The central piece of Rungu-Kutta method is the approximation of the increasement of the function. In 1st order ODE,
In a special case of Rungu-Kutta of order 4 (RK4), there are 2 array and , so that
where is ranging from 0 to 1, .
The is a must, otherwise, we have to define . There should be some methods to obtain an optimum values for and , but I don’t know.
For 2nd order ODE
These equation are the similar 1st order ODEs.
, where is the z for step.
The is using
This can be generalized to any order ODE by decoupling the ODE into
the equation for is
And for all the intermediate variable