Runge-Kutta Method

Theorem  (Runge-Kutta Method of order 4)  Assume that  f(t,y)  is continuous and satisfies a Lipschits condition in the variable  y,  and consider the  I. V. P. (initial value problem)

        
with
,  over the interval  
.
        
The Runge-Kutta method uses the formulas
,  and  

        
     for  
  
where
        
   

as an approximate solution to the differential equation using the discrete set of points  
.  

Theorem  (Precision  of the Runge-Kutta Method of Order 4)  Assume that  
  is the solution to the I.V.P.  
  with  
.  If  
  and  
  is the sequence of approximations generated by the Runge-Kutta method of order 4, then at each step, the local truncation error is of the order  
,  and the overall global truncation error  
is of the order

        
,  for  
.  


The error at the right end of the interval is called the final global error  

        
.  

Algorithm (Runge-Kutta Method). To compute a numerical approximation for the solution of the initial value problem  
with  
  over
  at a discrete set of points using the formula  

    
,  for  
  
where
    
,
    
,
    
, and
    
.

Mathematica Subroutine (Runge-Kutta Method of Order 4).