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.

Proof  Milne-Simpson's Method  Milne-Simpson's 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  

        .  

Proof  Milne-Simpson's Method  Milne-Simpson's Method  

Animations (Milne-Simpson's Method  Milne-Simpson's Method).  Internet hyperlinks to animations.

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  .  

Computer Programs  Milne-Simpson's Method  Milne-Simpson's Method   

Mathematica Subroutine (Milne-Simpson's Method).

Example 1.  Solve the I.V.P.  .  
Solution 1.

Example 2.  Use Mathematica to find the analytic solution and graph for the I.V.P.  .  
Solution 2.

Example 3.  Plot the error for Milne-Simpson's method.
Solution 3.

Example 4.  Reduce the step size by   and see what happens to the error.
Recalculate points for Milne-Simpson's method, and the analytic solution using twice as many subintervals.
Then Plot the error for Milne-Simpson's method.
Solution 4.

Example 5.   Solve    with    over  .
Solution 5.

Example 6.  Use Mathematica to find the analytic solution and graph for the I.V.P.  .  
Solution 6.

Example 7.  Plot the absolute value of the error for Milne-Simpson's method.
Solution 7.

Example 8.  Reduce the step size by   and see what happens to the error.
Recalculate points for Milne-Simpson's method, and the analytic solution using twice as many subintervals.
Then Plot the error for Milne-Simpson's method.
Solution 8.

Example 9.  Solve the I.V.P.  .  
Solution 9.

Example 10. Use Mathematica to find the analytic solution and graph for the I.V.P.  .  
Solution 10.

Various Scenarios and Animations for Milne-Simpson's Method for O.D.E's

Example 11.  Solve the I.V.P.  .    Compute the Milne-Simpson solution to the I.V.P.
Solution 11.

Animations (Milne-Simpson's Method  Milne-Simpson's Method).  Internet hyperlinks to animations.

Research Experience for Undergraduates

Milne-Simpson's Method  Milne-Simpson's Method  Internet hyperlinks to web sites and a bibliography of articles.  

Download this Mathematica Notebook Milne-Simpson's Method for O.D.E.'s

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(c) John H. Mathews 2004