Milne-Simpson's Method

   The methods of Euler, Heun, Taylor and Runge-Kutta are called single-step methods because they use only the information from one previous point to compute the successive point, that is, only the initial point  
  is used to compute  
  and in general  
  is needed to compute  
.  After several points have been found it is feasible to use several prior points in the calculation.  The Milne-Simpson method uses
  in the calculation of
.  This method is not self-starting;  four initial points  
,
,
,  and
must be given in advance in order to generate the points
.  

    A desirable feature of a multistep method is that the local truncation error (L. T. E.) can be determined and a correction term can be included, which improves the accuracy of the answer at each step.  Also, it is possible to determine if the step size is small enough to obtain an accurate value for  
, yet large enough so that unnecessary and time-consuming calculations are eliminated.  If the code for the subroutine is fine-tuned, then the combination of a  predictor and corrector requires only two function evaluations of  f(t,y)  per step.  

Theorem  (Milne-Simpson's Method)  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 Milne-Simpson method uses the formulas
,  and  

the predictor          
,  and

the corrector           
  for  
  

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

Remark.  The Milne-Simpson method is not a self-starting method.  Three additional starting values
  must be given.  They are usually computed using the Runge-Kutta method.

Theorem (Precision of the Milne-Simpson Method)  Assume that  
  is the solution to the I.V.P.  
  with  
.  If  
  and  
  is the sequence of approximations generated by Milne-Simpson method, 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 ((Milne-Simpson's Method).    To approximate the solution of the initial value problem  
  with  
  over  
  at a discrete set of points using the formulas:  

use the predictor      
,  

and  the corrector      
  for  
.  

Mathematica Subroutine (Milne-Simpson's Method).