Tuesday, February 2, 2010

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 [Graphics:Images/MilneSimpsonMod_gr_1.gif] is used to compute [Graphics:Images/MilneSimpsonMod_gr_2.gif] and in general [Graphics:Images/MilneSimpsonMod_gr_3.gif] is needed to compute [Graphics:Images/MilneSimpsonMod_gr_4.gif]. After several points have been found it is feasible to use several prior points in the calculation. The Milne-Simpson method uses [Graphics:Images/MilneSimpsonMod_gr_5.gif] in the calculation of [Graphics:Images/MilneSimpsonMod_gr_6.gif]. This method is not self-starting; four initial points [Graphics:Images/MilneSimpsonMod_gr_7.gif], [Graphics:Images/MilneSimpsonMod_gr_8.gif], [Graphics:Images/MilneSimpsonMod_gr_9.gif], and [Graphics:Images/MilneSimpsonMod_gr_10.gif] must be given in advance in order to generate the points [Graphics:Images/MilneSimpsonMod_gr_11.gif].

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 [Graphics:Images/MilneSimpsonMod_gr_12.gif], 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)

[Graphics:Images/MilneSimpsonMod_gr_13.gif] with [Graphics:Images/MilneSimpsonMod_gr_14.gif], over the interval [Graphics:Images/MilneSimpsonMod_gr_15.gif].

The Milne-Simpson method uses the formulas [Graphics:Images/MilneSimpsonMod_gr_16.gif], and

the predictor [Graphics:Images/MilneSimpsonMod_gr_17.gif], and

the corrector [Graphics:Images/MilneSimpsonMod_gr_18.gif] for [Graphics:Images/MilneSimpsonMod_gr_19.gif]

as an approximate solution to the differential equation using the discrete set of points [Graphics:Images/MilneSimpsonMod_gr_20.gif].

Remark: The Milne-Simpson method is not a self-starting method. Three additional starting values [Graphics:Images/MilneSimpsonMod_gr_21.gif] must be given. They are usually computed using the Runge-Kutta method.


Theorem (Precision of the Milne-Simpson Method:

Assume that [Graphics:Images/MilneSimpsonMod_gr_22.gif] is the solution to the I.V.P. [Graphics:Images/MilneSimpsonMod_gr_23.gif] with [Graphics:Images/MilneSimpsonMod_gr_24.gif]. If [Graphics:Images/MilneSimpsonMod_gr_25.gif] and [Graphics:Images/MilneSimpsonMod_gr_26.gif] is the sequence of approximations generated by Milne-Simpson method, then at each step, the local truncation error is of the order [Graphics:Images/MilneSimpsonMod_gr_27.gif], and the overall global truncation error [Graphics:Images/MilneSimpsonMod_gr_28.gif] is of the order

[Graphics:Images/MilneSimpsonMod_gr_29.gif], for [Graphics:Images/MilneSimpsonMod_gr_30.gif].


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

[Graphics:Images/MilneSimpsonMod_gr_31.gif].

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Introductory Methods of Numerical Analysis