeric solution method.

The (explicit) midpoint method is a second-order method with two stages (see also the implicit midpoint method below):

Heun's method is a second-order method with two stages (also known as explicit trapezoid rule):

Ralston's method is a second-order method with two stages and a minimum local error bound:

The "original" Runge–Kutta method.

This method doesn't have as much notoriety as the "classical" method, but is just as classical because it was proposed in the same paper (Kutta, 1901).

The embedded methods are designed to produce an estimate of the local truncation error of a single Runge-Kutta step, and as result, allow to control the error with adaptive stepsize. This is done by having two methods in the tableau, one with order p and one with order p-1.

The lower-order step is given by

where the ${\displaystyle k_{i}}$ are the same as for the higher order method. Then the error is

...
Wikipedia