Module

for

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 Adams-Bashforth-Moulton 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  (Adams-Bashforth-MoultonMethod)  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 Adams-Bashforth-Moulton 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 Adams-Bashforth-Moulton 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 Adams-Bashforth-MoultonMethod)  Assume that    is the solution to the I.V.P.    with  .  If    and    is the sequence of approximations generated by Adams-Bashforth-Moulton 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 (Adams-Bashforth-Moulton 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  .

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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 Adams-Bashforth-Moulton's method.
Solution 3.

Example 4.  Reduce the step size by   and see what happens to the error.
Recalculate points for Adams-Bashforth-Moulton's method, and the analytic solution using twice as many subintervals.
Then Plot the error for Adams-Bashforth-Moulton'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 Adams-Bashforth-Moulton's method.
Solution 7.

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

Example 11.  Solve the I.V.P.  .    Compute the Adams-Bashforth-Moulton solution to the I.V.P.
Solution 11.

(c) John H. Mathews 2004